LLM-generated corrections to the published research of P. M. Aronow

Journal article P012

Corrections to: Identification of homophily and preferential recruitment in respondent-driven sampling

Forrest W. Crawford, P. M. Aronow, Li Zeng, and Jianghong Li (2018). Identification of homophily and preferential recruitment in respondent-driven sampling. American Journal of Epidemiology 187(1):153–160.

DOI
10.1093/aje/kwx208
arXiv
1511.05397
Audited edition
American Journal of Epidemiology, 2018, 187(1):153–160; 8 pages in the audited file
SHA-256 of the audited file
dcff3579cb1914c3cc7bb6bbf7220fe05aa5c8afbe74eff8bfe0cafe3a7283dc
Permanent address
https://pmaronow.github.io/llm-audit/works/P012/

Summary

The paper defines homophily hh and preferential recruitment pp in respondent-driven sampling, shows why both generally fail to be point identified from the data such a study observes, and bounds both over all compatible graphs, with an application to injection drug users in Hartford. 11 entries are classified as errors (3 Moderate, 8 Minor), and 1 is a clarity note; 3 other concerns were not confirmed. Results 1–3 hold once their proofs are repaired. On the functionals' defined domain, the closed forms in P012-N007 replace the search and convergence argument for the four marginal endpoints, supply extremizing graphs and shorten the proofs of Results 2–3; the identification problem remains. The three lower homophily limits in Table 1 are too high by at least 0.0190.019 if the bounds use the nomination-network degrees; otherwise the comparison concerns different networks.

What was and was not checked

I checked the following, with the qualifications below.

  • Equation (1) is the correlation of the trait across the two ends of an edge; for a binary trait it reduces to (4e11e00−c2)/((2e11+c)(2e00+c))(4e_{11}e_{00}-c^2)/((2e_{11}+c)(2e_{00}+c)) in the numbers of edges of the three kinds, and it lies in [−1,1][-1,1] where it is defined. Each summand of equation (2) lies in (−1,1)(-1,1) and has conditional mean zero when recruitment is uniform among susceptible neighbors.
  • Results 1, 2 and 3, using the corrected proofs and the defined domains in P012-E006, by explicit construction and by exhaustive checks of small exact cases (104, 92 and 184 cases that meet the respective hypotheses); display (A1) for a swap that preserves the degrees.
  • Figure 1: the degrees (5,4,3,3,2)(5,4,3,3,2) implied by panel A, and the four extrema −5/8-5/8, 79/10579/105, −11/80-11/80 and 49/12049/120, which round to the printed values and are the extrema over all 26,18226{,}182 compatible pairs, counted up to the labeling of the unsampled vertices.
  • The arithmetic of the reported application counts: 2,626−527=2,0992{,}626-527=2{,}099, 3,307−1,180=2,1273{,}307-1{,}180=2{,}127 and 527−6=521527-6=521.
  • Table 1 against the text: the signs of the point values, each point value inside its printed interval, every printed interval covering zero.
  • The objective bound and harmonic series comparison in the proof of Proposition 1, after correcting the cooling direction and initial index, and the cited theorem of Hajek (1988) against the published article. Applying that theorem requires a fixed proposal on a finite state space, irreducibility and weak reversibility.
  • The proposals of Web Appendix 2 with the repairs in P012-E001: every proposal stays in the compatible set.
  • The remark that further correct information can only narrow the identification intervals, since it shrinks the compatible set.
  • The page numbers given with the quotations from references 7 and 22–26, which fall inside the page ranges of the reference list.

I could not recompute the Hartford point values or bounds, or reconstruct Figure 2, because no public copy of the Hartford RDS-net data or code for the bounds was found; the full-network values, the upper limits of IhI_h and the intervals IpI_p are neither confirmed nor shown wrong. The lower-limit inconsistency in P012-N002 follows from the printed network totals if the bounds use the nomination-network degrees; otherwise the comparison concerns different networks. It does not give the actual corrected endpoints or establish a failure of the unavailable implementation. The quotations from the empirical papers, and the cited sources other than those named in the corrections, were not compared with the originals. The RDSAT manual could not be obtained, and reference 6 of the Web Appendix was read in its preprint only. No search was made for corrections or supplements beyond the article and its official Web Appendix. That a passage has no correction here does not mean that it was found correct.

With a nonempty compatible set, positive total reported degree in each sampled trait group and at least one recruitment, I checked the closed forms in P012-N007 for all four sharp marginal endpoints and the graphs with private unsampled neighbors that attain them. These replace the appendix search and convergence argument for these endpoints and shorten the proofs of Results 2–3; the identification problem from incomplete networks remains. I did not address arbitrary objectives or additional graph constraints, or independently reproduce the Hartford bounds.

Headline claims and related corrections

  1. The paper defines homophily hh as the assortativity coefficient (correlation of a binary trait across the ends of edges) on the augmented recruitment-induced subgraph GSUG_{SU}, and preferential recruitment pp as the realized average deviation of same-trait recruitment from recruitment uniformly at random among susceptible neighbours.

    Definition 1, equation (1); Definition 2, equation (2); Figure 1

    See P012-E006, P012-N006.

  2. The recruitment-induced subgraph GSG_S and the augmented subgraph GSUG_{SU} are not identified by RDS data whenever two recruited vertices not joined by a recruitment edge both have reported degree above their recruitment degree.

    Result 1; Appendix, Proof of Result 1

    See P012-E003, P012-N005.

  3. Homophily hh is not point identified when two recruited vertices with different trait values both have reported degree above their recruitment degree.

    Result 2; Appendix, Proof of Result 2 and display (A1)

    See P012-E003, P012-N006.

  4. Preferential recruitment pp is not point identified when some vertex that recruited at least one other vertex has reported degree above its recruitment degree.

    Result 3; Appendix, Proof of Result 3 and display (A2)

    See P012-E002.

  5. The paper states that hh and pp are generally not point identified in RDS studies unless the recruitment tree coincides with the underlying subgraph.

    Results 2 and 3; abstract; introduction (printed p. 154); text after Result 3 (printed pp. 155-156); discussion (printed pp. 157-158)

  6. The paper defines identification regions for hh and pp as extrema over augmented subgraphs and traits compatible with the observed data (GR,dR,tR,ZR)(G_R,\mathbf d_R,\mathbf t_R,\mathbf Z_R). Its five-vertex example gives h∈[−0.625,0.752]h\in[-0.625,0.752] and p∈[−0.138,0.408]p\in[-0.138,0.408], and it states that a simulated-annealing search with a logarithmic cooling schedule converges in probability to an extremal compatible subgraph.

    Definitions 3-4; Figure 1; Web Appendix 1, Proposition 1; Web Appendix 2

    See P012-E001, P012-E007, P012-E006, P012-N007, P012-N008.

  7. In the Hartford RDS-net study (527 recruited, 6 seeds, nomination network of 2,626 people), the paper reports that the full-network values of hh and pp for sex, crack use and homelessness lie inside the intervals computed from RDS data alone, that hh is positive for sex and crack use and negative for homelessness, that recruitment deviations are small, and that every interval covers zero.

    Table 1; Figure 2

    See P012-N002, P012-E001.

  8. The paper argues that claims of homophily or recruitment bias measured from RDS data alone may not be credible, while bounds permit credible statements under three assumptions: the network exists, recruitment follows its edges, and nobody is recruited twice. It reports that the RDS-net regions are considerably smaller than [−1,1][-1,1].

    Results 1-3; Definition 4; Table 1; Discussion

    See P012-N002.

  9. Collecting additional network or recruitment information, with suitable assumptions, can tighten the identification intervals or permit point identification.

    Definitions 3-4; Figure 2; Table 1; introduction (printed p. 154); discussion (printed p. 158)

Corrections

Moderate (3)

A real substantive error with limited reach.

P012-E001 Moderate

Line 22 of the edge proposal lies outside the conditional opened at line 13, so the listing as printed can return a graph outside the compatible set; moving line 22 inside that conditional, with two boundary rules, repairs it.

Web Appendix 2, first proposal mechanism, lines 13–22 and 27, and second proposal mechanism, Web Appendix p. 3 (supplement PDF p. 3); the listing begins on Web Appendix p. 2

See also under Formal statements (Lean): P012-T1, P012-T2, P012-T19, P012-T24

My comment on this entry

As printed
Web Appendix 2, first proposal mechanism, line 2 on Web Appendix p. 2, lines 3–4, 12–23 and 27 on p. 3, as printed (indentation as in the listing).
LineStatement
2:Randomly choose i∈VRi\in V_R and j∈VSUj\in V_{SU} with i≠ji\neq j.
3:if {i,j}∈ESU\{i,j\}\in E_{SU} and {i,j}∉ER\{i,j\}\notin E_R then
4:   Remove {i,j}\{i,j\} from ESU∗E^*_{SU}
12:   Add {i,u}\{i,u\} to ESU∗E^*_{SU}
13:   if j∈VRj\in V_R then
14:      B∼Bernoulli⁡(1/2)B\sim\operatorname{Bernoulli}(1/2)
15:      if B<0.5B<0.5 and U−j≠∅U_{-j}\neq\emptyset then
16:         Randomly choose u∈U−ju\in U_{-j}
17:      else
18:         Add a new vertex uu to VSU∗V^*_{SU}
19:         Randomly choose a trait Zu∗∈{0,1}Z^*_u\in\{0,1\}
20:      end if
21:   end if
22:   Add {j,u}\{j,u\} to ESU∗E^*_{SU}
23:else if {i,j}∉ESU\{i,j\}\notin E_{SU} and ∃u1,u2∈U:{i,u1}∈ESU\exists u_1,u_2\in U:\{i,u_1\}\in E_{SU} and {j,u2}∈ESU\{j,u_2\}\in E_{SU} then
27:Remove any isolated vertices from VSU∗V^*_{SU}

Second proposal mechanism: 1: Choose u∈{u∈VSU:u∉VR}u\in\{u\in V_{SU}:u\notin V_R\}. 2: Set Zu∗=1−ZuZ^*_u=1-Z_u.

The problem

In the listing the conditional opened at line 13, “if j∈VRj\in V_R”, is closed at line 21, and line 22, which adds the edge {j,u}\{j,u\}, comes after it. Line 22 runs also when jj is an unsampled vertex. In that case uu is still the unsampled vertex attached to ii at line 12, and line 22 joins two unsampled vertices. The graph GSUG_{SU} has no such edges by definition. The step can also leave jj with no sampled neighbor, against condition 3 of Definition 3, and line 27 does not remove jj, because jj is not isolated. The listing is the paper’s only specification of the kernel KK of Web Appendix 1, which is defined on pairs of compatible states. Proposition 1 therefore does not cover the listing as printed, and bounds computed by an implementation that follows it literally need not be the bounds of Definition 4.

Two boundary cases also need a rule. Line 27 removes “any isolated vertices”, which would delete a sampled seed of reported degree zero. The second proposal mechanism has no vertex to choose when the current state has no unsampled vertex. The materials available to me do not include the code used for the Hartford bounds, and I do not know whether it followed the listing.

Evidence an exact counterexample

I checked an exact counterexample and a derivation for the corrected listing. Let VR={1,2}V_R=\{1,2\} with vertex 1 recruiting vertex 2, dR=(2,1)\mathbf d_R=(2,1) and ZR=(0,1)\mathbf Z_R=(0,1). The state with VSU={1,2,3}V_{SU}=\{1,2,3\}, ESU={{1,2},{1,3}}E_{SU}=\{\{1,2\},\{1,3\}\} and Z3=0Z_3=0 is compatible. Let i=1i=1 and j=3j=3. Line 4 removes {1,3}\{1,3\}. Since the set U−1U_{-1} is empty, lines 9 and 10 create a new vertex u=4u=4, say with Z4=1Z_4=1, and line 12 adds {1,4}\{1,4\}. The condition of line 13 is false. Line 22 adds {3,4}\{3,4\}, and line 27 removes nothing. The output has the edges {1,2}\{1,2\}, {1,4}\{1,4\} and {3,4}\{3,4\}: the edge {3,4}\{3,4\} joins two unsampled vertices, and vertex 3 has no sampled neighbor. With line 22 moved inside the conditional, the same choices produce the vertices {1,2,4}\{1,2,4\} and the edges {1,2}\{1,2\} and {1,4}\{1,4\}, a compatible state. A second example, with three sampled vertices, fails in the same way for every outcome of the coin and of the new trait.

For the corrected listing every execution returns a compatible pair. If an edge {i,j}\{i,j\} with jj unsampled is removed, ii loses one edge and gains {i,u}\{i,u\}, and jj is deleted if it becomes isolated. If jj is sampled, each of ii and jj loses one edge and gains one edge to an unsampled vertex that was not its neighbor. In lines 23 to 25 each of ii and jj loses one edge to an unsampled vertex and gains {i,j}\{i,j\}. In every case each remaining edge has a sampled endpoint. An unsampled vertex then has no sampled neighbor exactly when it is isolated, and line 27 removes it. The trait flip changes the trait of one unsampled vertex only. The sets U−iU_{-i} and U−jU_{-j} of the current state need no update between lines 12 and 16, because the replacement at ii does not change which existing unsampled vertices are adjacent to jj.

What survives

The definitions of hh, pp, the compatible set and the identification intervals do not depend on the listing, and neither do Results 1–3 or the full-network values of Table 1. I have not established a change to any reported number. Proposition 1 requires a compatible proposal and its other hypotheses (see P012-N008). The corrected listing keeps every proposal inside the compatible set; this alone does not establish convergence, and it does not show what the code behind the intervals of Table 1 did.

Corrected
Web Appendix 2, first proposal mechanism, lines 13–22 and 27, corrected.
LineStatement
13:   if j∈VRj\in V_R then
14:      B∼Bernoulli⁡(1/2)B\sim\operatorname{Bernoulli}(1/2)
15:      if B<0.5B<0.5 and U−j≠∅U_{-j}\neq\emptyset then
16:         Randomly choose u∈U−ju\in U_{-j}
17:      else
18:         Add a new vertex uu to VSU∗V^*_{SU}
19:         Randomly choose a trait Zu∗∈{0,1}Z^*_u\in\{0,1\}
20:      end if
21:      Add {j,u}\{j,u\} to ESU∗E^*_{SU}
22:   end if
27:Remove any isolated unsampled vertices from VSU∗V^*_{SU}

Second proposal mechanism: 1: If there is no unsampled vertex, return the current state. Otherwise choose u∈{u∈VSU:u∉VR}u\in\{u\in V_{SU}:u\notin V_R\}. 2: Set Zu∗=1−ZuZ^*_u=1-Z_u.

Screenshot of lines 12 to 23 of a numbered listing. Red boxes mark line 13, if j in V_R then; line 21, end if; and line 22, Add {j, u} to E*_SU, which is indented at the level of line 13 and comes after the end if.
Web Appendix 2, first proposal mechanism, lines 12–23, Web Appendix p. 3 (supplement PDF p. 3). Marked: the conditional opened at line 13, its end at line 21, and line 22, which follows the end.
Screenshot of the end of the first listing and the two-line second listing. Red boxes mark line 27, Remove any isolated vertices from V*_SU, and line 1 of the second listing, Choose u in the set of unsampled vertices.
Web Appendix 2, line 27 and the second proposal mechanism, Web Appendix p. 3 (supplement PDF p. 3). Marked: the removal of “any isolated vertices” and the choice of an unsampled vertex.
Three small graph drawings. Left: sampled vertices 1 and 2 joined by an arrow, and unsampled vertex 3 joined to vertex 1. Center: vertices 1 and 2 as before, a new unsampled vertex 4 joined to vertex 1, and a dashed orange edge between the unsampled vertices 3 and 4; vertex 3 has no edge to a sampled vertex. Right: vertices 1, 2 and 4 only, with vertex 4 joined to vertex 1.
The counterexample. Left: a compatible state. Center: the output of the listing as printed for i=1i=1, j=3j=3, with the edge {3,4}\{3,4\} between two unsampled vertices. Right: the output of the same choices with line 22 moved inside the conditional of line 13.
My comment and the history of this entry

Agreed.

  1. Classification before my review (the classification when the audit was first completed, before I reviewed it): Moderate error.
  2. I commented.
  3. Independent evaluation. No change was needed: my comment agrees with the entry.

P012-N002 Moderate

The three lower limits of IhI_h in Table 1 cannot be the lower limits of the identification intervals of Definition 4 for the network described on p. 156: a compatible graph has h≤−1462/1983≈−0.7373h\le-1462/1983\approx-0.7373, which is below all three.

Table 1, p. 158 (PDF p. 6), column IhI_h, the three lower limits; the network counts are on p. 156 (PDF p. 4)

See also under Formal statements (Lean): P012-T3, P012-T4, P012-T5, P012-T20

As printed
Table 1, homophily columns, as printed.
TraithhIhI_h
Sex0.26380.2638−0.7027\colorbox{#ffe58a}{$-0.7027$}, 0.82920.8292
Crack cocaine use0.27110.2711−0.6918\colorbox{#ffe58a}{$-0.6918$}, 0.82310.8231
Homelessness−0.2154-0.2154−0.7179\colorbox{#ffe58a}{$-0.7179$}, 0.82420.8242

The RDS-net study included ∣VSU∣=2,626|V_{SU}|=2{,}626 people, of whom n=∣VR∣=527n=|V_R|=527 were either seeds or recruited subjects, and 2,099 were nominated but never-recruited subjects. There are ∣ESU∣=3,307|E_{SU}|=3{,}307 edges in GSUG_{SU}, of which ∣ES∣=1,180|E_S|=1{,}180 link recruited subjects to recruited subjects, and 2,127 link recruited subjects to unrecruited subjects.

The problem

Definition 4 makes IhI_h the smallest interval that contains hh at every compatible pair. The paper identifies the nomination network with GSUG_{SU} and defines did_i as the number of edges at ii. The counts on p. 156 then fix the total of the reported degrees: 2(1,180)+2,127=4,4872(1{,}180)+2{,}127=4{,}487. The 527−6=521527-6=521 recruitment edges use 1,0421{,}042 of these edge-ends and leave S=3,445S=3{,}445. I attach each of the SS spare edge-ends to a new unsampled vertex of its own, and give that vertex the trait opposite to the trait of its sampled neighbor. The resulting graph satisfies the four conditions of Definition 3, and whatever the traits of the recruited subjects its homophily is at most (521−3,445)/(521+3,445)=−1462/1983≈−0.7373(521-3{,}445)/(521+3{,}445)=-1462/1983\approx-0.7373. Each printed lower limit exceeds this value, by at least 0.03450.0345 (sex), 0.04540.0454 (crack cocaine use) and 0.01930.0193 (homelessness), which rounding to four decimals cannot explain. Either the search that produced the limits stopped short of the minimum, or the degrees given to it were not those of the nomination network. In the second case the nomination network violates condition 4 of Definition 3 and is not in the compatible set, and the comparison of the point values with the intervals is not the one the paper describes. The published materials do not supply the Hartford data or analysis code, so I cannot say which case holds or give the correct limits.

Evidence a historical Lean attribution (Generated status: clause support not certified in this release)

The evidence is a derivation, with its algebra checked by exact arithmetic and by a Lean proof. For a graph with aa edges joining two vertices of trait 1, bb edges joining two vertices of trait 0 and cc edges joining unlike vertices, equation (1) reduces to

h=ϕ(a,b,c)=4ab−c2(2a+c)(2b+c).h=\phi(a,b,c)=\frac{4ab-c^2}{(2a+c)(2b+c)}.

In the graph constructed above let aa, bb and cRc_R count the recruitment edges of the three kinds; then a+b+cR=r=521a+b+c_R=r=521. All SS added edges join unlike vertices, and I have h=ϕ(a,b,c)h=\phi(a,b,c) with c=cR+Sc=c_R+S. I set T=r+S=3,966T=r+S=3{,}966. A direct computation gives (2a+c)(2b+c)=T2−(a−b)2>0(2a+c)(2b+c)=T^2-(a-b)^2>0 and

r−ST (2a+c)(2b+c)−(4ab−c2)=2T cR+2ST (a−b)2≥0,\frac{r-S}{T}\,(2a+c)(2b+c)-(4ab-c^2)=2T\,c_R+\frac{2S}{T}\,(a-b)^2\ge0,

and dividing by this positive product I obtain h≤(r−S)/T=−1462/1983h\le(r-S)/T=-1462/1983. The bound uses only the degree total 4,4874{,}487 and the fact that each did_i is at least the recruitment degree of ii; it does not depend on the unknown traits, individual degrees or recruitment graph. The constructed graph has 3,4453{,}445 unsampled vertices, more than the 2,0992{,}099 of the nomination network. Definition 3 allows this, since it does not fix the number of unsampled vertices, and Web Appendix 2 itself creates new ones.

Evaluating ϕ(a,b,cR+S)\phi(a,b,c_R+S) exactly at all 136,503136{,}503 triples of nonnegative integers with a+b+cR=521a+b+c_R=521, I find the largest value −178409/241987≈−0.73727-178409/241987\approx-0.73727, at (a,b,cR)=(260,261,0)(a,b,c_R)=(260,261,0) and (261,260,0)(261,260,0). A Lean proof covers the inequality above for nonnegative rational aa, bb, cRc_R with sum 521521 and its comparison with the three printed limits; it also covers the graph construction, its compatibility and the reduction of equation (1).

The printed lower limits of IhI_h against the bound −1462/1983-1462/1983 (exact differences).
TraitPrinted lower limitPrinted limit minus boundDecimal (rounded)
Sex−0.7027-0.7027685459/19830000685459/198300000.0345670.034567
Crack cocaine use−0.6918-0.6918450803/9915000450803/99150000.0454670.045467
Homelessness−0.7179-0.7179384043/19830000384043/198300000.0193670.019367
What survives

The point values of hh and pp, the upper limits of IhI_h and the intervals IpI_p are not shown wrong by this finding. What the paper reads from the table holds with any wider intervals: each point value lies inside its interval and every interval covers zero. Definitions 1–4 and Results 1–3 do not depend on the table. For nomination-network degrees, the three lower limits of IhI_h and the widths of those intervals change, and the remark on p. 158 that the regions “were considerably smaller than the outcome space [−1,1][-1,1]” has to be reassessed once the limits are recomputed; this finding does not show it false.

Corrected
Table 1, homophily columns, corrected as far as the published information allows for nomination-network degrees.
TraithhIhI_h
Sex0.26380.2638lower limit at most −1462/1983\colorbox{#ffe58a}{$\text{lower limit at most } -1462/1983$}, 0.82920.8292
Crack cocaine use0.27110.2711lower limit at most −1462/1983\colorbox{#ffe58a}{$\text{lower limit at most } -1462/1983$}, 0.82310.8231
Homelessness−0.2154-0.2154lower limit at most −1462/1983\colorbox{#ffe58a}{$\text{lower limit at most } -1462/1983$}, 0.82420.8242

To retain the stated nomination-network interpretation, the three lower limits are to be recomputed from that network’s sampled degrees and the traits of the recruited subjects. For those degrees, the value −1462/1983≈−0.7373-1462/1983\approx-0.7373 is an upper bound on each true lower limit and is not itself the limit. The degree vector dR\mathbf d_R used for the bounds, and its relation to the nomination network, should be stated. If a degree vector other than that of the nomination network was used, the intervals must be checked for that input, and the nomination network is not a member of the corresponding compatible set. The exact minimum has a closed form, given in P012-N007.

Screenshot of Table 1 with columns Trait, h, I_h, p and I_p and rows Sex, Crack cocaine use and Homelessness. Red boxes mark the three lower limits of I_h: -0.7027, -0.6918 and -0.7179.
Table 1, p. 158 (PDF p. 6). The three lower limits of IhI_h are marked.
Two-panel chart. Upper panel: three horizontal blue bars for sex, crack cocaine use and homelessness span the printed intervals from about -0.70 to about 0.83 on an axis from -1 to 1, with a dot at each reported point value of h; an orange dashed vertical line at -0.7373 lies to the left of all three bars. Lower panel: the same lower ends enlarged, with the printed limits -0.7027, -0.6918 and -0.7179 to the right of the dashed line and orange arrows labeled at least 0.0345, 0.0454 and 0.0193 spanning the gaps.
The printed intervals IhI_h of Table 1 on the whole range of hh (upper panel) and their lower ends enlarged (lower panel). The dots show the point values after excluding vertices with missing traits and their incident edges. The dashed line is the bound −1462/1983-1462/1983; for nomination-network degrees, the true lower limit of each interval lies at or to the left of it.

P012-N007 Moderate

The extrema of hh and pp over the compatible set have closed forms that are computed in one pass over the data; the opening sentence of Web Appendix 1 says that there are none.

Web Appendix 1, first sentence, Web Appendix p. 1 (supplement PDF p. 1)

See also under Formal statements (Lean): P012-T6, P012-T7, P012-T21

I disputed this classification, but as a rule, deferred to the models’ final judgment: see my comment and the history of this entry.

As printed

Unfortunately there are no general closed-form expressions for the extrema of hh and pp on C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R).

The problem

The sentence is the premise of the computational procedure of Web Appendixes 1 and 2, that is, of the simulated-annealing search and of Proposition 1. It is false under the conditions stated below. All four endpoints of the identification intervals are explicit functions of quantities read from the observed data: the recruitment edges counted by the traits of their ends, the spare degrees di−dird_i-d_i^r, and the order of recruitment. Each spare edge-end of a sampled vertex can be given its own unsampled neighbor, and the extreme values of hh and of pp are attained when the traits of these neighbors are all opposite to, or all equal to, the trait of the sampled vertex. A search is unnecessary for these four endpoints. For them, the appendix search, its convergence argument and the search for extremizing compatible graphs can all be replaced by the formulas and the two constructions with separate unsampled neighbors. A finite run of a search can return an interval that is too narrow. The formulas give the endpoints exactly. This finding does not by itself show any reported number to be wrong.

Evidence a derivation

The evidence is a complete derivation, with exact enumeration as a check. The notation is that of the corrected statement below.

Homophily. Let xx, yy and cc be the numbers of edges of a compatible graph that join two vertices of trait 1, two vertices of trait 0, and unlike vertices. Equation (1) reduces to

h=ϕ(x,y,c)=4xy−c2(2x+c)(2y+c)=1−c2x+c−c2y+c.h=\phi(x,y,c)=\frac{4xy-c^2}{(2x+c)(2y+c)}=1-\frac{c}{2x+c}-\frac{c}{2y+c}.

Every edge outside the recruitment graph uses at least one spare edge-end of a sampled vertex, of trait 1 if the edge joins two vertices of trait 1 and of trait 0 if it joins two vertices of trait 0. Then a≤x≤a+S1a\le x\le a+S_1, b≤y≤b+S0b\le y\le b+S_0 and cR≤c≤cR+Sc_R\le c\le c_R+S. The function ϕ\phi is nondecreasing in xx and in yy and nonincreasing in cc: for X≥xX\ge x and C≥cC\ge c,

ϕ(X,y,c)−ϕ(x,y,c)=2c(X−x)(2X+c)(2x+c)≥0,ϕ(x,y,c)−ϕ(x,y,C)=2x(C−c)(2x+c)(2x+C)+2y(C−c)(2y+c)(2y+C)≥0,\begin{aligned} \phi(X,y,c)-\phi(x,y,c)&=\frac{2c(X-x)}{(2X+c)(2x+c)}\ge0,\\ \phi(x,y,c)-\phi(x,y,C)&=\frac{2x(C-c)}{(2x+c)(2x+C)}+\frac{2y(C-c)}{(2y+c)(2y+C)}\ge0, \end{aligned}

and the first identity holds with xx and yy exchanged. Under the two conditions on the reported degrees every denominator that occurs is positive. I conclude that ϕ(a,b,cR+S)≤h≤ϕ(a+S1,b+S0,cR)\phi(a,b,c_R+S)\le h\le\phi(a+S_1,b+S_0,c_R). Both bounds are attained. To see this, I keep the recruitment edges and attach to each sampled vertex ii exactly sis_i new unsampled neighbors of degree one; the graph is compatible. If each new vertex has the trait opposite to that of its neighbor, the counts are (a,b,cR+S)(a,b,c_R+S); if each has the trait of its neighbor, they are (a+S1,b+S0,cR)(a+S_1,b+S_0,c_R).

Preferential recruitment. Fix a nonseed jj with recruiter r=rjr=r_j. The bjb_j vertices recruited by rr at or after tjt_j are neighbors of rr that are susceptible just before tjt_j, and they are the only susceptible neighbors of rr fixed by the recruitment graph; aja_j of them have the trait of rr. Any other susceptible neighbor of rr uses a spare edge-end of rr. Let yj≤sry_j\le s_r be the number of these and qj≤yjq_j\le y_j the number with the trait of rr. The fraction subtracted in the summand of jj in equation (2) is (aj+qj)/(bj+yj)(a_j+q_j)/(b_j+y_j), and by cross-multiplication I find

ajbj+sr≤aj+qjbj+yj≤aj+srbj+sr.\frac{a_j}{b_j+s_r}\le\frac{a_j+q_j}{b_j+y_j}\le\frac{a_j+s_r}{b_j+s_r}.

In the graph with one new unsampled neighbor for every spare edge-end, these neighbors are never recruited, and yj=srjy_j=s_{r_j} for every jj at once. With equal traits, qj=srjq_j=s_{r_j} for every jj, which is the minimum of pp; with opposite traits, qj=0q_j=0 for every jj, which is the maximum.

Checks. For the example of Figure 1, I have a=3a=3, b=0b=0, cR=1c_R=1, S1=7S_1=7, S0=2S_0=2 and S=9S=9, and from the formulas I obtain the four values in the table. They equal the extrema over all 26,18226{,}182 compatible pairs of that example, counted up to the labeling of the unsampled vertices and found by exhaustive enumeration with exact arithmetic. The formulas also agree with exhaustive enumeration on 5,0085{,}008 further small data sets, with 86,98886{,}988 compatible pairs in all.

Figure 1 of the paper: the closed forms against the printed extrema.
EndpointClosed formExact valuePrinted in Figure 1
Minimum of hhϕ(3,0,10)\phi(3,0,10)−5/8-5/8−0.625-0.625
Maximum of hhϕ(10,2,1)\phi(10,2,1)79/10579/1050.7520.752
Minimum of pp14(15+0−34+0)\tfrac14\bigl(\tfrac15+0-\tfrac34+0\bigr)−11/80-11/80−0.138-0.138
Maximum of pp14(45+13+0+12)\tfrac14\bigl(\tfrac45+\tfrac13+0+\tfrac12\bigr)49/12049/1200.4080.408
What survives

This correction leaves Definitions 1–4, the identification regions themselves and Results 1–3 unchanged; the four exact extrema round to the values printed in Figure 1. The formulas also give short proofs of the nonidentification conclusions in Results 2 and 3. The central motivation survives: incomplete network observation can leave hh and pp undetermined even when their sharp bounds are easy to compute. The full-network values in Table 1 do not use the search. Simulated annealing may still serve for another objective J(h,p)J(h,p) or for a compatible set with further constraints, and whether Proposition 1 is true is not at issue here. The premise of Web Appendixes 1 and 2 changes: the four endpoints should be computed from the formulas. The intervals of Table 1 can be recomputed in this way once the data are available; if the printed endpoints came from compatible graphs, the exact intervals are no narrower and still cover zero.

Corrected

The extrema of hh and pp on C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R) have closed-form expressions. For observed data with a nonempty compatible set, let si=di−dirs_i=d_i-d_i^r for i∈VRi\in V_R, let Sz=∑i∈VR:Zi=zsiS_z=\sum_{i\in V_R:Z_i=z}s_i for z=0,1z=0,1 and S=S0+S1S=S_0+S_1, and let aa, bb and cRc_R be the numbers of recruitment edges that join two vertices of trait 1, two vertices of trait 0, and unlike vertices. For a nonseed jj let bjb_j be the number of vertices recruited by rjr_j at or after time tjt_j, and aja_j the number of those with trait ZrjZ_{r_j}. Suppose that ∑i∈VR:Zi=zdi>0\sum_{i\in V_R:Z_i=z}d_i>0 for z=0z=0 and for z=1z=1, and that n>∣M∣n>|M|. Let ϕ(x,y,c)=(4xy−c2)/((2x+c)(2y+c))\phi(x,y,c)=(4xy-c^2)/((2x+c)(2y+c)). Then

min⁡Ch=ϕ(a,b,cR+S),max⁡Ch=ϕ(a+S1,b+S0,cR),min⁡Cp=1n−∣M∣∑j∉M[1{Zj=Zrj}−aj+srjbj+srj],max⁡Cp=1n−∣M∣∑j∉M[1{Zj=Zrj}−ajbj+srj].\begin{aligned} \min_{\mathcal C}h&=\phi(a,b,c_R+S),\\ \max_{\mathcal C}h&=\phi(a+S_1,b+S_0,c_R),\\ \min_{\mathcal C}p&=\frac{1}{n-|M|}\sum_{j\notin M}\Bigl[\mathbf 1\{Z_j=Z_{r_j}\}-\frac{a_j+s_{r_j}}{b_j+s_{r_j}}\Bigr],\\ \max_{\mathcal C}p&=\frac{1}{n-|M|}\sum_{j\notin M}\Bigl[\mathbf 1\{Z_j=Z_{r_j}\}-\frac{a_j}{b_j+s_{r_j}}\Bigr]. \end{aligned}

Each extremum is attained at a compatible pair in which every spare edge-end of a sampled vertex has its own unsampled neighbor: of the opposite trait for the minimum of hh and the maximum of pp, of the same trait for the maximum of hh and the minimum of pp. These four endpoints are computed directly from the data; simulated annealing remains available for other objectives or for additional constraints on the graph.

Screenshot of the heading Web Appendix 1 and its first three lines. A red box marks the words: there are no general closed-form expressions for the extrema of h and p.
Web Appendix 1, first sentence, Web Appendix p. 1 (supplement PDF p. 1). The marked words deny that the extrema of hh and pp have closed forms.
My comment and the history of this entry

I think this is actually a major correction, because it obviates much of the motivation of the paper. People should know there is a closed-form solution.

Suggested classification: Major error.

My position My comment above; I suggested a Major error.

The models’ position The same observed data still leave homophily and preferential recruitment unidentified over nontrivial intervals, and the identification results and the verified example survive. Readers need the direct formulas, but the consequence for the paper’s contribution is limited: Moderate.

  1. Classification before my review (the classification when the audit was first completed, before I reviewed it): Moderate error.
  2. I commented and suggested a Major error.
  3. Independent evaluation. Incorporated in part. The entry now says that, for these four endpoints, the closed-form formulas replace the search in the appendix and the convergence argument, and shorten the proofs of Results 2 and 3. My suggested rating (Major) was not adopted.

Minor (8)

Real but inconsequential, for example a misprint that does not change a result.

P012-E003 Minor

The proofs of Results 1 and 2 start from “any” compatible graph with the named edges, for which the described edge moves can fail; both proofs are repaired by starting from a graph built with new unsampled vertices, and the two results stand as printed.

Appendix, proofs of Results 1 and 2, p. 160 (PDF p. 8), right column; the results are stated on p. 155 (PDF p. 3)

See also under Formal statements (Lean): P012-T8, P012-T9, P012-T10, P012-T11

As printed

Proof of Result 1: Let GSU1=(VSU1,ESU1)G^1_{SU}=(V^1_{SU},E^1_{SU}) be any compatible subgraph with {i,u}∈ESU1\{i,u\}\in E^1_{SU}, {j,u}∈ESU1\{j,u\}\in E^1_{SU}, where u∉VRu\notin V_R is an unsampled vertex. Let GSU2=(VSU2,ESU2)G^2_{SU}=(V^2_{SU},E^2_{SU}) be identical to GSU1G^1_{SU} except that {i,j}∈ESU2\{i,j\}\in E^2_{SU}, so neither ii nor jj is connected to uu.

Proof of Result 2: Let (GSU1,ZSU1)(G^1_{SU},\mathbf Z^1_{SU}) be any compatible subgraph and trait set with the property that {i,u1}∈ESU1\{i,u_1\}\in E^1_{SU} and {j,u2}∈ESU1\{j,u_2\}\in E^1_{SU}, where u1u_1 and u2u_2 are unsampled vertices with Zu11=0Z^1_{u_1}=0 and Zu21=1Z^1_{u_2}=1. Let (GSU2,ZSU2)(G^2_{SU},\mathbf Z^2_{SU}) be identical to (GSU1,ZSU1)(G^1_{SU},\mathbf Z^1_{SU}) except that the edges connecting ii and jj to u1u_1 and u2u_2 are swapped: {i,u1}∉ESU2\{i,u_1\}\notin E^2_{SU}, {j,u2}∉ESU2\{j,u_2\}\notin E^2_{SU}, and {i,u2}∈ESU2\{i,u_2\}\in E^2_{SU} and {j,u1}∈ESU2\{j,u_1\}\in E^2_{SU}. Clearly (GSU2,ZSU2)∈C(GR,dR,ZR)(G^2_{SU},\mathbf Z^2_{SU})\in\mathrm{C}(G_R,\mathbf d_R,\mathbf Z_R).

The problem

Each proof takes as its starting graph any compatible graph that contains the named edges, changes some edges, and asserts that the result is compatible. This can fail. In the proof of Result 1 the edge {i,j}\{i,j\} may already belong to ESU1E^1_{SU}, since its absence from ERE_R does not exclude it. Removing {i,u}\{i,u\} and {j,u}\{j,u\} then lowers the degrees of ii and jj by one, and the later claim that {i,j}∉GS1\{i,j\}\notin G^1_S is false. In the proof of Result 2 the edges {i,u2}\{i,u_2\} and {j,u1}\{j,u_1\} may already belong to ESU1E^1_{SU}. The swap then lowers the degrees of ii and jj by one, the pair (GSU2,ZSU2)(G^2_{SU},\mathbf Z^2_{SU}) is not compatible, and display (A1), which uses one denominator for h1h_1 and h2h_2, does not apply.

Evidence an exact counterexample

The evidence is two exact counterexamples to the printed steps and a derivation of the corrected ones. Let VR={1,2,3}V_R=\{1,2,3\} with recruitments 1→2→31\to2\to3, reported degrees dR=(3,2,3)\mathbf d_R=(3,2,3) and traits ZR=(0,0,1)\mathbf Z_R=(0,0,1), and let i=1i=1, j=3j=3. I write an edge {k,l}\{k,l\} as klkl.

  • Result 1: with u=4u=4, the graph with edges 1212, 2323, 1313, 1414, 3434 is compatible and contains 1414 and 3434. The printed step removes 1414 and 3434, adds 1313, which is already present, and deletes vertex 4. The result has edges 1212, 2323, 1313 and sampled degrees (2,2,2)(2,2,2) in place of (3,2,3)(3,2,3).
  • Result 2: with u1=4u_1=4, u2=5u_2=5, Z4=0Z_4=0 and Z5=1Z_5=1, the graph with edges 1212, 2323, 1414, 3535, 1515, 3434 is compatible and contains 1414 and 3535. The swap removes 1414 and 3535 and adds 1515 and 3434, which are already present. The result has edges 1212, 2323, 1515, 3434 and sampled degrees (2,2,2)(2,2,2).

For the corrected steps let sv=dv−dvr≥0s_v=d_v-d_v^r\ge0. In the starting graph of the corrected proof of Result 1, the vertex ii has degree dir+1+(si−1)=did_i^r+1+(s_i-1)=d_i, likewise for jj, and every other sampled vv has degree dvr+sv=dvd_v^r+s_v=d_v. Each unsampled vertex has a sampled neighbor, and no two unsampled vertices are adjacent. The graph is then compatible. Its edges among sampled vertices are the recruitment edges, and since (i,j)∉ER(i,j)\notin E_R with i<ji<j, the edge {i,j}\{i,j\} is not one of them. Replacing {i,u}\{i,u\} and {j,u}\{j,u\} by {i,j}\{i,j\} and deleting uu leaves every sampled degree unchanged. In the corrected proof of Result 2 the vertices u1u_1 and u2u_2 are each adjacent to one sampled vertex only. Then {i,u2}\{i,u_2\} and {j,u1}\{j,u_1\} are absent, and the swap preserves every degree. Suppose that Zi=0Z_i=0 and Zj=1Z_j=1, as in the printed proof, and let m=∣ESU∣m=|E_{SU}| and Q=∑vdvZvQ=\sum_vd_vZ_v. The common denominator of h1h_1 and h2h_2 is D=Q(2m−Q)/(2m)D=Q(2m-Q)/(2m), and D>0D>0 because Q≥dj>0Q\ge d_j>0 and 2m−Q≥di>02m-Q\ge d_i>0. The numerators differ by 2, and I obtain h1−h2=2/D>0h_1-h_2=2/D>0, as (A1) states. On the instance VR={1,2}V_R=\{1,2\} with vertex 1 recruiting vertex 2, dR=(2,2)\mathbf d_R=(2,2) and ZR=(0,1)\mathbf Z_R=(0,1), the two graphs have h1=1/3h_1=1/3 and h2=−1h_2=-1, and 2/D=4/32/D=4/3.

What survives

Results 1 and 2 hold exactly as printed, with the same hypotheses, and display (A1) is correct for the corrected swap. Result 2 needs no condition that excludes a recruitment edge between ii and jj: in the last instance above ii recruits jj. The correction restricts the choice of a witness in two existence proofs and adds no assumption on the data or the network. Definitions 1–4, Result 3, Figure 1 and the application do not depend on these steps.

Corrected

Results 1 and 2 are unchanged. In both proofs write sv=dv−dvrs_v=d_v-d_v^r for v∈VRv\in V_R.

Proof of Result 1: Construct GSU1G^1_{SU} as follows. Take the recruitment edges as undirected edges. Add one new unsampled vertex uu with the edges {i,u}\{i,u\} and {j,u}\{j,u\}. Attach si−1s_i-1 further new unsampled vertices to ii, sj−1s_j-1 to jj, and svs_v to every other v∈VRv\in V_R, each adjacent to that one sampled vertex only. Then GSU1G^1_{SU} is compatible and {i,j}∉ESU1\{i,j\}\notin E^1_{SU}. Let GSU2G^2_{SU} be obtained from GSU1G^1_{SU} by removing {i,u}\{i,u\} and {j,u}\{j,u\}, adding {i,j}\{i,j\}, and deleting uu. The proof continues as printed.

Proof of Result 2: Construct (GSU1,ZSU1)(G^1_{SU},\mathbf Z^1_{SU}) as follows. Take the recruitment edges as undirected edges and attach svs_v new unsampled vertices to each v∈VRv\in V_R, each adjacent to vv only. Let u1u_1 be one of those attached to ii and u2u_2 one of those attached to jj, with Zu11=0Z^1_{u_1}=0 and Zu21=1Z^1_{u_2}=1; the traits of the other unsampled vertices are arbitrary. Then {i,u2}∉ESU1\{i,u_2\}\notin E^1_{SU} and {j,u1}∉ESU1\{j,u_1\}\notin E^1_{SU}. Let (GSU2,ZSU2)(G^2_{SU},\mathbf Z^2_{SU}) be identical to (GSU1,ZSU1)(G^1_{SU},\mathbf Z^1_{SU}) except that {i,u1}\{i,u_1\} and {j,u2}\{j,u_2\} are replaced by {i,u2}\{i,u_2\} and {j,u1}\{j,u_1\}. The swap preserves every degree, so (GSU2,ZSU2)∈C(GR,dR,ZR)(G^2_{SU},\mathbf Z^2_{SU})\in\mathcal C(G_R,\mathbf d_R,\mathbf Z_R), and the proof continues as printed.

Screenshot of the proof of Result 1. Red boxes mark the words: be any compatible subgraph, which run over a line break.
Appendix, proof of Result 1, p. 160 (PDF p. 8), right column. The marked words allow any compatible starting graph that contains {i,u}\{i,u\} and {j,u}\{j,u\}.
Screenshot of the proof of Result 2 up to the words The difference is. Red boxes mark the words: be any compatible subgraph, and the sentence: Clearly (G^2_SU, Z^2_SU) is in C(G_R, d_R, Z_R).
Appendix, proof of Result 2, p. 160 (PDF p. 8), right column. The marked words allow any compatible starting pair and assert that the pair obtained by the swap is compatible.

P012-E002 Minor

Display (A2) keeps one recruitment by ii; when ii recruited two or more subjects, p1−p2p_1-p_2 is a sum over all of them, with the same sign, and Result 3 is unchanged.

Appendix, proof of Result 3, display (A2), p. 160 (PDF p. 8), right column; Result 3 is stated on p. 155 (PDF p. 3)

See also under Formal statements (Lean): P012-T12, P012-T13

As printed
p1−p2  =  1n−∣M∣[(1{Zi=Zj}−K1∣Si(tj)∣)−(1{Zi=Zj}−K1−1∣Si(tj)∣)]=−1(n−∣M∣)∣Si(tj)∣≠0.(A2)\begin{aligned} p_1-p_2&\;\colorbox{#ffe58a}{$\displaystyle =$}\;\frac{1}{n-|M|}\left[\left(1\{Z_i=Z_j\}-\frac{K^1}{|S_i(t_j)|}\right)-\left(1\{Z_i=Z_j\}-\frac{K^1-1}{|S_i(t_j)|}\right)\right]\\ &=\frac{-1}{(n-|M|)|S_i(t_j)|}\ne0.\qquad\text{(A2)} \end{aligned}
The problem

The proof defines p1p_1 and p2p_2 as the whole functional of equation (2) and changes the trait of an unsampled neighbor uu of ii from 1 to 0. Since the vertex uu is never recruited, I have u∈Si(tℓ)u\in S_i(t_\ell) for every ℓ\ell that ii recruited, and the change of ZuZ_u alters every one of those summands of (2). Display (A2) keeps the summand of the single recruit jj. Its first equality is therefore false when ii has two or more recruits, which Result 3 allows (“at least 1 other vertex”). The omitted terms have the sign of the term kept, and p1−p2p_1-p_2 is still nonzero. The printed minus sign is correct.

Evidence an exact counterexample

The evidence is a derivation and an exact counterexample. Under the printed assumption Zi=1Z_i=1, the graph and the recruitment times are the same in the two pairs. Each set Srℓ(tℓ)S_{r_\ell}(t_\ell) and each indicator 1{Zℓ=Zrℓ}1\{Z_\ell=Z_{r_\ell}\} is then the same in both. If ℓ\ell was not recruited by ii, then u∉Srℓ(tℓ)u\notin S_{r_\ell}(t_\ell), because uu has no sampled neighbor other than ii, and the summand of ℓ\ell does not change. If ℓ\ell was recruited by ii, the number of vertices of Si(tℓ)S_i(t_\ell) with the trait of ii is smaller by one in the second pair. Subtracting the two averages term by term, I obtain the corrected display below. There exists a compatible pair of the kind the proof needs: it has the recruitment edges and, for each v∈VRv\in V_R, dv−dvrd_v-d_v^r new unsampled vertices adjacent to vv only.

For the counterexample let VR={1,2,3}V_R=\{1,2,3\}, let the seed 1 recruit 2 and then 3, let dR=(3,1,1)\mathbf d_R=(3,1,1) and Z1=Z2=Z3=1Z_1=Z_2=Z_3=1, and let u=4u=4 be the one unsampled neighbor of vertex 1. Then S1(t2)={2,3,4}S_1(t_2)=\{2,3,4\} and S1(t3)={3,4}S_1(t_3)=\{3,4\}. With Z4=1Z_4=1, p1=0p_1=0. With Z4=0Z_4=0,

p2=12[(1−23)+(1−12)]=512,p_2=\frac12\Bigl[\Bigl(1-\frac23\Bigr)+\Bigl(1-\frac12\Bigr)\Bigr]=\frac5{12},

and I find p1−p2=−5/12p_1-p_2=-5/12. Display (A2) gives −1/6-1/6 with j=2j=2 and −1/4-1/4 with j=3j=3.

What survives

Result 3 holds with its printed hypotheses, and display (A2) is exact when ii has exactly one recruit. Equation (2), Figure 1 and the application use the full functional pp, and no later formula uses the magnitude in (A2). Nothing else changes.

Corrected

Suppose, as in the printed proof, that Zi=1Z_i=1, that uu is an unsampled vertex adjacent to ii and to no other vertex of VRV_R, and that Zu=1Z_u=1 in ZSU1\mathbf Z^1_{SU} and Zu=0Z_u=0 in ZSU2\mathbf Z^2_{SU}. Then

p1−p2=−1n−∣M∣∑ℓ∈VR∖M: rℓ=i1∣Si(tℓ)∣<0.(A2)p_1-p_2=\colorbox{#ffe58a}{$\displaystyle -\frac{1}{n-|M|}\sum_{\ell\in V_R\setminus M:\,r_\ell=i}\frac{1}{|S_i(t_\ell)|}$}<0.\qquad\text{(A2)}

The sum is not empty, because ii recruited jj, and each ∣Si(tℓ)∣|S_i(t_\ell)| is a positive integer. Therefore preferential recruitment is not point identified.

Screenshot of the sentence that defines p_1 and p_2 and of display (A2). A red box marks the first equals sign, between p_1 minus p_2 and the bracket that contains the terms of the single recruit j.
Appendix, proof of Result 3, display (A2), p. 160 (PDF p. 8), right column. The marked equality holds only when ii has a single recruit.

P012-E006 Minor

Equations (1) and (2) are undefined at some compatible pairs, so Definition 4 needs the conditions under which hh and pp are defined on the whole compatible set.

Definitions 1, 2 and 4, equations (1) and (2), p. 155 (PDF p. 3); the same functionals are used in Web Appendix 1, p. 1

See also under Formal statements (Lean): P012-T14, P012-T15, P012-T24

As printed

Definition 1: Homophily is the correlation between trait values of vertices connected by an edge,

h(GSU,ZSU)=∑i,j∈VSU(Aij−didj/2∣ESU∣)ZiZj∑i,j∈VSU(diδij−didj/2∣ESU∣)ZiZj,(1)h(G_{SU},\mathbf Z_{SU})=\frac{\sum_{i,j\in V_{SU}}(A_{ij}-d_id_j/2|E_{SU}|)Z_iZ_j}{\colorbox{#ffe58a}{$\displaystyle \sum_{i,j\in V_{SU}}(d_i\delta_{ij}-d_id_j/2|E_{SU}|)Z_iZ_j$}},\qquad(1)

Definition 2: Preferential recruitment is the average deviation from proportional recruitment, given knowledge of GSUG_{SU}, GRG_R, tR\mathbf t_R, and ZSU\mathbf Z_{SU},

p(GSU,GR,tR,ZSU)=1n−∣M∣∑j∉M(1{Zj=Zrj}−∣{k∈Srj(tj):Zrj=Zk}∣∣Srj(tj)∣),(2)p(G_{SU},G_R,\mathbf t_R,\mathbf Z_{SU})=\colorbox{#ffe58a}{$\displaystyle \frac{1}{n-|M|}$}\sum_{j\notin M}\left(1\{Z_j=Z_{r_j}\}-\frac{|\{k\in S_{r_j}(t_j):Z_{r_j}=Z_k\}|}{|S_{r_j}(t_j)|}\right),\qquad(2)

where MM is the set of seeds.

Definition 4: Identification region: The identification regions for hh and pp are given by the smallest intervals that contain h(GSU,ZSU)h(G_{SU},\mathbf Z_{SU}) and p(GSU,GR,tR,ZSU)p(G_{SU},G_R,\mathbf t_R,\mathbf Z_{SU}) for every (GSU,ZSU)∈C(GR,dR,ZR)(G_{SU},\mathbf Z_{SU})\in\mathrm{C}(G_R,\mathbf d_R,\mathbf Z_R).

The problem

Definition 4 takes hh and pp at every compatible pair, but compatibility does not make the two formulas defined. Let m=∣ESU∣m=|E_{SU}| and q=∑i∈VSUdiZiq=\sum_{i\in V_{SU}}d_iZ_i. For m>0m>0 the denominator of (1) equals q(2m−q)/(2m)q(2m-q)/(2m). When all vertices of positive degree have the same trait (q=0q=0 or q=2mq=2m), this denominator and the numerator both vanish, and (1) reads 0/00/0. For m=0m=0 the terms didj/2∣ESU∣d_id_j/2|E_{SU}| are themselves undefined. The factor 1/(n−∣M∣)1/(n-|M|) in (2) is undefined when the sample consists of seeds only. The paper states no condition that excludes these cases.

Evidence an exact counterexample

The evidence is a derivation and three exact examples. The identities Zi2=ZiZ_i^2=Z_i and ∑idi=2m\sum_id_i=2m imply that

∑i,j∈VSU(diδij−didj2m)ZiZj=q−q22m=q(2m−q)2m.\sum_{i,j\in V_{SU}}\Bigl(d_i\delta_{ij}-\frac{d_id_j}{2m}\Bigr)Z_iZ_j=q-\frac{q^2}{2m}=\frac{q(2m-q)}{2m}.

I write D1=qD_1=q and D0=2m−qD_0=2m-q for the total degrees of the vertices with trait 11 and trait 00; the denominator is then D0D1/(2m)D_0D_1/(2m).

  1. Let vertex 1 be a seed that recruits vertex 2, with dR=(1,1)\mathbf d_R=(1,1) and ZR=(0,0)\mathbf Z_R=(0,0). The only compatible pair is the single edge {1,2}\{1,2\}. Here m=1m=1 and q=0q=0, and (1) is 0/00/0, while p=0p=0.
  2. Let vertices 1 and 2 both be seeds, with dR=(1,1)\mathbf d_R=(1,1) and ZR=(0,1)\mathbf Z_R=(0,1). With the edge {1,2}\{1,2\} the pair is compatible and h=−1h=-1, and (2) is an empty sum divided by n−∣M∣=0n-|M|=0 at every compatible pair.
  3. Let vertex 1 recruit vertex 2, with dR=(1,4)\mathbf d_R=(1,4) and ZR=(1,1)\mathbf Z_R=(1,1). Vertex 2 has three unsampled neighbors, and the eight assignments of their traits are the eight compatible pairs. Seven of them have h∈{−3/5,−1/3,−1/7}h\in\{-3/5,-1/3,-1/7\}; at the eighth, with all traits equal to 1, hh is undefined.

The condition in the corrected Definition 4 is the exact one for hh. If, for each trait, the reported degrees of the sampled vertices with that trait have a positive sum, then D0>0D_0>0 and D1>0D_1>0 at every compatible pair, because compatible graphs preserve the reported degrees. If all sampled vertices of positive degree share one trait, giving that trait to every unsampled vertex yields a compatible pair at which (1) is 0/00/0, because every edge of GSUG_{SU} has a sampled endpoint.

What survives

No result or number changes. The hypotheses of Result 2 imply that there exist sampled vertices of both traits with positive degree, and the hypothesis of Result 3 implies that at least one recruitment took place. Then hh and pp are defined on the whole compatible set where those results apply. Figure 1 meets both conditions, and the application has 527−6=521527-6=521 recruitments. What changes is the statement of Definition 4, the optimization of Web Appendix 1 over the compatible set, and the sentence on p. 158 that three assumptions suffice for the bounds: each needs the two conditions below.

Corrected

Definition 1 applies to pairs (GSU,ZSU)(G_{SU},\mathbf Z_{SU}) with ∣ESU∣>0|E_{SU}|>0 and D0D1>0D_0D_1>0, where Dz=∑i∈VSU:Zi=zdiD_z=\sum_{i\in V_{SU}:Z_i=z}d_i; at any other pair hh is undefined. Definition 2 applies when n>∣M∣n>|M|; otherwise pp is undefined.

Definition 4: Identification region. Suppose that ∑i∈VR:Zi=zdi>0\sum_{i\in V_R:Z_i=z}d_i>0 for z=0z=0 and for z=1z=1. Then hh is defined at every pair in C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R), and the identification region for hh is the smallest interval that contains h(GSU,ZSU)h(G_{SU},\mathbf Z_{SU}) for every (GSU,ZSU)∈C(GR,dR,ZR)(G_{SU},\mathbf Z_{SU})\in\mathcal C(G_R,\mathbf d_R,\mathbf Z_R). Suppose that n>∣M∣n>|M|. Then pp is defined at every pair in C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R), and the identification region for pp is the smallest interval that contains p(GSU,GR,tR,ZSU)p(G_{SU},G_R,\mathbf t_R,\mathbf Z_{SU}) for every such pair. If the condition for hh or for pp fails, the identification region of that functional is undefined.

Screenshot of Definition 1 with equation (1) for h as a ratio of two sums. A red box marks the denominator, the sum of (d_i delta_ij minus d_i d_j over 2|E_SU|) Z_i Z_j.
Definition 1 and equation (1), p. 155 (PDF p. 3), left column. The marked denominator is zero when all vertices of positive degree have the same trait.
Screenshot of equation (2) for p. A red box marks the leading factor 1 over (n minus |M|).
Equation (2), p. 155 (PDF p. 3), top of the right column. The marked factor 1/(n−∣M∣)1/(n-|M|) is undefined for a sample that consists of seeds only.
Screenshot of Definition 4. A red box marks the words: every (G_SU, Z_SU) in C(G_R, d_R, Z_R).
Definition 4, p. 155 (PDF p. 3), right column. The marked words take hh and pp at every compatible pair.

P012-E007 Minor

The cooling schedule is called “non-decreasing” where “non-increasing” is meant, and the schedule and the sums of display (2) start at t=1t=1, where log⁡t=0\log t=0.

Web Appendix 1, Web Appendix p. 1 (supplement PDF p. 1), sentence after the definition of the kernel KK; Proposition 1 and display (2), Web Appendix p. 2

See also under Formal statements (Lean): P012-T16, P012-T24

As printed

Let TtT_t be a positive non-decreasing sequence indexed by tt, with lim⁡t→∞Tt=0\lim_{t\to\infty}T_t=0. (Web Appendix p. 1.)

Proposition 1. Let the cooling schedule be given by Tt=(ϵlog⁡(t))−1T_t=\bigl(\epsilon\log(t)\bigr)^{-1} where ϵ>0\epsilon>0 is a constant. (Web Appendix p. 2.)

∑t=1∞exp⁡[−D∗/Tt]=∑t=1∞exp⁡[−D∗ϵlog⁡(t)]=∑t=1∞1tD∗ϵ≥∑t=1∞1t=∞(2)\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=1$}}^{\infty}\exp[-D^*/T_t]=\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=1$}}^{\infty}\exp[-D^*\epsilon\log(t)]=\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=1$}}^{\infty}\frac{1}{t^{D^*\epsilon}}\ge\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=1$}}^{\infty}\frac1t=\infty\qquad(2)
The problem

A positive non-decreasing sequence satisfies Tt≥T1>0T_t\ge T_1>0 and cannot tend to zero; the schedule of Proposition 1 is decreasing for t>1t>1, and “non-increasing” is meant. The schedule is also undefined at the first index, since log⁡1=0\log1=0, although the sums of display (2) start at t=1t=1.

Evidence a derivation

The evidence is a derivation. For t>1t>1, differentiation gives

ddt 1ϵlog⁡t=−1ϵ t (log⁡t)2<0.\frac{d}{dt}\,\frac{1}{\epsilon\log t}=-\frac{1}{\epsilon\,t\,(\log t)^2}<0.

For t≥2t\ge2, the bound D∗ϵ≤1D^*\epsilon\le1 implies that t−D∗ϵ≥t−1t^{-D^*\epsilon}\ge t^{-1}, and ∑t≥21/t=∞\sum_{t\ge2}1/t=\infty. The comparison in display (2) is unchanged when the first term is dropped.

What survives

The bound D∗ϵ≤1D^*\epsilon\le1 and the comparison with the harmonic series stand. This correction concerns the wording and the first index only; the other hypotheses of the convergence theorem are the subject of P012-N008, and the proposal kernel that of P012-E001.

Corrected

Let TtT_t be a positive non-increasing sequence indexed by tt, with lim⁡t→∞Tt=0\lim_{t\to\infty}T_t=0.

Proposition 1. Let the cooling schedule be given by a finite T1≥T2T_1\ge T_2 and Tt=(ϵlog⁡(t))−1T_t=\bigl(\epsilon\log(t)\bigr)^{-1} for t≥2t\ge2, where ϵ>0\epsilon>0 is a constant.

∑t=2∞exp⁡[−D∗/Tt]=∑t=2∞exp⁡[−D∗ϵlog⁡(t)]=∑t=2∞1tD∗ϵ≥∑t=2∞1t=∞(2)\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=2$}}^{\infty}\exp[-D^*/T_t]=\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=2$}}^{\infty}\exp[-D^*\epsilon\log(t)]=\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=2$}}^{\infty}\frac{1}{t^{D^*\epsilon}}\ge\sum_{\colorbox{#ffe58a}{$\displaystyle \scriptstyle t=2$}}^{\infty}\frac1t=\infty\qquad(2)
Screenshot of three lines of Web Appendix 1. A red box marks the word non-decreasing in the sentence: Let T_t be a positive non-decreasing sequence indexed by t, with limit zero.
Web Appendix 1, Web Appendix p. 1 (supplement PDF p. 1). The marked word should read “non-increasing”.
Screenshot of display (2), a chain of four infinite sums ending in a comparison with the sum of 1 over t. Red boxes mark the lower limit t=1 under each of the four summation signs.
Proof of Proposition 1, display (2), Web Appendix p. 2 (supplement PDF p. 2). The four marked sums start at t=1t=1, where Tt=(ϵlog⁡t)−1T_t=(\epsilon\log t)^{-1} is undefined.

P012-N001 Minor

The homophily estimators of RDS Analyst were publicly documented before the paper was accepted; the statement that the estimators of both programs “are not documented” is wrong for that program.

Introduction, p. 154 (PDF p. 2), right column, sentence beginning “Two software tools”; reference 39 is on p. 160 (PDF p. 8)

As printed

Two software tools for analysis of RDS data produce estimates of homophily or preferential recruitment—RDSAT (38) and RDS Analyst (39)—but the estimators used by these programs are not documented.

The problem

The sentence says of both programs that their estimators are not documented. For RDS Analyst this is not the case. Reference 39 is the RDS package, version 0.5, released on 28 November 2013. Its help page for the function homophily.estimates says that the function “computes an estimate of the population homophily and the recruitment homophily based on a categorical variable” and defines both ratios, and RDS Analyst obtains its homophily output by calling this function. The paper was accepted on 9 March 2017. I could not obtain the RDSAT 7.1 manual, and the half of the sentence that concerns RDSAT is not settled either way. The citation of reference 39 for RDS Analyst is apt and is not an error.

Evidence a check of the printed page

A check of public sources: the help file of homophily.estimates in RDS 0.5 (https://github.com/cran/RDS/blob/0.5/man/homophily.estimates.Rd), the function’s source in the same release, the help pages of RDS Analyst for recruitment homophily and population homophily (last modified in August and September 2013), and the source of the RDS Analyst interface, in which the two dialogs for homophily call homophily.estimates.

What survives

No definition, result or number of the paper depends on this sentence. That the estimators are documented does not mean that they identify hh or pp as the paper defines them; the paper’s argument on that point is untouched.

Corrected

Two software tools for analysis of RDS data produce estimates of homophily or preferential recruitment—RDSAT (38) and RDS Analyst (39); the RDS Analyst estimators are described in the help page of the function homophily.estimates of the RDS package (39).

Screenshot of four lines of the introduction. A red box marks the words: the estimators used by these programs are not documented.
Introduction, p. 154 (PDF p. 2), right column. The marked clause says that the estimators of RDSAT and RDS Analyst are not documented.

P012-N005 Minor

The last step of the proof of Result 1 cites Definition 3 where it needs the compatibility of recruitment-induced subgraphs defined at the start of the same proof.

Appendix, proof of Result 1, p. 160 (PDF p. 8), right column, sentence beginning “Clearly both”

See also under Formal statements (Lean): P012-T17

As printed

Clearly both GS1G^1_S and GS2G^2_S are compatible with the observed data under the conditions of definition 3, but {i,j}∉GS1\{i,j\}\notin G^1_S and {i,j}∈GS2\{i,j\}\in G^2_S.

The problem

Definition 3 concerns augmented graphs, and its condition 4 requires each sampled vertex to have exactly its reported degree. Removing the unsampled vertices lowers those degrees, and GS1G^1_S and GS2G^2_S need not satisfy Definition 3. They satisfy the definition given at the opening of the proof, which asks that the degree of ii in GSG_S be at most did_i, and this is what the step needs.

Evidence an exact counterexample

A check of the page and a small example. Let vertices 1 and 2 be seeds with d1=d2=1d_1=d_2=1 and no recruitment, and let the augmented graph have the edges {1,u}\{1,u\} and {2,u}\{2,u\}. Removing uu leaves degrees (0,0)(0,0), which fail condition 4 of Definition 3 and satisfy the opening definition.

What survives

The step is valid with the right reference, and Result 1 is unaffected.

Corrected

Clearly both GS1G^1_S and GS2G^2_S are compatible with the observed data in the sense defined at the start of this proof, but {i,j}∉GS1\{i,j\}\notin G^1_S and {i,j}∈GS2\{i,j\}\in G^2_S.

Screenshot of the end of the proof of Result 1. Red boxes mark the words: under the conditions of definition 3, which run over a line break.
Appendix, end of the proof of Result 1, p. 160 (PDF p. 8), right column. The marked words cite Definition 3.

P012-N006 Minor

In the proof of Result 2 the function hh is written four times with a redundant argument GRG_R, and on p. 155 once with an unsubscripted Z\mathbf Z; the function h(GSU,ZSU)h(G_{SU},\mathbf Z_{SU}) of equation (1) is meant.

Appendix, proof of Result 2, p. 160 (PDF p. 8), right column, four occurrences; and p. 155 (PDF p. 3), sentence beginning “To ease notation”

See also under Formal statements (Lean): P012-T18

As printed

[…] such that h(GSU1,GR, ZSU1)≠h(GSU2,GR, ZSU2)h(G^1_{SU},\colorbox{#ffe58a}{$G_R,$}\,\mathbf Z^1_{SU})\ne h(G^2_{SU},\colorbox{#ffe58a}{$G_R,$}\,\mathbf Z^2_{SU}). […] Let h1=h(GSU1,GR, ZSU1)h_1=h(G^1_{SU},\colorbox{#ffe58a}{$G_R,$}\,\mathbf Z^1_{SU}) and h2=h(GSU2,GR, ZSU2)h_2=h(G^2_{SU},\colorbox{#ffe58a}{$G_R,$}\,\mathbf Z^2_{SU}) be the calculated values of homophily. (p. 160.)

To ease notation, we will often use hh and pp to refer to h(GSU,Z)h(G_{SU},\colorbox{#ffe58a}{$\mathbf Z$}) and p(GSU,GR,tR,ZSU)p(G_{SU},G_R,\mathbf t_R,\mathbf Z_{SU}) respectively. (p. 155.)

The problem

Definition 1 gives hh two arguments, the augmented graph and its traits, and hh does not depend on the recruitment graph. The symbol Z\mathbf Z without a subscript is not defined in the paper. Both are slips of notation.

Evidence a check of the printed page

A check of the two pages against Definition 1, equation (1) and Definition 4. Display (A1) uses adjacency, degrees and traits only, in agreement with equation (1).

What survives

The calculation in (A1) and Result 2 are unaffected.

Corrected

[…] such that h(GSU1,ZSU1)≠h(GSU2,ZSU2)\colorbox{#ffe58a}{$h(G^1_{SU},\mathbf Z^1_{SU})$}\ne\colorbox{#ffe58a}{$h(G^2_{SU},\mathbf Z^2_{SU})$}. […] Let h1=h(GSU1,ZSU1)h_1=\colorbox{#ffe58a}{$h(G^1_{SU},\mathbf Z^1_{SU})$} and h2=h(GSU2,ZSU2)h_2=\colorbox{#ffe58a}{$h(G^2_{SU},\mathbf Z^2_{SU})$} be the calculated values of homophily.

To ease notation, we will often use hh and pp to refer to h(GSU,ZSU)h(G_{SU},\colorbox{#ffe58a}{$\mathbf Z_{SU}$}) and p(GSU,GR,tR,ZSU)p(G_{SU},G_R,\mathbf t_R,\mathbf Z_{SU}) respectively.

Screenshot of the proof of Result 2 up to the words The difference is. Four red boxes mark the argument G_R inside h(G^1_SU, G_R, Z^1_SU) and h(G^2_SU, G_R, Z^2_SU), each of which appears twice.
Appendix, proof of Result 2, p. 160 (PDF p. 8), right column. The four marked occurrences of GRG_R are redundant arguments of hh.
Screenshot of the sentence beginning To ease notation. A red box marks the bold Z in h(G_SU, Z), which has no subscript.
p. 155 (PDF p. 3), right column, below equation (2). The marked Z\mathbf Z lacks the subscript SUSU.

P012-N009 Minor

Reference [1] of the Web Appendix omits the second author, gives the first author the wrong initials and misspells “simulated”.

Web Appendix, References, entry [1], Web Appendix p. 3 (supplement PDF p. 3)

As printed

[1] Kirkpatrick C. D., Vecchi M. P.. Optimization by simmulated annealing Science. 1983;220:671–680.

The problem

The article is by S. Kirkpatrick, C. D. Gelatt Jr. and M. P. Vecchi. The entry omits Gelatt, attaches his initials to Kirkpatrick, and misspells the title.

Evidence a check of the printed page

A check of the published article: its first page carries the title “Optimization by Simulated Annealing” and the byline “S. Kirkpatrick, C. D. Gelatt, Jr., M. P. Vecchi”.

What survives

The journal, volume and pages identify the article, and nothing in the Web Appendix depends on the entry.

Corrected

[1] Kirkpatrick S, Gelatt CD Jr, Vecchi MP. Optimization by simulated annealing. Science. 1983;220(4598):671–680.

Screenshot of the heading References and entries 1 and 2. Red boxes mark the author list Kirkpatrick C. D., Vecchi M. P. and the word simmulated in entry 1.
Web Appendix, References, entry [1], Web Appendix p. 3 (supplement PDF p. 3). The author list and the misspelled word are marked.

Clarity notes

Imprecise or ambiguous wording whose intended meaning is correct; not an error.

P012-N008 Clarity note

Proposition 1 holds for the intended proposal, but its proof quotes Hajek’s theorem without the theorem’s standing hypotheses, restricts ϵ\epsilon to (0,1)(0,1) without need, and cites for connectivity an argument made for a smaller state space.

Web Appendix 1, Proposition 1 and its proof, Web Appendix p. 2 (supplement PDF p. 2); the sentence on connectivity is in Web Appendix 2, Web Appendix p. 3

As printed

Proposition 1. Let the cooling schedule be given by Tt=(ϵlog⁡(t))−1T_t=\bigl(\epsilon\log(t)\bigr)^{-1} where ϵ>0\epsilon>0 is a constant. Then lim⁡t→∞Pr⁡((GSU,ZSU)t∈M)=1\lim_{t\to\infty}\Pr\bigl((G_{SU},\mathbf Z_{SU})_t\in\mathcal M\bigr)=1.

Proof. Let J(h,p)=1/(1+ϵ+h)J(h,p)=1/(1+\epsilon+h) for 0<ϵ<10<\epsilon<1 and let M\mathcal M be the set of (GSU,ZSU)(G_{SU},\mathbf Z_{SU}) that achieve the global maximum of JJ on C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R). […] Theorem 1 of Hajek (3) states that if Tt→0T_t\to0 and ∑t=1∞exp⁡[−D∗/Tt]\sum_{t=1}^{\infty}\exp[-D^*/T_t] diverges, then the sequence (GSU,ZSU)t(G_{SU},\mathbf Z_{SU})_t converges in probability to an element of M\mathcal M.

The space C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R) is connected via proposals of this type (see 6, for explanation). (Web Appendix p. 3.)

Intended meaning

Proposition 1 applies Theorem 1 of Hajek (1988) to the annealing chain on the finite set C\mathcal C that the proposals of Web Appendix 2 define, with the proposal repairs in P012-E001 and with hh and pp defined throughout C\mathcal C (see P012-E006). A check of the published article shows that the theorem assumes a fixed proposal that is irreducible and weakly reversible, and positive temperatures that are nonincreasing with limit zero. Under these hypotheses the probability that the chain is in the set of global optima tends to one exactly when ∑kexp⁡(−d∗/Tk)=+∞\sum_k\exp(-d^*/T_k)=+\infty. The summary in the proof omits the hypotheses.

The intended chain satisfies them. From any compatible pair a sequence of proposals leads to the pair in which the recruitment edges are the only edges among sampled vertices and every spare edge-end has its own unsampled neighbor of trait 0. Each proposal can be undone by a path of proposals. These two facts imply that the chain is irreducible. The path can be chosen so that JJ stays at or above its smaller value at the two ends, which is weak reversibility. The schedule (ϵlog⁡t)−1(\epsilon\log t)^{-1} is decreasing for t≥2t\ge2. The bound D∗≤1/ϵD^*\le1/\epsilon and the comparison with ∑t1/t\sum_t1/t hold for every ϵ>0\epsilon>0, and the restriction to 0<ϵ<10<\epsilon<1 in the proof is unnecessary. The argument cited from reference 6, which I could check in the preprint only, concerns graphs on the sampled vertices with upper bounds on the degrees; this is a smaller space than the pairs (GSU,ZSU)(G_{SU},\mathbf Z_{SU}) with unsampled vertices and traits, and the argument above covers the larger one. The convergence is to the set M\mathcal M and not to one maximizing pair. The undefined T1T_1 is the subject of P012-E007.

Corrected

Proposition 1. Assume that hh and pp are defined throughout C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R) (see P012-E006). Let KK be the repaired proposal kernel of Web Appendix 2 on this finite set (see P012-E001). It is fixed, irreducible and weakly reversible in the sense of Hajek (3). Let the cooling schedule be given by a finite T1≥T2T_1\ge T_2 and Tt=(ϵlog⁡(t))−1T_t=\bigl(\epsilon\log(t)\bigr)^{-1} for t≥2t\ge2, where ϵ>0\epsilon>0 is a constant. Then lim⁡t→∞Pr⁡((GSU,ZSU)t∈M)=1\lim_{t\to\infty}\Pr\bigl((G_{SU},\mathbf Z_{SU})_t\in\mathcal M\bigr)=1.

Proof. Let J(h,p)=1/(1+ϵ+h)J(h,p)=1/(1+\epsilon+h) for ϵ>0\epsilon>0 and let M\mathcal M be the set of (GSU,ZSU)(G_{SU},\mathbf Z_{SU}) that achieve the global maximum of JJ on C(GR,dR,ZR)\mathcal C(G_R,\mathbf d_R,\mathbf Z_R). […] Theorem 1 of Hajek (3) states that, for a fixed proposal that is irreducible and weakly reversible and for positive nonincreasing temperatures Tt→0T_t\to0, if ∑t=2∞exp⁡[−D∗/Tt]\sum_{t=2}^{\infty}\exp[-D^*/T_t] diverges, then the probability that (GSU,ZSU)t(G_{SU},\mathbf Z_{SU})_t belongs to M\mathcal M tends to one.

Screenshot of Proposition 1 and the first paragraph of its proof. Red boxes mark epsilon greater than 0 in the statement, 0 less than epsilon less than 1 in the proof, and the sentence that begins: Theorem 1 of Hajek (3) states that.
Proposition 1 and the start of its proof, Web Appendix p. 2 (supplement PDF p. 2). Marked: the range of ϵ\epsilon in the statement, the narrower range in the proof, and the restatement of Hajek’s theorem.

Formal statements (Lean)

Formal results: 19 whole; 2 restricted; 2 refuted; 1 omitted.

Coverage concerns the identified claim under its stated assumptions. A checked proof establishes the formal statement; correspondence to the published text was reviewed by AI. Repository access is currently required to open the source links.

Each result links to its current formal statement and proof. “Whole” covers the identified claim under its assumptions; “restricted” covers the stated narrower scope; “refuted” gives a counterexample to the identified assertion; “omitted” has no current theorem. Any named published theorem used as a premise remains an assumption. Browse the formal results for this work.

P012-T1

Web Appendix 2, the two proposal mechanisms, as printed (said to propose another compatible pair)

Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel KK on C\mathcal{C}: Web Appendix p. 1; GSUG_{SU} and Definition 3: p. 155 (PDF p. 3)

The printed claim is false: P012-T1; Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel K: Web Appendix p. 1; augmented graphs and Definition 3: p. 155 (PDF p. 3). All four specified counterexamples hold, including their functional values. In (a) and (b), for the fixed chosen i,j and either coin outcome, every execution uses a fresh vertex of either trait and produces exactly the stated old graph with the removed sampled edge replaced by a sampled–fresh edge and an old-unsampled–fresh edge. Cleanup deletes nothing. The old unsampled vertex has no sampled neighbour; the output is incompatible, while conditions 1,2,4 remain true in (a). Every fresh name gives the named output up to renaming unsampled identities. In (c) both branches fail and cleanup deletes precisely sampled seed 3, violating vertex preservation and inclusion of the observed active trait assignment. In (d) every compatible pair is the displayed dyad on its active domain, so the trait proposal has no vertex to choose.

Historical component credits
  • Counterexample proved AronowAudit.P012.proposal_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The execution relation has no compatibility guard. Marginalizing Bernoulli draws to their support preserves this witness, and both fresh traits are explicitly quantified. Claim-stage verdict unchanged after reading the record.
  • Counterexample proved AronowAudit.P012.proposal_counterexample_b Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the fixed witness T1(b); no assertion about unobserved Hartford microdata or every compatible input is needed. Claim-stage verdict unchanged after reading the record.
  • Counterexample proved AronowAudit.P012.cleanup_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The isolated seed has permitted degree zero. The failure is loss of a sampled vertex, not an undefined functional or a refutation of the repaired rule. Claim-stage verdict unchanged after reading the record.
  • Counterexample proved AronowAudit.P012.empty_trait_pool_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The lack of a trait execution is the printed empty-choice defect; it does not assert that the graph mechanism has no execution. Claim-stage verdict unchanged after reading the record.

P012-T2

Web Appendix 2, the two proposal mechanisms, corrected

Web Appendix pp. 2–3 (supplement PDF pp. 2–3)

Web Appendix pp. 2–3 (supplement PDF pp. 2–3). Every execution of either corrected mechanism preserves compatibility. In the unsampled deletion branch exactly one sampled degree is restored, and cleanup deletes the old unsampled endpoint exactly when its only neighbour was the selected sampled vertex. In the sampled deletion branch both degrees are restored and the original second replacement pool remains nonadjacent. In the addition branch the second endpoint is sampled, two distinct non-recruitment edges are removed even for a shared unsampled endpoint, both sampled degrees are restored, and cleanup deletes exactly the selected vertices left isolated. If neither branch applies the state is unchanged. Recruitment edges, sampled vertices and traits, and the augmented graph rules are preserved throughout the returned states. The trait mechanism flips one unsampled trait or returns the same state when the unsampled set is empty.

Historical component credits
  • Proved AronowAudit.P012.corrected_proposal_preserves Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Assessed as the overall preservation component named by row_says. The additional branch details affect whole-target coverage, not this row's fidelity. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.corrected_trait_preserves Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not separately invoked; the selected aggregate declarations and supplied whole-statement proofs establish this component.
  • Proved AronowAudit.P012.degree_remove_step Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The raw-state extension changes no degree convention on the simple graphs needed by T2. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.degree_add_step Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The raw-state extension changes no degree convention on the simple graphs needed by T2. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.remove_edge_base Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The harmless absent-edge case does not replace the actual deletion step; present nonrecruitment edges are covered. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.add_edge_base Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Harmless extension to an already present edge; the required local invariant is fully covered. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.add_vertex_base Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not separately invoked; the selected aggregate declarations and supplied whole-statement proofs establish this component.
  • Proved AronowAudit.P012.replacement_properties Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The extra pending-state premises are appropriate for the explicitly auxiliary lemma; they must be discharged in the full branch argument and are not new restrictions on observed data. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.cleanup_compatible Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not separately invoked; the selected aggregate declarations and supplied whole-statement proofs establish this component.
  • Proved AronowAudit.P012.recruitment_edge_sampled Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The paper's directed recruitment graph is represented by its undirected skeleton, as expressly stipulated in S8(R2). Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.corrected_graph_preserves Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not separately invoked; the selected aggregate declarations and supplied whole-statement proofs establish this component.

P012-T3

Table 1, the three lower limits of IhI_h, as printed, read with the network counts of p. 156

Table 1: p. 158 (PDF p. 6); network counts and number of seeds: p. 156 (PDF p. 4); the bounds: pp. 156–157 (PDF pp. 4–5); Definitions 3–4: p. 155 (PDF p. 3)

The printed claim is false: P012-T3; Table 1: p. 158 (PDF p. 6); network counts and seeds: p. 156 (PDF p. 4); bounds: pp. 156–157 (PDF pp. 4–5); Definitions 3–4: p. 155 (PDF p. 3). For every admissible data set with 527 sampled vertices, six seeds and sampled degree total 4487, the opposite-trait private-leaf completion is compatible, has 521 recruitment edges, 3445 private degree-one leaves and 3972 vertices, and has positive equation-(1) denominator. Its edge counts and quotient value obey the uniform real bound -1462/1983 and the sharp integer bound -178409/241987, attained by the two stated integer triples. The displayed polynomial certificate is nonnegative. All three printed lower bounds are violated, with the exact stated gaps exceeding four-decimal rounding error; even a rounded lower endpoint cannot enclose every compatible homophily value. The larger completion is allowed because compatibility fixes no population size.

Historical component credits
  • Counterexample proved AronowAudit.P012.table_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the uniform counterexample component, with a defined homophily witness for every admissible data set with the published totals. It does not require the nomination-network vertex count to be fixed in the compatibility set, and makes no claim about table upper endpoints or preferential intervals.
  • Proved AronowAudit.P012.uniform_leaf_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This row covers the real upper-bound component; it does not purport to export the particular polynomial identity used to prove it.
  • Proved AronowAudit.P012.uniform_integer_leaf_bound Limited support; baseline fidelity: narrower; manifest: used; used in Solution; {'source': 'adjudication+second review', 'adjudicator': "P012_Identification.lean:1287 concludes only the uniform upper inequality, whereas T3(c) and the row say that the bound is the largest integer value and is attained at (260,261,0) and (261,260,0). Neither endpoint equality is in this declaration's conclusion. The omission is real, but its extent is minor: these are two direct numerical substitutions with positive denominator 3965*3967, and the endpoint evaluations are separately stated in table_numeric_gaps at P012_Identification.lean:1318. The other declaration does not supply this row's missing conclusion, but the reviewer overstates the severity of these mathematically trivial omitted cases.", 'suggested_fix': 'Conjoin the two equalities asserting that the displayed quotient at (260,261,0) and (261,260,0) equals -178409/241987; together with the existing upper bound these establish the claimed maximum and attainment.', 'second_review': "Target T3(c) says the largest integer value is -178409/241987, attained at (260,261,0) and (261,260,0), whereas P012_Identification.lean:1287 exports only the universal upper inequality. Neither attainment equality appears in its conclusion, so the target's maximum-and-attainment component needs a statement repair. The missing equalities are direct numerical substitutions with positive denominator 3965*3967, and equivalent phi evaluations appear separately at P012_Identification.lean:1318; conjoining those evaluations is mathematically trivial, making the omission minor.", 'severity': 'minor'}
  • Proved AronowAudit.P012.table_numeric_gaps Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary arithmetic only. Its last two equalities separately supply the scalar endpoint evaluations omitted by R079-08.
  • Proved AronowAudit.P012.leafCompletion_compatible Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary construction, not a separate refutation of every clause of T3. The fixed countable vertex universe and distinct natural time ranks represent arbitrary finite graphs and distinct finite recruitment times up to renaming/order, without excluding cases.
  • Proved AronowAudit.P012.recruitment_skeleton_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary forest-skeleton count; there is no additional graph-model premise beyond the stated admissibility conditions.
  • Proved AronowAudit.P012.spare_total_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.leaf_edge_card Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.opposite_leaf_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary count identity; the extra generality preserves the counterexample specialization.

P012-T4

Table 1, the three lower limits of IhI_h, corrected claim: the bound that each lower limit must satisfy

Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4)

Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4). The true lower endpoint obeys both certified bounds and all three gap bounds, and printed endpoints force a degree discrepancy with the nomination network.

Historical component credits
  • Proved AronowAudit.P012.corrected_table_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Assuming IsGLB characterizes the endpoint already named by the claim; it does not exclude a permissible dataset here because T=3445 gives a defined attaining completion. All rational gaps were checked exactly. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.table_degree_converse Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The inclusive rounding interval is slightly broader than any particular tie-breaking rule and includes every case the target needs. Claim-stage verdict unchanged after reading the record.

P012-T5

Table 1, the three lower limits of IhI_h, corrected: the exact lower limit in terms of the recruitment-edge counts

Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4); the computation described: pp. 156–157 (PDF pp. 4–5)

Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4); the computation described: pp. 156–157 (PDF pp. 4–5). The opposite-trait leaf completion attains the exact lower homophily endpoint on its stated domain.

Historical component credits
  • Proved AronowAudit.P012.exact_h_lower Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The endpoint-domain premise is exactly the target's, including one-trait sampled data with positive spare total. Undefined compatible states are excluded from hValues as required by T5. Claim-stage verdict unchanged after reading the record.

P012-T6

Web Appendix 1, first sentence, corrected: closed forms for the extrema of hh on C(GR,dR,ZR)\mathcal{C}(G_R, \mathbf{d}_R, \mathbf{Z}_R)

Web Appendix p. 1 (supplement PDF p. 1); Definitions 1, 3 and 4: p. 155 (PDF p. 3)

Web Appendix p. 1 (supplement PDF p. 1); Definitions 1, 3 and 4: p. 155 (PDF p. 3). For admissible observed data with positive sampled degree total in each trait group, every compatible graph has the positive displayed denominator and the displayed edge-count and homophily bounds. Opposite-trait and same-trait private degree-one leaf completions attain the bounds, so their endpoints give the smallest containing interval. An explicit algorithm scans each sampled-vertex record and each directed recruitment-edge record once, in any order within each stream, accumulating the two degree totals and three edge-type counts. It recovers the spare totals without truncated subtraction and returns the two exact rational endpoints. The instrumented scan has exactly n+|E_R| visits and accumulator additions, followed by 26 arithmetic operations, with indexed trait/degree/parent access and unit-cost exact arithmetic.

Historical component credits
  • Proved AronowAudit.P012.exact_h_region Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Full mathematical endpoint/interval component, with no hidden hypothesis carrying that conclusion. The target’s separate one-pass computation assertion is not exported by any credited statement here; see targets.json.
  • Proved AronowAudit.P012.phi_formula Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Nonzero factors prevent any misuse of total division in the target specialization.
  • Proved AronowAudit.P012.phi_mono_x Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.phi_mono_y Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.phi_anti_c Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.count_bounds_imply_h_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.trait_degree_edge_incidence Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.binary_edge_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary identity, not a replacement for the entire extrema theorem; arbitrary finite simple graphs are the declared scope of this helper.
  • Proved AronowAudit.P012.ordered_trait_adjacency Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.numerator_edge_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.graph_homophily_formula Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.extra_edge_budget Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.compatible_filtered_edge_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.compatible_edge_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.private_edge_card Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.same_leaf_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.

P012-T7

Web Appendix 1, first sentence, corrected: closed forms for the extrema of pp on C(GR,dR,ZR)\mathcal{C}(G_R, \mathbf{d}_R, \mathbf{Z}_R)

Web Appendix p. 1 (supplement PDF p. 1); Definitions 2, 3 and 4: p. 155 (PDF p. 3)

Web Appendix p. 1 (supplement PDF p. 1); Definitions 2–4: p. 155 (PDF p. 3). The observed recruitment history and spare degrees give the sharp attained preferential-recruitment interval. Obligatory child sets are nonempty; every compatible susceptible fraction has the stated decomposition and sharp bounds. Same-trait private degree-one neighbours attain p_min, and opposite-trait private degree-one neighbours attain p_max. The smallest containing interval is [p_min,p_max]. An exact rational forward algorithm computes both endpoints in one pass over the supplied chronological records, with positive event denominators, one read per record, at most n stored cells and at most 32n+4 unit arithmetic and indexed-cell operations, including the final divisions.

Historical component credits
  • Proved AronowAudit.P012.exact_p_region Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The row covers the corrected mathematical region and endpoint assertions, not a runtime guarantee. FullPreferentialRegion includes definedness for every compatible state and attained least/greatest values; it does not require interior values to be attainable.
  • Proved AronowAudit.P012.susceptible_fraction_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.later_children_susceptible Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.susceptible_recruitment_edges Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.matching_child_card Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.susceptible_decomposition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.leaf_susceptible_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.leaf_preferential_endpoints Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.

P012-T8

Proof of Result 1, the choice of GSU1G^1_{SU} as "any compatible subgraph" and the step to GSU2G^2_{SU}, as printed

Appendix, p. 160 (PDF p. 8); Result 1: p. 155 (PDF p. 3)

The printed claim is false: P012-T8; Appendix, p. 160 (PDF p. 8); Result 1: p. 155 (PDF p. 3). An allowed starting graph in the printed Result 1 proof yields an incompatible output with the same induced sampled graph.

Historical component credits
  • Counterexample proved AronowAudit.P012.result1_proof_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This refutes the universally permitted proof construction. It does not refute the existential assertion of Result 1. Claim-stage verdict unchanged after reading the record.

P012-T9

Proof of Result 2, the choice of the starting pair as "any compatible subgraph and trait set" and the sentence "Clearly (GSU2,ZSU2)∈C(GR,dR,ZR)(G^2_{SU}, \mathbf{Z}^2_{SU}) \in \mathrm{C}(G_R, \mathbf{d}_R, \mathbf{Z}_R)", as printed

Appendix, p. 160 (PDF p. 8); Result 2: p. 155 (PDF p. 3)

The printed claim is false: P012-T9; Appendix, p. 160 (PDF p. 8); Result 2: p. 155 (PDF p. 3). The specified compatible starting pair has both cross edges already present. The printed swap leaves degrees (2,2,2,1,1) instead of (3,2,3,2,2), and four edges instead of six, so its output is incompatible. Its equation-(1) denominator is 15/8, whereas the starting denominator is 35/12; both are positive and unequal. This refutes the printed proof step.

Historical component credits
  • Counterexample proved AronowAudit.P012.result2_proof_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This refutes the printed swap construction on an allowed input. It does not refute Result 2 itself. Claim-stage verdict unchanged after reading the record.

P012-T10

Result 1, with its proof corrected

Result 1: p. 155 (PDF p. 3); proof: Appendix, p. 160 (PDF p. 8)

Result 1: p. 155 (PDF p. 3); corrected proof: Appendix, p. 160 (PDF p. 8). Two time-ordered sampled vertices not joined by a recruitment edge, each with positive spare degree, give two distinct compatible augmented graphs and two distinct compatible sampled induced graphs. The first graph has exactly one shared unsampled neighbour of the two chosen vertices, s_i-1 and s_j-1 further private leaves there, and s_v private leaves at every other sampled vertex. The shared vertex has exactly the two chosen neighbours; every private leaf has degree one. For every binary assignment extending the observed traits both graphs are compatible. The second deletes precisely the shared vertex and its two edges, adds {i,j}, and keeps all remaining traits. The induced edge sets are B and B union {{i,j}}. The graphs remain distinct under every renaming of unsampled identities, so neither graph is identified.

Historical component credits
  • Proved AronowAudit.P012.result1 Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The covered component is Result 1 and its repaired shared-neighbour existence argument. The signature does not assert the full recipe's privacy of every other unsampled vertex or that the common neighbour has degree two; those details are unnecessary for the stated non-identification witnesses and are not claimed by row_says. Corrected_statement status is consistent with an unchanged Result 1 and a repaired proof.

P012-T11

Result 2 and display (A1), with the proof corrected

Result 2: p. 155 (PDF p. 3); proof and (A1): Appendix, p. 160 (PDF p. 8)

Result 2: p. 155 (PDF p. 3); corrected proof and (A1): Appendix, p. 160 (PDF p. 8). Opposite-trait sampled vertices with spare degree make homophily not point identified. Attach exactly s_v private degree-one neighbours to each sampled vertex. Any selected private neighbours of the normalized trait-0 and trait-1 vertices can be assigned traits 0 and 1, with all other unsampled traits arbitrary. The cross edges are absent and the swap is compatible and preserves vertices, traits, all degrees and edge count. With Q the trait-weighted degree sum, both denominators equal Q-Q^2/(2m)=Q(2m-Q)/(2m)>0, because Q≥d_j>0 and 2m-Q≥d_i>0. Both forms of (A1) give h_1-h_2=2/D≠0. The stated exact two-subject instance has m=3, Q=3, D=3/2, h_1=1/3, h_2=-1 and difference 4/3.

Historical component credits
  • Proved AronowAudit.P012.result2 Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The starting graph is existentially constructed; restricting that chosen witness to a permissible trait assignment does not restrict the universal observed-data claim. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.result2_swap_formula Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This row covers the arbitrary-other-leaf-trait formula component; R081-07 separately exports the final nonidentification assertion. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.a1_exact_instance Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Exact positive numerical specialization of the target, not a counterexample or a purported proof of its universal clauses. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.compatible_same_graph Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Auxiliary compatibility fact; sampled traits are fixed and unsampled traits remain unrestricted. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.private_leaf_swap_graph Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.same_degree_h_denominator Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Compatibility and positive denominators are unnecessary for this auxiliary equality; the application supplies the copied full trait function, so no case of the corrected swap is excluded. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.private_swap_true_edge_count Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.private_leaf_swap_value Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.
  • Proved AronowAudit.P012.private_swap_reversed Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected aggregate declaration supplies the needed component; this supporting row is not separately invoked.

P012-T12

Display (A2) in the proof of Result 3, as printed

Appendix, p. 160 (PDF p. 8); Result 3: p. 155 (PDF p. 3)

The printed claim is false: P012-T12; Appendix, p. 160 (PDF p. 8); Result 3: p. 155 (PDF p. 3). The two-recruit witness has preferential-recruitment difference minus five twelfths, disagreeing with both printed (A2) right-hand sides.

Historical component credits
  • Counterexample proved AronowAudit.P012.a2_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; A witness refuting display A2 for the entire functional, not a refutation of the positive non-identification conclusion of Result 3. Although the total trait functions differ at unused names, on the displayed vertex set only the private unsampled neighbour changes trait.

P012-T13

Result 3, with display (A2) corrected

Result 3: p. 155 (PDF p. 3); proof and (A2): Appendix, p. 160 (PDF p. 8)

Result 3, p. 155; corrected (A2), Appendix p. 160. A recruiter with spare degree admits two compatible completions with distinct preferential recruitment and the corrected signed full sum, sharp bound and sole-child equality criterion, which also hold at any compatible private-leaf start.

Historical component credits
  • Proved AronowAudit.P012.result3 Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Full corrected Result 3 is exported: witness graph, private leaf, both compatible states, event membership, the signed full sum, distinct defined values, nonidentification and the conditional bound/equality criterion. No converse is claimed by this target.
  • Proved AronowAudit.P012.private_trait_flip_difference Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Covers parts (b)–(d) for every compatible pair satisfying the private-neighbor hypotheses, without restricting other edges or traits. Definedness follows because j is an actual child and each recruit is susceptible at its event.
  • Proved AronowAudit.P012.unsampled_not_yet Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.flip_susceptible Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.private_susceptible_iff Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.flip_true_filter Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.flip_false_filter Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.private_flip_event_difference Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.private_flip_full_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.child_reciprocal_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.unsampled_trait_update_preserves Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.private_leaf_member Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.

P012-T14

Definitions 1, 2 and 4 (equations (1) and (2) evaluated at every compatible pair), as printed

p. 155 (PDF p. 3); the same use in Web Appendix 1, p. 1 (supplement PDF p. 1)

The printed claim is false: P012-T14; Definitions 1, 2 and 4, p. 155 (PDF p. 3); Web Appendix 1, p. 1. The four admissible examples A--D exhibit undefined printed quotients. A and C have unique displayed compatible pairs; B has raw numerator -1/2 and denominator 1/2 and an empty recruitment average at every pair. Every D graph is a dyad with three private leaves at vertex 2; all eight trait assignments are compatible, and their trait sum gives h=-3/5,-1/3,-1/7 or an undefined 0/0, while p=0. A and D have no whole-compatible-set h interval.

Historical component credits
  • Counterexample proved AronowAudit.P012.domain_counterexamples Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; These are witnesses to failure of the printed unqualified domain, not a claim that the displayed partial functions are themselves inconsistent. Uniqueness of A/C, graph-shape uniqueness of D and all permutations of D's selected numerical values are not exported, but the row claims the counterexample witnesses and does not need those stronger narrative facts.

P012-T15

Definitions 1, 2 and 4, with the domains of hh and pp stated

p. 155 (PDF p. 3); the same use in Web Appendix 1, p. 1 (supplement PDF p. 1)

Definitions 1, 2 and 4, p. 155 (PDF p. 3); Web Appendix 1, p. 1. The denominator identities give the exact individual domains of h and p. On admissible data the whole-compatible-set domains are exactly positive sampled degree in each trait class for h and at least one nonseed for p. Under each condition separately, unique attained endpoints give the smallest containing interval. If the h condition fails no whole-set h region exists; the h-defined-only set can be empty (example A), while p still has region {0}. The hypotheses of Results 2 and 3 imply the respective domain conditions.

Historical component credits
  • Proved AronowAudit.P012.full_domain_conditions Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The universal quantifier includes every compatible state, including undefined ones; admissible data admit the explicit leaf completion, preventing vacuous truth from an empty compatible family. Natural time ranks preserve every finite strict recruitment ordering, and vertex labels impose no finite graph restriction.
  • Proved AronowAudit.P012.homophily_domain Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This harmless strengthening specializes to the exact target component; it excludes no claimed case. The generalization is classified as broader_than_claim independently of its disclosure in the descriptions.
  • Proved AronowAudit.P012.full_p_domain_conditions Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The equivalence is over the entire compatible set and applies separately from the homophily domain; the nonseed average is undefined when all sampled vertices are seeds.
  • Proved AronowAudit.P012.recruit_is_susceptible Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This harmless strengthening specializes to the exact target component; it excludes no claimed case. The generalization is classified as broader_than_claim independently of its disclosure in the descriptions.
  • Proved AronowAudit.P012.denominator_sum_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.degree_sum_twice_edges Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This harmless strengthening specializes to the exact target component; it excludes no claimed case. The generalization is classified as broader_than_claim independently of its disclosure in the descriptions.
  • Proved AronowAudit.P012.trait_degree_partition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This harmless strengthening specializes to the exact target component; it excludes no claimed case. The generalization is classified as broader_than_claim independently of its disclosure in the descriptions.
  • Proved AronowAudit.P012.degreeTraitSum_eq Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This harmless strengthening specializes to the exact target component; it excludes no claimed case. The generalization is classified as broader_than_claim independently of its disclosure in the descriptions.
  • Proved AronowAudit.P012.denominator_domain_criterion Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.result_domain_conditions Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Checked the published Results 2 and 3 in the article, p. 155: opposite-trait sampled vertices with surplus degree, and a surplus-degree sampled recruiter with a child, respectively ([published article](https://pmc.ncbi.nlm.nih.gov/articles/PMC5860647/)). Under admissibility the latter is exactly a nonseed child whose parent has surplus degree. This row proves only their domain implications, as described.
  • Proved AronowAudit.P012.degree_eq_neighbor_card Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.adjacency_row_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.zero_degree_no_edge Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.zero_trait_degree Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.boundary_numerator_zero Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.sampled_trait_positive_transfer Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.leaf_same_zero_trait Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.

P012-T16

Web Appendix 1, the cooling schedule and display (2) of the proof of Proposition 1, corrected

Web Appendix p. 1 (the sentence after the display of KK) and p. 2 (Proposition 1 and display (2)); supplement PDF pp. 1–2

Web Appendix pp. 1-2, cooling schedule and display (2). The corrected logarithmic schedule is positive, nonincreasing, strictly decreasing on its tail and tends to zero, while its depth series diverges for 0 ≤ D*ε ≤ 1; the two printed schedule defects are also certified.

Historical component credits
  • Proved AronowAudit.P012.cooling_properties Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Zero-based indexing is a representation difference; the equality-allowed first step matches the corrected schedule. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.no_positive_nondecreasing_zero_limit Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The two hypotheses are jointly satisfiable; only their conjunction with convergence to zero is ruled out, exactly as corrected part (a) requires. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.printed_first_temperature_undefined Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This arithmetic nonexistence is the intended first-temperature obstruction, not vacuity caused by Lean’s convention for division by zero. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.cooling_term_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the termwise auxiliary component of the corrected display, not an assertion that the critical-depth constant exists or that the annealing chain converges. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.cooling_series_diverges Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The divergence statement is equivalent to the corrected positive-series assertion, including any permitted finite first-temperature contribution. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.harmonic_tail_log_bound Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.

P012-T17

Proof of Result 1, the compatibility of GS1G^1_S and GS2G^2_S, corrected

Appendix, p. 160 (PDF p. 8), last sentences of the proof of Result 1; Definition 3: p. 155 (PDF p. 3)

Appendix p. 160 (PDF p. 8), proof of Result 1; Definition 3, p. 155. Deleting unsampled vertices preserves compatibility with upper degree bounds. Exact degrees survive precisely when there is no sampled-to-unsampled edge, equivalently no unsampled vertex, equivalently the graph equals its induced graph. The two seeds with one common unsampled neighbour have degrees (1,1) but induced degrees (0,0); the edge-between-seeds comparison has both (1,1).

Historical component credits
  • Proved AronowAudit.P012.induced_compatibility Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; GraphCompatible omits trait preservation exactly as the opening of the printed proof does; InducedCompatible requires degree upper bounds, not full Definition 3 compatibility.

P012-T18

The arguments of hh in the proof of Result 2 and in the sentence after equation (2), corrected

Appendix, p. 160 (PDF p. 8), proof of Result 2 (four occurrences); p. 155 (PDF p. 3), sentence after equation (2)

Appendix p. 160 (PDF p. 8), proof of Result 2; p. 155 (PDF p. 3), after equation (2). Homophily depends only on the displayed augmented graph and its traits, including agreement of undefined partial values. Preferential recruitment can change with the recruiter map alone or with entry order alone: on the fixed four-vertex example the star, chain and reordered star histories give the defined values -1/12, 1/6 and -1/3 respectively.

Historical component credits
  • Proved AronowAudit.P012.homophily_arguments Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The row covers h-invariance. The target also contrasts this with p’s dependence on recruitment data, which this equality does not establish; see targets.json.

P012-T19

Web Appendix 2, the two proposal mechanisms, as printed

Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel KK: Web Appendix p. 1 (supplement PDF p. 1)

The printed claim is false: P012-T19; Web Appendix 2, pp. 2-3; kernel on Web Appendix p. 1. The literal printed proposal mechanisms do not preserve compatibility for every execution from admissible compatible data.

Historical component credits
  • Counterexample proved AronowAudit.P012.printed_proposal_preserves_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The positive printed target and counterexample_proved/exact_closed_contract_refutation metadata consistently identify a refutation. The execution relation contains no compatibility filter and matches the specified indentation and original-state replacement pools; connectivity and mixing are outside the target’s explicit formal preservation assertion.
  • Proved AronowAudit.P012.pair_edge_eq Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.

P012-T20

Table 1, the column IhI_h, as printed, with the text that presents it

Table 1: p. 158 (PDF p. 6); text: pp. 156–157 (PDF pp. 4–5)

The printed claim is false: P012-T20; Table 1, p. 158; study totals, pp. 156-157. No admissible shared study inputs with the reported totals can have all three printed homophily intervals, even allowing their four-decimal rounding tolerance.

Historical component credits
  • Counterexample proved AronowAudit.P012.Wave1.printed_table_intervals_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Exact full refutation of the joint homophily interval assertion, not a claim to determine the actual upper bounds, preferential intervals, point estimates, or corrected empirical endpoints. In particular -1462/1983 < -0.7027-1/20000, so any asserted least sex endpoint within the printed rounding band contradicts a compatible defined value. No target-kind/status inconsistency is present. Claim-stage verdict unchanged after reading the record.

P012-T21

Web Appendix 1, first sentence, as printed

Web Appendix p. 1 (supplement PDF p. 1)

No current theorem is supplied for this target.

Historical component credits
  • Not applicable Excluded association; baseline fidelity: None; manifest: not applicable; Solution use unverified; Historical excluded association.

P012-T22

The statement that hh and pp are not point identified unless the recruitment graph is identical to the underlying subgraph (abstract; introduction; text after Result 3)

Abstract: p. 153 (PDF p. 1); introduction: p. 154 (PDF p. 2); text after Result 3: pp. 155–156 (PDF pp. 3–4)

abstract p. 153, introduction p. 154, and text after Result 3, pp. 155--156 (PDF pp. 1--4). On the stated whole-set domain, joint point identification is equivalent to zero spare degree and to every compatible graph being the undirected recruitment graph with its observed traits. A same-trait private-leaf flip witnesses h variation, with the exact positive edge-count difference. Preferential recruitment alone is identified exactly when actual recruiters have no spare degree. In the (1,2)-degree, (0,1)-trait example every graph has one private leaf at vertex 2: p=0 and h is -1 or -1/3 according to its trait. Thus the exact value sets are {0} and {-1,-1/3}, and the smallest containing intervals are {0} and [-1,-1/3], with attained endpoints.

Historical component credits
  • Proved AronowAudit.P012.joint_identification Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P012_Identification.lean:3426 states the joint point-identification equivalence, equivalence with every compatible graph being the recruitment skeleton, non-identification of h when any sampled degree is spare, and uniqueness of graph and active traits when none is spare. These conclusions match the component of T22(i)-(iii) credited to this row, under exactly the admissibility, two positive sampled trait-degree totals, and nonempty recruitment assumptions in the claim's formal setup. The attainable-value sets and inhabited singleton predicate at P012_Identification.lean:240-251 give the same point-identification criterion as the claim's smallest containing intervals: such an interval is a singleton exactly when its nonempty attainable-value set is a singleton. Equality of traits only on the graph's vertices at P012_Identification.lean:3439 ignores values outside the mathematical graph, so it omits no part of the claimed compatible-pair uniqueness."}
  • Proved AronowAudit.P012.preferential_identification Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; {'source': 'adjudication', 'decision': 'rejected', 'what': "P012_Identification.lean:3291 states exactly T22(iv)'s equivalence: p is point identified if and only if every sampled vertex that recruited at least one child has reported degree equal to recruitment degree. Although the two sampled trait-degree positivity hypotheses at P012_Identification.lean:3292 are unnecessary for a theorem about p alone, the claim itself explicitly assumes both before listing (i)-(iv), so these hypotheses exclude no case of this target. The children set at P012_Identification.lean:1046 consists precisely of nonseeds with this recruiter, and pValues and PointIdentified at P012_Identification.lean:241 and P012_Identification.lean:251 express the claim's point-identification criterion on its nonempty, defined domain."}
  • Proved AronowAudit.P012.joint_leaf_flip_witness Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is a constructive witness for the positive non-identification result, not a refutation of a printed assertion. Although the conclusion names total hValue quotients rather than exporting HomophilyDefined, compatibility and the positive sampled trait degree totals imply positive trait-stub totals and a positive denominator for both displayed graphs.
  • Proved AronowAudit.P012.point_identified_iff_equal_extrema Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.preferential_endpoint_gap Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.preferential_endpoints_equal_iff Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.trait_spare_partition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.zero_spare_compatible_unique Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.homophily_endpoints_strict Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.
  • Proved AronowAudit.P012.private_edge_incidence Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Privacy is an explicit hypothesis of the auxiliary fact and is satisfied by the constructed witness; it is not a restriction on the target's class of compatible graphs. The extra sampled-membership binder is harmless in this setting.
  • Proved AronowAudit.P012.leaf_flip_edge_counts Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The private-leaf construction fixes the single affected edge; this count calculation is a supporting part of the witness, not a claim that every compatible graph has private neighbors.
  • Proved AronowAudit.P012.phi_strict_anti_c Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.

P012-T23

Figure 1: the extrema of hh and pp for the five-vertex example

Figure 1 and its caption: p. 156 (PDF p. 4)

Figure 1 and caption, p. 156 (PDF p. 4). For the five sampled vertices with degrees (5,4,3,3,2), recruitments 1 to 2, 2 to 3, 1 to 4, 2 to 5, and traits (1,1,1,0,1), the attained marginal extrema are -5/8, 79/105, -11/80 and 49/120. Every compatible pair lies in their rectangle inside [-1,1]^2. Some extremal augmented subgraphs are not unique, even after renaming unsampled vertices. The stated recruitment degrees, spare degrees, edge and stub counts, phi evaluations and individual preferential summands hold. The listed trait-specific edge and stub counts use the displayed coding (1,1,1,0,1). All arithmetic and statistical conclusions hold for every strictly increasing recruitment time order. The extrema, bounds and nonuniqueness conclusions hold for either coding of the binary traits; exchanging both labels preserves compatibility and both defined statistics.

Historical component credits
  • Proved AronowAudit.P012.figure1_extrema Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Expanded FullHomophilyRegion and FullPreferentialRegion assert each marginal extremum is attained and bound every compatible pair, without assuming the result or asserting joint interior attainability. Claim-stage verdict unchanged after reading the record.
  • Proved AronowAudit.P012.figure1_endpoint_arithmetic Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; An accurately labelled arithmetic auxiliary fact; no full target credit is assigned to it alone. Claim-stage verdict unchanged after reading the record.

P012-T24

Web Appendix 1, Proposition 1

Web Appendix p. 2 (supplement PDF p. 2); set-up on Web Appendix p. 1

Web Appendix 1, p. 2, display (1); set-up on p. 1. The objective is positive and has attained scalar maximum 1/epsilon on the square, and its value at any ordinary binary graph correlation is positive and bounded by 1/epsilon.

Historical component credits
  • Proved AronowAudit.P012.Wave1.annealing_convergence_proof Withdrawn support; baseline fidelity: wrong_credit; manifest: not used; not used in Solution; Excluded: proves convergence for a materially repaired proposal process, not the canonical printed process.
  • Proved AronowAudit.P012.objective_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Purpose: the scalar bound and attained maximum in display (1), not the asymptotic conclusion. Positivity of 1+epsilon+h avoids all total-division artifacts.
  • Proved AronowAudit.P012.unsampled_card_bound Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The incidence bound is not part of the retained objective estimate.
  • Proved AronowAudit.P012.Wave1.compatible_finite_up_to_names_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Quotient finiteness is not part of the retained objective estimate.
  • Proved AronowAudit.P012.critical_depth_bound Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Communication-depth assertions are omitted from the retained objective estimate.
  • Proved AronowAudit.P012.optimum_exact_zero_depth Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Communication-depth assertions are omitted from the retained objective estimate.
  • Proved AronowAudit.P012.critical_depth_singleton Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Communication-depth assertions are omitted from the retained objective estimate.
  • Proved AronowAudit.P012.homophily_global_range Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Purpose: the graph-correlation domain and range fact underlying display (1). The all-vertex positive-degree conditions are equivalent to the correlation’s ordinary domain in a finite simple graph and exclude no defined correlation case.
  • Proved AronowAudit.P012.compatible_objective_bounds Limited support; baseline fidelity: narrower; manifest: not used; not used in Solution; Not needed: the two ok rows give the graph estimate using all-vertex trait-degree positivity, without its stronger sampled-trait positivity premises.

Reproduction

What I recalculated or reran from public materials. "Agrees", "Agrees in part" or "Disagrees" compares my result with the printed one. "Not compared" means the comparison could not be made.

Status
Selected recalculation
Agreement
Agrees in part
Reported results that were targeted
TargetWhereMethodStatusAgreement
Figure 1p. 156 (PDF p. 4), panels A-F and captionown recalculation from printed diagrams: exact equations and exhaustive compatible-pair enumeration; four extrema agree to roundingRecalculatedTo rounding
Table 1p. 158 (PDF p. 6); interpretation on p. 157 (PDF p. 5) and discussion p. 158inspection of printed point values and interval arithmetic only; Hartford values and endpoints not recomputed because data and code are unavailable; P012-N002 gives a conditional lower-limit inconsistencyBlockedNot compared
Figure 2 and Hartford network countsFigure 2 p. 157 (PDF p. 5); in-text counts p. 156 (PDF p. 4)printed count arithmetic only; network layout and underlying counts not reproduced because Hartford data and layout code are unavailableBlockedNot compared