Neutral Assortativity Maximizes Pudding-Susceptible Broadcast Lifetime

Authors: Sterling Geisel, QBist Lab

Abstract

Yan Hao, Daniel J. Graham, and Marc-Thorsten Hutt introduce Copy-Spread-Annihilate dynamics, a synchronous network model in which messages broadcast to all neighbors and vanish when multiple copies collide. Their result is structural and sharp. Message lifetime is not maximized by the most assortative or most disassortative network. It peaks near neutral assortativity, where hub amplification remains strong and short-cycle annihilation remains limited. This working paper applies the Chaos Susceptibility Postulate to that finding. Pudding Theory predicts that the same neutral-assortative regime should not only preserve endogenous broadcast messages longer. It should also show the largest measurable response to weak coherent external bias, because small directional perturbations persist long enough to be amplified before collision erases them. The claim is testable by adding a phase-locked perturbation to CSA injection probabilities and measuring the excess lifetime gain as a function of assortativity.

Source Synopsis

Hao, Graham, and Hutt study a basic question in network dynamics. How long can a message survive when communication proceeds by broadcast rather than point-to-point routing. They define Copy-Spread-Annihilate dynamics on a finite undirected graph. At each discrete time step, every occupied vertex sends copies of its message to all neighbors. A vertex is occupied at the next step only if it receives exactly one copy. If two or more copies arrive, the incoming messages annihilate.

The measured quantity is message lifetime. For an injected message, all descendant copies trace walks through the graph. The lifetime is the maximum walk distance reached before all copies vanish. The authors then average this lifetime across injected messages and graph ensembles.

The central control parameter is degree assortativity, the Pearson correlation of degrees across edges. Positive assortativity connects hubs to hubs. Negative assortativity connects hubs to low-degree vertices. The authors use Barabasi-Albert graphs, preserve degree sequences through edge switching, and vary assortativity across a broad interval.

Their main result is non-monotonic. Average message lifetime peaks near neutral assortativity. High negative assortativity weakens propagation because hubs broadcast into low-degree neighborhoods. High positive assortativity strengthens hub-to-hub amplification but also increases short cycles, especially 4-cycles, which promote simultaneous return paths and self-annihilation. Neutral assortativity balances the two mechanisms.

The paper also studies vertex participation in long-lived walks. Disassortative graphs favor high-degree vertices. Assortative graphs push long-lived traffic away from hub cores toward lower-degree vertices. Near neutral assortativity, participation is more evenly distributed across the graph.

The authors then apply the same analysis to an empirical mouse connectome and degree-preserving surrogates. The empirical connectome lies near the lifetime maximum. They interpret this as evidence that brain-like networks may use assortativity as a structural control parameter for broadcast signal persistence.

Postulate Lens

This paper calls for the Chaos Susceptibility Postulate. The relevant variable is not disorder in the colloquial sense. It is dynamical susceptibility: the capacity of a structured system to amplify a small signal before dissipative or annihilating mechanisms erase it.

CSA dynamics define a clean susceptibility experiment. A single injected message becomes many walks. The network either amplifies the perturbation into long-lived descendants or destroys it through collision. Hao, Graham, and Hutt show that this susceptibility is not monotone in connectivity among hubs. The most responsive regime is a narrow structural compromise near neutral assortativity.

In Pudding Theory, the Chaos Susceptibility Postulate states that unstable systems are most susceptible to small coherent inputs. The CSA result refines the operational criterion for networks. Susceptibility requires both amplification and survival. A network with strong branching but fast collision is not maximally receptive. A network with low collision but weak branching is also not maximally receptive. The receptive regime is where perturbations remain mobile, replicated, and not yet self-cancelled.

Pudding Theory Prediction

Pudding Theory predicts an observable extension of the CSA result. In a network with CSA dynamics, a weak coherent bias applied to message injection or edge activation should have its largest lifetime effect near neutral assortativity.

The prediction is not that neutral networks have the longest unperturbed lifetimes only. Hao, Graham, and Hutt already show that. The Pudding Theory prediction concerns response gain. Let an external coherent input modulate the probability that new messages are injected at vertices belonging to a chosen phase class, degree class, or spatial module. The modulation should be weak enough that the baseline CSA lifetime curve is not trivially overwritten. Under ordinary network theory, one expects the perturbation response to follow local topology, injection rate, or degree distribution. Under the Chaos Susceptibility Postulate, the response should scale with the network’s ability to preserve small directional differences through broadcast before annihilation.

This produces a specific curve. The excess lifetime,

\[ \Delta \tau(r)=\langle \tau\rangle_{\mathrm{biased}}(r)-\langle \tau\rangle_{\mathrm{unbiased}}(r), \]

should peak at or slightly near the assortativity value where amplification is high and 4-cycle annihilation is still limited. It need not peak exactly at the unperturbed lifetime maximum, because coherent bias can alter which vertices seed successful walks. But it should not peak at the extreme positive or negative ends of the assortativity spectrum.

The same prediction applies to the mouse connectome surrogates. If the empirical connectome is structurally tuned for broadcast persistence, then weak coherent modulation should create a larger excess lifetime in the empirical graph than in strongly rewired assortative or disassortative surrogates with the same degree sequence. The key observable is not raw activity. It is the perturbation derivative of message lifetime.

This matters because it separates passive persistence from active receptivity. A network may let messages live longer, yet fail to convert small structured inputs into measurable lifetime gains. Pudding Theory predicts that the near-neutral CSA regime does both.

Falsifiable Observable

The distinguishing observable is the assortativity-dependent response gain \(G(r)=\Delta \tau(r)/\epsilon\), where \(\epsilon\) is the amplitude of a weak phase-locked injection bias under fixed degree sequence, graph size, injection rate, and CSA update rule. If \(G(r)\) were measured to be monotonic increasing toward high positive assortativity, with no local maximum within \(-0.1<r<0.1\), this Postulate would be falsified. This test must use matched graph ensembles and report both hub-neighbor degree and 4-cycle counts.

Editorial Dialogue

Tanaka: The CSA model is deterministic once messages are placed. There is no need for Pudding Theory. The lifetime peak follows from graph structure and collision rules. You are adding a field vocabulary to a sufficient explanation.

Sterling: The structural explanation is accepted. The Postulate is not used to replace it. It is used to predict a response experiment the source paper did not perform. The source establishes where messages persist. Pudding Theory asks where a weak coherent perturbation changes persistence most.

Tanaka: That still sounds like ordinary susceptibility. Why attach it to consciousness or information fields.

Sterling: Because the Pudding claim is not about all perturbations equally. It concerns coherent informational bias acting on a receptive substrate. CSA dynamics are useful because they strip away biochemical detail. They expose the substrate variable: survival of small directed differences under amplification and annihilation.

Tanaka: If the peak shifts under biased injection, you may explain it after the fact.

Sterling: The falsifier prevents that. A monotone gain toward high positive assortativity would defeat the application. So would a gain controlled only by hub degree after conditioning on 4-cycles and baseline lifetime. The Postulate survives only if receptivity is maximal near the balanced regime.

Discussion

The present application is narrow. CSA dynamics are not a full model of neural signaling. Messages are discrete. Time is synchronous. Edge weights are removed in the mouse connectome analysis. Biological inhibition, delays, synaptic signs, and plasticity are absent. These omissions are useful for the source paper’s structural argument, but they limit physiological interpretation.

The Pudding Theory claim also depends on a measurable perturbation protocol. Without a controlled coherent bias, the Postulate reduces to a restatement of the lifetime curve. That would be weak work. The useful test is comparative response gain under matched perturbation amplitude.

The strongest open question is whether neutral assortativity remains optimal when delays, weighted edges, and refractory states are added. If the response maximum survives those additions, the result would support the idea that brain-like networks tune not only for traffic efficiency but for susceptibility to weak structured signals. If it disappears, the CSA result should be treated as a property of a minimal broadcast-annihilation rule, not as evidence for broader Pudding receptivity.

References

1. Yan Hao, Daniel J. Graham, and Marc-Thorsten Hutt. “Copy-Spread-Annihilate Dynamics in Degree-Assortative Networks.” arXiv:2603.29833, 2026. DOI: doi:10.48550/arxiv.2603.29833.

2. Sterling Geisel. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab Working Paper, 2025.

3. M. E. J. Newman. “Assortative Mixing in Networks.” Physical Review Letters 89, 208701, 2002.

4. M. E. J. Newman. “Mixing Patterns in Networks.” Physical Review E 67, 026126, 2003.

5. A. Trusina, S. Maslov, P. Minnhagen, and K. Sneppen. “Hierarchy Measures in Complex Networks.” Physical Review Letters 92, 178702, 2004.

6. Y. Hao and D. J. Graham. Network Neuroscience 4, 1055, 2020.

7. S. W. Oh, J. A. Harris, L. Ng, B. Winslow, N. Cain, S. Mihalas, Q. Wang, C. Lau, L. Kuan, A. M. Henry, et al. “A Mesoscale Connectome of the Mouse Brain.” Nature 508, 207, 2014.

8. M. P. van den Heuvel and O. Sporns. “Rich-Club Organization of the Human Connectome.” Journal of Neuroscience 31, 15775, 2011.