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.
Assessing the structural network vulnerability has received increasing attention. For example, @cite_1 have proposed a Community Vulnerability Assessment (CVA) problem and suggested multiple heuristic based algorithms based on the modularity measure of communities in the network. These approaches are restricted to the scope of online social networks and do not cater to general network structures. Another work by @cite_16 explored the number of connected triplets in a network as they capture the strong connection of communities in social networks.
@cite_16 @cite_1
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We introduce Community structure Vulnerability Assessment (CVA) problem to assess the network vulnerability under a community structure point of view. Given a positive number k, CVA aims to find out the k most vulnerable nodes whose removals maximally transform the current network community structure to a different one. As the first attempt, we suggest an approximation algorithm for the special case k = 1, and propose multiple greedy algorithms for CVA problem. To certify the effectiveness of suggested approaches, we test them on not only synthesized networks with known community structures but also on real-world social traces. This paper develops a quantitative method to measure the vulnerability of community structure with emphasis on both internal and external connectivity characteristics of the community. In particular, the number of links between communities and the strength of links connecting two communities are considered as external factors, while the connection density, the degree of gateway nodes, as well as the strength of links within each community are treated as internal factors. A non-linear weighted function is used to combine the internal factors with external factors. Then the developed method is used to illustrate the vulnerability analysis of community structure of a power transmission grid, a karate club network, and an air transportation network. The results reveal that the proposed measure is effective in differentiating the vulnerability level of community structure in a variety of networks. We present a novel approach to quantify the vulnerability of a complex network, i.e., the capacity of a graph to maintain its functional performance under random damages or malicious attacks. The proposed measure represents a multiscale evaluation of vulnerability, and makes use of combined powers of the links’ betweenness. We show that the proposed approach is able to properly describe some cases for which earlier measures of vulnerability fail. The relevant applications of our method for technological network design are outlined. Most methods proposed to uncover communities in complex networks rely on their structural properties. Here we introduce the stability of a network partition, a measure of its quality defined in terms of the statistical properties of a dynamical process taking place on the graph. The time-scale of the process acts as an intrinsic parameter that uncovers community structures at different resolutions. The stability extends and unifies standard notions for community detection: modularity and spectral partitioning can be seen as limiting cases of our dynamic measure. Similarly, recently proposed multi-resolution methods correspond to linearisations of the stability at short times. The connection between community detection and Laplacian dynamics enables us to establish dynamically motivated stability measures linked to distinct null models. We apply our method to find multi-scale partitions for different networks and show that the stability can be computed efficiently for large networks with extended versions of current algorithms.