117,846 research outputs found
Randomness and Complexity in Networks
I start by reviewing some basic properties of random graphs. I then consider
the role of random walks in complex networks and show how they may be used to
explain why so many long tailed distributions are found in real data sets. The
key idea is that in many cases the process involves copying of properties of
near neighbours in the network and this is a type of short random walk which in
turn produce a natural preferential attachment mechanism. Applying this to
networks of fixed size I show that copying and innovation are processes with
special mathematical properties which include the ability to solve a simple
model exactly for any parameter values and at any time. I finish by looking at
variations of this basic model.Comment: Survey paper based on talk given at the workshop on ``Stochastic
Networks and Internet Technology'', Centro di Ricerca Matematica Ennio De
Giorgi, Matematica nelle Scienze Naturali e Sociali, Pisa, 17th - 21st
September 2007. To appear in proceeding
Natural Halting Probabilities, Partial Randomness, and Zeta Functions
We introduce the zeta number, natural halting probability and natural
complexity of a Turing machine and we relate them to Chaitin's Omega number,
halting probability, and program-size complexity. A classification of Turing
machines according to their zeta numbers is proposed: divergent, convergent and
tuatara. We prove the existence of universal convergent and tuatara machines.
Various results on (algorithmic) randomness and partial randomness are proved.
For example, we show that the zeta number of a universal tuatara machine is
c.e. and random. A new type of partial randomness, asymptotic randomness, is
introduced. Finally we show that in contrast to classical (algorithmic)
randomness--which cannot be naturally characterised in terms of plain
complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin
Around Kolmogorov complexity: basic notions and results
Algorithmic information theory studies description complexity and randomness
and is now a well known field of theoretical computer science and mathematical
logic. There are several textbooks and monographs devoted to this theory where
one can find the detailed exposition of many difficult results as well as
historical references. However, it seems that a short survey of its basic
notions and main results relating these notions to each other, is missing.
This report attempts to fill this gap and covers the basic notions of
algorithmic information theory: Kolmogorov complexity (plain, conditional,
prefix), Solomonoff universal a priori probability, notions of randomness
(Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff
dimension. We prove their basic properties (symmetry of information, connection
between a priori probability and prefix complexity, criterion of randomness in
terms of complexity, complexity characterization for effective dimension) and
show some applications (incompressibility method in computational complexity
theory, incompleteness theorems). It is based on the lecture notes of a course
at Uppsala University given by the author
Entanglement, quantum randomness, and complexity beyond scrambling
Scrambling is a process by which the state of a quantum system is effectively
randomized due to the global entanglement that "hides" initially localized
quantum information. In this work, we lay the mathematical foundations of
studying randomness complexities beyond scrambling by entanglement properties.
We do so by analyzing the generalized (in particular R\'enyi) entanglement
entropies of designs, i.e. ensembles of unitary channels or pure states that
mimic the uniformly random distribution (given by the Haar measure) up to
certain moments. A main collective conclusion is that the R\'enyi entanglement
entropies averaged over designs of the same order are almost maximal. This
links the orders of entropy and design, and therefore suggests R\'enyi
entanglement entropies as diagnostics of the randomness complexity of
corresponding designs. Such complexities form a hierarchy between information
scrambling and Haar randomness. As a strong separation result, we prove the
existence of (state) 2-designs such that the R\'enyi entanglement entropies of
higher orders can be bounded away from the maximum. However, we also show that
the min entanglement entropy is maximized by designs of order only logarithmic
in the dimension of the system. In other words, logarithmic-designs already
achieve the complexity of Haar in terms of entanglement, which we also call
max-scrambling. This result leads to a generalization of the fast scrambling
conjecture, that max-scrambling can be achieved by physical dynamics in time
roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published
versio
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Incremental evolution of cellular automata for random number generation
Cellular automata (CA) have been used in pseudorandom number generation for over a decade. Recent studies show that controllable CA (CCA) can generate better random sequences than conventional one-dimensional (1-d) CA and compete with two-dimensional (2-d) CA. Yet the structural complexity of CCA is higher than that of 1-d PCA. It would be good if CCA can attain good randomness quality with the least structural complexity. In this paper, we evolve PCA/CCA to their lowest complexity level using genetic algorithms (GAs). Meanwhile, the randomness quality and output efficiency of PCA/CCA are also evolved. The evolution process involves two algorithms a multi-objective genetic algorithm (MOGA) and an algorithm for incremental evolution. A set of PCA/CCA are evolved and compared in randomness, complexity, and efficiency. The results show that without any spacing, CCA could generate good random number sequences that could pass DIEHARD. And, to obtain the same randomness quality, the structural complexity of CCA is not higher than that of 1-d CA. Furthermore, the methodology developed could be used to evolve other CA or serve as a yardstick to compare different types of CA
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