Learning multifractal structure in large networks

Generating random graphs to model networks has a rich history. In this paper, we analyze and improve upon the multifractal network generator (MFNG) introduced by Palla et al. We provide a new result on the probability of subgraphs existing in graphs generated with MFNG. From this result it follows that we can quickly compute moments of an important set of graph properties, such as the expected number of edges, stars, and cliques. Specifically, we show how to compute these moments in time complexity independent of the size of the graph and the number of recursive levels in the generative model. We leverage this theory to a new method of moments algorithm for fitting large networks to MFNG. Empirically, this new approach effectively simulates properties of several social and information networks. In terms of matching subgraph counts, our method outperforms similar algorithms used with the Stochastic Kronecker Graph model. Furthermore, we present a fast approximation algorithm to generate graph instances following the multi- fractal structure. The approximation scheme is an improvement over previous methods, which ran in time complexity quadratic in the number of vertices. Combined, our method of moments and fast sampling scheme provide the first scalable framework for effectively modeling large networks with MFNG

Learning mixed kronecker product graph models with simulated method of moments

ABSTRACT There has recently been a great deal of work focused on developing statistical models of graph structure-with the goal of modeling probability distributions over graphs from which new, similar graphs can be generated by sampling from the estimated distributions. Although current graph models can capture several important characteristics of social network graphs (e.g., degree, path lengths), many of them do not generate graphs with sufficient variation to reflect the natural variability in real world graph domains. One exception is the mixed Kronecker Product Graph Model (mKPGM), a generalization of the Kronecker Product Graph Model, which uses parameter tying to capture variance in the underlying distribution In this work, we present the first learning algorithm for mKPGMs. The O(|E|) algorithm searches over the continuous parameter space using constrained line search and is based on simulated method of moments, where the objective function minimizes the distance between the observed moments in the training graph and the empirically estimated moments of the model. We evaluate the mKPGM learning algorithm by comparing it to several different graph models, including KPGMs. We use multi-dimensional KS distance to compare the generated graphs to the observed graphs and the results show mKPGMs are able to produce a closer match to real-world graphs (10-90% reduction in KS distance), while still providing natural variation in the generated graphs

Learning mixed kronecker product graph models with simulated method of moments

ABSTRACT There has recently been a great deal of work focused on developing statistical models of graph structure-with the goal of modeling probability distributions over graphs from which new, similar graphs can be generated by sampling from the estimated distributions. Although current graph models can capture several important characteristics of social network graphs (e.g., degree, path lengths), many of them do not generate graphs with sufficient variation to reflect the natural variability in real world graph domains. One exception is the mixed Kronecker Product Graph Model (mKPGM), a generalization of the Kronecker Product Graph Model, which uses parameter tying to capture variance in the underlying distribution In this work, we present the first learning algorithm for mKPGMs. The O(|E|) algorithm searches over the continuous parameter space using constrained line search and is based on simulated method of moments, where the objective function minimizes the distance between the observed moments in the training graph and the empirically estimated moments of the model. We evaluate the mKPGM learning algorithm by comparing it to several different graph models, including KPGMs. We use multi-dimensional KS distance to compare the generated graphs to the observed graphs and the results show mKPGMs are able to produce a closer match to real-world graphs (10-90% reduction in KS distance), while still providing natural variation in the generated graphs

BEBERAPA SIFAT HASIL KALI KRONECKER RANTAI MARKOV BERDIMENSI HINGGA

Pada paper ini akan dibahas tentang Rantai Markov yang diperoleh dari perkalian kronecker dua Rantai Markov berdimensi hingga. Pembahasan akan diawali dengan beberapa definisi Rantai Markov dengan Matriks Peluang transisinya dan diagram transisi antar keadaannya. Demikian pula dengan hasil kali kroneckernya, akan diperlihatkan bagaimana Ruang keadaan, Matriks peluang transisi dan diagram transisi antar keadaannya. Hasil utama pembahasannya adalah beberapa sifat Rantai Markov hasil perkalian kronecker yang mempertahankan semua sifat yang ada pada dua Rantai Markov awal

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference