23,819 research outputs found
Spread of Information and Diseases via Random Walks in Sparse Graphs
We consider a natural network diffusion process, modeling the spread of information or infectious diseases. Multiple mobile agents perform independent simple random walks on an n-vertex connected graph G. The number of agents is linear in n and the walks start from the stationary distribution. Initially, a single vertex has a piece of information (or a virus). An agent becomes informed (or infected) the first time it visits some vertex with the information (or virus); thereafter, the agent informs (infects) all vertices it visits. Giakkoupis et al. (PODC'19) have shown that the spreading time, i.e., the time before all vertices are informed, is asymptotically and w.h.p. the same as in the well-studied randomized rumor spreading process, on any d-regular graph with d=Ω(logn). The case of sub-logarithmic degree was left open, and is the main focus of this paper. First, we observe that the equivalence shown by Giakkoupis et al. does not hold for small d: We give an example of a 3-regular graph with logarithmic diameter for which the expected spreading time is Ω(log^2n/loglogn), whereas randomized rumor spreading is completed in time Î(logn), w.h.p. Next, we show a general upper bound of O~(dâ
diam(G)+log^3n/d), w.h.p., for the spreading time on any d-regular graph. We also provide a version of the bound based on the average degree, for non-regular graphs. Next, we give tight analyses for specific graph families. We show that the spreading time is O(logn), w.h.p., for constant-degree regular expanders. For the binary tree, we show an upper bound of O(lognâ
loglogn), w.h.p., and prove that this is tight, by giving a matching lower bound for the cover time of the tree by n random walks. Finally, we show a bound of O(diam(G)), w.h.p., for k-dimensional grids, by adapting a technique by Kesten and Sidoravicius.Supported in part by ANR Project PAMELA (ANR16-CE23-0016-01).
Gates Cambridge Scholarship programme.
Supported by the ERC Grant `Dynamic Marchâ
Estimating graph parameters with random walks
An algorithm observes the trajectories of random walks over an unknown graph
, starting from the same vertex , as well as the degrees along the
trajectories. For all finite connected graphs, one can estimate the number of
edges up to a bounded factor in
steps, where
is the relaxation time of the lazy random walk on and
is the minimum degree in . Alternatively, can be estimated in
, where is
the number of vertices and is the uniform mixing time on
. The number of vertices can then be estimated up to a bounded factor in
an additional steps. Our
algorithms are based on counting the number of intersections of random walk
paths , i.e. the number of pairs such that . This
improves on previous estimates which only consider collisions (i.e., times
with ). We also show that the complexity of our algorithms is optimal,
even when restricting to graphs with a prescribed relaxation time. Finally, we
show that, given either or the mixing time of , we can compute the
"other parameter" with a self-stopping algorithm
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
A macro-level model for investigating the effect of directional bias on network coverage
Random walks have been proposed as a simple method of efficiently searching,
or disseminating information throughout, communication and sensor networks. In
nature, animals (such as ants) tend to follow correlated random walks, i.e.,
random walks that are biased towards their current heading. In this paper, we
investigate whether or not complementing random walks with directional bias can
decrease the expected discovery and coverage times in networks.
To do so, we develop a macro-level model of a directionally biased random
walk based on Markov chains. By focussing on regular, connected networks, the
model allows us to efficiently calculate expected coverage times for different
network sizes and biases. Our analysis shows that directional bias can
significantly reduce coverage time, but only when the bias is below a certain
value which is dependent on the network size.Comment: 15 page
Switcher-random-walks: a cognitive-inspired mechanism for network exploration
Semantic memory is the subsystem of human memory that stores knowledge of
concepts or meanings, as opposed to life specific experiences. The organization
of concepts within semantic memory can be understood as a semantic network,
where the concepts (nodes) are associated (linked) to others depending on
perceptions, similarities, etc. Lexical access is the complementary part of
this system and allows the retrieval of such organized knowledge. While
conceptual information is stored under certain underlying organization (and
thus gives rise to a specific topology), it is crucial to have an accurate
access to any of the information units, e.g. the concepts, for efficiently
retrieving semantic information for real-time needings. An example of an
information retrieval process occurs in verbal fluency tasks, and it is known
to involve two different mechanisms: -clustering-, or generating words within a
subcategory, and, when a subcategory is exhausted, -switching- to a new
subcategory. We extended this approach to random-walking on a network
(clustering) in combination to jumping (switching) to any node with certain
probability and derived its analytical expression based on Markov chains.
Results show that this dual mechanism contributes to optimize the exploration
of different network models in terms of the mean first passage time.
Additionally, this cognitive inspired dual mechanism opens a new framework to
better understand and evaluate exploration, propagation and transport phenomena
in other complex systems where switching-like phenomena are feasible.Comment: 9 pages, 3 figures. Accepted in "International Journal of
Bifurcations and Chaos": Special issue on "Modelling and Computation on
Complex Networks
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