Partitioning networks into cliques: a randomized heuristic approach

In the context of community detection in social networks, the term community can be grounded in the strict way that simply everybody should know each other within the community. We consider the corresponding community detection problem. We search for a partitioning of a network into the minimum number of non-overlapping cliques, such that the cliques cover all vertices. This problem is called the clique covering problem (CCP) and is one of the classical NP-hard problems. For CCP, we propose a randomized heuristic approach. To construct a high quality solution to CCP, we present an iterated greedy (IG) algorithm. IG can also be combined with a heuristic used to determine how far the algorithm is from the optimum in the worst case. Randomized local search (RLS) for maximum independent set was proposed to find such a bound. The experimental results of IG and the bounds obtained by RLS indicate that IG is a very suitable technique for solving CCP in real-world graphs. In addition, we summarize our basic rigorous results, which were developed for analysis of IG and understanding of its behavior on several relevant graph classes

Finding Near-Optimal Independent Sets at Scale

The independent set problem is NP-hard and particularly difficult to solve in large sparse graphs. In this work, we develop an advanced evolutionary algorithm, which incorporates kernelization techniques to compute large independent sets in huge sparse networks. A recent exact algorithm has shown that large networks can be solved exactly by employing a branch-and-reduce technique that recursively kernelizes the graph and performs branching. However, one major drawback of their algorithm is that, for huge graphs, branching still can take exponential time. To avoid this problem, we recursively choose vertices that are likely to be in a large independent set (using an evolutionary approach), then further kernelize the graph. We show that identifying and removing vertices likely to be in large independent sets opens up the reduction space---which not only speeds up the computation of large independent sets drastically, but also enables us to compute high-quality independent sets on much larger instances than previously reported in the literature.Comment: 17 pages, 1 figure, 8 tables. arXiv admin note: text overlap with arXiv:1502.0168

Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?

Correlations in the signal observed via functional Magnetic Resonance Imaging (fMRI), are expected to reveal the interactions in the underlying neural populations through hemodynamic response. In particular, they highlight distributed set of mutually correlated regions that correspond to brain networks related to different cognitive functions. Yet graph-theoretical studies of neural connections give a different picture: that of a highly integrated system with small-world properties: local clustering but with short pathways across the complete structure. We examine the conditional independence properties of the fMRI signal, i.e. its Markov structure, to find realistic assumptions on the connectivity structure that are required to explain the observed functional connectivity. In particular we seek a decomposition of the Markov structure into segregated functional networks using decomposable graphs: a set of strongly-connected and partially overlapping cliques. We introduce a new method to efficiently extract such cliques on a large, strongly-connected graph. We compare methods learning different graph structures from functional connectivity by testing the goodness of fit of the model they learn on new data. We find that summarizing the structure as strongly-connected networks can give a good description only for very large and overlapping networks. These results highlight that Markov models are good tools to identify the structure of brain connectivity from fMRI signals, but for this purpose they must reflect the small-world properties of the underlying neural systems

Motif Clustering and Overlapping Clustering for Social Network Analysis

Motivated by applications in social network community analysis, we introduce a new clustering paradigm termed motif clustering. Unlike classical clustering, motif clustering aims to minimize the number of clustering errors associated with both edges and certain higher order graph structures (motifs) that represent "atomic units" of social organizations. Our contributions are two-fold: We first introduce motif correlation clustering, in which the goal is to agnostically partition the vertices of a weighted complete graph so that certain predetermined "important" social subgraphs mostly lie within the same cluster, while "less relevant" social subgraphs are allowed to lie across clusters. We then proceed to introduce the notion of motif covers, in which the goal is to cover the vertices of motifs via the smallest number of (near) cliques in the graph. Motif cover algorithms provide a natural solution for overlapping clustering and they also play an important role in latent feature inference of networks. For both motif correlation clustering and its extension introduced via the covering problem, we provide hardness results, algorithmic solutions and community detection results for two well-studied social networks

Cover-Encodings of Fitness Landscapes

The traditional way of tackling discrete optimization problems is by using local search on suitably defined cost or fitness landscapes. Such approaches are however limited by the slowing down that occurs when the local minima that are a feature of the typically rugged landscapes encountered arrest the progress of the search process. Another way of tackling optimization problems is by the use of heuristic approximations to estimate a global cost minimum. Here we present a combination of these two approaches by using cover-encoding maps which map processes from a larger search space to subsets of the original search space. The key idea is to construct cover-encoding maps with the help of suitable heuristics that single out near-optimal solutions and result in landscapes on the larger search space that no longer exhibit trapping local minima. We present cover-encoding maps for the problems of the traveling salesman, number partitioning, maximum matching and maximum clique; the practical feasibility of our method is demonstrated by simulations of adaptive walks on the corresponding encoded landscapes which find the global minima for these problems.Comment: 15 pages, 4 figure

A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems