  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.
