Quantifying the connectivity of a network: The network correlation function method
Quantifying the connectivity of a network: The network correlation function method
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DOI：
10.1103/physreve.80.046104
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发表时间：
2009-10-01
影响因子：
2.4
通讯作者：
Biham, Ofer
中科院分区：
文献类型：
--
作者：
Barzel, Baruch;Biham, Ofer
网络对于描述交互对象的系统很有用，其中节点表示对象，边表示它们之间的交互。这些应用包括化学和新陈代谢系统、食物网以及社交网络。最近发现，这些网络中的许多网络都表现出一些共同的拓扑特征，如高聚集度、小平均路径长度(小世界网络)和幂律度分布(无标度网络)。网络的拓扑特征通常与网络的功能有关。但是，单靠拓扑并不能说明网络中交互的性质及其强度。在这里，我们提出了一种评估网络中节点对之间的相关性的方法。这些关联既取决于网络的拓扑，也取决于网络的功能。具有高连通性的网络表现出其交互节点之间的强相关性，因此具有小世界功能。我们量化网络中所有节点对之间的相关性，并将它们表示为相关性矩阵中的矩阵元素。根据该信息，可以绘制网络的关联函数并提取关联长度。然后，网络的连通性被定义为该相关长度与网络的平均路径长度之间的比率。利用这种方法，我们区分了拓扑小世界和功能小世界，后者具有长程关联和高连通性的特点。显然，共享相同拓扑的网络可能具有不同的连接性，这取决于它们交互的性质和强度。该方法是在新陈代谢网络上演示的，但也可以很容易地推广到其他类型的网络。
Networks are useful for describing systems of interacting objects, where the nodes represent the objects and the edges represent the interactions between them. The applications include chemical and metabolic systems, food webs as well as social networks. Lately, it was found that many of these networks display some common topological features, such as high clustering, small average path length (small-world networks), and a power-law degree distribution (scale-free networks). The topological features of a network are commonly related to the network's functionality. However, the topology alone does not account for the nature of the interactions in the network and their strength. Here, we present a method for evaluating the correlations between pairs of nodes in the network. These correlations depend both on the topology and on the functionality of the network. A network with high connectivity displays strong correlations between its interacting nodes and thus features small-world functionality. We quantify the correlations between all pairs of nodes in the network, and express them as matrix elements in the correlation matrix. From this information, one can plot the correlation function for the network and to extract the correlation length. The connectivity of a network is then defined as the ratio between this correlation length and the average path length of the network. Using this method, we distinguish between a topological small world and a functional small world, where the latter is characterized by long-range correlations and high connectivity. Clearly, networks that share the same topology may have different connectivities, based on the nature and strength of their interactions. The method is demonstrated on metabolic networks, but can be readily generalized to other types of networks.