Identifying influential nodes in a network is a fundamental issue due to its wide applications, such as accelerating information diffusion or halting virus spreading. Many measures based on the network topology have emerged over the years to identify influential nodes such as Betweenness, Closeness, and Eigenvalue centrality. However, although most real-world networks are made of groups of tightly connected nodes which are sparsely connected with the rest of the network in a so-called modular structure, few measures exploit this property. Recent works have shown that it has a significant effect on the dynamics of networks. In a modular network, a node has two types of influence: a local influence (on the nodes of its community) through its intra-community links and a global influence (on the nodes in other communities) through its inter-community links. Depending on the strength of the community structure, these two components are more or less influential. Based on this idea, we propose to extend all the standard centrality measures defined for networks with no community structure to modular networks. The so-called “Modular centrality” is a two-dimensional vector. Its first component quantifies the local influence of a node in its community while the second component quantifies its global influence on the other communities of the network. In order to illustrate the effectiveness of the Modular centrality extensions, comparison with their scalar counterparts is performed in an epidemic process setting. Simulation results using the Susceptible-Infected-Recovered (SIR) model on synthetic networks with controlled community structure allows getting a clear idea about the relation between the strength of the community structure and the major type of influence (global/local). Furthermore, experiments on real-world networks demonstrate the merit of this approach.

中文翻译：

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模块化网络的集中度

由于其广泛的应用，例如加速信息传播或阻止病毒传播，识别网络中的有影响力的节点是一个基本问题。多年来，已经出现了许多基于网络拓扑的度量来识别有影响力的节点，例如介面性，紧密性和特征值中心性。但是，尽管大多数实际网络都是由紧密连接的节点组组成的，它们以所谓的模块化结构稀疏地与网络的其余部分连接，但是很少有措施可以利用此属性。最近的工作表明，它对网络的动力学有重大影响。在模块化网络中，节点具有两种类型的影响：通过社区内部链接在本地（在其社区的节点上）产生影响，通过社区之间链接在全球（在其他社区的节点上）产生影响。根据社区结构的强度，这两个部分或多或少具有影响力。基于此想法，我们建议将针对没有社区结构的网络定义的所有标准集中性度量扩展到模块化网络。所谓的“模块化中心性”是二维向量。它的第一部分量化节点在其社区中的局部影响，而第二部分量化其对网络其他社区的全局影响。为了说明模块化中心扩展的有效性，在流行过程中与标量扩展进行了比较。在具有受控社区结构的合成网络上使用“敏感感染恢复”（SIR）模型进行的仿真结果可以使人们对社区结构的强度与主要影响类型（全局/局部）之间的关系有一个清晰的认识。此外，在真实世界的网络上进行的实验证明了这种方法的优点。