Stochastic Kronecker Graph Research Articles

Scalable parallel simulation of dynamical processes on large stochastic Kronecker graphs

Complex networks are widely recognized today as a unified framework to model the dynamical processes in socio-technical systems at the level of interacting elements. A stochastic Kronecker graph (SKG) is a network generative model that allows reproducing real-world networks while keeping their important topological properties. When sizes of SKGs reach dozens of millions of nodes, there is a need to apply parallel computations to simulate processes on networks stored in a distributed manner. In general, parallel simulation of a dynamical process on a complex network implies all-to-all communication between subnetworks at each iteration. In this paper, we study the efficiency of different SKG partitioning algorithms and different data interchange algorithms for dynamical process simulation on large SKGs. We compare the theoretical efficiency given by parallel performance models with experimental results for different communication patterns. An experimental part of the study was carried out for sparse SKGs with a size up to one billion nodes using Lomonosov supercomputer (Moscow State University, Russian Federation). The results show that: (i) proposed algorithm of SKG partitioning provides highly balanced results, (ii) observed parallel performance is well agreed with presented theoretical models, (iii) the scheme with all-to-all-communications between subnetworks is the most efficient up to approximately one hundred cores, (iv) master–slave scheme with a single master per node outperforms all-to-all scheme for a large size of a communicator (for our experiments, it has achieved near-linear speedup for up to several hundred processes).

On Matchings in Stochastic Kronecker Graphs

The stochastic Kronecker graph is a random structure whose vertex set is a hypercube and the probability of an edge depends on the structure of its ends. We prove that a.a.s. as soon as the stochastic Kronecker graph becomes connected it contains a perfect matching.

A Parallel Algorithm for Modeling of Dynamical Processes on Large Stochastic Kronecker Graphs

A stochastic Kronecker graph (SKG) is one of the most widely used approaches to model complex networks due to the simplicity of generative procedure and the ability to reproduce the properties of real graphs. In this paper, we present a novel parallel algorithm for modeling dynamical processes on large Poisson SKGs (PSKGs). Proposed algorithm combines the modeling of spreading of a process with the generation of vertices and edges which are involved in it. An experimental study of an algorithm was carried out for different initiator matrices and graphs with the widely varying (from several million to one billion) number of vertices using the supercomputer from Lobachevsky State University of Nizhni Novgorod, Russia. The results confirmed the applicability of an algorithm for mid-size HPC clusters with the efficiency varying on average from 0.2 to 0.7.

Properties of stochastic Kronecker graphs

The stochastic Kronecker graph model introduced by Leskovec et al. is a random graph with vertex set Z n, where two vertices u and v are connected with probability � u·v(1 u)·(1 v) � n u·v (1 u)·(1 v) independently of the presence or absence of any other edge, for fixed parameters 0 < �,�, < 1. They have shown empirically that the degree sequence resembles a power law degree distribution. In this paper we show that the stochastic Kronecker graph a.a.s. does not feature a power law degree distribution for any parameters 0 < �,�, < 1. In addition, we analyze the number of subgraphs present in the stochastic Kronecker graph and study the typical neighborhood of any given vertex.

Robustness of Centrality Measures: The Stochastic Kronecker Graph Simulation Approach

This paper tests the robustness of four centrality measures – degree, betweenness, closeness, and eignvector centrality – using graphs simulating real world networks by Stochastic Kronecker Graphs approach. Existing studies have used Erdos-Renyi (1959) random graphs which are poor match for real world networks. Stochastic Kronecker Graphs have been found to display real world network properties. Results suggest revision of some conclusions arrived at using random graphs. Mistakes in recording nodes affect accuracy of centrality measures more than mistakes in recording connections (edges). This supports conventional wisdom, and seeks revision of Borgatti et al. (2006). Supporting some existing studies, betweenness was found to be most robust of these four centrality measures, and closeness the least robust. This paper also demonstrates how confidence intervals around centrality measures can be created using simulation to help arrive at robust analysis.

The spectra of multiplicative attribute graphs

A multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet Γ, and the probability of adjacency is a symmetric function Γt×Γt→[0,1]. These graphs are a generalization of stochastic Kronecker graphs, and both classes have been shown to exhibit several useful real world properties. We establish asymptotic bounds on the spectra of the adjacency matrix and the normalized Laplacian matrix for these two families of graphs under certain mild conditions. As an application we examine various properties of the stochastic Kronecker graph and the multiplicative attribute graph, including the diameter, clustering coefficient, chromatic number, and bounds on low-congestion routing.

An in-depth analysis of stochastic Kronecker graphs

Graph analysis is playing an increasingly important role in science and industry. Due to numerous limitations in sharing real-world graphs, models for generating massive graphs are critical for developing better algorithms. In this article, we analyze the stochastic Kronecker graph model (SKG), which is the foundation of the Graph500 supercomputer benchmark due to its favorable properties and easy parallelization. Our goal is to provide a deeper understanding of the parameters and properties of this model so that its functionality as a benchmark is increased. We develop a rigorous mathematical analysis that shows this model cannot generate a power-law distribution or even a lognormal distribution. However, we formalize an enhanced version of the SKG model that uses random noise for smoothing. We prove both in theory and in practice that this enhancement leads to a lognormal distribution. Additionally, we provide a precise analysis of isolated vertices, showing that the graphs that are produced by SKG might be quite different than intended. For example, between 50% and 75% of the vertices in the Graph500 benchmarks will be isolated. Finally, we show that this model tends to produce extremely small core numbers (compared to most social networks and other real graphs) for common parameter choices.

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.

Distance-Dependent Kronecker Graphs for Modeling Social Networks

This paper focuses on a generalization of stochastic Kronecker graphs, introducing a Kronecker-like operator and defining a family of generator matrices H dependent on distances between nodes in a specified graph embedding. We prove that any lattice-based network model with sufficiently small distance-dependent connection probability will have a Poisson degree distribution and provide a general framework to prove searchability for such a network. Using this framework, we focus on a specific example of an expanding hypercube and discuss the similarities and differences of such a model with recently proposed network models based on a hidden metric space. We also prove that a greedy forwarding algorithm can find very short paths of length O((log log n) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) on the hypercube with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> nodes, demonstrating that distance-dependent Kronecker graphs can generate searchable network models.

Connectivity and giant component of stochastic Kronecker graphs

Stochastic Kronecker graphs are a model for complex networks where each edge is present independently according the Kronecker (tensor) product of a fixed matrix k-by-k matrix P with entries in [0,1]. We develop a novel correspondence between the adjacencies in a general stochastic Kronecker graph and the action of a fixed Markov chain. Using this correspondence we are able to generalize the arguments of Horn and Radcliffe on the emergence of the giant component from the case where k = 2 to arbitrary k. We are also able to use this correspondence to completely analyze the connectivity of a general stochastic Kronecker graph.