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 lambdaA of the adjacency matrix A of a network has been considered and mean-field-type approximations to lambdaA have been developed for different kinds of networks. Here A is defined by Aij=1 (Aij=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 lambdaQ of the matrix Q, where Qij=qiAij and the "bias" qi may be different at each node i. (In the previously mentioned percolation and gene network contexts, qi is a probability and so lies in the range 0<or=qi<or=1.) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue lambdaA of A to lambdaQ, to develop suitable analytic approximations to lambdaQ, 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 qi 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.