An Optimization-based Approach To Node Role Discovery in Networks: Approximating Equitable Partitions

TL;DR

This paper introduces an optimization-based method for node role discovery in networks, leveraging equitable partitions and graph isomorphism concepts, validated through a novel benchmark with stochastic role assignments.

Contribution

It presents a new definition of node roles related to equitable partitions and graph isomorphism, along with optimization formulations and theoretical guarantees.

Findings

The proposed approach effectively identifies node roles in synthetic networks.

Theoretical guarantees support the optimization solutions.

Validation demonstrates the method's ability to distinguish roles in stochastic network models.

Abstract

Similar to community detection, partitioning the nodes of a network according to their structural roles aims to identify fundamental building blocks of a network. The found partitions can be used, e.g., to simplify descriptions of the network connectivity, to derive reduced order models for dynamical processes unfolding on processes, or as ingredients for various graph mining tasks. In this work, we offer a fresh look on the problem of role extraction and its differences to community detection and present a definition of node roles related to graph-isomorphism tests, the Weisfeiler-Leman algorithm and equitable partitions. We study two associated optimization problems (cost functions) grounded in ideas from graph isomorphism testing, and present theoretical guarantees associated to the solutions of these problems. Finally, we validate our approach via a novel "role-infused partition benchmark", a network model from which we can sample networks in which nodes are endowed with different roles in a stochastic way.