15,900 research outputs found
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
Convergent expansions for Random Cluster Model with q>0 on infinite graphs
In this paper we extend our previous results on the connectivity functions
and pressure of the Random Cluster Model in the highly subcritical phase and in
the highly supercritical phase, originally proved only on the cubic lattice
, to a much wider class of infinite graphs. In particular, concerning the
subcritical regime, we show that the connectivity functions are analytic and
decay exponentially in any bounded degree graph. In the supercritical phase, we
are able to prove the analyticity of finite connectivity functions in a smaller
class of graphs, namely, bounded degree graphs with the so called minimal
cut-set property and satisfying a (very mild) isoperimetric inequality. On the
other hand we show that the large distances decay of finite connectivity in the
supercritical regime can be polynomially slow depending on the topological
structure of the graph. Analogous analyticity results are obtained for the
pressure of the Random Cluster Model on an infinite graph, but with the further
assumptions of amenability and quasi-transitivity of the graph.Comment: In this new version the introduction has been revised, some
references have been added, and many typos have been corrected. 37 pages, to
appear in Communications on Pure and Applied Analysi
Standard imsets for undirected and chain graphical models
We derive standard imsets for undirected graphical models and chain graphical
models. Standard imsets for undirected graphical models are described in terms
of minimal triangulations for maximal prime subgraphs of the undirected graphs.
For describing standard imsets for chain graphical models, we first define a
triangulation of a chain graph. We then use the triangulation to generalize our
results for the undirected graphs to chain graphs.Comment: Published at http://dx.doi.org/10.3150/14-BEJ611 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Cycle-based Cluster Variational Method for Direct and Inverse Inference
We elaborate on the idea that loop corrections to belief propagation could be
dealt with in a systematic way on pairwise Markov random fields, by using the
elements of a cycle basis to define region in a generalized belief propagation
setting. The region graph is specified in such a way as to avoid dual loops as
much as possible, by discarding redundant Lagrange multipliers, in order to
facilitate the convergence, while avoiding instabilities associated to minimal
factor graph construction. We end up with a two-level algorithm, where a belief
propagation algorithm is run alternatively at the level of each cycle and at
the inter-region level. The inverse problem of finding the couplings of a
Markov random field from empirical covariances can be addressed region wise. It
turns out that this can be done efficiently in particular in the Ising context,
where fixed point equations can be derived along with a one-parameter log
likelihood function to minimize. Numerical experiments confirm the
effectiveness of these considerations both for the direct and inverse MRF
inference.Comment: 47 pages, 16 figure
Risk in a large claims insurance market with bipartite graph structure
We model the influence of sharing large exogeneous losses to the reinsurance
market by a bipartite graph. Using Pareto-tailed claims and multivariate
regular variation we obtain asymptotic results for the Value-at-Risk and the
Conditional Tail Expectation. We show that the dependence on the network
structure plays a fundamental role in their asymptotic behaviour. As is
well-known in a non-network setting, if the Pareto exponent is larger than 1,
then for the individual agent (reinsurance company) diversification is
beneficial, whereas when it is less than 1, concentration on a few objects is
the better strategy. An additional aspect of this paper is the amount of
uninsured losses which have to be convered by society. In the situation of
networks of agents, in our setting diversification is never detrimental
concerning the amount of uninsured losses. If the Pareto-tailed claims have
finite mean, diversification turns out to be never detrimental, both for
society and for individual agents. In contrast, if the Pareto-tailed claims
have infinite mean, a conflicting situation may arise between the incentives of
individual agents and the interest of some regulator to keep risk for society
small. We explain the influence of the network structure on diversification
effects in different network scenarios
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