748 research outputs found
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
Sampling and Inference for Beta Neutral-to-the-Left Models of Sparse Networks
Empirical evidence suggests that heavy-tailed degree distributions occurring
in many real networks are well-approximated by power laws with exponents
that may take values either less than and greater than two. Models based on
various forms of exchangeability are able to capture power laws with , and admit tractable inference algorithms; we draw on previous results to
show that cannot be generated by the forms of exchangeability used
in existing random graph models. Preferential attachment models generate power
law exponents greater than two, but have been of limited use as statistical
models due to the inherent difficulty of performing inference in
non-exchangeable models. Motivated by this gap, we design and implement
inference algorithms for a recently proposed class of models that generates
of all possible values. We show that although they are not exchangeable,
these models have probabilistic structure amenable to inference. Our methods
make a large class of previously intractable models useful for statistical
inference.Comment: Accepted for publication in the proceedings of Conference on
Uncertainty in Artificial Intelligence (UAI) 201
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