Approximating the largest eigenvalue of the modified adjacency matrix of networks with heterogeneous node biases

Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue λ_A of the adjacency matrix A of a network has been considered and mean-field-type approximations to λ_A 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 λ_Q of the matrix Q, where Q_ij=q_iA_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≤q_i≤1.) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue λ_A of A to λ_Q, to develop suitable analytic approximations to λ_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.