Large Sparse Graphs Research Articles

Parallel Maximum Clique Algorithms with Applications to Network Analysis

We present a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. The method exhibits a roughly linear runtime scaling over real-world networks ranging from a thousand to a hundred million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. At its heart the algorithm employs a branch-and-bound strategy with novel and aggressive pruning techniques. The pruning techniques include the combined use of core numbers of vertices along with a good initial heuristic solution to remove the vast majority of the search space. In addition, the exploration of the search tree is parallelized. During the search, processes immediately communicate changes to upper and lower bounds on the size of the maximum clique. This exchange of information occasionally results in a superlinear speedup because tasks with large search spaces can be pruned by other processes. We demonstrate the impact of the algorithm on applications using two different network analysis problems: computation of temporal strong components in dynamic networks and determination of compression-friendly ordering of nodes of massive networks.

On the Decay of Crossing Numbers of Sparse Graphs

AbstractRichter and Thomassen proved that every graph has an edge e such that the crossing number of is at least . Fox and Cs. Tóth proved that dense graphs have large sets of edges (proportional in the total number of edges) whose removal leaves a graph with crossing number proportional to the crossing number of the original graph; this result was later strenghthened by Černý, Kynčl, and G. Tóth. These results make our understanding of the decay of crossing numbers in dense graphs essentially complete. In this article we prove a similar result for large sparse graphs in which the number of edges is not artificially inflated by operations such as edge subdivisions. We also discuss the connection between the decay of crossing numbers and expected crossing numbers, a concept recently introduced by Mohar and Tamon.

Spectral density of the non-backtracking operator on random graphs

The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius , where ρ is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero spectral density. The fact that this phase transition is absent in the spectral density of other matrices commonly used for spectral clustering provides a physical justification of the performances of the non-backtracking operator in spectral clustering.

Solving the maximum edge-weight clique problem in sparse graphs with compact formulations

This paper studies the behavior of compact formulations for solving the maximum edge-weight clique (MEWC) problem in sparse graphs. The MEWC problem has long been discussed in the literature, but mostly addressing complete graphs, with or without a cardinality constraint on the clique. Yet, several real-world applications are defined on sparse graphs, where the missing edges are due to some threshold process or because they are not even supposed to be in the graph, at all. Such situations often arise in cell’s metabolic networks, where the amount of metabolites shared among reactions is an important issue to understand the cell’s prevalent elements. We propose new node-discretized formulations for the problem, which are more compact than other models known from the literature. Computational experiments on benchmark and real-world instances are conducted for discussing and comparing the models. These tests indicate that the node-discretized formulations are more efficient for solving large size sparse graphs. Additionally, we also address a new variant of the MEWC problem where the objective to be maximized includes the neighboring edges of the clique.

A minimum resource neural network framework for solving multiconstraint shortest path problems.

Characterized by using minimum hard (structural) and soft (computational) resources, a novel parameter-free minimal resource neural network (MRNN) framework is proposed for solving a wide range of single-source shortest path (SP) problems for various graph types. The problems are the k-shortest time path problems with any combination of three constraints: time, hop, and label constraints, and the graphs can be directed, undirected, or bidirected with symmetric and/or asymmetric traversal time, which can be real and time dependent. Isomorphic to the graph where the SP is to be sought, the network is activated by generating autowave at source neuron and the autowave travels automatically along the paths with the speed of a hop in an iteration. Properties of the network are studied, algorithms are presented, and computation complexity is analyzed. The framework guarantees globally optimal solutions of a series of problems during the iteration process of the network, which provides insight into why even the SP is still too long to be satisfied. The network facilitates very large scale integrated circuit implementation and adapt to very large scale problems due to its massively parallel processing and minimum resource utilization. When implemented in a sequentially processing computer, experiments on synthetic graphs, road maps of cities of the USA, and vehicle routing with time windows indicate that the MRNN is especially efficient for large scale sparse graphs and even dense graphs with some constraints, e.g., the CPU time taken and the iteration number used for the road maps of cities of the USA is even less than ∼2% and 0.5% that of the Dijkstra's algorithm.

Sampling for Conditional Inference on Network Data

Random graphs with given vertex degrees have been widely used as a model for many real-world complex networks. However, both statistical inference and analytic study of such networks present great challenges. In this article, we propose a new sequential importance sampling method for sampling networks with a given degree sequence. These samples can be used to approximate closely the null distributions of a number of test statistics involved in such networks and provide an accurate estimate of the total number of networks with given vertex degrees. We study the asymptotic behavior of the proposed algorithm and prove that the importance weight remains bounded as the size of the graph grows. This property guarantees that the proposed sampling algorithm can still work efficiently even for large sparse graphs. We apply our method to a range of examples to demonstrate its efficiency in real problems.

Listing All Maximal Cliques in Large Sparse Real-World Graphs

We implement a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Loffler, and Strash (ISAAC 2010) and analyze its performance on a large corpus of real-world graphs. Our analysis shows that this algorithm is the first to offer a practical solution to listing all maximal cliques in large sparse graphs. All other theoretically-fast algorithms for sparse graphs have been shown to be significantly slower than the algorithm of Tomita et al. (Theoretical Computer Science, 2006) in practice. However, the algorithm of Tomita et al. uses an adjacency matrix, which requires too much space for large sparse graphs. Our new algorithm opens the door for fast analysis of large sparse graphs whose adjacency matrix will not fit into working memory.

Mean first-passage time for random walks in general graphs with a deep trap

We provide an explicit formula for the global mean first-passage time (GMFPT) for random walks in a general graph with a perfect trap fixed at an arbitrary node, where GMFPT is the average of mean first-passage time to the trap over all starting nodes in the whole graph. The formula is expressed in terms of eigenvalues and eigenvectors of Laplacian matrix for the graph. We then use the formula to deduce a tight lower bound for the GMFPT in terms of only the numbers of nodes and edges, as well as the degree of the trap, which can be achieved in both complete graphs and star graphs. We show that for a large sparse graph, the leading scaling for this lower bound is proportional to the system size and the reciprocal of the degree for the trap node. Particularly, we demonstrate that for a scale-free graph of size N with a degree distribution P(d) ∼ d(-γ) characterized by γ, when the trap is placed on a most connected node, the dominating scaling of the lower bound becomes N(1-1∕γ), which can be reached in some scale-free graphs. Finally, we prove that the leading behavior of upper bounds for GMFPT on any graph is at most N(3) that can be reached in the bar-bell graphs. This work provides a comprehensive understanding of previous results about trapping in various special graphs with a trap located at a specific location.

Moment-Based Estimation of Stochastic Kronecker Graph Parameters

Stochastic Kronecker graphs supply a parsimonious model for large sparse real-world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have, however, proved difficult to choose in specific applications. This article looks at method-of-moments estimators that are computationally much simpler than maximum likelihood. The estimators are fast, and in our examples, they typically yield Kronecker parameters with expected feature counts closer to a given graph than we get from KronFit. The improvement is especially prominent for the number of triangles in the graph.

Network Similarity Decomposition (NSD): A Fast and Scalable Approach to Network Alignment

As graph-structured data sets become commonplace, there is increasing need for efficient ways of analyzing such data sets. These analyses include conservation, alignment, differentiation, and discrimination, among others. When defined on general graphs, these problems are considerably harder than their well-studied counterparts on sets and sequences. In this paper, we study the problem of global alignment of large sparse graphs. Specifically, we investigate efficient methods for computing approximations to the state-of-the-art IsoRank solution for finding pairwise topological similarity between nodes in two networks (or within the same network). Pairs of nodes with high similarity can be used to seed global alignments. We present a novel approach to this computationally expensive problem based on uncoupling and decomposing ranking calculations associated with the computation of similarity scores. Uncoupling refers to independent preprocessing of each input graph. Decomposition implies that pairwise similarity scores can be explicitly broken down into contributions from different link patterns traced back to a low-rank approximation of the initial conditions for the computation. These two concepts result in significant improvements, in terms of computational cost, interpretability of similarity scores, and nature of supported queries. We show over two orders of magnitude improvement in performance over IsoRank/Random Walk formulations, and over an order of magnitude improvement over constrained matrix-triple-product formulations, in the context of real data sets.

A Space-Filling Visualization Technique for Multivariate Small-World Graphs

We introduce an information visualization technique, known as GreenCurve, for large multivariate sparse graphs that exhibit small-world properties. Our fractal-based design approach uses spatial cues to approximate the node connections and thus eliminates the links between the nodes in the visualization. The paper describes a robust algorithm to order the neighboring nodes of a large sparse graph by solving the Fiedler vector of its graph Laplacian, and then fold the graph nodes into a space-filling fractal curve based on the Fiedler vector. The result is a highly compact visualization that gives a succinct overview of the graph with guaranteed visibility of every graph node. GreenCurve is designed with the power grid infrastructure in mind. It is intended for use in conjunction with other visualization techniques to support electric power grid operations. The research and development of GreenCurve was conducted in collaboration with domain experts who understand the challenges and possibilities intrinsic to the power grid infrastructure. The paper reports a case study on applying GreenCurve to a power grid problem and presents a usability study to evaluate the design claims that we set forth.

Efficiently Indexing Large Sparse Graphs for Similarity Search

The graph structure is a very important means to model schemaless data with complicated structures, such as protein-protein interaction networks, chemical compounds, knowledge query inferring systems, and road networks. This paper focuses on the index structure for similarity search on a set of large sparse graphs and proposes an efficient indexing mechanism by introducing the Q-Gram idea. By decomposing graphs to small grams (organized by κ-Adjacent Tree patterns) and pairing-up on those κ-Adjacent Tree patterns, the lower bound estimation of their edit distance can be calculated for candidate filtering. Furthermore, we have developed a series of techniques for inverted index construction and online query processing. By building the candidate set for the query graph before the exact edit distance calculation, the number of graphs need to proceed into exact matching can be greatly reduced. Extensive experiments on real and synthetic data sets have been conducted to show the effectiveness and efficiency of the proposed indexing mechanism.

Gibbs measures and phase transitions on sparse random graphs

Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.

Ramsey goodness and beyond

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.

A linear programming approach for estimating the structure of a sparse linear genetic network from transcript profiling data

BackgroundA genetic network can be represented as a directed graph in which a node corresponds to a gene and a directed edge specifies the direction of influence of one gene on another. The reconstruction of such networks from transcript profiling data remains an important yet challenging endeavor. A transcript profile specifies the abundances of many genes in a biological sample of interest. Prevailing strategies for learning the structure of a genetic network from high-dimensional transcript profiling data assume sparsity and linearity. Many methods consider relatively small directed graphs, inferring graphs with up to a few hundred nodes. This work examines large undirected graphs representations of genetic networks, graphs with many thousands of nodes where an undirected edge between two nodes does not indicate the direction of influence, and the problem of estimating the structure of such a sparse linear genetic network (SLGN) from transcript profiling data.ResultsThe structure learning task is cast as a sparse linear regression problem which is then posed as a LASSO (l1-constrained fitting) problem and solved finally by formulating a Linear Program (LP). A bound on the Generalization Error of this approach is given in terms of the Leave-One-Out Error. The accuracy and utility of LP-SLGNs is assessed quantitatively and qualitatively using simulated and real data. The Dialogue for Reverse Engineering Assessments and Methods (DREAM) initiative provides gold standard data sets and evaluation metrics that enable and facilitate the comparison of algorithms for deducing the structure of networks. The structures of LP-SLGNs estimated from the INSILICO1, INSILICO2 and INSILICO3 simulated DREAM2 data sets are comparable to those proposed by the first and/or second ranked teams in the DREAM2 competition. The structures of LP-SLGNs estimated from two published Saccharomyces cerevisae cell cycle transcript profiling data sets capture known regulatory associations. In each S. cerevisiae LP-SLGN, the number of nodes with a particular degree follows an approximate power law suggesting that its degree distributions is similar to that observed in real-world networks. Inspection of these LP-SLGNs suggests biological hypotheses amenable to experimental verification.ConclusionA statistically robust and computationally efficient LP-based method for estimating the topology of a large sparse undirected graph from high-dimensional data yields representations of genetic networks that are biologically plausible and useful abstractions of the structures of real genetic networks. Analysis of the statistical and topological properties of learned LP-SLGNs may have practical value; for example, genes with high random walk betweenness, a measure of the centrality of a node in a graph, are good candidates for intervention studies and hence integrated computational – experimental investigations designed to infer more realistic and sophisticated probabilistic directed graphical model representations of genetic networks. The LP-based solutions of the sparse linear regression problem described here may provide a method for learning the structure of transcription factor networks from transcript profiling and transcription factor binding motif data.

Less is More: Sparse Graph Mining with Compact Matrix Decomposition

Given a large sparse graph, how can we find patterns and anomalies? Several important applications can be modeled as large sparse graphs, e.g., network traffic monitoring, research citation network...

Less is More: Sparse Graph Mining with Compact Matrix Decomposition

AbstractGiven a large sparse graph, how can we find patterns and anomalies? Several important applications can be modeled as large sparse graphs, e.g., network traffic monitoring, research citation network analysis, social network analysis, and financial transactions. Low‐rank decompositions, such as singular value decomposition (SVD) and CUR, are powerful techniques for revealing latent/hidden variables and associated patterns from high dimensional data. However, those methods often ignore the sparsity property of the graph, and hence usually incur too high memory and computational cost to be practical.We propose a novel method, the Compact Matrix Decomposition (CMD), to compute sparse low‐rank approximations. CMD dramatically reduces both the computation cost and the space requirements over existing decomposition methods singular value decomposition (SVD) and CUR. Using CMD as the key building block, we further propose procedures to efficiently construct and analyze dynamic graphs from real‐time application data. We provide theoretical guarantee for our methods, and present results on two real, large datasets, one on network flow data (100 GB trace of 22K hosts over one month) and one on DBLP (200 MB over 25 years).We show that CMD is often an order of magnitude more efficient than the state of the art (SVD and CUR): it is over 10X faster, but requires less than 1/10 of the space, for the same reconstruction accuracy. Finally, we demonstrate how CMD is used for detecting anomalies and monitoring time‐evolving graphs, in which it successfully detects worm‐like hierarchical scanning patterns in real network data. Copyright © 2007 Wiley Periodicals, Inc., A Wiley Company Statistical Analy Data Mining 1: 000‐000, 2007

Phase transitions in the coloring of random graphs

We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Rényi and regular random graphs and determine their asymptotic values for a large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

CFinder: locating cliques and overlapping modules in biological networks

Most cellular tasks are performed not by individual proteins, but by groups of functionally associated proteins, often referred to as modules. In a protein association network modules appear as groups of densely interconnected nodes, also called communities or clusters. These modules often overlap with each other and form a network of their own, in which nodes (links) represent the modules (overlaps). We introduce CFinder, a fast program locating and visualizing overlapping, densely interconnected groups of nodes in undirected graphs, and allowing the user to easily navigate between the original graph and the web of these groups. We show that in gene (protein) association networks CFinder can be used to predict the function(s) of a single protein and to discover novel modules. CFinder is also very efficient for locating the cliques of large sparse graphs. CFinder (for Windows, Linux and Macintosh) and its manual can be downloaded from http://angel.elte.hu/clustering. Supplementary data are available on Bioinformatics online.

Quick k-Median, k-Center, and Facility Location for Sparse Graphs

We present $\tilde{O}(m)$ time and space constant factor approximation algorithms for the k-median, k-center, and facility location problems with assignment costs being shortest path distances in a weighted undirected graph with n vertices and m edges. For all of these location problems, $\tilde{O}(n^2)$ time and space algorithms were already known, but here we are addressing large sparse graphs. An application could be placement of content distributing servers on the Internet. The Internet is large and changes so frequently that an $\tilde{O}(n^2)$ time solution would likely be outdated long before completion.