Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its importance in many fields such as biology, social networks or network traffic analysis. The metrics proposed to shape communities are generic and follow two approaches: maximizing the internal density of such communities or reducing the connectivity of the internal vertices with those outside the community. However, these metrics take the edges as a set and do not consider the internal layout of the edges in the community. We define a set of properties oriented to social networks that ensure that communities are cohesive, structured and well defined. Then, we propose the Weighted Community Clustering (WCC), which is a community metric based on triangles. We proof that analyzing communities by triangles gives communities that fulfill the listed set of properties, in contrast to previous metrics. Finally, we experimentally show that WCC correctly captures the concept of community in social networks using real and syntethic datasets, and compare statistically some of the most relevant community detection algorithms in the state of the art.
The second type of metrics consists of those that focus on reducing the number of edges connecting communities. @cite_23 , introduce the conductance. Conductance, is the ratio between the edges going outside the community and the total number of edges between members of the community. However, conductance suffers from the fact that for any graph, the partition with a unique community containing all the vertices of the graph obtains the best conductance, making its direct optimization not viable. A recent survey @cite_19 of community metrics discusses the performance of many metrics on real networks: the cut ratio @cite_24 , the normalized cut @cite_18 , the Maximum-ODF (Out Degree Fraction), the Average-ODF and Flake-ODF @cite_10 . In this survey, showed that, among all these metrics, conductance is the metric that best captures the concept of community. Furthermore, their results reveal that the quality of communities decreases significantly for those of size greater than around 100 elements.
@cite_18 @cite_24 @cite_19 @cite_23 @cite_10
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We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging. 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. Detecting clusters or communities in large real-world graphs such as large social or information networks is a problem of considerable interest. In practice, one typically chooses an objective function that captures the intuition of a network cluster as set of nodes with better internal connectivity than external connectivity, and then one applies approximation algorithms or heuristics to extract sets of nodes that are related to the objective function and that "look like" good communities for the application of interest. In this paper, we explore a range of network community detection methods in order to compare them and to understand their relative performance and the systematic biases in the clusters they identify. We evaluate several common objective functions that are used to formalize the notion of a network community, and we examine several different classes of approximation algorithms that aim to optimize such objective functions. In addition, rather than simply fixing an objective and asking for an approximation to the best cluster of any size, we consider a size-resolved version of the optimization problem. Considering community quality as a function of its size provides a much finer lens with which to examine community detection algorithms, since objective functions and approximation algorithms often have non-obvious size-dependent behavior. We motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have poly-logarithmic worst-case guarantees under the new measure. The main result of the article is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worst-case guarantees; another finds a "good" clustering, if one exists. We de ne a communit y on the web as a set of sites that have more links (in either direction) to members of the community than to non-members. Members of such a community can be eAEciently iden ti ed in a maximum ow minim um cut framework, where the source is composed of known members, and the sink consists of well-kno wn non-members. A focused crawler that crawls to a xed depth can approximate community membership by augmenting the graph induced by the cra wl with links to a virtual sink node.The effectiveness of the approximation algorithm is demonstrated with several crawl results that iden tify hubs, authorities, w eb rings, and other link topologies that are useful but not easily categorized. Applications of our approach include focused cra wlers and search engines, automatic population of portal categories, and improved ltering.