非均匀节点偏置网络修正邻接矩阵最大特征值的逼近
Approximating the Largest Eigenvalue of the Modified Adjacency Matrix of Networks with Heterogeneous Node Biases
摘要：
由于网络系统的各种动力学行为与其相关性，本文研究了网络邻接矩阵$A$的最大特征值$λ_Q$，并对不同类型的网络给出了平均场型近似$λ_Q$.这里，如果存在（不存在）从$j$到$i$的定向网络链路，则$A$由$A_{ij} = 1$（$A_{ij} = 0$）定义。然而，在至少两个最近的问题，涉及网络的异质节点属性（有向网络上的渗流和基因网络的布尔模型的稳定性），一个类似的，但不同的特征值问题出现，即，找到最大的特征值$λ_Q$矩阵$Q$，其中$Q_{ij} = q_i A_{ij}$和“偏置”$q_i$可能是不同的，在每个节点$i$。(In在前面提到的渗流和基因网络背景下，$q_i$是概率，因此位于范围$0 \le q_i \le 1$中。）本文的目的是扩展以前的考虑$A$的最大特征值$λ_A$的$λ_Q$，开发合适的解析近似$λ_Q$，并测试这些近似与数值实验。特别是，考虑的三个问题是（i）相关性的影响（ii）网络链路两端两个节点的属性之间的相关性的影响（“包容性”）;以及（三）社区结构的影响，允许不同的$q$价值观与不同的社区相关联。
英文摘要：
Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue $λ_Q$ of the adjacency matrix $A$ of a network has been considered, and mean-field-type approximations to $λ_Q$ 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 $λ_Q$ 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 $λ_A$ of $A$ to $λ_Q$, to develop suitable analytic approximations to $λ_Q$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
