Communities in Networks

We survey some of the concepts, methods, and applications of community detection, which has become an increasingly important area of network science. To help ease newcomers into the field, we provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems. As a running theme, we emphasize the connections of community detection to problems in statistical physics and computational optimization.Comment: survey/review article on community structure in networks; published version is available at http://people.maths.ox.ac.uk/~porterm/papers/comnotices.pd

Decoding communities in networks

According to a recent information-theoretical proposal, the problem of defining and identifying communities in networks can be interpreted as a classical communication task over a noisy channel: memberships of nodes are information bits erased by the channel, edges and non-edges in the network are parity bits introduced by the encoder but degraded through the channel, and a community identification algorithm is a decoder. The interpretation is perfectly equivalent to the one at the basis of well-known statistical inference algorithms for community detection. The only difference in the interpretation is that a noisy channel replaces a stochastic network model. However, the different perspective gives the opportunity to take advantage of the rich set of tools of coding theory to generate novel insights on the problem of community detection. In this paper, we illustrate two main applications of standard coding-theoretical methods to community detection. First, we leverage a state-of-the-art decoding technique to generate a family of quasi-optimal community detection algorithms. Second and more important, we show that the Shannon's noisy-channel coding theorem can be invoked to establish a lower bound, here named as decodability bound, for the maximum amount of noise tolerable by an ideal decoder to achieve perfect detection of communities. When computed for well-established synthetic benchmarks, the decodability bound explains accurately the performance achieved by the best community detection algorithms existing on the market, telling us that only little room for their improvement is still potentially left.Comment: 9 pages, 5 figures + Appendi

Analyzing overlapping communities in networks using link communities

One way to analyze the structure of a network is to identify its communities, groups of related nodes that are more likely to connect to one another than to nodes outside the community. Commonly used algorithms for obtaining a network’s communities rely on clustering of the network’s nodes into a community structure that maximizes an appropriate objective function. However, defining communities as a partition of a network’s nodes, and thus stipulating that each node belongs to exactly one community, precludes the detection of overlapping communities that may exist in the network. Here we show that by defining communities as partition of a network’s links, and thus allowing individual nodes to appear in multiple communities, we can quantify the extent to which each pair of communities in a network overlaps. We define two measures of community overlap and apply them to the community structure of five networks from different disciplines. In every case, we note that there are many pairs of communities that share a significant number of nodes. This highlights a major advantage of using link partitioning, as opposed to node partitioning, when seeking to understand the community structure of a network. We also observe significant differences between overlap statistics in real-world networks as compared with randomly-generated null models. By virtue of their contexts, we expect many naturally-occurring networks to have very densely overlapping communities. Therefore, it is necessary to develop an understanding of how to use community overlap calculations to draw conclusions about the underlying structure of a network

Finding communities in sparse networks

Spectral algorithms based on matrix representations of networks are often used to detect communities but classic spectral methods based on the adjacency matrix and its variants fail to detect communities in sparse networks. New spectral methods based on non-backtracking random walks have recently been introduced that successfully detect communities in many sparse networks. However, the spectrum of non-backtracking random walks ignores hanging trees in networks that can contain information about the community structure of networks. We introduce the reluctant backtracking operators that explicitly account for hanging trees as they admit a small probability of returning to the immediately previous node unlike the non-backtracking operators that forbid an immediate return. We show that the reluctant backtracking operators can detect communities in certain sparse networks where the non-backtracking operators cannot while performing comparably on benchmark stochastic block model networks and real world networks. We also show that the spectrum of the reluctant backtracking operator approximately optimises the standard modularity function similar to the flow matrix. Interestingly, for this family of non- and reluctant-backtracking operators the main determinant of performance on real-world networks is whether or not they are normalised to conserve probability at each node.Comment: 11 pages, 4 figure

Measuring Significance of Community Structure in Complex Networks

Many complex systems can be represented as networks and separating a network into communities could simplify the functional analysis considerably. Recently, many approaches have been proposed for finding communities, but none of them can evaluate the communities found are significant or trivial definitely. In this paper, we propose an index to evaluate the significance of communities in networks. The index is based on comparing the similarity between the original community structure in network and the community structure of the network after perturbed, and is defined by integrating all the similarities. Many artificial networks and real-world networks are tested. The results show that the index is independent from the size of network and the number of communities. Moreover, we find the clear communities always exist in social networks, but don't find significative communities in proteins interaction networks and metabolic networks.Comment: 6 pages, 4 figures, 1 tabl