General Random Graphs Research Articles

A Modular Approach to Random Task Graph Generation

Validation of the robustness, efficiency of allocation and scheduling heuristics in large scale parallel and distributed systems is usually done using synthetic randomly generated workloads, represented by task graphs. We provide a modular approach to the problem of generating random Directed Acyclic Graphs(DAGs), called Modular Random Task Graph generator (MRTG), making it very flexible for the researchers to use. Modular open source approach provides a great advantage for future development as more modules can be added without disturbing the existing stable software. The task nodes are placement randomly using a layer-by-layer approach and then connected randomly. Paramount importance has been given to user-controlled randomness in developing this algorithm. The MRTG can generate task sets containing several different types of task graphs like rooted trees, isomorphic graphs and similar graphs with same node placement but different connections, with the flexibility to dictate the type of graph generated. In this paper, we also present a comparison of MRTG with existing solutions to the random task graph generation problem.

Scale-Free Property for Degrees and Weights in an N-Interactions Random Graph Model*

A general random graph evolution mechanism is defined. The evolution is based on the interactions of N vertices. Besides the interactions of the new vertex and the old ones, interactions among old vertices are also allowed. Moreover, both preferential attachment and uniform choice are possible. A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of interactions of the vertex. The asymptotic behavior of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2,∞) can be achieved. The proofs are based on discrete time martingale theory.

Annealed central limit theorems for the Ising model on random graphs

The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by √N of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature 0≤ βan n 0 and B ≠ 0. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters β and B, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.

Generator of 2D Geometric Constraint Graphs

ABSTRACTIn 2-dimensional geometric constraint solving, graph-based techniques are a dominant approach, particularly in CAD context. These methods transform the geometric problem into a graph which is decomposed into small sub-graphs. Each one is solved, separately, and the final solution is obtained by recomposing the solved sub-graphs. To the best of our knowledge, there is no random geometric constraint graph generator so far. In this paper, we introduce a simple, but efficient generator that produces any possible geometric configuration. It would be parameterized to generate graphs with some desirable proprieties, like highly or weakly decomposable graphs, or restricting the generated graph to a specific class of geometric configuration. Generated graphs can be used as a benchmark to make consistent tests, or to observe algorithm behaviour on the geometric constraint graphs with different sizes and structural properties. We prove that our generator is complete and suitable for two main classes of solvi...

Generating Realistic Labelled, Weighted Random Graphs

Generative algorithms for random graphs have yielded insights into the structure and evolution of real-world networks. Most networks exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Usually, random graph models consider only structural information, but many real-world networks also have labelled vertices and weighted edges. In this paper, we present a generative model for random graphs with discrete vertex labels and numeric edge weights. The weights are represented as a set of Beta Mixture Models (BMMs) with an arbitrary number of mixtures, which are learned from real-world networks. We propose a Bayesian Variational Inference (VI) approach, which yields an accurate estimation while keeping computation times tractable. We compare our approach to state-of-the-art random labelled graph generators and an earlier approach based on Gaussian Mixture Models (GMMs). Our results allow us to draw conclusions about the contribution of vertex labels and edge weights to graph structure.

General and exact approach to percolation on random graphs

We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite-size quenched or random multitype graphs. (ii) We define a very general random graph ensemble that encompasses most of the models published to this day and also makes it possible to model structural properties not yet included in a theoretical framework. Site and bond percolation on this ensemble is solved exactly in the infinite-size limit using probability generating functions [i.e., the percolation threshold, the size, and the composition of the giant (extensive) and small components]. Several examples and applications are also provided. (iii) Our approach can be adapted to model interdependent graphs-whose most striking feature is the emergence of an extensive component via a discontinuous phase transition-in an equally general fashion. We show how a graph can successively undergo a continuous then a discontinuous phase transition, and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition.

A Novel Simulation Method for Binary Discrete Exponential Families, With Application to Social Networks

Stochastic models for finite binary vectors are widely used in sociology, with examples ranging from social influence models on dichotomous behaviors or attitudes to models for random graphs. Exact sampling for such models is difficult in the presence of dependence, leading to the use of Markov chain Monte Carlo (MCMC) as an approximation technique. While often effective, MCMC methods have variable execution time, and the quality of the resulting draws can be difficult to assess. Here, we present a novel alternative method for approximate sampling from binary discrete exponential families having fixed execution time and well-defined quality guarantees. We demonstrate the use of this sampling procedure in the context of random graph generation, with an application to the simulation of a large-scale social network using both geographical covariates and dyadic dependence mechanisms.

Quenched Central Limit Theorems for the Ising Model on Random Graphs

The main goal of the paper is to prove central limit theorems for the magnetization rescaled by $$\sqrt{N}$$ for the Ising model on random graphs with N vertices. Both random quenched and averaged quenched measures are considered. We work in the uniqueness regime $$\beta >\beta _c$$ or $$\beta >0$$ and $$B\ne 0$$ , where $$\beta $$ is the inverse temperature, $$\beta _c$$ is the critical inverse temperature and B is the external magnetic field. In the random quenched setting our results apply to general tree-like random graphs (as introduced by Dembo, Montanari and further studied by Dommers and the first and third author) and our proof follows that of Ellis in $$\mathbb {Z}^d$$ . For the averaged quenched setting, we specialize to two particular random graph models, namely the 2-regular configuration model and the configuration model with degrees 1 and 2. In these cases our proofs are based on explicit computations relying on the solution of the one dimensional Ising models.

Sensible method for updating motif instances in an increased biological network

A network motif is defined as an over-represented subgraph pattern in a network. Network motif based techniques have been widely applied in analyses of biological networks such as transcription regulation networks (TRNs), protein–protein interaction networks (PPIs), and metabolic networks. The detection of network motifs involves the computationally expensive enumeration of subgraphs, NP-complete graph isomorphism testing, and significance testing through the generation of many random graphs to determine the statistical uniqueness of a given subgraph. These computational obstacles make network motif analysis unfeasible for many real-world applications. We observe that the fast growth of biotechnology has led to the rapid accretion of molecules (vertices) and interactions (edges) to existing biological network databases. Even with a small percentage of additions, revised networks can have a large number of differing motif instances. Currently, no existing algorithms recalculate motif instances in ‘updated’ networks in a practical manner. In this paper, we introduce a sensible method for efficiently recalculating motif instances by performing motif enumeration from only updated vertices and edges. Preliminary experimental results indicate that our method greatly reduces computational time by eliminating the repeated enumeration of overlapped subgraph instances detected in earlier versions of the network. The software program implementing this algorithm, defined as SUNMI (Sensible Update of Network Motif Instances), is currently a stand-alone java program and we plan to upgrade it as a web-interactive program that will be available through http://faculty.washington.edu/kimw6/research.htm in near future. Meanwhile it is recommended to contact authors to obtain the stand-alone SUNMI program.

The front of the epidemic spread and first passage percolation

We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments as n → ∞, with an appropriate X2log+X condition. We also study the epidemic trail between the source and typical vertices in the graph.

The front of the epidemic spread and first passage percolation

We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments asn→ ∞, with an appropriateX2log+Xcondition. We also study the epidemic trail between the source and typical vertices in the graph.

An Analytical Comparison of Social Network Measures

Network science spans many different fields of study, ranging from psychology to biology to the social sciences. A number of descriptive network measures have been identified for use within these fields; however, little research examines the relationships of these measures for possible statistical dependence. The research presented in this paper uses Spearman's rank correlation coefficient to examine the statistical dependence between pairs of 24 widely accepted social network measures. Confidence intervals are compared to determine whether computation times between measures in the same correlation group are significantly different. We use a three-factor, four-level, full-factorial experimental design to construct a test set of 64 unique network topologies. The three factors of interest are the network structural properties of size, cluster ability, and the scale-free parameter. A set of 320 networks are generated from a power law degree distribution using a random graph generation algorithm. Results indicate that there exists high correlation among 14 of the 24 tested network measures, many of which also exhibit statistically significant differences with respect to computation time. These findings are of interest to analysts seeking to identify measures that provide similar ranked outcomes and where computational efficiency is an important consideration.

Graph construction and random graph generation for modeling protein structures

AbstractResearchers often model folded protein structures as graphs with amino acids as the vertices and edges representing contacts between amino acids. The vertices in these graphs are naturally ordered in the amino acid sequence order. There are many different graph construction methods and there is no consensus about what construction to use or what the major issues are with each construction in the literature. We investigate different constructions and examine their effect on various graph measures. We also consider the small‐world network model for proteins, discuss its validity under the different constructions, and discuss random protein graph generation. We propose a new graph property for graphs with ordered vertices, the contact distribution, and propose a method of reciprocal attachment to merge neighborhoods for protein graphs. Statistical Analysis and Data Mining Statistical Analysis and Data Mining 2013 DOI: 10.1002/sam.11203

Bayesian variable selection for correlated covariates via colored cliques

We propose a Bayesian method to select groups of correlated explanatory variables in a linear regression framework. We do this by introducing in the prior distribution assigned to the regression coefficients a random matrix \(G\) that encodes the group structure. The groups can thus be inferred by sampling from the posterior distribution of \(G\). We then give a graph-theoretic interpretation of this random matrix \(G\) as the adjacency matrix of cliques. We discuss the extension of the groups from cliques to more general random graphs, so that the proposed approach can be viewed as a method to find networks of correlated covariates that are associated with the response.

Generating synthetic task graphs for simulating stream computing systems

Stream-computing is an emerging computational model for performing complex operations on and across multi-source, high-volume data flows. The pool of mature publicly available applications employing this model is fairly small, and therefore the availability of workloads for various types of applications is scarce. Thus, there is a need for synthetic generation of large-scale workloads to drive simulations and estimate the performance of stream-computing applications at scale. We identify the key properties shared by most task graphs of stream-computing applications and use them to extend known random graph generation concepts with stream computing specific features, providing researchers with realistic input stream graphs. Our graph generation techniques serve the purpose of covering a disparity of potential applications and user input. Our first “domain-specific” framework exhibits high user-controlled configurability while the second “application-agnostic” framework focuses solely on emulating the key properties of general stream-computing systems, at the loss of domain-specific fine-tuning.

A random graph generation algorithm for the analysis of social networks

Social network analysis (SNA) is a rapidly growing field with numerous applications in industry and government. However, the field still lacks means to generate random social networks with certain desired properties, thus inhibiting their ability to test new SNA algorithms and metrics. Available random graph generation algorithms suffer from tendencies to generate disconnected graphs and sometimes induce undesirable network properties. In this paper, we present an algorithm, the prescribed node degree, connected graph (PNDCG) algorithm, designed to generate weakly connected social networks. Extensions to the PNDCG algorithm allow one to create random graphs that control the clustering coefficient and degree correlation within the generated networks. Empirical test results demonstrate the capability of the PNDCG algorithm to produce networks with the desired properties.

An effective and efficient parallel approach for random graph generation over GPUs

The widespread usage of random graphs has been highlighted in the context of database applications for several years. This because such data structures turn out to be very useful in a large family of database applications ranging from simulation to sampling, from analysis of complex networks to study of randomized algorithms, and so forth. Amongst others, Erdős–Rényi Γv,p is the most popular model to obtain and manipulate random graphs. Unfortunately, it has been demonstrated that classical algorithms for generating Erdős–Rényi based random graphs do not scale well in large instances and, in addition to this, fail to make use of the parallel processing capabilities of modern hardware. Inspired by this main motivation, in this paper we propose and experimentally assess a novel parallel algorithm for generating random graphs under the Erdős–Rényi model that is designed and implemented in a Graphics Processing Unit (GPU), called PPreZER. We demonstrate the nice amenities due to our solution via a succession of several intermediary algorithms, both sequential and parallel, which show the limitations of classical approaches and the benefits due to the PPreZER algorithm. Finally, our comprehensive experimental assessment and analysis brings to light a relevant average speedup gain of PPreZER over baseline algorithms.

An Efficient behavioural analysis of Graph Clustering Algorithms via Random Graphs

proposed last research entitled An Effective Data Comparison of Graph Clustering Algorithms via Random Graphs compared two mostly used algorithms for graph clustering i.e. restricted neighborhood search and markov clustering algorithms via random graph generators i.e. Erdos- Renyi and power law graphs. This paper is an extension to our last research work. In this we have examined an efficient behavioral analysis of both algorithms via random graphs. This paper mainly shows the behavior of both the algorithms under certain parameters which we have used. Previously in case of Erdos-renyii we used graphs with 1000 nodes with variable edge densities, while in this paper we have modified the number of nodes from 1000 to 15000 with variable edge densities ranging from 0.1 to 0.5 while in case of Power-law we have variable number of nodes ranging from 1000 to 15000. This paper also depicts as to which algorithm works more efficient, whether in case of Erdos-Renyi or Power-Law graphs. Our last research showed that in case of Erdos-Renyi graph run time of RNSC algorithm is better as compared to MCL for graph having nodes less than 2000 but as nodes keep on increasing the run time of RNSC increases drastically while run time of MCL doesn't increase, so MCL is better in case of Erdos-Renyi graph having more than 2000 nodes and having high connectivity between the nodes (12) while in this paper it is clearly visible that both RNSC and MCL works better in case of Power-Law graph as compared to Erdos- Renyi graph which clearly states that both algorithm shows some similar characteristics in graphs where edge connectedness is not very high for all vertices. Furthermore we studied the behaviour of RNSC in case of Erdos-Renyi individually also.

Propagation of trust and distrust for the detection of trolls in a social network

Trust and Reputation Systems constitute an essential part of many social networks due to the great expansion of these on-line communities in the past few years. As a consequence of this growth, some users try to disturb the normal atmosphere of these communities, or even to take advantage of them in order to obtain some kind of benefits. Therefore, the concept of trust is a key point in the performance of on-line systems such as on-line marketplaces, review aggregators, social news sites, and forums. In this work we propose a method to compute a ranking of the users in a social network, regarding their trustworthiness. The aim of our method is to prevent malicious users from illicitly gaining high reputation in the network by demoting them in the ranking of users. We propose a novel system intended to propagate both positive and negative opinions of the users through a network, in such way that the opinions from each user about others influence their global trust score. Our proposal has been evaluated in different challenging situations. The experiments include the generation of random graphs, the use of a real-world dataset extracted from a social news site, and a combination of both a real dataset and generation techniques, in order to test our proposals in different environments. The results show that our method performs well in every situations, showing the propagation of trust and distrust to be a reliable mechanism in a Trust and Reputation System.

Topological pattern selection in recurrent networks

The impact of adding correlation to a population of neurons on the information and the activity of the population is one of the fundamental questions in recent system neuroscience. In this paper, we would like to introduce topology-based correlation at the level of storing patterns in a recurrent network. We then study the effects of topological patterns on the activity and memory capacity of the network. The general aim of the present work is to show how the repertoire of possible stored patterns is determined by the underlying network topology. Two topological probability rules for pattern selection in recurrent network are introduced. The first one selects patterns according to a Gibbs-type distribution. We start with a Hopfield-type dynamics on a ring model and then a Langevin model on a general random graph is treated. The phenomenon of phase transition in pattern selection motivated us to introduce an alternative topological rule for pattern selection. In a network of N neurons on a random d-regular graph, two asymptotic cases, d/N→0 and d/N→1, have been discussed for the new rule, and it is shown that capacity of the network grows considerably as d/N→1. By introducing the notions of asymptotic eigenvector, we will be able to study the behaviour of the discrete model in the limit d/N→0. It will be proved that for degree d less than a critical value there is a positive role for noise, in which case increasing the number of patterns will improve the storage capacity.