# Influence measures in subnetworks using vertex centrality

## TL;DR

This paper introduces a new framework for measuring node influence in networks and subnetworks using adapted vertex centrality, with empirical testing on financial asset networks to analyze influence over time.

## Contribution

It proposes a general definition of relative vertex centrality that accounts for subnetwork influence, extending classical centrality measures.

## Key findings

- Relative centrality decomposes influence into network-wide and subnetwork-specific components.
- Empirical analysis on S&P 100 assets reveals sector-specific influence patterns.
- Time-based analysis captures dynamic influence changes over periods.

## Abstract

This work deals with the issue of assessing the influence of a node in the entire network and in the subnetwork to which it belongs as well, adapting the classical idea of vertex centrality. We provide a general definition of relative vertex centrality measure with respect to the classical one, referred to the whole network. Specifically, we give a decomposition of the relative centrality measure by including also the relative influence of the single node with respect to a given subgraph containing it. The proposed measure of relative centrality is tested in the empirical networks generated by collecting assets of the $S\&P$ 100, focusing on two specific centrality indices: betweenness and eigenvector centrality. The analysis is performed in a time perspective, capturing the assets influence, with respect to the characteristics of the analysed measures, in both the entire network and the specific sectors to which the assets belong.