4,386 research outputs found
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
Tight estimates for convergence of some non-stationary consensus algorithms
The present paper is devoted to estimating the speed of convergence towards
consensus for a general class of discrete-time multi-agent systems. In the
systems considered here, both the topology of the interconnection graph and the
weight of the arcs are allowed to vary as a function of time. Under the
hypothesis that some spanning tree structure is preserved along time, and that
some nonzero minimal weight of the information transfer along this tree is
guaranteed, an estimate of the contraction rate is given. The latter is
expressed explicitly as the spectral radius of some matrix depending upon the
tree depth and the lower bounds on the weights.Comment: 17 pages, 5 figure
Diameter of the stochastic mean-field model of distance
We consider the complete graph \cK_n on vertices with exponential mean
edge lengths. Writing for the weight of the smallest-weight path
between vertex , Janson showed that converges in probability to 3. We extend this result by showing
that converges in distribution to a
limiting random variable that can be identified via a maximization procedure on
a limiting infinite random structure. Interestingly, this limiting random
variable has also appeared as the weak limit of the re-centered graph diameter
of the barely supercritical Erd\H{o}s-R\'enyi random graph in work by Riordan
and Wormald.Comment: 27 page
- …