362 research outputs found
Information Spreading in Stationary Markovian Evolving Graphs
Markovian evolving graphs are dynamic-graph models where the links among a
fixed set of nodes change during time according to an arbitrary Markovian rule.
They are extremely general and they can well describe important dynamic-network
scenarios.
We study the speed of information spreading in the "stationary phase" by
analyzing the completion time of the "flooding mechanism". We prove a general
theorem that establishes an upper bound on flooding time in any stationary
Markovian evolving graph in terms of its node-expansion properties.
We apply our theorem in two natural and relevant cases of such dynamic
graphs. "Geometric Markovian evolving graphs" where the Markovian behaviour is
yielded by "n" mobile radio stations, with fixed transmission radius, that
perform independent random walks over a square region of the plane.
"Edge-Markovian evolving graphs" where the probability of existence of any edge
at time "t" depends on the existence (or not) of the same edge at time "t-1".
In both cases, the obtained upper bounds hold "with high probability" and
they are nearly tight. In fact, they turn out to be tight for a large range of
the values of the input parameters. As for geometric Markovian evolving graphs,
our result represents the first analytical upper bound for flooding time on a
class of concrete mobile networks.Comment: 16 page
Not Always Sparse: Flooding Time in Partially Connected Mobile Ad Hoc Networks
In this paper we study mobile ad hoc wireless networks using the notion of
evolving connectivity graphs. In such systems, the connectivity changes over
time due to the intermittent contacts of mobile terminals. In particular, we
are interested in studying the expected flooding time when full connectivity
cannot be ensured at each point in time. Even in this case, due to finite
contact times durations, connected components may appear in the connectivity
graph. Hence, this represents the intermediate case between extreme cases of
fully mobile ad hoc networks and fully static ad hoc networks. By using a
generalization of edge-Markovian graphs, we extend the existing models based on
sparse scenarios to this intermediate case and calculate the expected flooding
time. We also propose bounds that have reduced computational complexity.
Finally, numerical results validate our models
Distributed Community Detection in Dynamic Graphs
Inspired by the increasing interest in self-organizing social opportunistic
networks, we investigate the problem of distributed detection of unknown
communities in dynamic random graphs. As a formal framework, we consider the
dynamic version of the well-studied \emph{Planted Bisection Model}
\sdG(n,p,q) where the node set of the network is partitioned into two
unknown communities and, at every time step, each possible edge is
active with probability if both nodes belong to the same community, while
it is active with probability (with ) otherwise. We also consider a
time-Markovian generalization of this model.
We propose a distributed protocol based on the popular \emph{Label
Propagation Algorithm} and prove that, when the ratio is larger than
(for an arbitrarily small constant ), the protocol finds the right
"planted" partition in time even when the snapshots of the dynamic
graph are sparse and disconnected (i.e. in the case ).Comment: Version I
Time-Varying Graphs and Dynamic Networks
The past few years have seen intensive research efforts carried out in some
apparently unrelated areas of dynamic systems -- delay-tolerant networks,
opportunistic-mobility networks, social networks -- obtaining closely related
insights. Indeed, the concepts discovered in these investigations can be viewed
as parts of the same conceptual universe; and the formal models proposed so far
to express some specific concepts are components of a larger formal description
of this universe. The main contribution of this paper is to integrate the vast
collection of concepts, formalisms, and results found in the literature into a
unified framework, which we call TVG (for time-varying graphs). Using this
framework, it is possible to express directly in the same formalism not only
the concepts common to all those different areas, but also those specific to
each. Based on this definitional work, employing both existing results and
original observations, we present a hierarchical classification of TVGs; each
class corresponds to a significant property examined in the distributed
computing literature. We then examine how TVGs can be used to study the
evolution of network properties, and propose different techniques, depending on
whether the indicators for these properties are a-temporal (as in the majority
of existing studies) or temporal. Finally, we briefly discuss the introduction
of randomness in TVGs.Comment: A short version appeared in ADHOC-NOW'11. This version is to be
published in Internation Journal of Parallel, Emergent and Distributed
System
Evolving graphs: dynamical models, inverse problems and propagation
Applications such as neuroscience, telecommunication, online social networking,
transport and retail trading give rise to connectivity patterns that change over time.
In this work, we address the resulting need for network models and computational
algorithms that deal with dynamic links. We introduce a new class of evolving
range-dependent random graphs that gives a tractable framework for modelling and
simulation. We develop a spectral algorithm for calibrating a set of edge ranges from
a sequence of network snapshots and give a proof of principle illustration on some
neuroscience data. We also show how the model can be used computationally and
analytically to investigate the scenario where an evolutionary process, such as an
epidemic, takes place on an evolving network. This allows us to study the cumulative
effect of two distinct types of dynamics
Bistability through triadic closure
We propose and analyse a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, on-line social behaviour and information processing in neuroscience. We model the evolving network as a discrete time Markov chain, and study a very general framework where, conditioned on the current state, edges appear or disappear independently at the next timestep. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean field theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure effect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean field theory predicts bistable dynamics, and computational results confirm this prediction. We also discuss the calibration issue for a set of real cell phone data, and find support for a stratified model, where individuals are assigned to one of two distinct groups having different within-group and across-group dynamics
Competing edge networks
We introduce a model for a pair of nonlinear evolving networks, defined over a common set of vertices, sub ject to edgewise competition. Each network may grow new edges spontaneously or through triad closure. Both networks
inhibit the other’s growth and encourage the other’s demise. These nonlinear stochastic competition equations yield to a mean field analysis resulting in a nonlinear deterministic system. There may be multiple equilibria; and bifurcations of different types are shown to occur within a reduced parameter space. This situation models competitive peer-to-peer communication networks such as BlackBerry Messenger displacing SMS; or instant messaging
displacing emails
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