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

MEDUSA - New Model of Internet Topology Using k-shell Decomposition

The k-shell decomposition of a random graph provides a different and more insightful separation of the roles of the different nodes in such a graph than does the usual analysis in terms of node degrees. We develop this approach in order to analyze the Internet's structure at a coarse level, that of the "Autonomous Systems" or ASes, the subnetworks out of which the Internet is assembled. We employ new data from DIMES (see http://www.netdimes.org), a distributed agent-based mapping effort which at present has attracted over 3800 volunteers running more than 7300 DIMES clients in over 85 countries. We combine this data with the AS graph information available from the RouteViews project at Univ. Oregon, and have obtained an Internet map with far more detail than any previous effort. The data suggests a new picture of the AS-graph structure, which distinguishes a relatively large, redundantly connected core of nearly 100 ASes and two components that flow data in and out from this core. One component is fractally interconnected through peer links; the second makes direct connections to the core only. The model which results has superficial similarities with and important differences from the "Jellyfish" structure proposed by Tauro et al., so we call it a "Medusa." We plan to use this picture as a framework for measuring and extrapolating changes in the Internet's physical structure. Our k-shell analysis may also be relevant for estimating the function of nodes in the "scale-free" graphs extracted from other naturally-occurring processes.Comment: 24 pages, 17 figure

Simplifying complex networks:from a clustering to a coarse graining strategy

The framework of complex networks has been shown to describe adequately a wide class of complex systems made up of a large number of interacting units. In this framework, a node is associated to each unit and two nodes are connected by an edge if the two units interact with each other. Examples of such systems can be found in living organisms—the map of interactions between proteins or the network of neurons in the brain. Moreover, artificial systems such as the WWW, electrical grids or airplane connections have been studied using the tools of complex networks. Finally networks have found many applications in social sciences to characterize for instance human interactions of different kinds underlying the spreading of an epidemic. For most of these systems, the complexity arises because of the large number of units and their intricate connection patterns. A natural approach is therefore to simplify the systems by decreasing their size. Different schemes can indeed be designed for each particular system, leading to effective but case-dependent methods. From a more global and statistical perspective, a promising alternative is to reduce the complexity of the corresponding networks. In order to simplify complex networks, two strategies are presented in this Thesis. The first approach refers to the well-known clustering paradigm. It aims at identifying groups of nodes densely connected between each other and much less to the rest of the network. Those groups are referred to as clusters or communities. For most real systems, nodes within a community share some similarity or common feature. For instance, in a synonymy network where nodes are words and edges connect synonymous words, we have shown that finding communities allowed us to identify words corresponding to a single concept. We have also studied a network describing the dynamics of a peptide by associating a node to a microscopic configuration and an edge to a transition. The community structure of the network was shown to provide a new methodology to explore the main characteristics of the peptide dynamics and to unravel the large-scale features of the underlying free-energy landscape. Finally we have designed a new technique to probe the robustness of the community structure against external perturbations of the network topology. This method allows us, among else, to assess whether communities correspond to a real structure of the network, or are simple artifacts of the clustering algorithms. Community detection techniques have found a large number of practical applications as a method to simplify networks since the number of clusters is often much smaller than the number of nodes. However, a crucial issue has often been disregarded: is the network of clusters truly representative of the initial one? In this Thesis, we show that this is indeed not verified for most networks. For example we have considered the evolution of random walks on the network of clusters and found that it behaves quite differently than in the initial network. This observation led us to develop a new strategy to simplify complex networks, ensuring that the reduced network is representative of the initial one. It is based on the idea of grouping nodes, akin to community detection. However, the aim is no longer to identify the "correct" clusters, but to find a smaller network which preserves the relevant features of the initial one, and especially the spectral properties. We therefore refer to our method as Spectral Coarse Graining, by analogy with the coarse graining framework used in Statistical Physics. Applying this method to various kinds of networks, we have shown that the coarse-grained network provides an excellent approximation of the initial one, while the size could be easily reduced by a factor of ten. Therefore, the Spectral Coarse Graining provides a well-defined way of studying large networks and their dynamics considering a much smaller coarse-grained version. Overall, we first discuss the use and the limits of the usual clustering approach to reduce the complexity of networks, and apply it to several real-world systems. In a second part, we develop a new coarse graining strategy to approximate large networks by smaller ones and provide several examples to illustrate the power and the novelty of the method

Node dynamics on graphs: dynamical heterogeneity in consensus, synchronisation and final value approximation for complex networks

Here we consider a range of Laplacian-based dynamics on graphs such as dynamical invariance and coarse-graining, and node-specific properties such as convergence, observability and consensus-value prediction. Firstly, using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterise convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions. We show that the EEP with respect to a node can reveal nodes in the graph with increased rate of asymptotic convergence to the consensus value as characterised by the second smallest eigenvalue of the quotient Laplacian. Secondly, we extend this characterisation of the relationship between the EEP and Laplacian based dynamics to study the synchronisation of coupled oscillator dynamics on networks. We show that the existence of a non-trivial EEP describes partial synchronisation dynamics for nodes within cells of the partition. Considering linearised stability analysis, the existence of a nontrivial EEP with respect to an individual node can imply an increased rate of asymptotic convergence to the synchronisation manifold, or a decreased rate of de-synchronisation, analogous to the linear consensus case. We show that high degree 'hub' nodes in large complex networks such as Erdős-Rényi, scale free and entangled graphs are more likely to exhibit such dynamical heterogeneity under both linear consensus and non-linear coupled oscillator dynamics. Finally, we consider a separate but related problem concerning the ability of a node to compute the final value for discrete consensus dynamics given only a finite number of its own state values. We develop an algorithm to compute an approximation to the consensus value by individual nodes that is ϵ close to the true consensus value, and show that in most cases this is possible for substantially less steps than required for true convergence of the system dynamics. Again considering a variety of complex networks we show that, on average, high degree nodes, and nodes belonging to graphs with fast asymptotic convergence, approximate the consensus value employing fewer steps.Open Acces

Nonparametric Message Passing Methods for Cooperative Localization and Tracking

The objective of this thesis is the development of cooperative localization and tracking algorithms using nonparametric message passing techniques. In contrast to the most well-known techniques, the goal is to estimate the posterior probability density function (PDF) of the position of each sensor. This problem can be solved using Bayesian approach, but it is intractable in general case. Nevertheless, the particle-based approximation (via nonparametric representation), and an appropriate factorization of the joint PDFs (using message passing methods), make Bayesian approach acceptable for inference in sensor networks. The well-known method for this problem, nonparametric belief propagation (NBP), can lead to inaccurate beliefs and possible non-convergence in loopy networks. Therefore, we propose four novel algorithms which alleviate these problems: nonparametric generalized belief propagation (NGBP) based on junction tree (NGBP-JT), NGBP based on pseudo-junction tree (NGBP-PJT), NBP based on spanning trees (NBP-ST), and uniformly-reweighted NBP (URW-NBP). We also extend NBP for cooperative localization in mobile networks. In contrast to the previous methods, we use an optional smoothing, provide a novel communication protocol, and increase the efficiency of the sampling techniques. Moreover, we propose novel algorithms for distributed tracking, in which the goal is to track the passive object which cannot locate itself. In particular, we develop distributed particle filtering (DPF) based on three asynchronous belief consensus (BC) algorithms: standard belief consensus (SBC), broadcast gossip (BG), and belief propagation (BP). Finally, the last part of this thesis includes the experimental analysis of some of the proposed algorithms, in which we found that the results based on real measurements are very similar with the results based on theoretical models