Rigorous Analysis of Kronecker Graphs and their Algorithms

Real world graphs have been observed to display a number of surprising properties. These properties include heavy-tails for inand out-degree distributions, small diameters, and a densification law [5]. These features do not arise from the classical Erdos-Renyi random graph model [1]. To address these difficulties, Kronecker Graphs were first introduced in [5] as a new method of generating graphs which match network data. In this literature survey, we review two works on Kronecker Graphs [8], [6] which respectively study the theoretical properties of Kronecker Graphs and introduce new learning algorithms to fit Kronecker Graphs to real Network Data. We also review the classical work by Erdos and Renyi [1] to gain inspiration from the techniques of the old masters. The first paper we consider is Kronecker Graphs: An Approach to modeling networks [6]. This paper reviews the use of Kronecker graphs as useful tools for modeling networks (the results covered were first introduced in earlier works such as [5]). Kronecker graphs provide methods of “multiplying” two graphs together via the tensor (kronecker) product on the adjacency matrix. The paper also covers the formalism of Stochastic Kronecker graphs. Such graphs are constructed not from adjacency matrices, but rather from kronecker products of probability matrices, where the (i, j)-th entry is the probability of an edge between i and j existing. The probability matrix associated with a Kronecker graph is also called its initiator matrix. Note that each such probability matrix defines a distribution over graphs and not a single graph. A fast approximate algorithm for generating Kronecker Graphs is presented. The theoretical guarantees of this algorithm are not studied. This work also considers the problem of modeling real-world network data with Kronecker graphs. It presents a MLE algorithm KronFit that fits an initiator matrix to match a Kronecker graph to a real world network. An interesting aspect of this algorithm is that it deals inference upon an enormous, combinatorial space (the generated Kronecker graph). However, by clever exploitation of the recursive structure of the graph generation, efficient algorithms are achieved. One interesting challenge is that the likelihood in this space cannot be evaluated explicitly (for doing so would require a summation over an exponentially large set), but is rather evaluated through clever use of Markov Chain Monte Carlo methods and approximate gradient descent. A Markov chain is used to sample random permutations which can permute the elements of a proposed Kronecker Graph. These permutations are necessary since graphs which only differ in the labelling of their vertices should not be considered as different. The convergence of this chain is mainly evaluated through empirical methods, so some interesting work might involve studying the mixing time of this Markov Chain using more analytical methods. The evaluation of likelihood functions is sped up through use of a Taylor Approximation. Explicit error bounds are not presented for this construction. The size of the initiator matrix for the fitted Kronecker graph is chosen through use of the Bayesian Information Criterion.