36,824 research outputs found
Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks
We show that numerical approximations of Kolmogorov complexity (K) applied to
graph adjacency matrices capture some group-theoretic and topological
properties of graphs and empirical networks ranging from metabolic to social
networks. That K and the size of the group of automorphisms of a graph are
correlated opens up interesting connections to problems in computational
geometry, and thus connects several measures and concepts from complexity
science. We show that approximations of K characterise synthetic and natural
networks by their generating mechanisms, assigning lower algorithmic randomness
to complex network models (Watts-Strogatz and Barabasi-Albert networks) and
high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these
results via two different Kolmogorov complexity approximation methods applied
to the adjacency matrices of the graphs and networks. The methods used are the
traditional lossless compression approach to Kolmogorov complexity, and a
normalised version of a Block Decomposition Method (BDM) measure, based on
algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical
Mechanics and its Application
The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy
The principle of maximum entropy (Maxent) is often used to obtain prior
probability distributions as a method to obtain a Gibbs measure under some
restriction giving the probability that a system will be in a certain state
compared to the rest of the elements in the distribution. Because classical
entropy-based Maxent collapses cases confounding all distinct degrees of
randomness and pseudo-randomness, here we take into consideration the
generative mechanism of the systems considered in the ensemble to separate
objects that may comply with the principle under some restriction and whose
entropy is maximal but may be generated recursively from those that are
actually algorithmically random offering a refinement to classical Maxent. We
take advantage of a causal algorithmic calculus to derive a thermodynamic-like
result based on how difficult it is to reprogram a computer code. Using the
distinction between computable and algorithmic randomness we quantify the cost
in information loss associated with reprogramming. To illustrate this we apply
the algorithmic refinement to Maxent on graphs and introduce a Maximal
Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a
generalisation over previous approaches. We discuss practical implications of
evaluation of network randomness. Our analysis provides insight in that the
reprogrammability asymmetry appears to originate from a non-monotonic
relationship to algorithmic probability. Our analysis motivates further
analysis of the origin and consequences of the aforementioned asymmetries,
reprogrammability, and computation.Comment: 30 page
Sigma models for quantum chaotic dynamics
We review the construction of the supersymmetric sigma model for unitary
maps, using the color- flavor transformation. We then illustrate applications
by three case studies in quantum chaos. In two of these cases, general Floquet
maps and quantum graphs, we show that universal spectral fluctuations arise
provided the pertinent classical dynamics are fully chaotic (ergodic and with
decay rates sufficiently gapped away from zero). In the third case, the kicked
rotor, we show how the existence of arbitrarily long-lived modes of excitation
(diffusion) precludes universal fluctuations and entails quantum localization
Detecting Activations over Graphs using Spanning Tree Wavelet Bases
We consider the detection of activations over graphs under Gaussian noise,
where signals are piece-wise constant over the graph. Despite the wide
applicability of such a detection algorithm, there has been little success in
the development of computationally feasible methods with proveable theoretical
guarantees for general graph topologies. We cast this as a hypothesis testing
problem, and first provide a universal necessary condition for asymptotic
distinguishability of the null and alternative hypotheses. We then introduce
the spanning tree wavelet basis over graphs, a localized basis that reflects
the topology of the graph, and prove that for any spanning tree, this approach
can distinguish null from alternative in a low signal-to-noise regime. Lastly,
we improve on this result and show that using the uniform spanning tree in the
basis construction yields a randomized test with stronger theoretical
guarantees that in many cases matches our necessary conditions. Specifically,
we obtain near-optimal performance in edge transitive graphs, -nearest
neighbor graphs, and -graphs
Numerical Investigation of Graph Spectra and Information Interpretability of Eigenvalues
We undertake an extensive numerical investigation of the graph spectra of
thousands regular graphs, a set of random Erd\"os-R\'enyi graphs, the two most
popular types of complex networks and an evolving genetic network by using
novel conceptual and experimental tools. Our objective in so doing is to
contribute to an understanding of the meaning of the Eigenvalues of a graph
relative to its topological and information-theoretic properties. We introduce
a technique for identifying the most informative Eigenvalues of evolving
networks by comparing graph spectra behavior to their algorithmic complexity.
We suggest that extending techniques can be used to further investigate the
behavior of evolving biological networks. In the extended version of this paper
we apply these techniques to seven tissue specific regulatory networks as
static example and network of a na\"ive pluripotent immune cell in the process
of differentiating towards a Th17 cell as evolving example, finding the most
and least informative Eigenvalues at every stage.Comment: Forthcoming in 3rd International Work-Conference on Bioinformatics
and Biomedical Engineering (IWBBIO), Lecture Notes in Bioinformatics, 201
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