  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 modular,
few measures exploit this property. Recent works have shown that it has a
significant effect on the dynamics on 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 of 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 counterpart are 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.
