返回概率和k步测度
Return probability and k-step measures
摘要：
返回概率的概念--最著名的是乔治·波利亚（George Pólya）在d维格子上的探索--有可能作为网络分析的一种度量。我们提出了一个有效的方法来寻找返回概率分布的连接无向图。我们认为，回报概率具有相同的歧视权力，现有的k-步骤措施-特别是，β中心（负β），图论的权力指数（GPI），和子图中心。我们比较了我们的算法的运行时间beta中心和子图中心，发现它是显着更快。当返回概率被用来测量与贝塔中心性相同的现象时，它在线性时间内运行- O（n+m），其中n和m分别是节点和边的数量-这比计算贝塔中心性的精确或近似形式所需的矩阵求逆或矩阵乘法序列所需的时间要少得多。我们将这种形式的收益概率称为波利亚幂指数（Pólya Power Index，PPI）。计算子图中心性需要对邻接矩阵进行昂贵的特征分解;在2000个节点的网络上，返回概率仅为特征分解时间的一半。这些性能改进很重要，因为为了分析大型网络，计算效率的措施是必要的。
英文摘要：
The notion of return probability -- explored most famously by George Pólya on d-dimensional lattices -- has potential as a measure for the analysis of networks. We present an efficient method for finding return probability distributions for connected undirected graphs. We argue that return probability has the same discriminatory power as existing k-step measures -- in particular, beta centrality (with negative beta), the graph-theoretical power index (GPI), and subgraph centrality. We compare the running time of our algorithm to beta centrality and subgraph centrality and find that it is significantly faster. When return probability is used to measure the same phenomena as beta centrality, it runs in linear time -- O(n+m), where n and m are the number of nodes and edges, respectively -- which takes much less time than either the matrix inversion or the sequence of matrix multiplications required for calculating the exact or approximate forms of beta centrality, respectively. We call this form of return probability the Pólya power index (PPI). Computing subgraph centrality requires an expensive eigendecomposition of the adjacency matrix; return probability runs in half the time of the eigendecomposition on a 2000-node network. These performance improvements are important because computationally efficient measures are necessary in order to analyze large networks.
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