In this paper, we study the crucial elements of complex networks, namely nodes, and edges and their properties such as their community structure, which play an important role in dictating the robustness of the network towards structural perturbations. Specifically, we want to identify all vital nodes, which when removed would lead to a large change in the underlying community structure of the network. This problem is extremely important because the community structure of a network allows deep underlying insights into how function of a network and its topology affect each other. Moreover, it even provides a way to condense large graphs into smaller graphs where each community acts as a meta node and hence aids in easier network analysis. If this community structure was to be compromised by either accidental or intentional perturbations to the network, that would make such analysis difficult. Since the problem of identifying such vital nodes is computationally intractable, we propose some heuristics that allow us to find solutions close to the optimal solution. To certify the effectiveness of our approach, we first test these heuristics on small networks, and then move to larger networks to show that we achieve similar results. The results reveal that the proposed approaches are effective to analyze the vulnerability of communities in graphs irrespective of their size and scale. From the application point of view, we show that the proposed algorithm is scalable and can be applied to the information diffusion task to curtail the spread of active nodes which was observed empirically. Additionally, we show the performance of our algorithm through an extrinsic evaluation -- we employ two tasks, like prediction and information diffusion, and show that the effect of our algorithm on these tasks is higher than the other baselines.
@cite_26: Uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents. Many algorithms have been proposed so far, but none of them has been subjected to strict tests to evaluate their performance. Most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure, which is very uncommon in real systems. Here we test several methods against a recently introduced class of benchmark graphs, with heterogeneous distributions of degree and community size. The methods are also tested against the benchmark by Girvan and Newman [Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002)] and on random graphs. As a result of our analysis, three recent algorithms introduced by Rosvall and Bergstrom [Proc. Natl. Acad. Sci. U.S.A. 104, 7327 (2007); Proc. Natl. Acad. Sci. U.S.A. 105, 1118 (2008)], [J. Stat. Mech.: Theory Exp. (2008), P10008], and Ronhovde and Nussinov [Phys. Rev. E 80, 016109 (2009)] have an excellent performance, with the additional advantage of low computational complexity, which enables one to analyze large systems.
@cite_7: A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.
This work involves around two broad strategies, namely community structure analysis and structural network vulnerability assessment. Numerous approaches have been developed and applied to detect community structure. For instance, a hierarchical agglomerative algorithm was proposed by @cite_7 . A much more extensive literature can be found in an excellent survey which describes all approaches derived to solve the problem @cite_26 .