48,740 research outputs found
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called âq-state Potts unique-nessâ and relate it to chromatic and Tutte
uniqueness, and also to âchromaticâflow uniquenessâ, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, LovĂĄsz and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the âedge elimination polynomialâ of Averbouch, Godlin and
Makowsky and the âinduced subgraph poly-nomialâ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de EducaciĂłn y Ciencia MTM2008-05866-C03-01Junta de AndalucĂa FQM- 0164Junta de AndalucĂa P06-FQM-0164
Note on the smallest root of the independence polynomial
One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real
Differential Calculus on Graphon Space
Recently, the theory of dense graph limits has received attention from
multiple disciplines including graph theory, computer science, statistical
physics, probability, statistics, and group theory. In this paper we initiate
the study of the general structure of differentiable graphon parameters . We
derive consistency conditions among the higher G\^ateaux derivatives of
when restricted to the subspace of edge weighted graphs .
Surprisingly, these constraints are rigid enough to imply that the multilinear
functionals satisfying the
constraints are determined by a finite set of constants indexed by isomorphism
classes of multigraphs with edges and no isolated vertices. Using this
structure theory, we explain the central role that homomorphism densities play
in the analysis of graphons, by way of a new combinatorial interpretation of
their derivatives. In particular, homomorphism densities serve as the monomials
in a polynomial algebra that can be used to approximate differential graphon
parameters as Taylor polynomials. These ideas are summarized by our main
theorem, which asserts that homomorphism densities where has at
most edges form a basis for the space of smooth graphon parameters whose
st derivatives vanish. As a consequence of this theory, we also extend
and derive new proofs of linear independence of multigraph homomorphism
densities, and characterize homomorphism densities. In addition, we develop a
theory of series expansions, including Taylor's theorem for graph parameters
and a uniqueness principle for series. We use this theory to analyze questions
raised by Lov\'asz, including studying infinite quantum algebras and the
connection between right- and left-homomorphism densities.Comment: Final version (36 pages), accepted for publication in Journal of
Combinatorial Theory, Series
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
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
Bayesian Networks for Max-linear Models
We study Bayesian networks based on max-linear structural equations as
introduced in Gissibl and Kl\"uppelberg [16] and provide a summary of their
independence properties. In particular we emphasize that distributions for such
networks are generally not faithful to the independence model determined by
their associated directed acyclic graph. In addition, we consider some of the
basic issues of estimation and discuss generalized maximum likelihood
estimation of the coefficients, using the concept of a generalized likelihood
ratio for non-dominated families as introduced by Kiefer and Wolfowitz [21].
Finally we argue that the structure of a minimal network asymptotically can be
identified completely from observational data.Comment: 18 page
- âŚ