8,717 research outputs found
Estimating and Sampling Graphs with Multidimensional Random Walks
Estimating characteristics of large graphs via sampling is a vital part of
the study of complex networks. Current sampling methods such as (independent)
random vertex and random walks are useful but have drawbacks. Random vertex
sampling may require too many resources (time, bandwidth, or money). Random
walks, which normally require fewer resources per sample, can suffer from large
estimation errors in the presence of disconnected or loosely connected graphs.
In this work we propose a new -dimensional random walk that uses
dependent random walkers. We show that the proposed sampling method, which we
call Frontier sampling, exhibits all of the nice sampling properties of a
regular random walk. At the same time, our simulations over large real world
graphs show that, in the presence of disconnected or loosely connected
components, Frontier sampling exhibits lower estimation errors than regular
random walks. We also show that Frontier sampling is more suitable than random
vertex sampling to sample the tail of the degree distribution of the graph
Ornstein-Uhlenbeck limit for the velocity process of an -particle system interacting stochastically
An -particle system with stochastic interactions is considered.
Interactions are driven by a Brownian noise term and total energy conservation
is imposed. The evolution of the system, in velocity space, is a diffusion on a
-dimensional sphere with radius fixed by the total energy. In the
limit, a finite number of velocity components are shown to
evolve independently and according to an Ornstein-Uhlenbeck process.Comment: 19 pages ; streamlined notations ; new section on many particles with
momentum conservation ; new appendix on Kac syste
Multiple Random Walks to Uncover Short Paths in Power Law Networks
Consider the following routing problem in the context of a large scale
network , with particular interest paid to power law networks, although our
results do not assume a particular degree distribution. A small number of nodes
want to exchange messages and are looking for short paths on . These nodes
do not have access to the topology of but are allowed to crawl the network
within a limited budget. Only crawlers whose sample paths cross are allowed to
exchange topological information. In this work we study the use of random walks
(RWs) to crawl . We show that the ability of RWs to find short paths bears
no relation to the paths that they take. Instead, it relies on two properties
of RWs on power law networks: 1) RW's ability observe a sizable fraction of the
network edges; and 2) an almost certainty that two distinct RW sample paths
cross after a small percentage of the nodes have been visited. We show
promising simulation results on several real world networks
Arithmetic of positive characteristic L-series values in Tate algebras
The second author has recently introduced a new class of L-series in the
arithmetic theory of function fields over finite fields. We show that the value
at one of these L-series encode arithmetic informations of certain Drinfeld
modules defined over Tate algebras. This enables us to generalize Anderson's
log-algebraicity Theorem and Taelman's Herbrand-Ribet Theorem.Comment: final versio
Bayesian Inference of Online Social Network Statistics via Lightweight Random Walk Crawls
Online social networks (OSN) contain extensive amount of information about
the underlying society that is yet to be explored. One of the most feasible
technique to fetch information from OSN, crawling through Application
Programming Interface (API) requests, poses serious concerns over the the
guarantees of the estimates. In this work, we focus on making reliable
statistical inference with limited API crawls. Based on regenerative properties
of the random walks, we propose an unbiased estimator for the aggregated sum of
functions over edges and proved the connection between variance of the
estimator and spectral gap. In order to facilitate Bayesian inference on the
true value of the estimator, we derive the approximate posterior distribution
of the estimate. Later the proposed ideas are validated with numerical
experiments on inference problems in real-world networks
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