Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

Communities and bottlenecks: Trees and treelike networks have high modularity

Much effort has gone into understanding the modular nature of complex networks. Communities, also known as clusters or modules, are typically considered to be densely interconnected groups of nodes that are only sparsely connected to other groups in the network. Discovering high quality communities is a difficult and important problem in a number of areas. The most popular approach is the objective function known as modularity, used both to discover communities and to measure their strength. To understand the modular structure of networks it is then crucial to know how such functions evaluate different topologies, what features they account for, and what implicit assumptions they may make. We show that trees and treelike networks can have unexpectedly and often arbitrarily high values of modularity. This is surprising since trees are maximally sparse connected graphs and are not typically considered to possess modular structure, yet the nonlocal null model used by modularity assigns low probabilities, and thus high significance, to the densities of these sparse tree communities. We further study the practical performance of popular methods on model trees and on a genealogical data set and find that the discovered communities also have very high modularity, often approaching its maximum value. Statistical tests reveal the communities in trees to be significant, in contrast with known results for partitions of sparse, random graphs.Comment: 9 pages, 5 figure

Gaussian Approximation of Collective Graphical Models

The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous work has explored Markov Chain Monte Carlo (MCMC) and MAP approximations for learning and inference. This paper studies Gaussian approximations to the CGM. As the population grows large, we show that the CGM distribution converges to a multivariate Gaussian distribution (GCGM) that maintains the conditional independence properties of the original CGM. If the observations are exact marginals of the CGM or marginals that are corrupted by Gaussian noise, inference in the GCGM approximation can be computed efficiently in closed form. If the observations follow a different noise model (e.g., Poisson), then expectation propagation provides efficient and accurate approximate inference. The accuracy and speed of GCGM inference is compared to the MCMC and MAP methods on a simulated bird migration problem. The GCGM matches or exceeds the accuracy of the MAP method while being significantly faster.Comment: Accepted by ICML 2014. 10 page version with appendi

Graph Theory

This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results

Hypergraph reconstruction from network data

Networks can describe the structure of a wide variety of complex systems by specifying how pairs of nodes interact. This choice of representation is flexible, but not necessarily appropriate when joint interactions between groups of nodes are needed to explain empirical phenomena. Networks remain the de facto standard, however, as relational datasets often fail to include higher-order interactions. Here, we introduce a Bayesian approach to reconstruct these missing higher-order interactions, from pairwise network data. Our method is based on the principle of parsimony and only includes higher-order structures when there is sufficient statistical evidence for them.Comment: 12 pages, 6 figures. Code is available at https://graph-tool.skewed.de

Diffusion Adaptation Strategies for Distributed Estimation over Gaussian Markov Random Fields

The aim of this paper is to propose diffusion strategies for distributed estimation over adaptive networks, assuming the presence of spatially correlated measurements distributed according to a Gaussian Markov random field (GMRF) model. The proposed methods incorporate prior information about the statistical dependency among observations, while at the same time processing data in real-time and in a fully decentralized manner. A detailed mean-square analysis is carried out in order to prove stability and evaluate the steady-state performance of the proposed strategies. Finally, we also illustrate how the proposed techniques can be easily extended in order to incorporate thresholding operators for sparsity recovery applications. Numerical results show the potential advantages of using such techniques for distributed learning in adaptive networks deployed over GMRF.Comment: Submitted to IEEE Transactions on Signal Processing. arXiv admin note: text overlap with arXiv:1206.309