322 research outputs found
Mixing and relaxation time for Random Walk on Wreath Product Graphs
Suppose that G and H are finite, connected graphs, G regular, X is a lazy
random walk on G and Z is a reversible ergodic Markov chain on H. The
generalized lamplighter chain X* associated with X and Z is the random walk on
the wreath product H\wr G, the graph whose vertices consist of pairs (f,x)
where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H
and x is a vertex in G. In each step, X* moves from a configuration (f,x) by
updating x to y using the transition rule of X and then independently updating
both f_x and f_y according to the transition probabilities on H; f_z for z
different of x,y remains unchanged. We estimate the mixing time of X* in terms
of the parameters of H and G. Further, we show that the relaxation time of X*
is the same order as the maximal expected hitting time of G plus |G| times the
relaxation time of the chain on H.Comment: 30 pages, 1 figur
Weighted distances in scale-free configuration models
In this paper we study first-passage percolation in the configuration model
with empirical degree distribution that follows a power-law with exponent . We assign independent and identically distributed (i.i.d.)\ weights
to the edges of the graph. We investigate the weighted distance (the length of
the shortest weighted path) between two uniformly chosen vertices, called
typical distances. When the underlying age-dependent branching process
approximating the local neighborhoods of vertices is found to produce
infinitely many individuals in finite time -- called explosive branching
process -- Baroni, Hofstad and the second author showed that typical distances
converge in distribution to a bounded random variable. The order of magnitude
of typical distances remained open for the case when the
underlying branching process is not explosive. We close this gap by determining
the first order of magnitude of typical distances in this regime for arbitrary,
not necessary continuous edge-weight distributions that produce a non-explosive
age-dependent branching process with infinite mean power-law offspring
distributions. This sequence tends to infinity with the amount of vertices,
and, by choosing an appropriate weight distribution, can be tuned to be any
growing function that is , where is the number of vertices
in the graph. We show that the result remains valid for the the erased
configuration model as well, where we delete loops and any second and further
edges between two vertices.Comment: 24 page
First passage percolation on the Newman-Watts small world model
The Newman-Watts model is given by taking a cycle graph of n vertices and
then adding each possible edge with probability
for some constant. In this paper we add i.i.d. exponential
edge weights to this graph, and investigate typical distances in the
corresponding random metric space given by the least weight paths between
vertices. We show that typical distances grow as for a
and determine the distribution of smaller order terms in terms of
limits of branching process random variables. We prove that the number of edges
along the shortest weight path follows a Central Limit Theorem, and show that
in a corresponding epidemic spread model the fraction of infected vertices
follows a deterministic curve with a random shift.Comment: 29 pages, 4 figure
Topics in Markov chains: mixing and escape rate
These are the notes for the minicourse on Markov chains delivered at the
Saint Petersburg Summer School, June 2012. The main emphasis is on methods for
estimating mixing times (for finite chains) and escape rates (for infinite
chains). Lamplighter groups are key examples in both topics and the
Varopolous-Carne long range estimate is useful in both settings.Comment: 28 pages, 1 figur
Fixed speed competition on the configuration model with infinite variance degrees: equal speeds
We study competition of two spreading colors starting from single sources on
the configuration model with i.i.d. degrees following a power-law distribution
with exponent . In this model two colors spread with a fixed and
equal speed on the unweighted random graph.
We analyse how many vertices the two colors paint eventually. We show that
coexistence sensitively depends on the initial local neighborhoods of the
source vertices: if these neighborhoods are `dissimilar enough', then there is
no coexistence, and the `loser' color paints a polynomial fraction of the
vertices with a random exponent.
If the local neighborhoods of the starting vertices are `similar enough',
then there is coexistence, i.e., both colors paint a strictly positive
proportion of vertices. We give a quantitative characterization of `similar'
local neighborhoods: two random variables describing the double exponential
growth of local neighborhoods of the source vertices must be within a factor
of each other. Both of the two outcomes happen with positive
probability with asymptotic value that is explicitly computable.
This picture reinforces the common belief that location is an important
feature in advertising.
This paper is a follow-up of the similarly named paper that handles the case
when the speeds of the two colors are not equal. There, we have shown that the
faster color paints almost all vertices, while the slower color paints only a
random sub-polynomial fraction of the vertices.Comment: 62 pages, 11 figure
Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds
We study competition of two spreading colors starting from single sources on
the configuration model with i.i.d. degrees following a power-law distribution
with exponent tau in (2,3). In this model two colors spread with a fixed but
not necessarily equal speed on the unweighted random graph. We show that if the
speeds are not equal, then the faster color paints almost all vertices, while
the slower color can paint only a random subpolynomial fraction of the
vertices. We investigate the case when the speeds are equal and typical
distances in a follow-up paper.Comment: 44 pages, 9 picture
The front of the epidemic spread and first passage percolation
In this paper we establish a connection between epidemic models on random
networks with general infection times considered in Barbour and Reinert 2013
and first passage percolation. Using techniques developed in Bhamidi, van der
Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods,
we extend results on the epidemic curve in Barbour Reinert 2013 from bounded
degree graphs to general sparse random graphs with degrees having finite third
moments as the number of vertices tends to infinity. We also study the epidemic
trail between the source and typical vertices in the graph. This connection to
first passage percolation can be also be used to study epidemic models with
general contagious periods as in Barbour Reinert 2013 without bounded degree
assumptions.Comment: 14 page
- β¦