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

Jianru Shen

cs.LGOct 1, 2026

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

Authors: Jianru Shen

Organizations: University of Montana Missoula, MT, USA

Abstract

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.

Figures & tables

Fig. 1: Effective resistance in tissue-specific interactomes. (a) On a representative network the log Laplacian pseudoinverse diagonal is almost determined by log degree, and the structural score of a node is its distance from the fitted line. The fitted slope is steeper than the reference value of $-1$ that a pure inverse-degree relationship would give asymptotically, which is why a fixed-coefficient correction leaves degree dependence behind. (b) The score is recomputed and the model retrained in three worlds: the real interactome, a degree-preserving rewired null, and a near-tree geodesic control where resistance reduces to graph distance. Networks are schematic and the annotated values are corpus medians. (c) Each increment is read against a permutation floor measured in its own world, giving factors of $5.6$ and $1.5$ above that floor.

Graph family or anchor	Graphs	Nodes	Mean degree	$\lambda_{2}$	$D_{\deg}$	$D_{\mathrm{geo}}$
Tissue-specific interactomes	24	591–3480	13.0–32.1	0.16–0.53	0.038	0.931
Degree-preserving null	72	591–3480	13.0–32.1	0.30–0.85	0.032	0.931
Peptides-func	15535	5–444	1.6–2.2	0.000–0.519	0.931	0.000
Path	1	300	1.99	0.0001	0.989	0.000
Random tree	1	300	1.99	0.0006	0.940	0.000
Ring lattice, $k=4$	1	300	4.00	0.0022	0.957	0.697

TABLE I: Resistance regimes. Medians across graphs for the three families, ranges for size and spectrum. Density does not set the regime: the barbell is five times denser than the dense random graph yet sits at the opposite corner, and its small

\lambda_{2}

identifies the bottleneck responsible.

Model	Micro-F1
Feature-only perceptron	$0.5546\pm 0.0038$
GraphSAGE, 2 layers	$0.8399\pm 0.0054$
GraphSAGE, 4 layers	$0.8799\pm 0.0055$
GraphSAGE, 6 layers	$0.9026\pm 0.0057$
GraphSAGE, 6 layers, null world	$0.5972\pm 0.0037$

TABLE II: Base model quality. Validation micro-F1 over four folds and five seeds, with the standard deviation across runs.

Score and setting	$\Delta R^{2}$ under $Z_{0}$	$\Delta R^{2}$ under $Z_{0}^{+}$	Interactomes
Permutation floor	—	$+0.00021$ [ $+0.00017$ , $+0.00025$ ]	—
Primary score $s$ , by depth
2 layers	$+0.00237$ [ $+0.00167$ , $+0.00315$ ]	$+0.00084$ [ $+0.00052$ , $+0.00119$ ]	24/24
4 layers	$+0.00306$ [ $+0.00225$ , $+0.00400$ ]	$+0.00094$ [ $+0.00062$ , $+0.00132$ ]	24/24
6 layers	$\mathbf{+0.00330}$ [ $+0.00232$ , $+0.00444$ ]	$\mathbf{+0.00115}$ [ $+0.00074$ , $+0.00164$ ]	24/24
floor subtracted	—	$+0.00095$ [ $+0.00052$ , $+0.00143$ ]	19/24

TABLE III: Incremental variance in per-node loss from each structural score, per graph, averaged over seeds, with

95\%

confidence intervals from resampling the 24 interactomes. All rows outside the depth block use six message-passing layers. The permutation floor, estimated from one thousand draws per graph, is the scale against which every other row should be read. Since

\Delta R^{2}

cannot be negative in sample, the count of positive interactomes is informative only once that floor is subtracted, which leaves 19 of 24 for the primary score at six layers.

Fig. 2: Variance of the log pseudoinverse diagonal left unexplained after degree, neighbour degree, clustering and two-hop neighbourhood size, for each interactome against its own degree-preserving null. Every network lies above the diagonal. The largest departure belongs to the smallest network, matching the size trend in

D_{\deg}

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

	Peptides	Interactomes
	12 worst cond.	4 largest
Nodes	401–444	3195–3480
$\lambda_{2}$	$1.5\times 10^{-4}$	$0.32$ – $0.38$
Condition number	$3.5\times 10^{4}$	$2.1\times 10^{3}$
Eigendecomp. vs SVD	$2.5\times 10^{-15}$	$1.2\times 10^{-14}$
Eigendecomp. vs ( 9 )	$1.4\times 10^{-11}$	$5.4\times 10^{-14}$

Appendix

TABLE IV: Largest deviation observed between three routes to

L^{+}

, on the worst-conditioned graphs of either corpus. Route differences are relative to the scale of the diagonal.

	Real		Null
Covariate added	Unexpl.	Incr.	Unexpl.	Incr.
Degree	$0.935\%$	—	$0.675\%$	—
$+$ neighbour degree	$0.731\%$	$0.00204$	$0.596\%$	$0.00078$
$+$ clustering	$0.710\%$	$0.00022$	$0.587\%$	$0.00009$
$+$ 2-hop ball	$0.303\%$	$0.00407$	$0.107\%$	$0.00481$
$+$ 3-hop ball	$0.260\%$	$0.00042$	$0.098\%$	$0.00009$

Appendix

TABLE V: Variance of

\log L^{+}_{ii}

left unexplained as covariates are added, averaged over graphs. Coverage is the mean fraction of the network inside the ball.

	Message-passing depth
	2	4	6
Entropy vs loss	$+0.844$	$+0.868$	$+0.891$
Margin vs loss	$-0.853$	$-0.884$	$-0.911$
Dropout variance vs loss	$+0.300$	$+0.449$	$+0.529$
Dropout variance vs entropy	$+0.492$	$+0.616$	$+0.668$

Appendix

TABLE VI: Spearman correlation of each predictive uncertainty score with per-node loss, on validation graphs. A usable score correlates positively; margin is a confidence score and so correlates negatively.

Source	Within	Between	Ratio
Training seeds	$0.0104$	$0.0305$	$2.9$
Rewiring replicas, structural	$0.0125$	$0.0367$	$2.9$
Rewiring replicas, downstream	$0.00030$	$0.00033$	$1.1$

Appendix

TABLE VII: Variation between interactomes relative to variation within one. A large ratio means the sample of graphs, not the number of repetitions, is the binding constraint.

Split and index	Nodes	Mean degree	$D_{\deg}$	Resid. real (%)	Resid. null (%)	Ratio	$\Delta R^{2}$
train[8]	591	13.04	0.0851	0.863	0.253	3.41	$0.00105$
train[5]	1021	17.84	0.0571	0.427	0.143	2.99	$0.00280$
train[1]	1377	21.57	0.0513	0.337	0.141	2.38	$0.00024$
train[0]	1546	20.90	0.0491	0.554	0.143	3.89	$0.00014$
train[4]	1578	22.92	0.0449	0.284	0.107	2.66	$0.00011$
train[12]	1819	25.16	0.0418	0.259	0.101	2.55	$0.00473$

Appendix

TABLE VIII: All 24 tissue interactomes, ordered by size. Residual columns give the variance of

\log L^{+}_{ii}

left unexplained after the full covariate ladder, for the real network and for the mean of its three degree-preserving replicas.

\Delta R^{2}

is the incremental variance in per-node loss under

Z_{0}^{+}

at six layers, averaged over five seeds. The permutation floor,

+0.00021

, is given for comparison.

Degree block	Coverage	Mean deg.	Nodes	Corrected $\Delta R^{2}$
Low	$0.052$	$3.3$	414	$+0.00232$
Low	$0.202$	$5.8$	413	$+0.00231$
Middle	$0.231$	$13.6$	381	$+0.00022$
Middle	$0.387$	$17.7$	378	$-0.00017$
High	$0.444$	$35.4$	388	$+0.00011$
High	$0.653$	$86.5$	386	$+0.00094$

Appendix

TABLE IX: Corrected increment within degree blocks, split by two-hop coverage. Each increment has its own permutation floor subtracted, since that floor grows as the subset shrinks.

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ORBIT-FMIB: Tracking Order-Resolved Epistatic Information Through ESM-2

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Protein foundation models support mutation-effect and structural prediction, but predictive performance alone does not reveal which forms of biological interaction information remain accessible through model depth. We ask whether ESM-2 retains higher-order epistatic information as strongly as first- and second-order information across its representation hierarchy, introducing ORBIT-FMIB, a diagnostic framework combining Walsh-based interaction decomposition with subset-conditioned neural dependence estimation. The method is validated on synthetic landscapes with known interaction structure before being applied to the dense four-site GB1 fitness landscape using frozen ESM-2 representations. An initial production run suggested ESM-2 retains higher-order epistatic information less well than lower-order information (

Δ_{\mathrm{HO-LO}}=-0.107

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Δ_{\mathrm{HO-LO}}=-0.017

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Abstract

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