33,237 research outputs found
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Discriminating different classes of biological networks by analyzing the graphs spectra distribution
The brain's structural and functional systems, protein-protein interaction,
and gene networks are examples of biological systems that share some features
of complex networks, such as highly connected nodes, modularity, and
small-world topology. Recent studies indicate that some pathologies present
topological network alterations relative to norms seen in the general
population. Therefore, methods to discriminate the processes that generate the
different classes of networks (e.g., normal and disease) might be crucial for
the diagnosis, prognosis, and treatment of the disease. It is known that
several topological properties of a network (graph) can be described by the
distribution of the spectrum of its adjacency matrix. Moreover, large networks
generated by the same random process have the same spectrum distribution,
allowing us to use it as a "fingerprint". Based on this relationship, we
introduce and propose the entropy of a graph spectrum to measure the
"uncertainty" of a random graph and the Kullback-Leibler and Jensen-Shannon
divergences between graph spectra to compare networks. We also introduce
general methods for model selection and network model parameter estimation, as
well as a statistical procedure to test the nullity of divergence between two
classes of complex networks. Finally, we demonstrate the usefulness of the
proposed methods by applying them on (1) protein-protein interaction networks
of different species and (2) on networks derived from children diagnosed with
Attention Deficit Hyperactivity Disorder (ADHD) and typically developing
children. We conclude that scale-free networks best describe all the
protein-protein interactions. Also, we show that our proposed measures
succeeded in the identification of topological changes in the network while
other commonly used measures (number of edges, clustering coefficient, average
path length) failed
Quantum Graphical Models and Belief Propagation
Belief Propagation algorithms acting on Graphical Models of classical
probability distributions, such as Markov Networks, Factor Graphs and Bayesian
Networks, are amongst the most powerful known methods for deriving
probabilistic inferences amongst large numbers of random variables. This paper
presents a generalization of these concepts and methods to the quantum case,
based on the idea that quantum theory can be thought of as a noncommutative,
operator-valued, generalization of classical probability theory. Some novel
characterizations of quantum conditional independence are derived, and
definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and
Bayesian Networks are proposed. The structure of Quantum Markov Networks is
investigated and some partial characterization results are obtained, along the
lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation
algorithm is presented and is shown to converge on 1-Bifactor Networks and
Markov Networks when the underlying graph is a tree. The use of Quantum Belief
Propagation as a heuristic algorithm in cases where it is not known to converge
is discussed. Applications to decoding quantum error correcting codes and to
the simulation of many-body quantum systems are described.Comment: 58 pages, 9 figure
Properties of dense partially random graphs
We study the properties of random graphs where for each vertex a {\it
neighbourhood} has been previously defined. The probability of an edge joining
two vertices depends on whether the vertices are neighbours or not, as happens
in Small World Graphs (SWGs). But we consider the case where the average degree
of each node is of order of the size of the graph (unlike SWGs, which are
sparse). This allows us to calculate the mean distance and clustering, that are
qualitatively similar (although not in such a dramatic scale range) to the case
of SWGs. We also obtain analytically the distribution of eigenvalues of the
corresponding adjacency matrices. This distribution is discrete for large
eigenvalues and continuous for small eigenvalues. The continuous part of the
distribution follows a semicircle law, whose width is proportional to the
"disorder" of the graph, whereas the discrete part is simply a rescaling of the
spectrum of the substrate. We apply our results to the calculation of the
mixing rate and the synchronizability threshold.Comment: 14 pages. To be published in Physical Review
(Quantum) Space-Time as a Statistical Geometry of Lumps in Random Networks
In the following we undertake to describe how macroscopic space-time (or
rather, a microscopic protoform of it) is supposed to emerge as a
superstructure of a web of lumps in a stochastic discrete network structure. As
in preceding work (mentioned below), our analysis is based on the working
philosophy that both physics and the corresponding mathematics have to be
genuinely discrete on the primordial (Planck scale) level. This strategy is
concretely implemented in the form of \tit{cellular networks} and \tit{random
graphs}. One of our main themes is the development of the concept of
\tit{physical (proto)points} or \tit{lumps} as densely entangled subcomplexes
of the network and their respective web, establishing something like
\tit{(proto)causality}. It may perhaps be said that certain parts of our
programme are realisations of some early ideas of Menger and more recent ones
sketched by Smolin a couple of years ago. We briefly indicate how this
\tit{two-story-concept} of \tit{quantum} space-time can be used to encode the
(at least in our view) existing non-local aspects of quantum theory without
violating macroscopic space-time causality.Comment: 35 pages, Latex, under consideration by CQ
Faster and Simpler Distributed Algorithms for Testing and Correcting Graph Properties in the CONGEST-Model
In this paper we present distributed testing algorithms of graph properties
in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error
testing algorithms in the general graph model.
We first describe a general procedure for converting -testers with
a number of rounds , where denotes the diameter of the graph, to
rounds, where is the number of
processors of the network. We then apply this procedure to obtain an optimal
tester, in terms of , for testing bipartiteness, whose round complexity is
, which improves over the -round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for
cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects
the graph so that the corrected graph is acyclic. Note that, unlike a tester, a
corrector needs to mend the graph in many places in the case that the graph is
far from having the property.
In the second part of the paper we design algorithms for testing whether the
network is -free for any connected of size up to four with round
complexity of . This improves over the
-round algorithms for testing triangle freeness by
Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size by
Fraigniaud et al. (DISC 2016).
In the last part we generalize the global tester by Iwama and Yoshida (ITCS
2014) of testing -path freeness to testing the exclusion of any tree of
order . We then show how to simulate this algorithm in the CONGEST-model in
rounds
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