10,342 research outputs found
Parameter identifiability in a class of random graph mixture models
We prove identifiability of parameters for a broad class of random graph
mixture models. These models are characterized by a partition of the set of
graph nodes into latent (unobservable) groups. The connectivities between nodes
are independent random variables when conditioned on the groups of the nodes
being connected. In the binary random graph case, in which edges are either
present or absent, these models are known as stochastic blockmodels and have
been widely used in the social sciences and, more recently, in biology. Their
generalizations to weighted random graphs, either in parametric or
non-parametric form, are also of interest in many areas. Despite a broad range
of applications, the parameter identifiability issue for such models is
involved, and previously has only been touched upon in the literature. We give
here a thorough investigation of this problem. Our work also has consequences
for parameter estimation. In particular, the estimation procedure proposed by
Frank and Harary for binary affiliation models is revisited in this article
On the phase transitions of graph coloring and independent sets
We study combinatorial indicators related to the characteristic phase
transitions associated with coloring a graph optimally and finding a maximum
independent set. In particular, we investigate the role of the acyclic
orientations of the graph in the hardness of finding the graph's chromatic
number and independence number. We provide empirical evidence that, along a
sequence of increasingly denser random graphs, the fraction of acyclic
orientations that are `shortest' peaks when the chromatic number increases, and
that such maxima tend to coincide with locally easiest instances of the
problem. Similar evidence is provided concerning the `widest' acyclic
orientations and the independence number
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erd\H{o}s--R\'enyi random graph
asks to estimate the probability that the number of
copies of a graph in exceeds its expectation by a factor .
Chatterjee and Dembo showed that in the sparse regime of as
with for an explicit ,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
is a clique. Here we extend the latter work to any fixed graph and
determine a function such that, for as above and any fixed
, the upper tail probability is , where is the maximum degree of . As it turns out, the
leading order constant in the large deviation rate function, , is
governed by the independence polynomial of , defined as where is the number of independent sets of size in . For
instance, if is a regular graph on vertices, then is the
minimum between and the unique positive solution of
Upper Tail Estimates with Combinatorial Proofs
We study generalisations of a simple, combinatorial proof of a Chernoff bound
similar to the one by Impagliazzo and Kabanets (RANDOM, 2010).
In particular, we prove a randomized version of the hitting property of
expander random walks and apply it to obtain a concentration bound for expander
random walks which is essentially optimal for small deviations and a large
number of steps. At the same time, we present a simpler proof that still yields
a "right" bound settling a question asked by Impagliazzo and Kabanets.
Next, we obtain a simple upper tail bound for polynomials with input
variables in which are not necessarily independent, but obey a certain
condition inspired by Impagliazzo and Kabanets. The resulting bound is used by
Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number
of calls in a black-box construction of a pseudorandom generator from a one-way
function.
We then show that the same technique yields the upper tail bound for the
number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph,
matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math,
2002).Comment: Full version of the paper from STACS 201
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