Random incidence matrices: moments of the spectral density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semi-circle of "small" eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit), we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e=2.72... is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix. Keywords: random graphs, random matrices, sparse matrices, incidence matrices spectrum, momentsComment: 39 pages, 9 figures, Latex2e, [v2: ref. added, Sect. 4 modified

ConnectIt: A Framework for Static and Incremental Parallel Graph Connectivity Algorithms

Connected components is a fundamental kernel in graph applications due to its usefulness in measuring how well-connected a graph is, as well as its use as subroutines in many other graph algorithms. The fastest existing parallel multicore algorithms for connectivity are based on some form of edge sampling and/or linking and compressing trees. However, many combinations of these design choices have been left unexplored. In this paper, we design the ConnectIt framework, which provides different sampling strategies as well as various tree linking and compression schemes. ConnectIt enables us to obtain several hundred new variants of connectivity algorithms, most of which extend to computing spanning forest. In addition to static graphs, we also extend ConnectIt to support mixes of insertions and connectivity queries in the concurrent setting. We present an experimental evaluation of ConnectIt on a 72-core machine, which we believe is the most comprehensive evaluation of parallel connectivity algorithms to date. Compared to a collection of state-of-the-art static multicore algorithms, we obtain an average speedup of 37.4x (2.36x average speedup over the fastest existing implementation for each graph). Using ConnectIt, we are able to compute connectivity on the largest publicly-available graph (with over 3.5 billion vertices and 128 billion edges) in under 10 seconds using a 72-core machine, providing a 3.1x speedup over the fastest existing connectivity result for this graph, in any computational setting. For our incremental algorithms, we show that our algorithms can ingest graph updates at up to several billion edges per second. Finally, to guide the user in selecting the best variants in ConnectIt for different situations, we provide a detailed analysis of the different strategies in terms of their work and locality

Community Recovery in the Geometric Block Model

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model builds on the random geometric graphs (Gilbert, 1961), one of the basic models of random graphs for spatial networks, in the same way that the well-studied stochastic block model builds on the Erd\H{o}s-R\'{en}yi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancements in community detection. To analyze the geometric block model, we first provide new connectivity results for random annulus graphs which are generalizations of random geometric graphs. The connectivity properties of geometric graphs have been studied since their introduction, and analyzing them has been more difficult than their Erd\H{o}s-R\'{en}yi counterparts due to correlated edge formation. We then use the connectivity results of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for the geometric block model. We show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. For this we consider the following two regimes of graph density. In the regime where the average degree of the graph grows logarithmically with the number of vertices, we show that our algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model in the logarithmic degree regime. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.Comment: 53 pages, 18 figures. Accepted at the Journal of Machine Learning Research (JMLR). Shorter versions accepted in AAAI 2018 (see arXiv:1709.05510) and RANDOM 2019 (see arXiv:1804.05013). arXiv admin note: text overlap with arXiv:1804.0501

Graph measures and network robustness

Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness

Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one