We propose and study a set of algorithms for discovering community structure in networks-natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

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Finding and evaluating community structure in networks.

Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets.

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Modularity and community structure in networks

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.

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Community detection in graphs

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

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Finding community structure in networks using the eigenvectors of matrices

Technological progress in integrated, low-power, CMOS communication devices and sensors makes a rich design space of networked sensors viable. They can be deeply embedded in the physical world and spread throughout our environment like smart dust. The missing elements are an overall system architecture and a methodology for systematic advance. To this end, we identify key requirements, develop a small device that is representative of the class, design a tiny event-driven operating system, and show that it provides support for efficient modularity and concurrency-intensive operation. Our operating system fits in 178 bytes of memory, propagates events in the time it takes to copy 1.25 bytes of memory, context switches in the time it takes to copy 6 bytes of memory and supports two level scheduling. The analysis lays a groundwork for future architectural advances.

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System architecture directions for networked sensors

A vast body of theoretical research has focused either on overly simplistic models of parallel computation, notably the PRAM, or overly specific models that have few representatives in the real world. Both kinds of models encourage exploitation of formal loopholes, rather than rewarding development of techniques that yield performance across a range of current and future parallel machines. This paper offers a new parallel machine model, called LogP, that reflects the critical technology trends underlying parallel computers. it is intended to serve as a basis for developing fast, portable parallel algorithms and to offer guidelines to machine designers. Such a model must strike a balance between detail and simplicity in order to reveal important bottlenecks without making analysis of interesting problems intractable. The model is based on four parameters that specify abstractly the computing bandwidth, the communication bandwidth, the communication delay, and the efficiency of coupling communication and computation. Portable parallel algorithms typically adapt to the machine configuration, in terms of these parameters. The utility of the model is demonstrated through examples that are implemented on the CM-5.

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LogP: towards a realistic model of parallel computation

The design challenge for large-scale multiprocessors is (1) to minimize communication overhead, (2) allow communication to overlap computation, and (3) coordinate the two without sacrificing processor cost/performance. We show that existing message passing multiprocessors have unnecessarily high communication costs. Research prototypes of message driven machines demonstrate low communication overhead, but poor processor cost/performance. We introduce a simple communication mechanism, Active Messages, show that it is intrinsic to both architectures, allows cost effective use of the hardware, and offers tremendous flexibility. Implementations on nCUBE/2 and CM-5 are described and evaluated using a split-phase shared-memory extension to C, Split-C. We further show that active messages are sufficient to implement the dynamically scheduled languages for which message driven machines were designed. With this mechanism, latency tolerance becomes a programming/compiling concern. Hardware support for active messages is desirable and we outline a range of enhancements to mainstream processors.

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Active messages: a mechanism for integrated communication and computation

技術解説 IEEE Computer

enough to be generally useful and to keep the algorithm analysis tractable. Ideally, producing a better algorithm under the model should yield a better program in practice. The Parallel Random Access Machine (PRAM) [8] is the most popular model for representing and analyzing the complexity of parallel algorithms. A LogP A Practic

LogP: a practical model of parallel computation

With the spread of the Internet the computing model on server systems is undergoing several important changes. Recent research ideas concerning dynamic operating system extensibility are finding their way into the commercial domain, resulting in designs of extensible databases and Web servers. In addition, both ordinary users and service providers must deal with untrusted downloadable executable code of unknown origin and intentions.Across the board, Java has emerged as the language of choice for Internet-oriented software. We argue that, in order to realize its full potential in applications dealing with untrusted code, Java needs a flexible resource accounting interface. The design and prototype implementation of such an interface --- JRes --- is presented in this paper. The interface allows to account for heap memory, CPU time, and network resources consumed by individual threads or groups of threads. JRes allows limits to be set on resources available to threads and it can invoke callbacks when these limits are exceeded. The JRes prototype described in this paper is implemented on top of standard Java virtual machines and requires only a small amount of native code.

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Thorsten von Eicken

Papers

LogP: towards a realistic model of parallel computation

Active messages: a mechanism for integrated communication and computation

技術解説 IEEE Computer

LogP: a practical model of parallel computation

JRes: a resource accounting interface for Java