10,755 research outputs found
Random Walks on Stochastic Temporal Networks
In the study of dynamical processes on networks, there has been intense focus
on network structure -- i.e., the arrangement of edges and their associated
weights -- but the effects of the temporal patterns of edges remains poorly
understood. In this chapter, we develop a mathematical framework for random
walks on temporal networks using an approach that provides a compromise between
abstract but unrealistic models and data-driven but non-mathematical
approaches. To do this, we introduce a stochastic model for temporal networks
in which we summarize the temporal and structural organization of a system
using a matrix of waiting-time distributions. We show that random walks on
stochastic temporal networks can be described exactly by an
integro-differential master equation and derive an analytical expression for
its asymptotic steady state. We also discuss how our work might be useful to
help build centrality measures for temporal networks.Comment: Chapter in Temporal Networks (Petter Holme and Jari Saramaki
editors). Springer. Berlin, Heidelberg 2013. The book chapter contains minor
corrections and modifications. This chapter is based on arXiv:1112.3324,
which contains additional calculations and numerical simulation
Steady state and mean recurrence time for random walks on stochastic temporal networks
Random walks are basic diffusion processes on networks and have applications
in, for example, searching, navigation, ranking, and community detection.
Recent recognition of the importance of temporal aspects on networks spurred
studies of random walks on temporal networks. Here we theoretically study two
types of event-driven random walks on a stochastic temporal network model that
produces arbitrary distributions of interevent-times. In the so-called active
random walk, the interevent-time is reinitialized on all links upon each
movement of the walker. In the so-called passive random walk, the
interevent-time is only reinitialized on the link that has been used last time,
and it is a type of correlated random walk. We find that the steady state is
always the uniform density for the passive random walk. In contrast, for the
active random walk, it increases or decreases with the node's degree depending
on the distribution of interevent-times. The mean recurrence time of a node is
inversely proportional to the degree for both active and passive random walks.
Furthermore, the mean recurrence time does or does not depend on the
distribution of interevent-times for the active and passive random walks,
respectively.Comment: 5 figure
Classes of random walks on temporal networks with competing timescales
Random walks find applications in many areas of science and are the heart of
essential network analytic tools. When defined on temporal networks, even basic
random walk models may exhibit a rich spectrum of behaviours, due to the
co-existence of different timescales in the system. Here, we introduce random
walks on general stochastic temporal networks allowing for lasting
interactions, with up to three competing timescales. We then compare the mean
resting time and stationary state of different models. We also discuss the
accuracy of the mathematical analysis depending on the random walk model and
the structure of the underlying network, and pay particular attention to the
emergence of non-Markovian behaviour, even when all dynamical entities are
governed by memoryless distributions.Comment: 16 pages, 5 figure
Generalized Master Equations for Non-Poisson Dynamics on Networks
The traditional way of studying temporal networks is to aggregate the
dynamics of the edges to create a static weighted network. This implicitly
assumes that the edges are governed by Poisson processes, which is not
typically the case in empirical temporal networks. Consequently, we examine the
effects of non-Poisson inter-event statistics on the dynamics of edges, and we
apply the concept of a generalized master equation to the study of
continuous-time random walks on networks. We show that the equation reduces to
the standard rate equations when the underlying process is Poisson and that the
stationary solution is determined by an effective transition matrix whose
leading eigenvector is easy to calculate. We discuss the implications of our
work for dynamical processes on temporal networks and for the construction of
network diagnostics that take into account their nontrivial stochastic nature
Temporal-varying failures of nodes in networks
We consider networks in which random walkers are removed because of the
failure of specific nodes. We interpret the rate of loss as a measure of the
importance of nodes, a notion we denote as failure-centrality. We show that the
degree of the node is not sufficient to determine this measure and that, in a
first approximation, the shortest loops through the node have to be taken into
account. We propose approximations of the failure-centrality which are valid
for temporal-varying failures and we dwell on the possibility of externally
changing the relative importance of nodes in a given network, by exploiting the
interference between the loops of a node and the cycles of the temporal pattern
of failures. In the limit of long failure cycles we show analytically that the
escape in a node is larger than the one estimated from a stochastic failure
with the same failure probability. We test our general formalism in two
real-world networks (air-transportation and e-mail users) and show how
communities lead to deviations from predictions for failures in hubs.Comment: 7 pages, 3 figure
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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