4,806 research outputs found
Epidemic Threshold of an SIS Model in Dynamic Switching Networks
In this paper, we analyze dynamic switching networks, wherein the networks
switch arbitrarily among a set of topologies. For this class of dynamic
networks, we derive an epidemic threshold, considering the SIS epidemic model.
First, an epidemic probabilistic model is developed assuming independence
between states of nodes. We identify the conditions under which the epidemic
dies out by linearizing the underlying dynamical system and analyzing its
asymptotic stability around the origin. The concept of joint spectral radius is
then used to derive the epidemic threshold, which is later validated using
several networks (Watts-Strogatz, Barabasi-Albert, MIT reality mining graphs,
Regular, and Gilbert). A simplified version of the epidemic threshold is
proposed for undirected networks. Moreover, in the case of static networks, the
derived epidemic threshold is shown to match conventional analytical results.
Then, analytical results for the epidemic threshold of dynamic networksare
proved to be applicable to periodic networks. For dynamic regular networks, we
demonstrate that the epidemic threshold is identical to the epidemic threshold
for static regular networks. An upper bound for the epidemic spread probability
in dynamic Gilbert networks is also derived and verified using simulation.Comment: Published in IEEE Transactions on Systems, Man and Cybernetic
Reproducibility and Pseudo-Determinism in Log-Space
A curious property of randomized log-space search algorithms is that their
outputs are often longer than their workspace. This leads to the question: how
can we reproduce the results of a randomized log space computation without
storing the output or randomness verbatim? Running the algorithm again with new
random bits may result in a new (and potentially different) output.
We show that every problem in search-RL has a randomized log-space algorithm
where the output can be reproduced. Specifically, we show that for every
problem in search-RL, there are a pair of log-space randomized algorithms A and
B where for every input x, A will output some string t_x of size O(log n), such
that B when running on (x, t_x) will be pseudo-deterministic: that is, running
B multiple times on the same input (x, t_x) will result in the same output on
all executions with high probability. Thus, by storing only O(log n) bits in
memory, it is possible to reproduce the output of a randomized log-space
algorithm.
An algorithm is reproducible without storing any bits in memory (i.e.,
|t_x|=0) if and only if it is pseudo-deterministic. We show
pseudo-deterministic algorithms for finding paths in undirected graphs and
Eulerian graphs using logarithmic space. Our algorithms are substantially
faster than the best known deterministic algorithms for finding paths in such
graphs in log-space.
The algorithm for search-RL has the additional property that its output, when
viewed as a random variable depending on the randomness used by the algorithm,
has entropy O(log n)
Summoning Information in Spacetime, or Where and When Can a Qubit Be?
One of the most important properties of quantum information, and the one
ultimately responsible for its cryptographic applications, is that it can't be
copied. That statement, however, is not completely accurate. While the
no-cloning theorem of quantum mechanics prevents quantum information from being
copied in space, the reversibility of microscopic physics actually requires
that the information be copied in time. In spacetime as a whole, therefore,
quantum information is widely replicated but in a restricted fashion. We fully
characterize which regions of spacetime can all hold the same quantum
information. Because quantum information can be delocalized through quantum
error correction and teleportation, it need not follow well-defined
trajectories. Instead, replication of the information in any configuration of
spacetime regions not leading to violations of causality or the no-cloning
principle is allowed. To demonstrate this, we answer the operational question
of exactly when the information can be summoned to a set of spacetime points,
showing how to do so efficiently using a combination of teleportation and
codeword-stabilized quantum codes. This provides a simple and complete
description of where and when a qubit can be located in spacetime, revealing a
remarkable variety of possibilities.Comment: v1: 5 pages, 1.2 figures per page on average. v2: 2 words and one
arrow added. v3: now includes an efficient construction v4: bug fix in
construction, new abstract v5, v6: cosmetic change
Physics Informed Topology Learning in Networks of Linear Dynamical Systems
Learning influence pathways of a network of dynamically related processes
from observations is of considerable importance in many disciplines. In this
article, influence networks of agents which interact dynamically via linear
dependencies are considered. An algorithm for the reconstruction of the
topology of interaction based on multivariate Wiener filtering is analyzed. It
is shown that for a vast and important class of interactions, that respect flow
conservation, the topology of the interactions can be exactly recovered. The
class of problems where reconstruction is guaranteed to be exact includes power
distribution networks, dynamic thermal networks and consensus networks. The
efficacy of the approach is illustrated through simulation and experiments on
consensus networks, IEEE power distribution networks and thermal dynamics of
buildings.Comment: 14 pages, 10 figure
Correlation between graphs with an application to brain networks analysis
The global functional brain network (graph) is more suitable for
characterizing brain states than local analysis of the connectivity of brain
regions. Therefore, graph-theoretic approaches are the natural methods to study
the brain. However, conventional graph theoretical analyses are limited due to
the lack of formal statistical methods for estimation and inference for random
graphs. For example, the concept of correlation between two vectors of graphs
is yet not defined. The aim of this article to introduce a notion of
correlation between graphs. In order to develop a framework to infer
correlation between graphs, we assume that they are generated by mathematical
models and that the parameters of the models are our random variables. Then, we
define that two vectors of graphs are independent whether their parameters are
independent. The problem is that, in real world, the model is rarely known, and
consequently, the parameters cannot be estimated. By analyzing the graph
spectrum, we showed that the spectral radius is highly associated with the
parameters of the graph model. Based on it, we constructed a framework for
correlation inference between graphs and illustrate our approach in a
functional magnetic resonance imaging data composed of 814 subjects comprising
529 controls and 285 individuals diagnosed with autism spectrum disorder (ASD).
Results show that correlations between default-mode and control, default-mode
and somatomotor, and default-mode and visual sub-networks are higher ()
in ASD than in controls
Effective noise channels for encoded quantum systems
We investigate effective noise channels for encoded quantum systems with and
without active error correction. Noise acting on physical qubits forming a
logical qubit is thereby described as a logical noise channel acting on the
logical qubits, which leads to a significant decrease of the effective system
dimension. This provides us with a powerful tool to study entanglement features
of encoded quantum systems. We demonstrate this framework by calculating lower
bounds on the lifetime of distillable entanglement and the negativity for
encoded multipartite qubit states with different encodings. At the same time,
this approach leads to a simple understanding of the functioning of
(concatenated) error correction codes.Comment: 10 pages, 6 figure
Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)
These are the lecture notes for the DIMACS Tutorial "Limits of Approximation
Algorithms: PCPs and Unique Games" held at the DIMACS Center, CoRE Building,
Rutgers University on 20-21 July, 2009. This tutorial was jointly sponsored by
the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus
on Algorithmic Foundations of the Internet, and the Center for Computational
Intractability with support from the National Security Agency and the National
Science Foundation.
The speakers at the tutorial were Matthew Andrews, Sanjeev Arora, Moses
Charikar, Prahladh Harsha, Subhash Khot, Dana Moshkovitz and Lisa Zhang. The
sribes were Ashkan Aazami, Dev Desai, Igor Gorodezky, Geetha Jagannathan,
Alexander S. Kulikov, Darakhshan J. Mir, Alantha Newman, Aleksandar Nikolov,
David Pritchard and Gwen Spencer.Comment: 74 pages, lecture note
Synchronization Clustering based on a Linearized Version of Vicsek model
This paper presents a kind of effective synchronization clustering method
based on a linearized version of Vicsek model. This method can be represented
by an Effective Synchronization Clustering algorithm (ESynC), an Improved
version of ESynC algorithm (IESynC), a Shrinking Synchronization Clustering
algorithm based on another linear Vicsek model (SSynC), and an effective
Multi-level Synchronization Clustering algorithm (MSynC). After some analysis
and comparisions, we find that ESynC algorithm based on the Linearized version
of the Vicsek model has better synchronization effect than SynC algorithm based
on an extensive Kuramoto model and a similar synchronization clustering
algorithm based on the original Vicsek model. By simulated experiments of some
artificial data sets, we observe that ESynC algorithm, IESynC algorithm, and
SSynC algorithm can get better synchronization effect although it needs less
iterative times and less time than SynC algorithm. In some simulations, we also
observe that IESynC algorithm and SSynC algorithm can get some improvements in
time cost than ESynC algorithm. At last, it gives some research expectations to
popularize this algorithm.Comment: 37 pages, 9 figures, 3 tabels, 27 conferenc
Got the Flu (or Mumps)? Check the Eigenvalue!
For a given, arbitrary graph, what is the epidemic threshold? That is, under
what conditions will a virus result in an epidemic? We provide the super-model
theorem, which generalizes older results in two important, orthogonal
dimensions. The theorem shows that (a) for a wide range of virus propagation
models (VPM) that include all virus propagation models in standard literature
(say, [8][5]), and (b) for any contact graph, the answer always depends on the
first eigenvalue of the connectivity matrix. We give the proof of the theorem,
arithmetic examples for popular VPMs, like flu (SIS), mumps (SIR), SIRS and
more. We also show the implications of our discovery: easy (although sometimes
counter-intuitive) answers to `what-if' questions; easier design and evaluation
of immunization policies, and significantly faster agent-based simulations.Comment: 26 pages, 12 figures, uses Tik
Comparison of mean-field based theoretical analysis methods for SIS model
Epidemic spreading has been intensively studied in SIS epidemic model.
Although the mean-field theory of SIS model has been widely used in the
research, there is a lack of comparative results between different theoretical
calculations, and the differences between them should be systematically
explained. In this paper, we have compared different theoretical solutions for
mean-field theory and explained the underlying reason. We first describe the
differences between different equations for mean-field theory in different
networks. The results show that the difference between mean-field reaction
equations is due to the different probability consideration for the infection
process. This finding will help us to design better theoretical solutions for
epidemic models.Comment: 11 pages, 5 figure
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