On the Asymptotic Convergence of Subgraph Generated Models

TL;DR

This paper proves that the adjacency matrices of subgraph generated models (SUGMs) converge to their expected matrices as networks grow, enabling prediction of node centralities without full network data.

Contribution

It establishes the asymptotic convergence of SUGMs' adjacency matrices and applies this to predict centrality measures in large networks.

Findings

Adjacency matrices of SUGMs converge to expected matrices as network size increases.

Centrality measures in sampled networks concentrate around their expected values.

Node importance can be predicted from the model without exact network data.

Abstract

We study a family of random graph models - termed subgraph generated models (SUGMs) - initially developed by Chandrasekhar and Jackson in which higher-order structures are explicitly included in the network formation process. We use matrix concentration inequalities to show convergence of the adjacency matrix of networks realized from such SUGMs to the expected adjacency matrix as a function of the network size. We apply this result to study concentration of centrality measures (such as degree, eigenvector, and Katz centrality) in sampled networks to the corresponding centralities in the expected network, thus proving that node importance can be predicted from knowledge of the random graph model without the need of exact network data.