The Correlation Hole

Excite a single node in a random network. Watch the excitation decay. The survival probability — the chance the excitation remains at its origin — drops through three regimes: fast initial decay, power-law intermediate behavior, eventual saturation.

Peralta-Martinez and Méndez-Bermúdez find that the power-law exponent is not arbitrary. Both the decay rate and the time-averaged survival probability scale with the correlation dimension of the network’s eigenstates. The eigenstate structure — how the network’s modes are distributed across nodes — determines how quickly a localized excitation spreads.

The correlation hole is the most structurally interesting feature. Before saturation, the survival probability dips below its long-time average — the excitation temporarily becomes less likely to be found at its origin than it will be at equilibrium. The depth of this dip scales with the average network degree. Dense networks produce deep correlation holes; sparse networks produce shallow ones. The network’s connectivity determines how far below equilibrium the excitation overshoots.

The eigenstates themselves display multifractal properties. They are neither fully extended (uniform across all nodes) nor fully localized (concentrated on a single node). The same eigenstate has different fractal dimensions at different scales — a hallmark of multifractality. The randomly weighted adjacency matrices of Erdős-Rényi networks produce this multifractal structure naturally.

The structural insight: a delta excitation on a random network is a probe of eigenstate geometry. The temporal evolution of the survival probability encodes the fractal structure of the modes. Watching how an excitation dies reveals how the network’s internal structure is organized — not its topology, but its spectral architecture.