197 research outputs found
Properties of stochastic Kronecker graphs
The stochastic Kronecker graph model introduced by Leskovec et al. is a
random graph with vertex set , where two vertices and
are connected with probability
independently of the presence or absence of any other edge, for fixed
parameters . They have shown empirically that the
degree sequence resembles a power law degree distribution. In this paper we
show that the stochastic Kronecker graph a.a.s. does not feature a power law
degree distribution for any parameters . In addition,
we analyze the number of subgraphs present in the stochastic Kronecker graph
and study the typical neighborhood of any given vertex.Comment: 37 pages, 2 figure
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
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