Renormalization of Interacting Random Graph Models

TL;DR

This work embeds exponential random graphs within a generalized statistical-mechanical framework, treating the adjacency structure via a moment-generating Hamiltonian and studying how pairwise conditioning evolves under renormalization. For bilinear interactions on the line graph, the authors derive exact RG decimations (CN=2), which drive the system toward an Erdős–Rényi fixed point, while higher coordination numbers lose closure and generate more complex couplings that are largely irrelevant at long wavelengths. Introducing disorder leads to RG-induced probability flows on the disorder manifold, interpreted as time-reversed drift-diffusion on the statistical manifold; sparse disorder converges to the fixed line $g=0$, whereas uniform disorder exhibits gradient-flow dynamics in the corresponding distributions. The framework provides a principled way to model peer effects and preferential attachment, and it offers a path to network inference under limited data by marginalizing over hidden degrees of freedom and constraining microscopic parameters through RG flow, with future work extending to higher-order interactions and weighted/directed graphs.

Abstract

Random graphs offer a useful mathematical representation of a variety of real world complex networks. Exponential random graphs, for example, are particularly suited towards generating random graphs constrained to have specified statistical moments. In this investigation, we elaborate on a generalization of the former where link probabilities are conditioned on the appearance of other links, corresponding to the introduction of interactions in an effective generalized statistical mechanical formalism. When restricted to the simplest non-trivial case of pairwise interactions, one can derive a closed form renormalization group transformation for maximum coordination number two on the corresponding line graph. Higher coordination numbers do not admit exact closed form renormalization group transformations, a feature that paraphrases the usual absence of exact transformations in two or more dimensional lattice systems. We introduce disorder and study the induced renormalization group flow on its probability assignments, highlighting its formal equivalence to time reversed anisotropic drift-diffusion on the statistical manifold associated with the effective Hamiltonian. We discuss the implications of our findings, stressing the long wavelength irrelevance of certain classes of pair-wise conditioning on random graphs, and conclude with possible applications. These include modeling the scaling behavior of preferential effects on social networks, opinion dynamics, and reinforcement effects on neural networks, as well as how our findings offer a systematic framework to deal with data limitations in inference and reconstruction problems.

Figures (8)

• Figure 1: Schematic rendering of maximum coordination two couplings (top) and a single bare coordination number three coupling (bottom). More generally, successive RG steps increase the maximum coordination number, and flow towards all-to-all couplings (bottom most diagram).
• Figure 2: RG flow for homogeneous coordination number two interactions towards an Erdős-Renyi fixed line. Left panel: A shearing behavior for positively correlated conditioning is apparent, along with a separatrix along the line $\varepsilon = -g$ for $g>0$. This corresponds to the vanishing of the beta function of external magnetic field $h := \varepsilon + g$ along the line $h \equiv 0$ for the analog spin glass system, illustrated by the right panel.
• Figure 3: Top panel: Renormalization group induced deformation of the bare probability assignment $\Pi(y)$ given by Eq. \ref{['bareP']} for $\alpha_\pi = 5$, $\lambda_\pi = e^{-1}$ (red line) for $k$ from 1 to 5 (gray dashed lines with successively coarser dashing). Bottom panel: The same, but for $\Pi(y)$ given with parameters $\alpha_\pi = 4$, $\lambda_\pi = e^{0}$.
• Figure 4: Renormalization group induced deformation of the uniform probability assignments $C^{(k)}(\varepsilon)$ (top) and $P^{(k)}(h)$ (bottom). Bold red lines denote the bare probability assignments, taken to be Gaussians with means and variances given by $\mu_\varepsilon = -3, \sigma_\varepsilon = 1$ and $\mu_h = -3, \sigma_h = 1$, respectively, and the gray dashed lines represent successive RG convolutions running from $k=1$ to $3$ with increasingly coarse dashing.
• Figure 5: Schematic representation of a random graph, where each dot represents a possible link (i.e. an off diagonal entry of the adjacency matrix). The realization of a given link in a specific draw corresponding to the relevant entry of the adjacency matrix are denoted by circles.