Functional Central Limit Theorem for the simultaneous subgraph count of dynamic Erdős-Rényi random graphs
Rajat Subhra Hazra, Nikolai Kriukov, Michel Mandjes
TL;DR
The paper develops a functional central limit theorem for the joint evolution of multiple subgraph counts in dynamic Erdős-Rényi graphs with edge processes evolving independently over time. By placing mild Lipschitz-type conditions on edge switching and carefully scaling the subgraph counts, it proves convergence to a multidimensional Gaussian process whose components are linear combinations of independent orbit-subgraph processes, with an explicit covariance structure governed by limiting edge-probability and covariance functions. A key innovation is the use of common subgraph patterns (OCS) and combinatorial constants to express the limiting covariance, together with a general framework that encompasses regimes where edge probabilities vanish with the graph size or remain size-free. The results imply, in particular, asymptotic independence of subgraph-count processes when the least common subgraph differs, and they provide concrete guidance for understanding fluctuations of network motifs in temporal graphs. The methodology combines finite-dimensional convergence, tightness, and detailed combinatorial analysis of subgraph overlaps to establish a robust process-level limit for dynamic random graphs.
Abstract
In this paper we consider a dynamic Erdős-Rényi random graph with independent identically distributed edge processes. Our aim is to describe the joint evolution of the entries of a subgraph count vector. The main result of this paper is a functional central limit theorem: we establish, under an appropriate centering and scaling, the joint functional convergence of the vector of subgraph counts to a specific multidimensional Gaussian process. The result holds under mild assumptions on the edge processes, most notably a Lipschitz-type condition.