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

Networked Computing in Wireless Sensor Networks for Structural Health Monitoring

This paper studies the problem of distributed computation over a network of wireless sensors. While this problem applies to many emerging applications, to keep our discussion concrete we will focus on sensor networks used for structural health monitoring. Within this context, the heaviest computation is to determine the singular value decomposition (SVD) to extract mode shapes (eigenvectors) of a structure. Compared to collecting raw vibration data and performing SVD at a central location, computing SVD within the network can result in significantly lower energy consumption and delay. Using recent results on decomposing SVD, a well-known centralized operation, into components, we seek to determine a near-optimal communication structure that enables the distribution of this computation and the reassembly of the final results, with the objective of minimizing energy consumption subject to a computational delay constraint. We show that this reduces to a generalized clustering problem; a cluster forms a unit on which a component of the overall computation is performed. We establish that this problem is NP-hard. By relaxing the delay constraint, we derive a lower bound to this problem. We then propose an integer linear program (ILP) to solve the constrained problem exactly as well as an approximate algorithm with a proven approximation ratio. We further present a distributed version of the approximate algorithm. We present both simulation and experimentation results to demonstrate the effectiveness of these algorithms

Correlation Clustering Generalized

We present new results for LambdaCC and MotifCC, two recently introduced variants of the well-studied correlation clustering problem. Both variants are motivated by applications to network analysis and community detection, and have non-trivial approximation algorithms. We first show that the standard linear programming relaxation of LambdaCC has a Theta(log n) integrality gap for a certain choice of the parameter lambda. This sheds light on previous challenges encountered in obtaining parameter-independent approximation results for LambdaCC. We generalize a previous constant-factor algorithm to provide the best results, from the LP-rounding approach, for an extended range of lambda. MotifCC generalizes correlation clustering to the hypergraph setting. In the case of hyperedges of degree 3 with weights satisfying probability constraints, we improve the best approximation factor from 9 to 8. We show that in general our algorithm gives a 4(k-1) approximation when hyperedges have maximum degree k and probability weights. We additionally present approximation results for LambdaCC and MotifCC where we restrict to forming only two clusters

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Morphology of Mock SDSS Catalogues

We measure the geometry, topology and morphology of the superclusters in mock SDSS catalogues prepared by Cole et al.(1998). The mock catalogues refer to $\tau$CDM and \LCDM {\em flat} cosmological models and are populated by galaxies so that these act as biased tracers of mass, conforming with the correlation function measured using APM catalogue. We compute the Minkowski Functionals (MFs) for the cosmic density fields using SURFGEN (Sheth et al.2003) and use the available 10 realizations of $\tau$CDM to study the effect of cosmic variance in estimation of MFs and Shapefinders, which we find to be extremely well constrained statistics. Although all the mock catalogues of galaxies have the same two-point correlation function and similar clustering amplitude, the global MFs due to $\tau$CDM show systematically lower amplitude compared to those due to \LCDM; an indirect, but detectable effect due to nonzero, higher order correlation functions. The characteristic thickness (T), breadth (B) and length (L) of the superclusters are measured using the available 10 realizations of $\tau$CDM. While T$\le$B and T, B$\in$[1,17] h$^{-1}$Mpc, we find the top 10 superclusters to be as long as 90 h$^{-1}$Mpc, with the longest superclusters identified at percolation to be rare objects with their length as large as 150 h$^{-1}$Mpc. The $\tau$CDM superclusters are found to be significantly longer than those in \LCDM. Thickness (T), breadth (B), planarity (P) and mass/volume$-$weighted planarity and filamentarity of the superclusters are found to be useful to compare the two models (abridged).Comment: 23 Pages, 12 Figures, MNRAS Style. Minor modifications to the text. New references adde