Stability of graph communities across time scales

The complexity of biological, social and engineering networks makes it desirable to find natural partitions into communities that can act as simplified descriptions and provide insight into the structure and function of the overall system. Although community detection methods abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We show here that the quality of a partition can be measured in terms of its stability, defined in terms of the clustered autocovariance of a Markov process taking place on the graph. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser clusterings. Within our framework we can then provide a unifying view of several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler's spectral clustering emerges at long times. We apply our method to characterize the relevance and persistence of partitions over time for constructive and real networks, including hierarchical graphs and social networks. We also obtain reduced descriptions for atomic level protein structures over different time scales.Comment: submitted; updated bibliography from v

Survey of partitioning techniques in silicon compilation

In the silicon compilation design process, partitioning is usually the first problem to be investigated because partitioning algorithms form the backbone of many algorithms including: system synthesis, processor synthesis, floorplanning, and placement. In this survey, several partitioning techniques will be examined. In addition, this paper will review the partitioning algorithms used by synthesis systems at different design levels

Defining and identifying communities in networks

The investigation of community structures in networks is an important issue in many domains and disciplines. This problem is relevant for social tasks (objective analysis of relationships on the web), biological inquiries (functional studies in metabolic, cellular or protein networks) or technological problems (optimization of large infrastructures). Several types of algorithm exist for revealing the community structure in networks, but a general and quantitative definition of community is still lacking, leading to an intrinsic difficulty in the interpretation of the results of the algorithms without any additional non-topological information. In this paper we face this problem by introducing two quantitative definitions of community and by showing how they are implemented in practice in the existing algorithms. In this way the algorithms for the identification of the community structure become fully self-contained. Furthermore, we propose a new local algorithm to detect communities which outperforms the existing algorithms with respect to the computational cost, keeping the same level of reliability. The new algorithm is tested on artificial and real-world graphs. In particular we show the application of the new algorithm to a network of scientific collaborations, which, for its size, can not be attacked with the usual methods. This new class of local algorithms could open the way to applications to large-scale technological and biological applications.Comment: Revtex, final form, 14 pages, 6 figure

Multi-Scale Link Prediction

The automated analysis of social networks has become an important problem due to the proliferation of social networks, such as LiveJournal, Flickr and Facebook. The scale of these social networks is massive and continues to grow rapidly. An important problem in social network analysis is proximity estimation that infers the closeness of different users. Link prediction, in turn, is an important application of proximity estimation. However, many methods for computing proximity measures have high computational complexity and are thus prohibitive for large-scale link prediction problems. One way to address this problem is to estimate proximity measures via low-rank approximation. However, a single low-rank approximation may not be sufficient to represent the behavior of the entire network. In this paper, we propose Multi-Scale Link Prediction (MSLP), a framework for link prediction, which can handle massive networks. The basis idea of MSLP is to construct low rank approximations of the network at multiple scales in an efficient manner. Based on this approach, MSLP combines predictions at multiple scales to make robust and accurate predictions. Experimental results on real-life datasets with more than a million nodes show the superior performance and scalability of our method.Comment: 20 pages, 10 figure

Statistical Mechanics of Community Detection

Starting from a general \textit{ansatz}, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the \textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the modularity $Q$ as defined by Newman and Girvan \cite{Girvan03} as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the \textit{ansatz} may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure

Construction of a Pragmatic Base Line for Journal Classifications and Maps Based on Aggregated Journal-Journal Citation Relations

A number of journal classification systems have been developed in bibliometrics since the launch of the Citation Indices by the Institute of Scientific Information (ISI) in the 1960s. These systems are used to normalize citation counts with respect to field-specific citation patterns. The best known system is the so-called "Web-of-Science Subject Categories" (WCs). In other systems papers are classified by algorithmic solutions. Using the Journal Citation Reports 2014 of the Science Citation Index and the Social Science Citation Index (n of journals = 11,149), we examine options for developing a new system based on journal classifications into subject categories using aggregated journal-journal citation data. Combining routines in VOSviewer and Pajek, a tree-like classification is developed. At each level one can generate a map of science for all the journals subsumed under a category. Nine major fields are distinguished at the top level. Further decomposition of the social sciences is pursued for the sake of example with a focus on journals in information science (LIS) and science studies (STS). The new classification system improves on alternative options by avoiding the problem of randomness in each run that has made algorithmic solutions hitherto irreproducible. Limitations of the new system are discussed (e.g. the classification of multi-disciplinary journals). The system's usefulness for field-normalization in bibliometrics should be explored in future studies.Comment: accepted for publication in the Journal of Informetrics, 20 July 201

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

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