Spectral radius, criticality & dynamic range

Research summary

One paper, summarised for reference. Predicting Criticality and Dynamic Range in Complex Networks: Effects of Topology — Daniel B. Larremore, Woodrow L. Shew & Juan G. Restrepo, Phys. Rev. Lett. (2011) 106, 058101. doi:10.1103/PhysRevLett.106.058101.

The result

For a network of coupled excitable nodes, criticality and maximal dynamic range are governed by one quantity: the largest eigenvalue $\lambda$ of the weighted adjacency matrix (the spectral radius). $\lambda=1$ is the critical point with maximum dynamic range, universally — for random, scale-free, assortative, and degree-correlated topologies alike. This generalizes the earlier branching-ratio / mean-degree criterion $\sigma=1$, which fails for heterogeneous degree distributions.

Setup and the old criterion

In the Kinouchi–Copelli model, a resting node is excited by an active neighbour $j$ with probability $A_{ij}$, or by external stimulus with probability $\eta$; the response $F$ is the time-averaged excited fraction, and dynamic range $\Delta = 10\log_{10}(\eta_{\text{high}}/\eta_{\text{low}})$ measures the span of stimuli that produce distinguishable responses. The prior theory said criticality occurs at branching ratio $\sigma = \tfrac1N\sum_{ij}A_{ij} = \langle d\rangle = 1$ (one excitation begets one), where avalanches are power-law — but this holds only for homogeneous (Erdős–Rényi) networks.

The derivation

Linearising the excitation dynamics about the quiescent fixed point $p=0$ gives $\lambda u_i = \sum_j u_j A_{ij}$ — so stability is set by the largest eigenvalue $\lambda$ ($\lambda<1$ quiescent, $\lambda>1$ active; Perron–Frobenius makes $\lambda$ real and positive). Near critical, the excitation profile aligns with the dominant eigenvector $u$, and a weakly-nonlinear expansion (using the left/right eigenvectors $u,v$) yields the response in terms of a few spectral properties of $A$, e.g.

$$F_{\eta\to0} = \frac{\lambda-1}{\lambda+\tfrac12\lambda^2}\,\frac{\langle uv\rangle\langle u\rangle}{\langle u^2 v\rangle}.$$

The bridge to the old rule: $\lambda \approx \rho\,\langle d^{\text{in}}d^{\text{out}}\rangle/\langle d\rangle$ (with $\rho$ the assortativity), which reduces to $\lambda\approx\langle d\rangle$ only when the network is homogeneous and uncorrelated. Simulations across six network families confirm the transition sits at $\lambda=1$, not $\langle d\rangle=1$ — including cases tuned through criticality by changing assortativity alone at fixed mean degree.

Homogeneity raises dynamic range

The peak dynamic range carries a topology term, $\Lambda_{\max} = 10\log_{10}\!\frac{2}{3F_*^2} - 10\log_{10}\frac{\langle vu^2\rangle}{\langle v\rangle\langle u\rangle^2}$; for an uncorrelated undirected network the second term is $\approx -10\log_{10}(\langle d^3\rangle/\langle d\rangle^3)$, maximised when the degree is uniform. So homogeneous topologies achieve higher dynamic range than heterogeneous (scale-free) ones — confirmed by scale-free nets with larger exponent $\gamma$ (more homogeneous) reaching higher range. The authors close with the conjecture that if wide dynamic range is adaptive, evolution may homogenize brain topology, and that real brains operate at $\lambda\approx1$.

Why it’s collected here

This is the most directly actionable paper for our node/reservoir scale. The controlling quantity is literally the spectral radius — the spectral_radius measure already in the harness — and the paper says $\lambda=1$ is the correct, topology-general critical point, of which the branching-ratio $\sigma=1$ (what our node-branching estimator targets) is only the homogeneous special case. That reframes our own observations cleanly: in the N-sweep the spectral radius rising through the performance peak, and $m_{\text{node}}\approx1$ under homeostasis, are two views of $\lambda\to1$; and the “homogeneous topology → higher dynamic range” result is a concrete, testable prediction for the connectome variants (link density, degree distribution) we’ve been sweeping. Dynamic range is also a candidate functional readout — a principled, criticality-linked alternative to the raw forage score.

References

Larremore, Shew & Restrepo (2011), Phys. Rev. Lett. 106, 058101 · Kinouchi & Copelli (2006), Nat. Phys. 2, 348 · Shew et al. (2009), J. Neurosci. 29, 15595 · Beggs & Plenz (2003), J. Neurosci. 23, 11167 · Restrepo, Ott & Hunt (2006), Phys. Rev. Lett. 97, 094102 · Restrepo, Ott & Hunt (2007), Phys. Rev. E 76, 056119 · Copelli & Campos (2007), Eur. Phys. J. B 56, 273 · Newman (2003), Phys. Rev. E 67, 026126.