Random Graph Generator for Bipartite Networks Modeling

The purpose of this article is to introduce a new iterative algorithm with properties resembling real life bipartite graphs. The algorithm enables us to generate wide range of random bigraphs, which features are determined by a set of parameters.We adapt the advances of last decade in unipartite complex networks modeling to the bigraph setting. This data structure can be observed in several situations. However, only a few datasets are freely available to test the algorithms (e.g. community detection, influential nodes identification, information retrieval) which operate on such data. Therefore, artificial datasets are needed to enhance development and testing of the algorithms. We are particularly interested in applying the generator to the analysis of recommender systems. Therefore, we focus on two characteristics that, besides simple statistics, are in our opinion responsible for the performance of neighborhood based collaborative filtering algorithms. The features are node degree distribution and local clustering coeficient

Kronecker Graphs: An Approach to Modeling Networks

How can we model networks with a mathematically tractable model that allows for rigorous analysis of network properties? Networks exhibit a long list of surprising properties: heavy tails for the degree distribution; small diameters; and densification and shrinking diameters over time. Most present network models either fail to match several of the above properties, are complicated to analyze mathematically, or both. In this paper we propose a generative model for networks that is both mathematically tractable and can generate networks that have the above mentioned properties. Our main idea is to use the Kronecker product to generate graphs that we refer to as "Kronecker graphs". First, we prove that Kronecker graphs naturally obey common network properties. We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. We then present KronFit, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks. A naive approach to fitting would take super- exponential time. In contrast, KronFit takes linear time, by exploiting the structure of Kronecker matrix multiplication and by using statistical simulation techniques. Experiments on large real and synthetic networks show that KronFit finds accurate parameters that indeed very well mimic the properties of target networks. Once fitted, the model parameters can be used to gain insights about the network structure, and the resulting synthetic graphs can be used for null- models, anonymization, extrapolations, and graph summarization

Mixing Properties of CSMA Networks on Partite Graphs

We consider a stylized stochastic model for a wireless CSMA network. Experimental results in prior studies indicate that the model provides remarkably accurate throughput estimates for IEEE 802.11 systems. In particular, the model offers an explanation for the severe spatial unfairness in throughputs observed in such networks with asymmetric interference conditions. Even in symmetric scenarios, however, it may take a long time for the activity process to move between dominant states, giving rise to potential starvation issues. In order to gain insight in the transient throughput characteristics and associated starvation effects, we examine in the present paper the behavior of the transition time between dominant activity states. We focus on partite interference graphs, and establish how the magnitude of the transition time scales with the activation rate and the sizes of the various network components. We also prove that in several cases the scaled transition time has an asymptotically exponential distribution as the activation rate grows large, and point out interesting connections with related exponentiality results for rare events and meta-stability phenomena in statistical physics. In addition, we investigate the convergence rate to equilibrium of the activity process in terms of mixing times.Comment: Valuetools, 6th International Conference on Performance Evaluation Methodologies and Tools, October 9-12, 2012, Carg\`ese, Franc

Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite, and related graphs

We show that the infinitesimal generator of the symmetric simple exclusion process, recast as a quantum spin-1/2 ferromagnetic Heisenberg model, can be solved by elementary techniques on the complete, complete bipartite, and related multipartite graphs. Some of the resulting infinitesimal generators are formally identical to homogeneous as well as mixed higher spins models. The degeneracies of the eigenspectra are described in detail, and the Clebsch-Gordan machinery needed to deal with arbitrary spin-s representations of the SU(2) is briefly developed. We mention in passing how our results fit within the related questions of a ferromagnetic ordering of energy levels and a conjecture according to which the spectral gaps of the random walk and the interchange process on finite simple graphs must be equal.Comment: Final version as published, 19 pages, 4 figures, 40 references given in full forma