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

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

Tied Kronecker product graph models to capture variance in network populations

Abstract—Much of the past work on mining and modeling networks has focused on understanding the observed properties of single example graphs. However, in many real-life applications it is important to characterize the structure of populations of graphs. In this work, we investigate the distributional properties of Kronecker product graph models (KPGMs) [1]. Specifically, we examine whether these models can represent the natural variability in graph properties observed across multiple networks and find surprisingly that they cannot. By considering KPGMs from a new viewpoint, we can show the reason for this lack of variance theoretically—which is primarily due to the generation of each edge independently from the others. Based on this understanding we propose a generalization of KPGMs that uses tied parameters to increase the variance of the model, while preserving the expectation. We then show experimentally, that our mixed-KPGM can adequately capture the natural variability across a population of networks. I