Approximating the Largest Eigenvalue of the Modified Adjacency Matrix of Networks with Heterogeneous Node Biases

TL;DR

This paper develops and tests analytical approximations for the largest eigenvalue of a modified adjacency matrix in networks with heterogeneous node biases, extending previous models to account for node-specific properties and correlations.

Contribution

It introduces new methods to approximate the largest eigenvalue of the matrix Q in networks with node-dependent biases, considering correlations, assortativity, and community structure.

Findings

Analytic approximations accurately predict eigenvalues in various network models.

Node-bias correlations significantly influence the eigenvalue.

Community structure impacts the eigenvalue distribution.

Abstract

Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue $\lambda_Q$ of the adjacency matrix $A$ of a network has been considered, and mean-field-type approximations to $\lambda_Q$ have been developed for different kinds of networks. Here $A$ is defined by $A_{ij} = 1$ ($A_{ij} = 0$) if there is (is not) a directed network link to $i$ from $j$. However, in at least two recent problems involving networks with heterogeneous node properties (percolation on a directed network and the stability of Boolean models of gene networks), an analogous but different eigenvalue problem arises, namely, that of finding the largest eigenvalue $\lambda_Q$ of the matrix $Q$, where $Q_{ij} = q_i A_{ij}$ and the `bias' $q_i$ may be different at each node $i$. (In the previously mentioned percolation and gene network contexts, $q_i$ is a probability and so lies in the range $0 \le q_i \le 1$.) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue $\lambda_A$ of $A$ to $\lambda_Q$, to develop suitable analytic approximations to $\lambda_Q$, and to test these approximations with numerical experiments. In particular, three issues considered are (i) the effect of the correlation (or anticorrelation) between the value of $q_i$ and the number of links to and from node $i$; (ii) the effect of correlation between the properties of two nodes at either end of a network link (`assortativity'); and (iii) the effect of community structure allowing for a situation in which different $q$-values are associated with different communities.