Near-Critical Network Plasticity Amplifies Small Coherent Biases

Authors — Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Branchi proposes that plasticity can be defined before change occurs, rather than inferred after it. The source paper formalizes plasticity as the ratio of system size to aggregate connectivity strength. This places plasticity between two failures. Strong connectivity yields rigidity. Weak connectivity yields instability. Intermediate connectivity yields a critical regime in which a system can both change and settle. This working paper applies Chaos Susceptibility to that framework. The Postulate predicts that the same near-critical regime that maximizes effective plasticity should also maximize response to weak coherent inputs, provided those inputs are phase-consistent with available transitions. The distinguishing observable is not plasticity itself, but an excess transition probability under controlled coherent perturbation, normalized by spontaneous transition rate and branching value. A null result at criticality would cut against this application of Pudding Theory.

Source Synopsis

Igor Branchi’s paper, “Quantifying plasticity: a network-based framework linking structure to dynamical regimes,” argues that plasticity should be treated as a prospective property of complex systems. The paper rejects the common habit of inferring plasticity only from later change in morphology, behavior, symptoms, or activity. Instead, plasticity is operationalized from network structure.

The central formula is simple. Plasticity is proportional to the number of nodes, \(N\), and inversely proportional to aggregate edge strength, \(\sum_{e \in E}|w(e)|\). System size defines the dimensionality of the accessible state space. Connectivity strength constrains motion through that space. A large system has more possible configurations. A tightly connected system resists local change because each element is constrained by the rest.

Branchi distinguishes transition plasticity from configurational plasticity. Transition plasticity concerns how easily a system moves between states within a fixed state space. Configurational plasticity concerns the dimensionality of the state space itself. Their combination yields a structural measure of plasticity as \(P=N/\sum |w(e)|\).

The paper’s main claim is that optimal plasticity is not maximal plasticity. Very low plasticity produces rigidity. Very high plasticity produces instability. The useful regime lies between them. This intermediate regime coincides with criticality, described through a branching value near \(\sigma=1\). The paper therefore defines effective plasticity as a normalized measure of proximity to this critical point.

Branchi also argues that plasticity drives criticality, rather than merely accompanying it. Connectivity and system size are treated as causal constraints on dynamic regimes. This matters for mental health, ecology, economics, neural systems, and social systems. Plasticity becomes a structural regulator of possible transitions, not a verdict about whether a transition will be beneficial.

Postulate Lens

This paper applies Chaos Susceptibility.

The fit is direct. Branchi’s framework identifies intermediate connectivity and near-critical branching as the regime in which a system is most able to change without losing coherence. Chaos Susceptibility states that systems with high dynamical sensitivity amplify small coherent inputs into macroscopic outcomes. The source paper gives a structural map of where that amplification should be greatest: not in the rigid regime, and not in the unstable regime, but near effective plasticity maximum.

In this reading, plasticity is not the Pudding effect. It is the receptivity parameter. A coherent perturbation should have the largest measurable influence where \(P_{\mathrm{eff}}\) is high and \(\sigma\) is close to one. It should have little effect when connectivity is too strong, because the perturbation is damped. It should also have little organized effect when connectivity is too weak, because the perturbation cannot consolidate into a stable transition.

Pudding Theory Prediction

Pudding Theory predicts a nonmonotonic coupling between coherent perturbation and state transition probability across Branchi’s plasticity gradient.

Let a finite network be prepared with fixed node count \(N\), tunable aggregate connectivity \(W=\sum |w(e)|\), and measured branching value \(\sigma\). Let the network possess two or more stable or metastable attractor states. Under ordinary dynamics, transitions among those states occur at a baseline rate \(r_0(P,\sigma)\). Under a weak coherent input aligned with one target transition, Pudding Theory predicts a modified rate \(r_c(P,\sigma)\).

The relevant prediction is not that all plastic systems change more. It is that coherent bias is selectively amplified near the effective plasticity peak. Define

\[ R_c(P,\sigma)=\frac{r_c(P,\sigma)-r_0(P,\sigma)}{r_0(P,\sigma)}. \]

The predicted \(R_c\) should peak near \(\sigma=1\), with lower values in both the rigid and unstable regimes. The curve should resemble Branchi’s effective plasticity curve more than the raw plasticity curve. This distinction matters. If raw plasticity rises as connectivity weakens, Pudding Theory does not predict a monotonic rise in coherent influence. In the unstable regime, changes occur, but they should not preserve target-specific information. They should look like noise, drift, or transient fragmentation.

In neural or psychopathology networks, the prediction is that high effective plasticity should increase sensitivity to coherent contextual structure. A therapeutic context, repeated task demand, or controlled attentional perturbation should shift transition probabilities more strongly when symptom or activity networks are near criticality. In artificial networks, the same claim can be tested more cleanly by varying edge weights and measuring target-state occupancy under identical coherent inputs.

Current consensus can accommodate critical sensitivity without invoking Pudding Theory. The additional claim here is target-selective amplification by weak coherent input after controlling for baseline transition rate, noise level, and network topology. The observable must therefore separate mere lability from directed susceptibility.

Falsifiable Observable

The distinguishing observable is \(R_c(\sigma)\), the normalized excess target-transition rate under weak coherent input, measured across matched networks spanning \(\sigma<1\), \(\sigma\approx1\), and \(\sigma>1\). Pudding Theory predicts \(R_c(\sigma\approx1)>R_c(\sigma<1)\) and \(R_c(\sigma\approx1)>R_c(\sigma>1)\), with target-specific information retained after transition. If near-critical susceptibility ratio \(R_c(\sigma\approx1)\) were measured to be 0.00 with a preregistered 95% confidence interval contained within \([-0.01,0.01]\), this Postulate would be falsified.

Editorial Dialogue

Tanaka: Branchi’s paper already explains the result without Pudding Theory. Critical systems are sensitive. That is the entire point. A weak input has a larger effect near the critical point because dynamic range is maximal. Nothing else is needed.

Sterling: That objection is correct against a vague reading. It is not correct against the present claim. The claim is not increased dynamic range alone. The claim is target-selective excess transition under weak coherent input, normalized by baseline transition rate and branching value.

Tanaka: But coherence is still an ordinary control variable. A periodic or structured input can entrain a critical system. That is standard nonlinear dynamics.

Sterling: Then the experiment must remove that refuge. The input amplitude should be below the entrainment threshold determined by control trials. The analysis should ask whether target-state occupancy still shifts when ordinary forcing is too weak to account for the effect.

Tanaka: If the effect disappears under blinding and matched noise, the Postulate fails.

Sterling: Yes. That is why the falsifiable observable is framed as a null value at criticality. The source framework gives the correct structural axis. Pudding Theory adds a directed susceptibility term. It must earn that term.

Discussion

The source paper supplies a useful discipline for Pudding Theory. It prevents a loose claim that “plastic systems are easier to influence.” Branchi’s framework separates change capacity from stable adaptation. Pudding Theory should do the same. A highly labile system is not necessarily a receptive one. It may simply fail to retain structure.

The main limitation is measurement. Branching value, node definition, and edge weights depend on the modeler’s scale. Branchi notes this explicitly. A symptom network, cortical network, ecological network, and financial network may all admit \(P=N/\sum |w(e)|\), but homologous nodes are required for comparison.

The strongest test is computational or in vitro. A network can be tuned across connectivity regimes, held at constant noise, and exposed to weak coherent perturbations. Human studies are less clean because context, expectation, and measurement alter the system being measured.

The conclusion would change if near-critical networks showed no target-selective excess transition under inputs below ordinary forcing thresholds. It would also change if the effect rose monotonically with raw plasticity rather than peaking at effective plasticity. Either outcome would favor Branchi’s structural account without the Pudding extension.

References

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2. Geisel, S. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab Working Paper, September 10, 2025.

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