Ultrametric Graphons and Hierarchical Community Networks: Spectral Theory and Applications

TL;DR

This paper develops a spectral theory for ultrametric graphons representing hierarchical community networks, providing explicit formulas and applications in community detection, network analysis, and epidemic modeling.

Contribution

It introduces a new class of ultrametric graphons, derives explicit spectral formulas, and applies these to community detection, network dynamics, and epidemic thresholds.

Findings

Eigenvalues and spectral projectors of the Laplacian are explicitly characterized.

Spectral approximations enable analysis of large hierarchical networks.

Stability conditions for epidemic models depend on community structure.

Abstract

We develop a theory of ultrametric graphons as limiting objects for random networks with nested hierarchical community structure. A graphon $W:[0,1]^2\to[0,1]$ is called ultrametric if $W(x,y)=w(d(x,y))$, where $d$ is an ultrametric on $[0,1]$ induced by a family of nested partitions and $w$ is a positive kernel. The resulting random graphs exhibit a nested hierarchical community structure in which the density of connections is governed by the ultrametric distance between vertices. The Laplacian $L_d^k$ of the deterministic graph sampled from an ultrametric graphon is itself an ultrametric Laplacian, whose eigenvalues and spectral projectors admit completely explicit closed-form expressions in terms of the community sizes and inter-community connection densities. We show that the normalized eigenvalues and spectral projectors of the random Laplacian $L_r^k$ are arbitrarily close to those of $L_d^k$ with high probability as $k\to\infty$, so that the explicit formulas for $L_d^k$ provide closed-form approximations for the spectrum and spectral projectors of $L_r^k$. As applications: a sign structure theorem generalizes the Fiedler vector criterion to hierarchical networks with arbitrarily many communities; a detectability threshold $p^*=\min_i\rho_i$ governs spectral community detection for one-level hierarchical graphons; the pseudo-inverse Laplacian $L_W^+$ is constructed and shown to be the almost sure limit of the pseudo-inverse of $L_r^k$, implying that hitting and commute times collapse to quantities depending only on the expected degrees of the endpoints; and explicit closed-form stability conditions for the SIS disease-free equilibrium reveal a fundamental tension between homogeneous and heterogeneous community structures, confirmed by numerical experiments.