955 research outputs found
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
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
Predicting the speed of epidemics spreading on networks
Global transport and communication networks enable information, ideas and
infectious diseases now to spread at speeds far beyond what has historically
been possible. To effectively monitor, design, or intervene in such
epidemic-like processes, there is a need to predict the speed of a particular
contagion in a particular network, and to distinguish between nodes that are
more likely to become infected sooner or later during an outbreak. Here, we
study these quantities using a message-passing approach to derive simple and
effective predictions which are validated against epidemic simulations on a
variety of real-world networks with good agreement. In addition to
individualized predictions for different nodes, we find an overall sudden
transition from low density to almost full network saturation as the contagion
develops in time. Our theory is developed and explained in the setting of
simple contagions on tree-like networks, but we are also able to show how the
method extends remarkably well to complex contagions and highly clustered
networks
Optimal modularity and memory capacity of neural reservoirs
The neural network is a powerful computing framework that has been exploited
by biological evolution and by humans for solving diverse problems. Although
the computational capabilities of neural networks are determined by their
structure, the current understanding of the relationships between a neural
network's architecture and function is still primitive. Here we reveal that
neural network's modular architecture plays a vital role in determining the
neural dynamics and memory performance of the network of threshold neurons. In
particular, we demonstrate that there exists an optimal modularity for memory
performance, where a balance between local cohesion and global connectivity is
established, allowing optimally modular networks to remember longer. Our
results suggest that insights from dynamical analysis of neural networks and
information spreading processes can be leveraged to better design neural
networks and may shed light on the brain's modular organization
Predicting the epidemic threshold of the susceptible-infected-recovered model
Researchers have developed several theoretical methods for predicting
epidemic thresholds, including the mean-field like (MFL) method, the quenched
mean-field (QMF) method, and the dynamical message passing (DMP) method. When
these methods are applied to predict epidemic threshold they often produce
differing results and their relative levels of accuracy are still unknown. We
systematically analyze these two issues---relationships among differing results
and levels of accuracy---by studying the susceptible-infected-recovered (SIR)
model on uncorrelated configuration networks and a group of 56 real-world
networks. In uncorrelated configuration networks the MFL and DMP methods yield
identical predictions that are larger and more accurate than the prediction
generated by the QMF method. When compared to the 56 real-world networks, the
epidemic threshold obtained by the DMP method is closer to the actual epidemic
threshold because it incorporates full network topology information and some
dynamical correlations. We find that in some scenarios---such as networks with
positive degree-degree correlations, with an eigenvector localized on the high
-core nodes, or with a high level of clustering---the epidemic threshold
predicted by the MFL method, which uses the degree distribution as the only
input parameter, performs better than the other two methods. We also find that
the performances of the three predictions are irregular versus modularity
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