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.
Deep Dive into 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 l
What do real graphs look like? How do they evolve over time? How can we generate synthetic, but realistic looking, time-evolving graphs? Recently, network analysis has been attracting much interest, with an emphasis on finding patterns and abnormalities in social networks, computer networks, e-mail interactions, gene regulatory networks, and many more. Most of the work focuses on static snapshots of graphs, where fascinating "laws" have been discovered, including small diameters and heavy-tailed degree distributions.
In parallel with discoveries of such structural “laws” there has been effort to find mechanisms and models of network formation that generate networks with such structures. So, a good realistic network generation model is important for at least two reasons. The first is that it can generate graphs for extrapolations, hypothesis testing, “what-if” scenarios, and simulations, when real graphs are difficult or impossible to collect. For example, how well will a given protocol run on the Internet five years from now? Accurate network models can produce more realistic models for the future Internet, on which simulations can be run. The second reason is more subtle. It forces us to think about network properties that generative models should obey to be realistic.
In this paper we introduce Kronecker graphs, a generative network model which obeys all the main static network patterns that have appeared in the literature (Faloutsos et al., 1999;Albert et al., 1999;Chakrabarti et al., 2004;Farkas et al., 2001;Mihail and Papadimitriou, 2002;Watts and Strogatz, 1998). Our model also obeys recently discovered temporal evolution patterns (Leskovec et al., 2005b(Leskovec et al., , 2007a)). And, contrary to other models that match this combination of network properties (as for example, (Bu and Towsley, 2002;Klemm and Eguíluz, 2002;Vázquez, 2003;Leskovec et al., 2005b;Zheleva et al., 2009)), Kronecker graphs also lead to tractable analysis and rigorous proofs. Furthermore, the Kronecker graphs generative process also has a nice natural interpretation and justification.
Our model is based on a matrix operation, the Kronecker product. There are several known theorems on Kronecker products. They correspond exactly to a significant portion of what we want to prove: heavy-tailed distributions for in-degree, out-degree, eigenvalues, and eigenvectors. We also demonstrate how a Kronecker graphs can match the behavior of several real networks (social networks, citations, web, internet, and others). While Kronecker products have been studied by the algebraic combinatorics community (see, e.g., (Chow, 1997;Imrich, 1998;Imrich and Klavžar, 2000;Hammack, 2009)), the present work is the first to employ this operation in the design of network models to match real data.
Then we also make a step further and tackle the following problem: Given a large real network, we want to generate a synthetic graph, so that the resulting synthetic graph matches the properties of the real network as well as possible.
Ideally we would like: (a) A graph generation model that naturally produces networks where many properties that are also found in real networks naturally emerge. (b) The model parameter estimation should be fast and scalable, so that we can handle networks with millions of nodes. (c) The resulting set of parameters should generate realistic-looking networks that match the statistical properties of the target, real networks.
In general the problem of modeling network structure presents several conceptual and engineering challenges: Which generative model should we choose, among the many in the literature? How do we measure the goodness of the fit? (Least squares don’t work well for power laws, for subtle reasons!) If we use likelihood, how do we estimate it faster than in time quadratic on the number of nodes? How do we solve the node correspondence problem, i.e., which node of the real network corresponds to what node of the synthetic one?
To answer the above questions we present KRONFIT, a fast and scalable algorithm for fitting Kronecker graphs by using the maximum likelihood principle. When calculating the likelihood there are two challenges: First, one needs to solve the node correspondence problem by matching the nodes of the real and the synthetic network. Essentially, one has to consider all mappings of nodes of the network to the rows and columns of the graph adjacency matrix. This becomes intractable for graphs with more than tens of nodes. Even when given the “true” node correspondences, just evaluating the likelihood is still prohibitively expensive for large graphs that we consider, as one needs to evaluate the probability of each possible edge. We present solutions to both of these problems: We develop a Metropolis sampling algorithm for sampling node correspondences, and approximate the likelihood to obtain a linear time algorithm for Kronecker graph model parameter estimation that scales to large networks wi
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
