21,512 research outputs found
Stochastic Weighted Graphs: Flexible Model Specification and Simulation
In most domains of network analysis researchers consider networks that arise
in nature with weighted edges. Such networks are routinely dichotomized in the
interest of using available methods for statistical inference with networks.
The generalized exponential random graph model (GERGM) is a recently proposed
method used to simulate and model the edges of a weighted graph. The GERGM
specifies a joint distribution for an exponential family of graphs with
continuous-valued edge weights. However, current estimation algorithms for the
GERGM only allow inference on a restricted family of model specifications. To
address this issue, we develop a Metropolis--Hastings method that can be used
to estimate any GERGM specification, thereby significantly extending the family
of weighted graphs that can be modeled with the GERGM. We show that new
flexible model specifications are capable of avoiding likelihood degeneracy and
efficiently capturing network structure in applications where such models were
not previously available. We demonstrate the utility of this new class of
GERGMs through application to two real network data sets, and we further assess
the effectiveness of our proposed methodology by simulating non-degenerate
model specifications from the well-studied two-stars model. A working R version
of the GERGM code is available in the supplement and will be incorporated in
the gergm CRAN package.Comment: 33 pages, 6 figures. To appear in Social Network
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
Private Graphon Estimation for Sparse Graphs
We design algorithms for fitting a high-dimensional statistical model to a
large, sparse network without revealing sensitive information of individual
members. Given a sparse input graph , our algorithms output a
node-differentially-private nonparametric block model approximation. By
node-differentially-private, we mean that our output hides the insertion or
removal of a vertex and all its adjacent edges. If is an instance of the
network obtained from a generative nonparametric model defined in terms of a
graphon , our model guarantees consistency, in the sense that as the number
of vertices tends to infinity, the output of our algorithm converges to in
an appropriate version of the norm. In particular, this means we can
estimate the sizes of all multi-way cuts in .
Our results hold as long as is bounded, the average degree of grows
at least like the log of the number of vertices, and the number of blocks goes
to infinity at an appropriate rate. We give explicit error bounds in terms of
the parameters of the model; in several settings, our bounds improve on or
match known nonprivate results.Comment: 36 page
Phase transition in the two star Exponential Random Graph Model
This paper gives a way to simulate from the two star probability distribution
on the space of simple graphs via auxiliary variables. Using this simulation
scheme, the model is explored for various domains of the parameter values, and
the phase transition boundaries are identified, and shown to be similar as that
of the Curie-Weiss model of statistical physics. Concentration results are
obtained for all the degrees, which further validate the phase transition
predictions.Comment: 21 pages, 7 figure
Projective, Sparse, and Learnable Latent Position Network Models
When modeling network data using a latent position model, it is typical to
assume that the nodes' positions are independently and identically distributed.
However, this assumption implies the average node degree grows linearly with
the number of nodes, which is inappropriate when the graph is thought to be
sparse. We propose an alternative assumption---that the latent positions are
generated according to a Poisson point process---and show that it is compatible
with various levels of sparsity. Unlike other notions of sparse latent position
models in the literature, our framework also defines a projective sequence of
probability models, thus ensuring consistency of statistical inference across
networks of different sizes. We establish conditions for consistent estimation
of the latent positions, and compare our results to existing frameworks for
modeling sparse networks.Comment: 51 pages, 2 figure
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