The Linking Probability of Deep Spider-Web Networks

We consider crossbar switching networks with base $b$ (that is, constructed from $b\times b$ crossbar switches), scale $k$ (that is, with $b^k$ inputs, $b^k$ outputs and $b^k$ links between each consecutive pair of stages) and depth $l$ (that is, with $l$ stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least $k$ stages. We assume that each vertex is independently idle with probability $q$, the vacancy probability. We assume that $b\ge 2$ and the vacancy probability $q$ are fixed, and that $k$ and $l = ck$ tend to infinity with ratio a fixed constant $c>1$. We consider the linking probability $Q$ (the probability that there exists at least one idle path between a given idle input and a given idle output). In a previous paper it was shown that if $c\le 2$, then the linking probability $Q$ tends to 0 if $0<q<q_c$ (where $q_c = 1/b^{(c-1)/c}$ is the critical vacancy probability), and tends to $(1-\xi)^2$ (where $\xi$ is the unique solution of the equation $(1-q (1-x))^b=x$ in the range $0<x<1$) if $q_c<q<1$. In this paper we extend this result to all rational $c>1$. This is done by using generating functions and complex-variable techniques to estimate the second moments of various random variables involved in the analysis of the networks.Comment: i+21 p

Non-perturbative corrections to mean-field behavior: spherical model on spider-web graph

We consider the spherical model on a spider-web graph. This graph is effectively infinite-dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of $\delta$-functions, and all the modes are localized. The fractional number of modes with frequency less than $\omega$ varies as $\exp (-C/\omega)$ for $\omega$ tending to zero, where $C$ is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time $t$ varies as $\exp(- C' t^{1/3})$, for large $t$, where $C'$ is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free-energy per site at temperature $T$, near and above the critical temperature $T_c$, also has an essential singularity of the type $\exp[ -K {(T - T_c)}^{-1/2}]$.Comment: substantially revised, a section adde

Efficiency Analysis of Swarm Intelligence and Randomization Techniques

Swarm intelligence has becoming a powerful technique in solving design and scheduling tasks. Metaheuristic algorithms are an integrated part of this paradigm, and particle swarm optimization is often viewed as an important landmark. The outstanding performance and efficiency of swarm-based algorithms inspired many new developments, though mathematical understanding of metaheuristics remains partly a mystery. In contrast to the classic deterministic algorithms, metaheuristics such as PSO always use some form of randomness, and such randomization now employs various techniques. This paper intends to review and analyze some of the convergence and efficiency associated with metaheuristics such as firefly algorithm, random walks, and L\'evy flights. We will discuss how these techniques are used and their implications for further research.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1212.0220, arXiv:1208.0527, arXiv:1003.146

L\'evy walks

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The L\'{e}vy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to L\'{e}vy walks, surveys their existing applications, including latest advances, and outlines further perspectives.Comment: 50 page

Toward enhancement of deep learning techniques using fuzzy logic: a survey

Deep learning has emerged recently as a type of artificial intelligence (AI) and machine learning (ML), it usually imitates the human way in gaining a particular knowledge type. Deep learning is considered an essential data science element, which comprises predictive modeling and statistics. Deep learning makes the processes of collecting, interpreting, and analyzing big data easier and faster. Deep neural networks are kind of ML models, where the non-linear processing units are layered for the purpose of extracting particular features from the inputs. Actually, the training process of similar networks is very expensive and it also depends on the used optimization method, hence optimal results may not be provided. The techniques of deep learning are also vulnerable to data noise. For these reasons, fuzzy systems are used to improve the performance of deep learning algorithms, especially in combination with neural networks. Fuzzy systems are used to improve the representation accuracy of deep learning models. This survey paper reviews some of the deep learning based fuzzy logic models and techniques that were presented and proposed in the previous studies, where fuzzy logic is used to improve deep learning performance. The approaches are divided into two categories based on how both of the samples are combined. Furthermore, the models' practicality in the actual world is revealed