text: To find interesting structure in networks, community detection algorithms have to take into account not only the network topology, but also dynamics of interactions between nodes. We investigate this claim using the paradigm of synchronization in a network of coupled oscillators. As the network evolves to a global steady state, nodes belonging to the same community synchronize faster than nodes belonging to different communities. Traditionally, nodes in network synchronization models are coupled via one-to-one, or conservative interactions. However, social interactions are often one-to-many, as for example, in social media, where users broadcast messages to all their followers. We formulate a novel model of synchronization in a network of coupled oscillators in which the oscillators are coupled via one-to-many, or non-conservative interactions. We study the dynamics of different interaction models and contrast their spectral properties. To find multi-scale community structure in a network of interacting nodes, we define a similarity function that measures the degree to which nodes are synchronized and use it to hierarchically cluster nodes. We study real-world social networks, including networks of two social media providers. To evaluate the quality of the discovered communities in a social media network we propose a community quality metric based on user activity. We find that conservative and non-conservative interaction models lead to dramatically different views of community structure even within the same network. Our work offers a novel mathematical framework for exploring the relationship between network structure, topology and dynamics.
  @cite_4: 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.
summary: Community detection is an extremely active research area, with a variety of methods proposed, including spectral clustering, graph partitioning and modularity maximization @cite_4 . We show in this paper that these methods can be expressed in terms of the generalized linear interaction model that assumes a specific type of interaction. We also demonstrate that to get the full picture of network's emergent structure, community detection method must account for the dynamic process occurring on the network.