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- W2023872024 abstract "Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue ${ensuremath{lambda}}_{A}$ of the adjacency matrix $A$ of a network has been considered and mean-field-type approximations to ${ensuremath{lambda}}_{A}$ 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 ${ensuremath{lambda}}_{Q}$ 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 $0ensuremath{le}{q}_{i}ensuremath{le}1$.) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue ${ensuremath{lambda}}_{A}$ of $A$ to ${ensuremath{lambda}}_{Q}$, to develop suitable analytic approximations to ${ensuremath{lambda}}_{Q}$, 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." @default.
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- W2023872024 date "2009-05-22" @default.
- W2023872024 modified "2023-09-23" @default.
- W2023872024 title "Approximating the largest eigenvalue of the modified adjacency matrix of networks with heterogeneous node biases" @default.
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- W2023872024 doi "https://doi.org/10.1103/physreve.79.056111" @default.
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