In this paper, we use a partition of the links of a network in order to uncover its community structure. This approach allows for communities to overlap at nodes, so that nodes may be in more than one community. We do this by making a node partition of the line graph of the original network. In this way we show that any algorithm which produces a partition of nodes can be used to produce a partition of links. We discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this.

Many complex systems are organized in the form of a network embedded in space. Important examples include the physical Internet infrastucture, road networks, flight connections, brain functional networks, and social networks. The effect of space on network topology has recently come under the spotlight because of the emergence of pervasive technologies based on geolocalization, which constantly fill databases with people's movements and thus reveal their trajectories and spatial behavior. Extracting patterns and regularities from the resulting massive amount of human mobility data requires the development of appropriate tools for uncovering information in spatially embedded networks. In contrast with most works that tend to apply standard network metrics to any type of network, we argue in this paper for a careful treatment of the constraints imposed by space on network topology. In particular, we focus on the problem of community detection and propose a modularity function adapted to spatial networks. We show that it is possible to factor out the effect of space in order to reveal more clearly hidden structural similarities between the nodes. Methods are tested on a large mobile phone network and computer-generated benchmarks where the effect of space has been incorporated. complex networks | social systems U nderstanding the principles driving the organization of complex networks is crucial for a broad range of fields including information and social sciences, economics, biology, and neuroscience (1). In networks where nodes occupy positions in an Euclidian space, spatial constraints may have a strong effect on their connectivity patterns (2). Edges may either be spatially embedded, such as in roads or railway lines in transportation networks or cables in a power grid, or abstract entities, such as friendship relations in online and offline social networks or functional connectivity in brain networks. In either case, space plays a crucial role by affecting, directly or indirectly, network connectivity and making its architecture radically different from that of random networks (3). A crucial difference stems from the cost associated to long-distance links (4-12), which restricts the existence of hubs (i.e., high-degree nodes), and thus the observation of fat-tailed degree distributions in spatial networks.From a modeling viewpoint, gravity models (13-15) have long been used to model flows in spatial networks. These models focus on the intensity of interaction between locations i and j separated by a certain physical distance d ij . It has been shown for systems as diverse as the International Trade Market (16), human migration (17), traffic flows (18), or mobile communication between cities (19, 20) that the volume of interaction between distant locations is successfully modeled bywhere N i measures the importance of location i, e.g., its population, and the deterrence function f describes the influence of space. Eq. 1 emphasizes that the number of interactions between two locations is proportional to the numb...

The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and also explore their dual consensus dynamics.

It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.

The authors raise spatial analysis to a new level of sophistication -and insight -