Extremal Optimization for Graph Partitioning

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the NP-hard graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of runtime and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. Our numerical results demonstrate that on random graphs, extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over runtime roughly as a power law t^(-0.4). On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal, with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.Comment: 34 pages, RevTex4, 1 table and 20 ps-figures included, related papers available at http://www.physics.emory.edu/faculty/boettcher

Extremal Optimization of Graph Partitioning at the Percolation Threshold

The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called Extremal Optimization, is compared to Simulated Annealing in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold. At this threshold, the relative error of Simulated Annealing for large graphs is found to diverge relative to Extremal Optimization at equalized runtime. On the other hand, Extremal Optimization, based on the extremal dynamics of self-organized critical systems, reproduces known results about optimal partitions at this critical point quite well.Comment: 7 pages, RevTex, 9 ps-figures included, as to appear in Journal of Physics

Extremal optimization for sensor report pre-processing

We describe the recently introduced extremal optimization algorithm and apply it to target detection and association problems arising in pre-processing for multi-target tracking. Here we consider the problem of pre-processing for multiple target tracking when the number of sensor reports received is very large and arrives in large bursts. In this case, it is sometimes necessary to pre-process reports before sending them to tracking modules in the fusion system. The pre-processing step associates reports to known tracks (or initializes new tracks for reports on objects that have not been seen before). It could also be used as a pre-process step before clustering, e.g., in order to test how many clusters to use. The pre-processing is done by solving an approximate version of the original problem. In this approximation, not all pair-wise conflicts are calculated. The approximation relies on knowing how many such pair-wise conflicts that are necessary to compute. To determine this, results on phase-transitions occurring when coloring (or clustering) large random instances of a particular graph ensemble are used.Comment: 10 page

Modularity and community structure in networks

Many networks of interest in the sciences, including a variety of social and biological networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure has attracted considerable recent attention. One of the most sensitive detection methods is optimization of the quality function known as "modularity" over the possible divisions of a network, but direct application of this method using, for instance, simulated annealing is computationally costly. Here we show that the modularity can be reformulated in terms of the eigenvectors of a new characteristic matrix for the network, which we call the modularity matrix, and that this reformulation leads to a spectral algorithm for community detection that returns results of better quality than competing methods in noticeably shorter running times. We demonstrate the algorithm with applications to several network data sets.Comment: 7 pages, 3 figure

Correlation Clustering with Low-Rank Matrices

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data

Network Community Detection on Metric Space

Community detection in a complex network is an important problem of much interest in recent years. In general, a community detection algorithm chooses an objective function and captures the communities of the network by optimizing the objective function, and then, one uses various heuristics to solve the optimization problem to extract the interesting communities for the user. In this article, we demonstrate the procedure to transform a graph into points of a metric space and develop the methods of community detection with the help of a metric defined for a pair of points. We have also studied and analyzed the community structure of the network therein. The results obtained with our approach are very competitive with most of the well-known algorithms in the literature, and this is justified over the large collection of datasets. On the other hand, it can be observed that time taken by our algorithm is quite less compared to other methods and justifies the theoretical findings