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

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.

15 pages, 3 figures

References in corpus (6)

• The onset of synchronization in large networks of coupled oscillators
• Resilience to damage of graphs with degree correlations
• Approximating the largest eigenvalue of network adjacency matrices
• The effect of network topology on the stability of discrete state models of genetic control
• Weighted percolation on directed networks
• The emergence of coherence in complex networks of heterogeneous dynamical systems

Cited by in corpus (6)

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