Morrison Greg, Dudte Levi H, Mahadevan L
Department of Physics, University of Houston, Houston, TX 77204, USA.
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.
R Soc Open Sci. 2018 Aug 29;5(8):172281. doi: 10.1098/rsos.172281. eCollection 2018 Aug.
The identification of relationships in complex networks is critical in a variety of scientific contexts. This includes the identification of globally central nodes and analysing the importance of pairwise relationships between nodes. In this paper, we consider the concept of topological proximity (or 'closeness') between nodes in a weighted network using the generalized Erdős numbers (GENs). This measure satisfies a number of desirable properties for networks with nodes that share a finite resource. These include: (i) real-valuedness, (ii) non-locality and (iii) asymmetry. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to new methods to measure centrality. We show that the square of the leading eigenvector of an importance matrix defined using the GENs is strongly correlated with well-known measures such as PageRank, and define a personalized measure of centrality that is also well correlated with other existing measures. The utility of this measure of topological proximity is demonstrated by showing the asymmetries in both the dynamics of random walks and the mean infection time in epidemic spreading are better predicted by the topological definition of closeness provided by the GENs than they are by other measures.
在各种科学背景下，识别复杂网络中的关系至关重要。这包括识别全局中心节点以及分析节点之间成对关系的重要性。在本文中，我们使用广义厄多斯数（GENs）来考虑加权网络中节点之间的拓扑接近度（或“紧密性”）概念。对于共享有限资源的节点的网络，该度量满足许多理想的属性。这些属性包括：（i）实值性，（ii）非局部性和（iii）不对称性。我们表明，它们可用于定义网络中节点重要性的个性化度量，其具有自然的解释，从而产生了测量中心性的新方法。我们表明，使用GENs定义的重要性矩阵的主特征向量的平方与诸如PageRank等知名度量高度相关，并定义了一种与其他现有度量也高度相关的中心性个性化度量。通过表明GENs提供的接近度的拓扑定义比其他度量更好地预测了随机游走动力学和流行病传播中的平均感染时间的不对称性，证明了这种拓扑接近度度量的实用性。
