121 research outputs found
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
Message-Passing Methods for Complex Contagions
Message-passing methods provide a powerful approach for calculating the
expected size of cascades either on random networks (e.g., drawn from a
configuration-model ensemble or its generalizations) asymptotically as the
number of nodes becomes infinite or on specific finite-size networks. We
review the message-passing approach and show how to derive it for
configuration-model networks using the methods of (Dhar et al., 1997) and
(Gleeson, 2008). Using this approach, we explain for such networks how to
determine an analytical expression for a "cascade condition", which determines
whether a global cascade will occur. We extend this approach to the
message-passing methods for specific finite-size networks (Shrestha and Moore,
2014; Lokhov et al., 2015), and we derive a generalized cascade condition.
Throughout this chapter, we illustrate these ideas using the Watts threshold
model.Comment: 14 pages, 3 figure
A framework for epidemic spreading in multiplex networks of metapopulations
We propose a theoretical framework for the study of epidemics in structured
metapopulations, with heterogeneous agents, subjected to recurrent mobility
patterns. We propose to represent the heterogeneity in the composition of the
metapopulations as layers in a multiplex network, where nodes would correspond
to geographical areas and layers account for the mobility patterns of agents of
the same class. We analyze both the classical Susceptible-Infected-Susceptible
and the Susceptible-Infected-Removed epidemic models within this framework, and
compare macroscopic and microscopic indicators of the spreading process with
extensive Monte Carlo simulations. Our results are in excellent agreement with
the simulations. We also derive an exact expression of the epidemic threshold
on this general framework revealing a non-trivial dependence on the mobility
parameter. Finally, we use this new formalism to address the spread of diseases
in real cities, specifically in the city of Medellin, Colombia, whose
population is divided into six socio-economic classes, each one identified with
a layer in this multiplex formalism.Comment: 13 pages, 11 figure
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