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