20,404 research outputs found
Viral processes by random walks on random regular graphs
We study the SIR epidemic model with infections carried by particles
making independent random walks on a random regular graph. Here we assume
, where is the number of vertices in the random graph,
and is some sufficiently small constant. We give an edge-weighted
graph reduction of the dynamics of the process that allows us to apply standard
results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In
particular, we show how the parameters of the model give two thresholds: In the
subcritical regime, particles are infected. In the supercritical
regime, for a constant determined by the parameters of the
model, get infected with probability , and get
infected with probability . Finally, there is a regime in which all
particles are infected. Furthermore, the edge weights give information
about when a particle becomes infected. We exploit this to give a completion
time of the process for the SI case.Comment: Published in at http://dx.doi.org/10.1214/13-AAP1000 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The end time of SIS epidemics driven by random walks on edge-transitive graphs
Network epidemics is a ubiquitous model that can represent different
phenomena and finds applications in various domains. Among its various
characteristics, a fundamental question concerns the time when an epidemic
stops propagating. We investigate this characteristic on a SIS epidemic induced
by agents that move according to independent continuous time random walks on a
finite graph: Agents can either be infected (I) or susceptible (S), and
infection occurs when two agents with different epidemic states meet in a node.
After a random recovery time, an infected agent returns to state S and can be
infected again. The End of Epidemic (EoE) denotes the first time where all
agents are in state S, since after this moment no further infections can occur
and the epidemic stops.
For the case of two agents on edge-transitive graphs, we characterize EoE as
a function of the network structure by relating the Laplace transform of EoE to
the Laplace transform of the meeting time of two random walks. Interestingly,
this analysis shows a separation between the effect of network structure and
epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically
in the size of the graph) under different parameter scalings, identifying
regimes where EoE converges in distribution to a proper random variable or to
infinity. We also highlight the impact of different graph structures on EoE,
characterizing it under complete graphs, complete bipartite graphs, and rings
Mean field conditions for coalescing random walks
The main results in this paper are about the full coalescence time
of a system of coalescing random walks over a finite graph .
Letting denote the mean meeting time of two such walkers, we
give sufficient conditions under which and has approximately the same law
as in the "mean field" setting of a large complete graph. One of our theorems
is that mean field behavior occurs over all vertex-transitive graphs whose
mixing times are much smaller than ; this nearly solves an open
problem of Aldous and Fill and also generalizes results of Cox for discrete
tori in dimensions. Other results apply to nonreversible walks and
also generalize previous theorems of Durrett and Cooper et al. Slight
extensions of these results apply to voter model consensus times, which are
related to coalescing random walks via duality. Our main proof ideas are a
strengthening of the usual approximation of hitting times by exponential random
variables, which give results for nonstationary initial states; and a new
general set of conditions under which we can prove that the hitting time of a
union of sets behaves like a minimum of independent exponentials. In
particular, this will show that the first meeting time among random walkers
has mean \approx\mathsf{m}(G)/\bigl({\matrix{k 2}}\bigr).Comment: Published in at http://dx.doi.org/10.1214/12-AOP813 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs
We analyze a class of distributed quantized consensus algorithms for
arbitrary static networks. In the initial setting, each node in the network has
an integer value. Nodes exchange their current estimate of the mean value in
the network, and then update their estimation by communicating with their
neighbors in a limited capacity channel in an asynchronous clock setting.
Eventually, all nodes reach consensus with quantized precision. We analyze the
expected convergence time for the general quantized consensus algorithm
proposed by Kashyap et al \cite{Kashyap}. We use the theory of electric
networks, random walks, and couplings of Markov chains to derive an upper bound for the expected convergence time on an arbitrary graph of size
, improving on the state of art bound of for quantized consensus
algorithms. Our result is not dependent on graph topology. Example of complete
graphs is given to show how to extend the analysis to graphs of given topology.Comment: to appear in IEEE Trans. on Automatic Control, January, 2015. arXiv
admin note: substantial text overlap with arXiv:1208.078
Exact Single-Source SimRank Computation on Large Graphs
SimRank is a popular measurement for evaluating the node-to-node similarities
based on the graph topology. In recent years, single-source and top- SimRank
queries have received increasing attention due to their applications in web
mining, social network analysis, and spam detection. However, a fundamental
obstacle in studying SimRank has been the lack of ground truths. The only exact
algorithm, Power Method, is computationally infeasible on graphs with more than
nodes. Consequently, no existing work has evaluated the actual
trade-offs between query time and accuracy on large real-world graphs. In this
paper, we present ExactSim, the first algorithm that computes the exact
single-source and top- SimRank results on large graphs. With high
probability, this algorithm produces ground truths with a rigorous theoretical
guarantee. We conduct extensive experiments on real-world datasets to
demonstrate the efficiency of ExactSim. The results show that ExactSim provides
the ground truth for any single-source SimRank query with a precision up to 7
decimal places within a reasonable query time.Comment: ACM SIGMOD 202
On the coalescence time of reversible random walks
Consider a system of coalescing random walks where each individual performs
random walk over a finite graph G, or (more generally) evolves according to
some reversible Markov chain generator Q. Let C be the first time at which all
walkers have coalesced into a single cluster. C is closely related to the
consensus time of the voter model for this G or Q.
We prove that the expected value of C is at most a constant multiple of the
largest hitting time of an element in the state space. This solves a problem
posed by Aldous and Fill and gives sharp bounds in many examples, including all
vertex-transitive graphs. We also obtain results on the expected time until
only k>1 clusters remain. Our proof tools include a new exponential inequality
for the meeting time of a reversible Markov chain and a deterministic
trajectory, which we believe to be of independent interest.Comment: 29 pages in 11pt font with 3/2 line spacing. v2 has an extra
reference and corrects a minor error in the proof of the last claim. To
appear in Transactions of the AM
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