9,125 research outputs found
Characterizing the Initial Phase of Epidemic Growth on some Empirical Networks
A key parameter in models for the spread of infectious diseases is the basic
reproduction number , which is the expected number of secondary cases a
typical infected primary case infects during its infectious period in a large
mostly susceptible population. In order for this quantity to be meaningful, the
initial expected growth of the number of infectious individuals in the
large-population limit should be exponential.
We investigate to what extent this assumption is valid by performing repeated
simulations of epidemics on selected empirical networks, viewing each epidemic
as a random process in discrete time. The initial phase of each epidemic is
analyzed by fitting the number of infected people at each time step to a
generalised growth model, allowing for estimating the shape of the growth. For
reference, similar investigations are done on some elementary graphs such as
integer lattices in different dimensions and configuration model graphs, for
which the early epidemic behaviour is known.
We find that for the empirical networks tested in this paper, exponential
growth characterizes the early stages of the epidemic, except when the network
is restricted by a strong low-dimensional spacial constraint, such as is the
case for the two-dimensional square lattice. However, on finite integer
lattices of sufficiently high dimension, the early development of epidemics
shows exponential growth.Comment: To be included in the conference proceedings for SPAS 2017
(International Conference on Stochastic Processes and Algebraic Structures),
October 4-6, 201
Variational Walkback: Learning a Transition Operator as a Stochastic Recurrent Net
We propose a novel method to directly learn a stochastic transition operator
whose repeated application provides generated samples. Traditional undirected
graphical models approach this problem indirectly by learning a Markov chain
model whose stationary distribution obeys detailed balance with respect to a
parameterized energy function. The energy function is then modified so the
model and data distributions match, with no guarantee on the number of steps
required for the Markov chain to converge. Moreover, the detailed balance
condition is highly restrictive: energy based models corresponding to neural
networks must have symmetric weights, unlike biological neural circuits. In
contrast, we develop a method for directly learning arbitrarily parameterized
transition operators capable of expressing non-equilibrium stationary
distributions that violate detailed balance, thereby enabling us to learn more
biologically plausible asymmetric neural networks and more general non-energy
based dynamical systems. The proposed training objective, which we derive via
principled variational methods, encourages the transition operator to "walk
back" in multi-step trajectories that start at data-points, as quickly as
possible back to the original data points. We present a series of experimental
results illustrating the soundness of the proposed approach, Variational
Walkback (VW), on the MNIST, CIFAR-10, SVHN and CelebA datasets, demonstrating
superior samples compared to earlier attempts to learn a transition operator.
We also show that although each rapid training trajectory is limited to a
finite but variable number of steps, our transition operator continues to
generate good samples well past the length of such trajectories, thereby
demonstrating the match of its non-equilibrium stationary distribution to the
data distribution. Source Code: http://github.com/anirudh9119/walkback_nips17Comment: To appear at NIPS 201
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