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Effective Resistance and Graph Neural Network Reliability in Tissue-Specific Interactomes

Jianru Shen

arXiv:2610.02175v1cs.LGq-bio.MN

TL;DR

Protein annotation needs to identify which model predictions to distrust, but confidence scores do not use a protein’s position in the interactome. The paper tests corrected effective resistance as a structural signal for per-node failure, finding an additional association with loss under controls but a small practical contribution bounded by degree degeneration.

  • Problem

    Protein function predictions need reliability information because incorrect labels can propagate through downstream analyses, while model-output reliability scores ignore a protein’s interactome position.

    Source: §I · p. 1

  • Method

    The study removes degree dependence from effective resistance and tests its residual association with per-node loss under confidence, degree, annotation and structural controls, using permutation floors and degree-preserving nulls.

    Source: §I · p. 1, D. Three worlds · p. 3, C. Permutation floor · p. 3

  • Results

    The corrected residual explains additional per-node loss in 19 of 24 held-out networks after controls, and real interactomes retain more corrected resistance variation than degree-matched nulls in all 24 networks.

    Source: §I · p. 1, Appendix D · p. 8

  • Takeaways & Limitations

    The room for resistance-aware reliability on these interactomes is bounded by their small departure from the inverse-degree limit.

    Source: §VI · p. 6

    The risk experiment uses a single benchmark, and the degree-preserving null does not separate local clustering from larger-scale organization.

    Source: B. Limitations · p. 6

Abstract

from arXiv · show

Protein function annotation needs to know which predictions to distrust, not only what a model predicts. We ask whether tissue-specific interaction structure carries that information. Our candidate signal is effective resistance, used previously to relieve over-squashing by rewiring. Across 24 tissue-specific interactomes it is dominated by inverse degree, and the degeneration deepens as the co-expression filtered network grows, with a Spearman correlation of -0.955. The residual departure from that limit exceeds degree-preserving null graphs in all 24 networks. Controlling for predictive entropy, degree, annotation cardinality, local structure and feature-only difficulty, the residual explains additional per-node loss in 19 of 24 held-out networks once a permutation floor is subtracted, at every depth, and the effect strengthens monotonically with depth. The increment reaches 0.37% of the variance the controls leave unexplained, 5.6 times a permutation floor, against 1.5 times when the model is retrained in a degree-preserving null world. Selective prediction improves negligibly. The signal is reproducible; degree degeneration bounds it.

I. INTRODUCTION

The paper tests whether tissue-specific interactome structure can identify protein predictions to distrust beyond model confidence, focusing on effective resistance and its degeneration toward inverse degree. It finds that a corrected structural signal remains associated with per-node loss after controls, while its practical contribution is small.

A. Effective resistance

  • Effective resistance is nearly determined by inverse degree in tissue-specific interactomes, limiting the global structural information it can add.It measures resistance between nodes when each edge is a unit resistor and is used here as a node-level indicator of structural remoteness.Source: §I · p. 1, A. Effective resistance · p. 2

I. INTRODUCTION

  • The study computes exact Laplacian pseudoinverses for all 24 benchmark networks and tests residual resistance against prediction failure beyond confidence and elementary structure.The stated aim is to bound what resistance-aware reliability methods can achieve on this data class.Source: §I · p. 1
  • The inverse-degree degeneration deepens as co-expression-filtered interactomes grow, while near-tree molecular graphs provide a contrasting geodesic regime.The introduction frames the interactome and near-tree comparison as a test of the structural regimes.Source: §I · p. 1
  • Real interactomes retain more corrected resistance variation than degree-preserving null graphs in all 24 tissues.The comparison distinguishes the real-network residual from variation retained by degree-preserving rewiring.Source: §I · p. 1
  • Corrected resistance explains additional per-node loss in 19 of 24 held-out networks after controlling for predictive entropy, degree, annotation cardinality, local structure and feature-only difficulty.The contribution is reported at every depth, while the increment is small and does not improve selective prediction.Source: §I · p. 1

B. Two limiting regimes

The section distinguishes two regimes where effective resistance becomes reducible to simpler graph quantities: inverse degree in well-mixed graphs and shortest-path distance on trees. It measures departures from both limits because available finite-graph bounds do not settle the interactome case.

B. Two limiting regimes

  • In well-mixed graphs, resistance converges to the sum of the endpoint inverse degrees, leaving no information beyond those degrees.The cited finite-graph deviation bound scales as 1/dmin^2 and is uninformative for the studied networks, all of which contain degree-one nodes.Source: B. Two limiting regimes · p. 2
  • On a tree, resistance equals shortest-path distance because the unique path places edge resistances in series.In this regime resistance reduces to a shortest-path statistic.Source: B. Two limiting regimes · p. 2
  • Ddeg and Dgeo measure normalized departures from the inverse-degree and geodesic limits, respectively, with each normalized by the quantity exact in its own regime.Ddeg is restricted to non-adjacent pairs, making its numerator a non-negative excess; both quantities are evaluated over all pairs.Source: B. Two limiting regimes · p. 2

C. Effective resistance and message passing

The paper repurposes effective resistance from graph rewiring as a fixed descriptor for testing where a trained model fails. It removes degree-related signal from resistance and tests whether the residual explains loss beyond confidence and structural controls.

C. Effective resistance and message passing

  • Resistance bounds over-squashing and has been used to select edges for rewiring, but here it is tested as a descriptor of failure locations in an unmodified network.The depth analysis is motivated by the prediction that deeper models fail more systematically at high-resistance nodes.Source: C. Effective resistance and message… · p. 2

A. Structural score

  • The structural score is the within-graph standardized residual from regressing log Laplacian-pseudoinverse diagonal on log degree.The score is label-free and model-free; fitting the slope rather than fixing it avoids leaving degree dependence when the fitted slope departs from the inverse-degree reference.Source: A. Structural score · pp. 2–3

B. Risk target and nested models

  • The nested loss models test whether the score adds explanatory power beyond predictive entropy, degree and annotation cardinality, with local structure added in the stricter design.For selective prediction, annotation cardinality is dropped because it requires labels unavailable when deciding whether to trust a prediction.Source: B. Risk target and nested models · p. 3

C. Permutation floor

The section corrects the apparent gain from adding a score for the increase in R2 expected from an uninformative variable. It also compares real, degree-preserving null, and near-tree geodesic settings, with a separate permutation floor for each world.

C. Permutation floor

  • The corrected increment subtracts the average gain from 1,000 random permutations that preserve the score distribution but break its node-to-loss correspondence.The floor is re-estimated in every subset; a single permutation's standard deviation exceeds the floor itself.Source: C. Permutation floor · p. 3

D. Three worlds

  • The degree-preserving null rewires edges by double-edge swaps while retaining every degree, and both the score and model are recomputed in that world.Each world is compared against its own permutation floor because model quality differs between worlds.Source: D. Three worlds · p. 3
  • The near-tree geodesic control tests specificity because resistance reduces to shortest-path distance on those graphs.The passage states that a pipeline manufacturing signal would report an effect even where theory says none exists.Source: D. Three worlds · p. 3

E. Unit of analysis

The study evaluates 24 tissue-specific interactomes, a near-tree peptide control, and GraphSAGE models under a held-out-network protocol. Structural measurements and model training are designed to support comparisons across graphs and against degree-preserving nulls.

E. Unit of analysis

A. Tissue-specific interactomes

  • The tissue-specific networks average 2,373 nodes, while the peptide control is mostly near-tree and has median relative resistance–distance discrepancy 0.0103.Three interactomes are disconnected; structural quantities use their largest connected components, retaining at least 87.5% of nodes.Source: A. Tissue-specific interactomes · pp. 3–4, Table I · p. 4
  • Degree-preserving null graphs use connectivity-preserving double-edge swaps, which halve mean clustering but do not eliminate it.The null comparison therefore supports claims beyond degree sequence, but cannot separate local clustering from larger-scale organization.Source: A. Tissue-specific interactomes · p. 4

B. Model and protocol

  • GraphSAGE uses 2, 4, or 6 layers, and a matching feature-only perceptron controls for node difficulty unrelated to the graph.Four outer folds hold out every interactome once; five seeds are used per fold, and validation micro-F1 is reported across runs.Source: B. Model and protocol · p. 4, Table II · p. 4

V. RESULTS

Resistance in tissue-specific interactomes is dominated by the inverse-degree limit, with stronger degeneration in larger networks. Corrected resistance variation exceeds degree-preserving nulls across all 24 networks, while the peptide control behaves as expected for near-tree graphs.

A. Interactomes lie in the inverse-degree regime

  • The Spearman correlation between network size and Ddeg is −0.955; Ddeg falls from 0.085 in the smallest network to 0.033 in the largest.Typical resistance departures from 1/di + 1/dj are under one tenth of their value in interactomes.Source: A. Interactomes lie in the… · p. 4

B. Real interactomes exceed their degree-matched nulls

  • 0.30% of log pseudoinverse-diagonal variance remains unexplained on real networks versus 0.10% on degree-preserving nulls, with real networks higher in all 24 pairs.The mean real-to-null ratio is 2.80 (range 1.93 to 3.89); rewiring shifts Ddeg by +0.0078 on average and Dgeo by +0.00005.Source: B. Real interactomes exceed their… · pp. 4–5, Figure 2 · p. 5

C. The geodesic control behaves as theory requires

  • On exact peptide trees, the largest resistance–distance discrepancy is 1.7 × 10−10; across the corpus, the median relative discrepancy is 0.0103.Geodesic summaries leave 0.77% of variance in mean resistance, with cycle rank contributing 0.0002 versus 0.175 for diameter.Source: C. The geodesic control behaves as… · p. 5

D. The residual explains per-node loss

The corrected structural score explains additional per-node loss beyond the strict controls, and its increment grows with model depth. Tests of additional controls and node-level mechanism evidence qualify how that association should be interpreted.

D. The residual explains per-node loss

  • +0.00115 of explained variance in per-node loss is added under the strict design, with a confidence interval from +0.00074 to +0.00164; 19 of 24 networks remain positive after floor subtraction.Removing local structure from the score and adding it to the design both yield +0.00115; the six-layer model reaches 0.9026 micro-F1 versus 0.5546 for the matching feature-only perceptron.Source: D. The residual explains per-node loss · p. 5

E. The effect strengthens with depth

  • +0.00084 at two layers rises to +0.00115 at six layers, positive in all 24 interactomes at every depth.The six-to-two-layer ratio is 1.37 under Z+0 and 1.39 under Z0, indicating that depth sensitivity remains under the strict control.Source: E. The effect strengthens with depth · p. 5

F. Controls

  • Adding feature-only per-node loss leaves the increment at +0.00339, positive in all 24 interactomes, but does not establish a message-passing-specific mechanism.Across 120 fits, the score coefficient is +0.00616 and positive in 113; structurally remote proteins carry higher loss.Source: D. The residual explains per-node loss · p. 5, F. Controls · p. 6

G. A mechanism test that fails

  • Two-hop coverage is collinear with degree at ρ = 0.917, and blocking on degree removes its trend; the node-level test does not demonstrate the proposed mechanism.The pooled difference is −0.00014 [−0.00110, +0.00102], positive in 31 of 72 cells, and the score-by-coverage interaction has the opposite sign to the prediction.Source: G. A mechanism test that fails · p. 6

H. The practical ceiling

The structural score adds only a small increment to risk prediction, while resistance’s information is bounded by degree degeneration. The study also limits its claims to standalone risk scoring on one benchmark and identifies further tests for other networks and rewiring.

H. The practical ceiling

  • 0.37% of the variance left unexplained by the strict design is claimed by the structural score; selective prediction changes little, with risk-coverage area shifting from 0.0405 to 0.0404.Precision-recall area rises from 0.7954 to 0.7958, a gain of +0.00034 [+0.00016, +0.00053].Source: H. The practical ceiling · p. 6

VI. DISCUSSION AND CONCLUSION

  • The residual association survives controls and both correction routes at every depth, but is small enough to change no deferral decision; resistance remains close to the inverse-degree limit.The departure from that limit shrinks as co-expression-filtered networks grow, leaving most potential contribution absorbed by degree.Source: §VI · p. 6

A. Scope

  • The evaluation covers standalone node-level risk scoring on one benchmark, one architecture family, and one uncertainty target, not resistance-based rewiring.The authors state that these results do not bear on rewiring, and note that networks farther from the degree limit may behave differently.Source: A. Scope · p. 6

B. Limitations

  • The null comparison supports claims beyond degree sequence, but because rewiring only halves clustering, it cannot separate local clustering from larger-scale organization.Per-graph consistency drops from 24 of 24 to 19 of 24 after floor subtraction; the mechanism evidence is model-level, and the risk experiment uses one benchmark.Source: B. Limitations · p. 6

C. Outlook

  • Future tests would measure resistance on biological graphs farther from the inverse-degree limit and compare its diagnostic ceiling with gains from rewiring before training.Both directions should retain degree-matched nulls and estimate a permutation floor within each tested subset.Source: C. Outlook · p. 6

APPENDIX A NUMERICAL VERIFICATION

The appendix verifies the numerical resistance computation and tests whether structural residuals persist after covariate saturation. It also explains why predictive entropy, rather than dropout variance, is the stricter uncertainty control.

APPENDIX A NUMERICAL VERIFICATION

  • No two numerical routes to L+ disagree by more than 1.4 × 10−11 relative, and the worst-conditioned trees depart from exact resistance by at most 1.9 × 10−12.For exact trees, resistance equals graph distance in closed form.Source: Appendix A · p. 7

APPENDIX B STRUCTURAL QUANTITIES

  • The score’s remaining structural increment is not an unfinished covariate ladder: the degree-matched null leaves 0.00009 unexplained against 0.00042 in the real-network comparison.The three-hop ball already covers 89% of a real interactome, making that covariate close to constant.Source: Appendix B · p. 7
  • 0.303% of variance remains unexplained after the two-hop ball on real interactomes, versus 0.107% on degree-matched nulls.The three-hop ball covers 89% of a real interactome, indicating the covariate sequence is close to saturated rather than prematurely truncated.Source: Appendix B · p. 7

A. Choice of predictive uncertainty

  • Predictive entropy tracks per-node loss more closely than dropout variance, with the gap widening as depth increases.Dropout variance peaks near predicted probability one half, whereas the highest losses occur at confident but incorrect nodes.Source: A. Choice of predictive uncertainty · p. 7

B. Number of seeds and replicas

Variation between the 24 tissue graphs, rather than additional seeds or rewiring replicas, is the binding constraint on uncertainty intervals. Per-graph structural and risk effects are directionally consistent but heterogeneous, while the proposed node-level coverage mechanism disappears after degree is fixed.

B. Number of seeds and replicas

  • The 24 graphs, not more seeds or replicas, are the binding constraint because between-interactome variation dominates within-interactome variation.The rewiring randomness is largely absorbed by the trained model by the time it reaches downstream measurements.Source: B. Number of seeds and replicas · pp. 7–8

APPENDIX D PER-GRAPH RESULTS

  • The real interactome leaves more unexplained structural variance than its degree-matched null in all 24 cases, with ratios from 1.93 to 3.89.Ddeg declines from 0.085 in the smallest network to 0.033 in the largest, corresponding to Spearman correlation −0.955.Source: Appendix D · p. 8
  • ∆R2 is positive in every interactome but ranges from 0.00005 to 0.00473, and five networks fall below the +0.00021 permutation floor.The aggregate interval excludes zero because the effect is directionally consistent, not because it is strong in every tissue.Source: Appendix D · p. 8

APPENDIX E A MECHANISM TEST THAT FAILS

  • After degree is fixed, low- versus high-coverage nodes show no consistent loss-increment difference; pooled across blocks the difference is −0.00014 [−0.00110, +0.00102].Both the degree-block comparison and the continuous score-by-coverage interaction indicate coverage carries nothing once degree is fixed.Source: Appendix E · p. 8

APPENDIX F DERIVATIONS

The appendix derives a node-level score as an average departure from the inverse-degree limit and establishes a nonnegative resistance excess for non-adjacent pairs. It also describes a degree-block analysis that subtracts a permutation floor from each increment.

A. The inverse-degree decomposition

  • The theory-aligned score is the node-averaged departure from the inverse-degree limit, up to a graph-dependent constant.The passage presents this as an algebraic rearrangement.Source: A. The inverse-degree decomposition · p. 8

B. The shorting bound

  • For non-adjacent pairs, the resistance excess is non-negative; observed minima are zero to machine precision across all 24 interactomes and their degree-preserving nulls.The reported minima are −1.1 × 10−13 for interactomes and −7.5 × 10−14 for nulls, with equality reached by pairs of low-degree nodes.Source: B. The shorting bound · p. 8

VARIANCE IN PER-NODE LOSS UNDER Z+

  • The degree-block analysis splits corrected increments by two-hop coverage and subtracts a separate permutation floor for each subset.The table caption specifies that the floor grows as subset size shrinks.Source: Table IX · p. 9

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