Influence and Connectivity in Networks: A Generating Function Approach

TL;DR

This paper presents a systematic generating function framework for enumerating walks in networks, enabling analysis of structural interventions, centrality measures, and link strategies with broad applications.

Contribution

It introduces a novel walk decomposition method and interconnected equations to analyze network walks and their responses to structural changes.

Findings

Framework unifies walk enumeration methods

Enables evaluation of structural interventions

Generalizes target centrality to multiple receivers

Abstract

Many widely used network centralities are based on counting walks that meet specific criteria. This paper introduces a systematic framework for walk enumeration using generating functions. We introduce a first-passage decomposition that uniquely divides any walk passing through specified nodes or links into two components: a first-reaching walk and a subsequent walk. This decomposition yields a system of interconnected equations that relate three disjoint categories of walks: unrestricted walks, walks that avoid specific elements, and walks that pass through designated sets. The framework offers a range of applications, including evaluating the effects of structural interventions, such as node or link modifications, on network walks, generalizing target centrality to multi-receiver scenarios in information networks, and comparing different strategies for adding links.