In this article we present Miuz, a robustness index for complex networks. Miuz measures the impact of disconnecting a node from the network while comparing the sizes of the remaining connected components. Strictly speaking, Miuz for a node is defined as the inverse of the size of the largest connected component divided by the sum of the sizes of the remaining ones. We tested our index in attack strategies where the nodes are disconnected in decreasing order of a specified metric. We considered Miuz and other well-known centrality measures such as betweenness, degree, and harmonic centrality. All of these metrics were compared regarding the behavior of the robustness (R- index) during the attacks. In an attempt to simulate the internet backbone, the attacks were performed in complex networks with power-law degree distributions (scale-free networks). Preliminary results show that attacks based on disconnecting a few number of nodes Miuz are more dangerous (decreasing the robustness) than the same attacks based on other centrality measures. We believe that Miuz, as well as other measures based on the size of the largest connected component, provides a good addition to other robustness metrics for complex networks.
Over the last decade, there has been a huge interest in the analysis of complex networks and their connectivity properties @cite_13 . During the last years, networks and in particular social networks have gained significant popularity. An in-depth understanding of the graph structure is key to convert data into information. To do so, complex networks tools have emerged @cite_9 to classify networks @cite_0 , detect communities @cite_3 , determine important features and measure them @cite_14 .
@cite_14 @cite_9 @cite_3 @cite_0 @cite_13
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Networks have recently emerged as a powerful tool to describe and quantify many complex systems, with applications in engineering, communications, ecology, biochemistry and genetics. A general technique to divide network vertices in groups and sub-groups is reported. Revealing such underlying hierarchies in turn allows the predicting of missing links from partial data with higher accuracy than previous methods. The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i.e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e.g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks. There has been a quickly growing interest in networks, since they can represent the structure of a wide class of complex systems occurring from the level of cells to society. Data obtained on real networks show that the corresponding graphs exhibit unexpected nontrivial properties, e.g., anomalous degree distributions, diameter, spreading phenomena, clustering coefficient, and correlations [1]. Very recently great attention has been paid to the local structural units of networks. Small and well defined subgraphs have been introduced as ‘‘motifs’’ [2]. Their distribution and clustering properties [2,3] can be used to interpret global features as well. Somewhat larger units, made up of vertices that are more densely connected to each other than to the rest of the network, are often referred to as communities [4], and have been considered to be the essential structural units of real networks. They have no obvious definition, and most of the recent methods for their identification rely on dividing the network into smaller pieces. The biggest drawback of these methods is that they do not allow for overlapping communities, although overlaps are generally assumed to be crucial features of communities. In this Letter we lay down the fundamentals of a kind of percolation phenomenon on graphs, which can also be used as an effective and deterministic method for uniquely identifying overlapping communities in large real networks [5]. Meanwhile, the various aspects of the classical Erdo ýs Many researchers have studied complex networks such as the World Wide Web, social networks, and the protein interaction network. They have found scale-free characteristics, the small-world effect, the property of high-clustering coefficient, and so on. One hot topic in this area is community detection. For example, the community shows a set of web pages about a certain topic in the WWW. The community structure is unquestionably a key characteristic of complex networks. In this paper, we propose a new method for finding communities in complex networks. Our proposed method considers the overlaps between communities using the concept of the intersection graph. Additionally, we address the problem of edge in homogeneity by weighting edges using the degree of overlaps and the similarity of content information between sets. Finally, we conduct clustering based on modularity. And then, we evaluate our method on a real SNS network.