2,697 research outputs found
Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality
Conventionally used exponential random graphs cannot directly model weighted
networks as the underlying probability space consists of simple graphs only.
Since many substantively important networks are weighted, this limitation is
especially problematic. We extend the existing exponential framework by
proposing a generic common distribution for the edge weights. Minimal
assumptions are placed on the distribution, that is, it is non-degenerate and
supported on the unit interval. By doing so, we recognize the essential
properties associated with near-degeneracy and universality in edge-weighted
exponential random graphs.Comment: 15 pages, 4 figures. This article extends arXiv:1607.04084, which
derives general formulas for the normalization constant and characterizes
phase transitions in exponential random graphs with uniformly distributed
edge weights. The present article places minimal assumptions on the
edge-weight distribution, thereby recognizing essential properties associated
with near-degeneracy and universalit
On the lower tail variational problem for random graphs
We study the lower tail large deviation problem for subgraph counts in a
random graph. Let denote the number of copies of in an
Erd\H{o}s-R\'enyi random graph . We are interested in
estimating the lower tail probability for fixed .
Thanks to the results of Chatterjee, Dembo, and Varadhan, this large
deviation problem has been reduced to a natural variational problem over
graphons, at least for (and conjecturally for a larger
range of ). We study this variational problem and provide a partial
characterization of the so-called "replica symmetric" phase. Informally, our
main result says that for every , and for some
, as slowly, the main contribution to the lower tail
probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted
edge density. On the other hand, this is false for non-bipartite and
close to 1.Comment: 15 pages, 5 figures, 1 tabl
Ground States for Exponential Random Graphs
We propose a perturbative method to estimate the normalization constant in
exponential random graph models as the weighting parameters approach infinity.
As an application, we give evidence of discontinuity in natural parametrization
along the critical directions of the edge-triangle model.Comment: 12 pages, 3 figures, 1 tabl
A detailed investigation into near degenerate exponential random graphs
The exponential family of random graphs has been a topic of continued
research interest. Despite the relative simplicity, these models capture a
variety of interesting features displayed by large-scale networks and allow us
to better understand how phases transition between one another as tuning
parameters vary. As the parameters cross certain lines, the model
asymptotically transitions from a very sparse graph to a very dense graph,
completely skipping all intermediate structures. We delve deeper into this near
degenerate tendency and give an explicit characterization of the asymptotic
graph structure as a function of the parameters.Comment: 15 pages, 3 figures, 2 table
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