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_27 proposed an approach where the resilience of the community structure is quantified by first introducing disruption in the original network and measuring the change in the community structure temporally i.e., after the disconnection and during the restoration process. @cite_20 provide a review of various approaches that make use of the facility importance concept to understand the system wide vulnerability. @cite_24 proposed a new vulnerability metric where they considered a combination of external and internal factors such as connection density etc. These methods allow us to quantify the vulnerability of a community, but does not provide us with a set of nodes that contribute to the vulnerability of the community itself.
@cite_24 @cite_27 @cite_20
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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. In this paper, we propose two metrics, i.e., the optimal repair time and the resilience reduction worth, to measure the criticality of the components of a network system from the perspective of their contribution to system resilience. Specifically, the two metrics quantify: 1) the priority with which a failed component should be repaired and re-installed into the network and 2) the potential loss in the optimal system resilience due to a time delay in the recovery of a failed component, respectively. Given the stochastic nature of disruptive events on infrastructure networks, a Monte Carlo-based method is proposed to generate probability distributions of the two metrics for all of the components of the network; then, a stochastic ranking approach based on the Copeland's pairwise aggregation is used to rank components importance. Numerical results are obtained for the IEEE 30-bus test network and a comparison is made with three classical centrality measures. 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. We empirically analyze five online communities: Friendster, Livejournal, Facebook, Orkut, and Myspace, to study how social networks decline. We define social resilience as the ability of a community to withstand changes. We do not argue about the cause of such changes, but concentrate on their impact. Changes may cause users to leave, which may trigger further leaves of others who lost connection to their friends. This may lead to cascades of users leaving. A social network is said to be resilient if the size of such cascades can be limited. To quantify resilience, we use the k-core analysis, to identify subsets of the network in which all users have at least k friends. These connections generate benefits (b) for each user, which have to outweigh the costs (c) of being a member of the network. If this difference is not positive, users leave. After all cascades, the remaining network is the k-core of the original network determined by the cost-to-benefit (c b) ratio. By analysing the cumulative distribution of k-cores we are able to calculate the number of users remaining in each community. This allows us to infer the impact of the c b ratio on the resilience of these online communities. We find that the different online communities have different k-core distributions. Consequently, similar changes in the c b ratio have a different impact on the amount of active users. Further, our resilience analysis shows that the topology of a social network alone cannot explain its success of failure. As a case study, we focus on the evolution of Friendster. We identify time periods when new users entering the network observed an insufficient c b ratio. This measure can be seen as a precursor of the later collapse of the community. Our analysis can be applied to estimate the impact of changes in the user interface, which may temporarily increase the c b ratio, thus posing a threat for the community to shrink, or even to collapse.