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.

Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems.

Detecting community structure is fundamental for uncovering the links between structure and function in complex networks and for practical applications in many disciplines such as biology and sociology. A popular method now widely used relies on the optimization of a quantity called modularity, which is a quality index for a partition of a network into communities. We find that modularity optimization may fail to identify modules smaller than a scale which depends on the total size of the network and on the degree of interconnectedness of the modules, even in cases where modules are unambiguously defined. This finding is confirmed through several examples, both in artificial and in real social, biological, and technological networks, where we show that modularity optimization indeed does not resolve a large number of modules. A check of the modules obtained through modularity optimization is thus necessary, and we provide here key elements for the assessment of the reliability of this community detection method.complex networks ͉ modular structure ͉ metabolic networks ͉ social networks C ommunity detection in complex networks has attracted a lot of attention in recent years (for a review, see refs. 1 and 2). The main reason is that complex networks (3-7) are made of a large number of nodes and most previous quantitative investigations focused on statistical properties disregarding the roles played by specific subgraphs. Detecting communities (or modules) can be a way to identify substructures which could correspond to important functions. This is, for example, confirmed in the case of the World Wide Web, where communities are sets of Web pages dealing with the same topic (8). In biological networks, it is widely believed that the modular structure results from evolutionary constraints and plays a crucial role in biological functions (9-11), which makes community detection very relevant (12)(13)(14). Relevant community structures were also found in social networks (15-17), the Internet (18), food webs (19,20), and in networks of sexual contacts (21,22).Loosely speaking, a community is a subgraph of a network whose nodes are more tightly connected with each other than with nodes outside the subgraph. A decisive advance in community detection was made by Newman and Girvan (23), who introduced a quantitative measure for the quality of a partition of a network into communities, the modularity. This measure essentially compares the number of links inside a given module with the expected value for a randomized graph of the same size and same degree sequence. If one chooses modularity as the relevant quality function, the problem of community detection becomes equivalent to modularity optimization. The latter is not trivial, as the number of possible partitions of a network into clusters increases at least exponentially with the size of the network, making exhaustive optimization computationally unfeasible even for relatively small graphs. Therefore, a number of algorithms have been devised to find a good optimization t...