Infinite-Mean Fitness Produces Measurable Signal-Dominant Clustering Without Latent Geometry

Sterling Geisel, QBist Lab, Dr. Hideo Tanaka

Abstract

Catanzaro, van der Hofstad, and Garlaschelli prove that sparse random graphs with conditionally independent edges can sustain finite local clustering without metric geometry or higher-order edge dependence. Their mechanism is not triangle inequality. It is infinite-mean node fitness under a node aggregation invariant connection rule. This Working Paper applies Signal Dominance to that result. The Pudding Theory claim is narrow. If network-generating fitness is a physical proxy for persistent informational signal strength, then strong clustering should arise when the empirical fitness distribution has stable-law tails with exponent \(0<\alpha<1\), even when edge formation is dyadic and geometry-free. The predicted discriminator is not clustering alone. It is the joint occurrence of finite local clustering, power-law degree, and non-self-averaging of low-degree node fractions across repeated realizations. A finite-mean truncation of fitness should suppress this joint signature. This gives a falsifiable test of Signal Dominance in network data.

Source Synopsis

The source paper addresses a central problem in random network theory. Many real networks are sparse, broad-tailed in degree, and locally clustered. Sparsity means link density vanishes as network size grows. Local clustering means that the average fraction of closed triangles around eligible nodes remains positive. Standard independent-edge sparse graphs have trouble producing both. The Erdős-Rényi model makes density and clustering scale together, so clustering vanishes when density vanishes.

Previous work offered two main routes. One route abandons edge independence through triangles, cliques, bipartite projections, hypergraphs, or other higher-order dependencies. The second route keeps conditional edge independence but introduces node variables. In most finite-mean hidden-variable models, clustering still vanishes. Random geometric graphs are the famous exception. There edge probability depends on metric distance, and triangle inequality supplies triadic closure.

Catanzaro, van der Hofstad, and Garlaschelli show a third route. They study a multi-scale model in which each node has a positive weight \(w_i\), sampled independently from a heavy-tailed distribution. Edges are conditionally independent with probability \[ p_{ij}=1-e^{-\delta_n w_iw_j}. \] For the aggregation-invariant regime, \(\delta_n=n^{-1/\alpha}\), and weights are Pareto or one-sided stable with \(0<\alpha<1\). The mean fitness diverges.

Their main result is an asymptotic clustering function. With reduced degree \(a=k/\sqrt{n}\), the annealed clustering function tends to an explicit double integral. Leaf nodes have clustering tending to \(1\). Hub clustering decays as \[ \bar C(a)\sim 2\Gamma(1-\alpha)\frac{\log a}{a^2}. \] The average local clustering tends to \(1\) when nodes of degree \(0\) and \(1\) are excluded. If those nodes are assigned clustering zero, the limiting average remains random through the fluctuating fraction \(r_{0/1}\). This is a breakdown of self-averaging. The paper’s compact conclusion is that “clustering does not imply geometry.”

Postulate Lens

This paper applies Signal Dominance. The source model gives every node a scalar fitness. Edge formation is independent conditional on those scalar values, yet infinite-mean signal heterogeneity organizes a sparse graph into strongly clustered local neighborhoods. That is exactly the formal situation in which a persistent informational signal dominates local probability structure without requiring a metric embedding.

No second Postulate is needed. The source paper is about scalar dominance under aggregation invariance, not about proximity, time ordering, object persistence, or observer boundary structure.

In the Pudding Theory vocabulary, \(w_i\) is read as a coarse observable for node-level signal strength. This reading does not change the mathematics of the source model. It adds an empirical constraint. If signal strength is the organizing variable, then its distribution must be heavy enough that ordinary averaging fails. The important transition is not broad degree alone. It is the infinite-mean regime \(0<\alpha<1\). In that regime, a finite set of high-signal nodes does not merely become hubs. It changes the local closure statistics of low-degree nodes.

Signal Dominance therefore does not predict that all clustered networks hide geometry. It predicts the opposite conditional statement. A network can be clustered without geometry when the latent signal variable has no finite first moment and remains stable under aggregation. The organizing cause is scalar intensity, not metric nearness.

Pudding Theory Prediction

Pudding Theory predicts a measurable triplet in systems where the latent fitness variable is an informational signal rather than a geometric coordinate. The triplet is finite local clustering, sparse density, and non-self-averaging of low-degree fractions under repeated sampling or coarse graining.

The source paper already proves the mathematical possibility. The Pudding Theory addition is physical interpretation and experimental ordering. In a real network, one should first estimate whether observed degrees are compatible with a conditionally independent model whose latent fitness has a stable-law tail with \(0<\alpha<1\). If that estimate is supported, then local clustering should persist after controlling for measured geometry. The persistence should be strongest in low reduced-degree nodes, not in hubs. This is a distinctive prediction because many geometric accounts tie clustering to metric neighborhoods, while this account places the closure in signal-dominant leaves attached through extreme latent weights.

The second prediction concerns aggregation. If nodes are aggregated into blocks and block signal is approximately additive, then the distribution of block fitness should remain in the same stable family. Under correct aggregation, the estimated \(\alpha\) should be stable, the sparse scaling should remain close to \(\log n/n\), and clustering among eligible low-degree nodes should remain near unity. Under arbitrary aggregation, this invariance should weaken.

The third prediction concerns truncation. If a real system has an upper cutoff in signal strength, or if the analyst imposes one by censoring high-fitness nodes, clustering should fall toward the finite-mean hidden-variable expectation. The largest change should occur when the cutoff crosses the scale of the largest order statistics. The fall need not be smooth, because the source mechanism itself is non-self-averaging.

The clean empirical test is thus a perturbation test. Fit the model. Remove or cap inferred high-signal nodes. Recompute local clustering, \(r_0\), \(r_1\), and their replicate variance. Signal Dominance predicts that the clustering mechanism collapses when the infinite-mean tail is destroyed.

Falsifiable Observable

The distinguishing observable is the finite-size scaling of the eligible-node average local clustering after tail truncation of inferred node fitness. If the eligible-node average local clustering under fitted geometry-free independent edges with inferred \(0<\alpha<1\) were measured to converge to the same positive limit after replacing all inferred \(w_i\) above \(n^{1/\alpha-\epsilon}\) by that cutoff, this Postulate would be falsified. The cutoff should restore finite effective moments and remove the signal-dominant source of closure.

Editorial Dialogue

Tanaka: The source paper is a theorem about random graphs. It does not contain consciousness, vacuum structure, or intent. Why should a Pudding Theory Postulate be applied at all?

Sterling: Because the application is to the structural role of a scalar field, not to a human observer. The source model assigns each node a latent scalar. Conditional on that scalar, edges are independent. Yet the scalar distribution alone produces finite clustering. Signal Dominance is the only Postulate needed because it says that a persistent informational signal can organize probability structure.

Tanaka: But “fitness” in a network model is a mathematical parameter. Calling it signal strength may add no content.

Sterling: It adds content only if it predicts a discriminator. Here it does. The discriminator is the infinite-mean tail and the loss of self-averaging. A finite-mean scalar field would not do the same work.

Tanaka: Geometry-free clustering has already been proven by the source authors. Is Pudding Theory merely relabeling it?

Sterling: No. The source paper proves existence. This paper specifies what to look for in empirical systems if signal dominance is the physical mechanism: stable-law aggregation, leaf clustering near one, and collapse under high-fitness truncation.

Discussion

The interpretation is deliberately narrow. The source paper does not establish Pudding Theory. It establishes that scalar heterogeneity can replace geometry as a clustering mechanism in sparse independent-edge networks. Pudding Theory treats that scalar as a possible observable of informational signal strength only when three conditions are met: aggregation invariance, infinite-mean behavior, and persistent clustering without metric dependence.

There are limitations. Real networks are rarely sampled from a clean conditional independent-edge ensemble. Edges may have history, semantics, sampling bias, institutional constraints, or explicit triadic rules. A measured power law is not enough. Many finite samples mimic stable tails. The proposed truncation test is useful because it acts on the mechanism rather than the appearance.

The most serious open question is identifiability. A geometric model and an infinite-mean signal model may both fit degree and clustering. The source paper shows that triangles are not sufficient evidence for geometry. The reverse also matters. Lack of known geometry is not sufficient evidence for signal dominance. The decisive evidence would be the joint finite-size behavior of clustering, aggregation stability, and non-self-averaging across independently sampled comparable networks.

References

1. Alessio Catanzaro, Remco van der Hofstad, and Diego Garlaschelli. “Clustering without geometry in sparse networks with independent edges.” arXiv:2603.13159. DOI: doi:10.48550/arxiv.2603.13159, 2026.

2. Sterling Geisel. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab canonical paper, September 10, 2025.

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