115 research outputs found
Network Discovery by Generalized Random Walks
We investigate network exploration by random walks defined via stationary and
adaptive transition probabilities on large graphs. We derive an exact formula
valid for arbitrary graphs and arbitrary walks with stationary transition
probabilities (STP), for the average number of discovered edges as function of
time. We show that for STP walks site and edge exploration obey the same
scaling as function of time . Therefore, edge exploration
on graphs with many loops is always lagging compared to site exploration, the
revealed graph being sparse until almost all nodes have been discovered. We
then introduce the Edge Explorer Model, which presents a novel class of
adaptive walks, that perform faithful network discovery even on dense networks.Comment: 23 pages, 7 figure
Brownian-Vacancy Mediated Disordering Dynamics
The disordering of an initially phase segregated system of finite size,
induced by the presence of highly mobile vacancies, is shown to exhibit dynamic
scaling in its late stages.
A set of characteristic exponents is introduced and computed analytically, in
excellent agreement with Monte Carlo data. In particular, the characteristic
time scale, controlling the crossover between increasing disorder and
saturation, is found to depend on the exponent scaling the number of vacancies
in the sample.Comment: 6 pages, typeset using Euro-LaTex, 6 figures, compresse
A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix
A joint degree matrix (JDM) specifies the number of connections between nodes
of given degrees in a graph, for all degree pairs and uniquely determines the
degree sequence of the graph. We consider the space of all balanced
realizations of an arbitrary JDM, realizations in which the links between any
two degree groups are placed as uniformly as possible. We prove that a swap
Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced
realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the
relaxation time of the chain is bounded from above by a polynomial in the
number of nodes . To prove fast mixing, we first prove a general
factorization theorem similar to the Martin-Randall method for disjoint
decompositions (partitions). This theorem can be used to bound from below the
spectral gap with the help of fast mixing subchains within every partition and
a bound on an auxiliary Markov chain between the partitions. Our proof of the
general factorization theorem is direct and uses conductance based methods
(Cheeger inequality).Comment: submitted, 18 pages, 4 figure
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