Do more graph steps give more information? When averages erase differences

Sharing information can make nodes indistinguishable

Graph models use relations among components, document entities or scientific objects. Neighbor communication gathers context, but repeated exchange can erase task-relevant differences. A fully calculable example uses four nodes, one value each and repeated local averaging. We identify preserved, attenuated and immediately lost signal components.

Abstract. Four-node undirected cycle uses P=.5 I+.25 A, adjacency A, eigenvalues 1,.5,.5,0. Alternating(1,−1,1,−1)vanishes after one step; smooth(1,0,−1,0)halves amplitude and quarters variance per step. Study constant component, slower mixing variant and no-self-weight oscillation. Linear experiments without trained GNN do not prove every deep architecture fails.

The graph and update rule

Edges 1–2,2–3,3–4,4–1 are bidirectional, two neighbors each. Adjacency A is 1 on edges,0 otherwise; I identity. Updates keep half self-value plus a quarter from each neighbor. Nonnegative row weights sum 1: averaging weights, not class probabilities. No learned weights, nonlinearities or node-ID features.

h is a four-entry dimensionless column, k step count. P@h simultaneously computes averages from prior-step values; in-place sequential updates would be order-dependent and a different algorithm. Matrix multiplication avoids that ambiguity. This regular graph’s P is symmetric, column sums also 1, preserving global mean.

A signal that disappears immediately

Start with h⁽⁰⁾=(1,−1,1,−1). Node 1 becomes.5 × 1+.25 ×(−1)+.25 ×(−1)=0; all others likewise. One step erases alternating contrast despite connectivity and messages. If those differences matter, more context did not enrich them. Output alone cannot reconstruct input: zero input gives the same output.

For a finite representation argument, assign labels(1,0,1,0) and allow a deterministic classifier only the averaged value, no IDs or features. Four zeros force one answer for all, matching at most 2/4. This is a constructed-set limitation, not estimated test accuracy. Preserved inputs or additional information change the premise.

Eigenvalues reveal which differences survive

An eigenvector is a signal direction scaled, not mixed into other directions, by an eigenvalue. Repeating P multiplies its component by that eigenvalue’s power. Here direct multiplication verifies constant preservation, two half-amplitude modes and zero alternating mode. Four independent vectors form a basis for any input.

Constant component contains global mean; half-eigenvalue modes vanish as k grows and u₄ is already lost, so nodes approach one mean. This isolates excessive smoothing. Other GNNs need not use this matrix; irregular graphs or normalizations need not converge to arithmetic mean. The proof applies to declared P, not the GNN label alone.

Measuring contrast loss

For smooth h⁽⁰⁾=(1,0,−1,0), h⁽ᵏ⁾=(.5)ᵏh⁽⁰⁾. Across-node variance averages squared deviations, dividing by 4 because these are all nodes, not a random population sample. Mean zero gives initial variance.5; halving deviations quarters squares. The formula quantifies contrast loss.

Two cautions: constant(1,1,1,1)has zero variance yet remains, not erased. Smooth-signal positive/negative signs persist at finite steps in ideal arithmetic, so declining variance alone proves no declining accuracy curve. Alternating signal instead suffers exact irreversible loss. Analysis must identify task-relevant information, not merely a shrinking metric.

Inverting the smooth mode after 10 steps multiplies by 2¹⁰=1024, also amplifying any additive representation error by 1024. This is algebra, not measured device noise; code records gain without simulating noise. A zero-eigenvalue mode cannot be inverted at all. Attenuation and exact cancellation differ.

Does changing self-weight change the conclusion?

Generalize to P_α=(1−α)I+αA/2, controlling neighbor versus self information. Original α=.5. Same eigenvectors have smooth factors 1−α and alternating 1−2α. Code also runs α=.25 and 1 on alternating input, saving six states. This changes one linear averaging rule, not a new trained network.

At α=.25 alternating contrast halves rather than vanishing. For 0<α<1 all nonconstant factors have magnitude below 1, so slowing does not eliminate eventual attenuation. α=0 exchanges nothing. α=1 removes self-weight; the bipartite cycle’s alternating mode flips sign forever, variance 1. Thus convergence depends on graph, self-links and operator, not averaging as a vague slogan.

Connecting the example to GNN research

Li, Han and Wu’s Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning, arXivv 1,22 January 2018, analyzes Laplacian smoothing. Derivations, theorem conditions and experiments were read. Cora, CiteSeer, PubMed protocols report 50 runs,10 for PubMed with specific labels/settings. Our cycle reproduces none of their datasets, co-training or results. It independently illustrates a foundational linear mechanism, not a 2026 state-of-art survey.

Real networks add learned transforms, nonlinearities, input-preserving links or aggregations. Concatenating original input keeps that signal available, but does not guarantee training uses it well. We choose no best architecture. Depth explanations should separate communication reach, contrast loss and task quality, each with suitable evidence.

Executed package uses NumPy 2.5.3, Python 3.14.0, no randomness. Assertions check products; eigensolver’s order 10⁻¹⁶residual near zero is handled numerically, distinct from algebraic zero. Sparse averaging of dfeatures, nodes n, edges e costs O((n+e)d)per step; dense 4 × 4 here aids inspection, not scalability measurement. No EL-AI accuracy/performance measured.

More steps move signals but do not automatically add useful information. One contrast vanishes immediately, others attenuate by calculable eigenvalues; preserving mean does not preserve distinctions. Graph-model benefits must be shown on task-relevant information, not inferred from message count as a universal understanding index.

Qimai Li, Zhichao Han, Xiao-Ming Wu. Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning, arXiv 1801.07606 v1,22 January 2018. Derivations, theorem conditions, experiments read 17 October 2026; not reproduced..

from experiment import P
import numpy as np
h=np.array([1.,-1.,1.,-1.])
print(P@h, np.linalg.eigvalsh(P))

Code, data, and instructions · JSON. Educational calculations executed with Python 3.14.0; figures with Matplotlib 3.11.2. AI-assisted analysis, without claiming peer review or human review. Original illustrative ImageGen cover: it does not document EL-AI people, premises, or installations. Sources accessed 17 October 2026.