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.
Ranking the nodes according to their centrality constitutes the standard deterministic approach to uncover the most influential nodes in a network. These measures rely usually on various network topological properties. However, the community structure of the network is rarely taken into consideration. Few researchers have payed attention to this property encountered in many real-world networks @cite_9 @cite_12 @cite_28 @cite_25 @cite_22 @cite_3 @cite_14 @cite_23 @cite_19 @cite_18 @cite_2 . In this section, we give a brief overview of the main deterministic methods that motivates our proposition.
@cite_18 @cite_14 @cite_22 @cite_28 @cite_9 @cite_3 @cite_19 @cite_23 @cite_2 @cite_25 @cite_12
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Measuring the centrality of nodes in real-world networks has remained an important task in the technological, social, and biological network paradigms carrying implications on their analysis and applications. Exact inference of centrality values is infeasible in large networks due to the need to solve the all-pairs shortest path problem. We introduce a framework to approximate node centralities in real-world networks that are known to exhibit modularity, i.e., the presence of dense subgraphs or communities, which are themselves sparsely connected. We also propose a novel centrality measure known as Community Inbetweenness that ranks nodes based solely on community information. In a modular network of size n with √n(1 2) evenly sized communities and m edges, our framework requires linear time O(m) and O(√nm) time for the approximation of closeness and betweenness respectively. Utilizing a recently proposed linear time method in community detection, our approximation techniques are faster than traditional sampling algorithms, applicable in real-time distributed environments, and offer highly comparable results. Since full coverage treatment is not feasible due to limited resources, we need to utilize an immunization strategy to effectively distribute the available vaccines. On the other hand, the structure of contact network among people has a significant impact on epidemics of infectious diseases (such as SARS and influenza) in a population. Therefore, network-based immunization strategies aim to reduce the spreading rate by removing the vaccinated nodes from contact network. Such strategies try to identify more important nodes in epidemics spreading over a network. In this paper, we address the effect of overlapping nodes among communities on epidemics spreading. The proposed strategy is an optimized random-walk based selection of these nodes. The whole process is local, i.e. it requires contact network information in the level of nodes. Thus, it is applicable to large-scale and unknown networks in which the global methods usually are unrealizable. Our simulation results on different synthetic and real networks show that the proposed method outperforms the existing local methods in most cases. In particular, for networks with strong community structures, high overlapping membership of nodes or small size communities, the proposed method shows better performance. Understanding the epidemic dynamics, and finding out efficient techniques to control it, is a challenging issue. A lot of research has been done on targeted immunization strategies, exploiting various global network topological properties. However, in practice, information about the global structure of the contact network may not be available. Therefore, immunization strategies that can deal with a limited knowledge of the network structure are required. In this paper, we propose targeted immunization strategies that require information only at the community level. Results of our investigations on the SIR epidemiological model, using a realistic synthetic benchmark with controlled community structure, show that the community structure plays an important role in the epidemic dynamics. An extensive comparative evaluation demonstrates that the proposed strategies are as efficient as the most influential global centrality based immunization strategies, despite the fact that they use a limited amount of information. Furthermore, they outperform alternative local strategies, which are agnostic about the network structure, and make decisions based on random walks. We investigate the effects of modular and temporal connectivity patterns on epidemic spreading. To this end, we introduce and analytically characterise a model of time-varying networks with tunable modularity. Within this framework, we study the epidemic size of Susceptible-Infected-Recovered, SIR, models and the epidemic threshold of Susceptible-Infected-Susceptible, SIS, models. Interestingly, we find that while the presence of tightly connected clusters inhibits SIR processes, it speeds up SIS phenomena. In this case, we observe that modular structures induce a reduction of the threshold with respect to time-varying networks without communities. We confirm the theoretical results by means of extensive numerical simulations both on synthetic graphs as well as on a real modular and temporal network. We propose a community structure-based approach that does not require community labels of nodes, for network immunization. Social networks have been widely used as daily communication infrastructures these days. However, fast spreading of information over networks may have downsides such as computer viruses or epidemics of diseases. Because contamination is propagated among subgraphs communities along links in a network, use of community structure of the network would be effective for network immunization. However, despite various research efforts, it is still difficult to identify ground-truth community labels of nodes in a network. Because communities are often interwoven through intermediate nodes, we propose to identify such nodes based on the community structure of a network without requiring community labels. By regarding the community structure in terms of nodes, we construct a vector representation of nodes based on a quality measure of communities. The distribution of the constructed vectors is used for immunizing intermediate nodes among communities, through the hybrid use of the norm and the relation in the vector representation. Experiments are conducted over both synthetic and real-world networks, and our approach is compared with other network centrality-based approaches. The results are encouraging and indicate that it is worth pursuing this path. Understanding how the network topology affects the spread of an epidemic is a main concern in order to develop efficient immunization strategies. While there is a great deal of work dealing with the macroscopic topological properties of the networks, few studies have been devoted to the influence of the community structure. Furthermore, while in many real-world networks communities may overlap, in these studies non-overlapping community structures are considered. In order to gain insight about the influence of the overlapping nodes in the epidemic process we conduct an empirical evaluation of basic deterministic immunization strategies based on the overlapping nodes. Using the classical SIR model on a real-world network with ground truth overlapping community structure we analyse how immunization based on the membership number of overlapping nodes (which is the number of communities the node belongs to) affect the largest connected component size. Comparison with random immunization strategies designed for networks with non-overlapping community structure show that overlapping nodes play a major role in the epidemic process. Several centrality measures have been formulated to quantify the notion of 'importance' of actors in social networks. Current measures scrutinize either local or global connectivity of the nodes and have been found to be inadequate for social networks. Ignoring hierarchy and community structure, which are inherent in all human social networks, is the primary cause of this inadequacy. Positional hierarchy and embeddedness of an actor in the community are intuitively crucial determinants of his importance. The theory of social capital asserts that an actor's importance is derived from his position in network hierarchy as well as from the potential to mobilize resources through intra-community (bonding) and inter-community (bridging) ties. Inspired by this idea, we propose a novel centrality measure SC (Social Centrality) for actors in social networks. Our measure accounts for - i) an individual's propensity to socialize, and ii) his connections within and outside the community. These two factors are suitably aggregated to produce social centrality score. Comparative analysis of SC measure with classical and recent centrality measures using large public networks shows that it consistently produces a more realistic ranking of nodes. The inference is based on the available ground truth for each tested networks. Extensive analysis of rankings delivered by SC measure and mapping with known facts in well-studied networks justifies its effectiveness in diverse social networks. Scalability evaluation of SC measure justifies its efficacy for real-world large networks.