116,222 research outputs found
Lumping of Degree-Based Mean Field and Pair Approximation Equations for Multi-State Contact Processes
Contact processes form a large and highly interesting class of dynamic
processes on networks, including epidemic and information spreading. While
devising stochastic models of such processes is relatively easy, analyzing them
is very challenging from a computational point of view, particularly for large
networks appearing in real applications. One strategy to reduce the complexity
of their analysis is to rely on approximations, often in terms of a set of
differential equations capturing the evolution of a random node, distinguishing
nodes with different topological contexts (i.e., different degrees of different
neighborhoods), like degree-based mean field (DBMF), approximate master
equation (AME), or pair approximation (PA). The number of differential
equations so obtained is typically proportional to the maximum degree kmax of
the network, which is much smaller than the size of the master equation of the
underlying stochastic model, yet numerically solving these equations can still
be problematic for large kmax. In this paper, we extend AME and PA, which has
been proposed only for the binary state case, to a multi-state setting and
provide an aggregation procedure that clusters together nodes having similar
degrees, treating those in the same cluster as indistinguishable, thus reducing
the number of equations while preserving an accurate description of global
observables of interest. We also provide an automatic way to build such
equations and to identify a small number of degree clusters that give accurate
results. The method is tested on several case studies, where it shows a high
level of compression and a reduction of computational time of several orders of
magnitude for large networks, with minimal loss in accuracy.Comment: 16 pages with the Appendi
Deleting edges to restrict the size of an epidemic in temporal networks.
Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information or behaviour spread over social networks, biological diseases spreading over contact or trade networks, and the potential flow of goods over logistical infrastructure. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with graphs whose structure is subject to discrete changes over time, i.e. with temporal graphs. Here, we consider temporal graphs in which edges are available at specified timesteps, and study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path (i.e. a path, along which the times of the edge availabilities increase). We introduce a natural deletion problem for temporal graphs and we provide positive and negative results on its computational complexity, both in the traditional and the parameterised sense (subject to various natural parameters), as well as addressing the approximability of this problem
Dynamic message-passing equations for models with unidirectional dynamics
Understanding and quantifying the dynamics of disordered out-of-equilibrium
models is an important problem in many branches of science. Using the dynamic
cavity method on time trajectories, we construct a general procedure for
deriving the dynamic message-passing equations for a large class of models with
unidirectional dynamics, which includes the zero-temperature random field Ising
model, the susceptible-infected-recovered model, and rumor spreading models. We
show that unidirectionality of the dynamics is the key ingredient that makes
the problem solvable. These equations are applicable to single instances of the
corresponding problems with arbitrary initial conditions, and are
asymptotically exact for problems defined on locally tree-like graphs. When
applied to real-world networks, they generically provide a good analytic
approximation of the real dynamics.Comment: Final versio
Gossip in a Smartphone Peer-to-Peer Network
In this paper, we study the fundamental problem of gossip in the mobile
telephone model: a recently introduced variation of the classical telephone
model modified to better describe the local peer-to-peer communication services
implemented in many popular smartphone operating systems. In more detail, the
mobile telephone model differs from the classical telephone model in three
ways: (1) each device can participate in at most one connection per round; (2)
the network topology can undergo a parameterized rate of change; and (3)
devices can advertise a parameterized number of bits about their state to their
neighbors in each round before connection attempts are initiated. We begin by
describing and analyzing new randomized gossip algorithms in this model under
the harsh assumption of a network topology that can change completely in every
round. We prove a significant time complexity gap between the case where nodes
can advertise bits to their neighbors in each round, and the case where
nodes can advertise bit. For the latter assumption, we present two
solutions: the first depends on a shared randomness source, while the second
eliminates this assumption using a pseudorandomness generator we prove to exist
with a novel generalization of a classical result from the study of two-party
communication complexity. We then turn our attention to the easier case where
the topology graph is stable, and describe and analyze a new gossip algorithm
that provides a substantial performance improvement for many parameters. We
conclude by studying a relaxed version of gossip in which it is only necessary
for nodes to each learn a specified fraction of the messages in the system.Comment: Extended Abstract to Appear in the Proceedings of the ACM Conference
on the Principles of Distributed Computing (PODC 2017
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