842 research outputs found
The graph bottleneck identity
A matrix is said to determine a
\emph{transitional measure} for a digraph on vertices if for all
the \emph{transition inequality} holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in from to contains
. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
is \emph{graph-geodetic}, that is,
holds if and only if every path in connecting and contains .
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic
Spanning Forests and the Golden Ratio
For a graph G, let f_{ij} be the number of spanning rooted forests in which
vertex j belongs to a tree rooted at i. In this paper, we show that for a path,
the f_{ij}'s can be expressed as the products of Fibonacci numbers; for a
cycle, they are products of Fibonacci and Lucas numbers. The {\em doubly
stochastic graph matrix} is the matrix F=(f_{ij})/f, where f is the total
number of spanning rooted forests of G and n is the number of vertices in G. F
provides a proximity measure for graph vertices. By the matrix forest theorem,
F^{-1}=I+L, where L is the Laplacian matrix of G. We show that for the paths
and the so-called T-caterpillars, some diagonal entries of F (which provides a
measure of the self-connectivity of vertices) converge to \phi^{-1} or to
1-\phi^{-1}, where \phi is the golden ratio, as the number of vertices goes to
infinity. Thereby, in the asymptotic, the corresponding vertices can be
metaphorically considered as "golden introverts" and "golden extroverts,"
respectively. This metaphor is reinforced by a Markov chain interpretation of
the doubly stochastic graph matrix, according to which F equals the overall
transition matrix of a random walk with a random number of steps on G.Comment: 12 pages, 2 figures, 25 references. As accepted by Disc. Appl. Math.
(2007
Simple expressions for the long walk distance
The walk distances in graphs are defined as the result of appropriate
transformations of the proximity measures, where
is the weighted adjacency matrix of a connected weighted graph and is a
sufficiently small positive parameter. The walk distances are graph-geodetic,
moreover, they converge to the shortest path distance and to the so-called long
walk distance as the parameter approaches its limiting values. In this
paper, simple expressions for the long walk distance are obtained. They involve
the generalized inverse, minors, and inverses of submatrices of the symmetric
irreducible singular M-matrix where is the Perron
root of Comment: 7 pages. Accepted for publication in Linear Algebra and Its
Application
Extending Utility Representations of Partial Orders
The problem is considered as to whether a monotone function defined on a
subset P of a Euclidean space can be strictly monotonically extended to the
whole space. It is proved that this is the case if and only if the function is
{\em separably increasing}. Explicit formulas are given for a class of
extensions which involves an arbitrary bounded increasing function. Similar
results are obtained for monotone functions that represent strict partial
orders on arbitrary abstract sets X. The special case where P is a Pareto
subset is considered.Comment: 15 page
Do logarithmic proximity measures outperform plain ones in graph clustering?
We consider a number of graph kernels and proximity measures including
commute time kernel, regularized Laplacian kernel, heat kernel, exponential
diffusion kernel (also called "communicability"), etc., and the corresponding
distances as applied to clustering nodes in random graphs and several
well-known datasets. The model of generating random graphs involves edge
probabilities for the pairs of nodes that belong to the same class or different
predefined classes of nodes. It turns out that in most cases, logarithmic
measures (i.e., measures resulting after taking logarithm of the proximities)
perform better while distinguishing underlying classes than the "plain"
measures. A comparison in terms of reject curves of inter-class and intra-class
distances confirms this conclusion. A similar conclusion can be made for
several well-known datasets. A possible origin of this effect is that most
kernels have a multiplicative nature, while the nature of distances used in
cluster algorithms is an additive one (cf. the triangle inequality). The
logarithmic transformation is a tool to transform the first nature to the
second one. Moreover, some distances corresponding to the logarithmic measures
possess a meaningful cutpoint additivity property. In our experiments, the
leader is usually the logarithmic Communicability measure. However, we indicate
some more complicated cases in which other measures, typically, Communicability
and plain Walk, can be the winners.Comment: 11 pages, 5 tables, 9 figures. Accepted for publication in the
Proceedings of 6th International Conference on Network Analysis, May 26-28,
2016, Nizhny Novgorod, Russi
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