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Motivated by its relevance to various types of dynamical behavior of networksystems, the maximum eigenvalue $\lambda_Q$ of the adjacency matrix $A$ of anetwork 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 withheterogeneous node properties (percolation on a directed network and thestability of Boolean models of gene networks), an analogous but differenteigenvalue 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 andgene network contexts, $q_i$ is a probability and so lies in the range $0 \leq_i \le 1$.) The purposes of this paper are to extend the previousconsiderations of the maximum eigenvalue $\lambda_A$ of $A$ to $\lambda_Q$, todevelop suitable analytic approximations to $\lambda_Q$, and to test theseapproximations with numerical experiments. In particular, three issuesconsidered are (i) the effect of the correlation (or anticorrelation) betweenthe value of $q_i$ and the number of links to and from node $i$; (ii) theeffect of correlation between the properties of two nodes at either end of anetwork link (`assortativity'); and (iii) the effect of community structureallowing for a situation in which different $q$-values are associated withdifferent communities.

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