Narrow Results

It was proved in [4] that the knotting probability of a Gaussian random polygon goes to 1 as the length of the polygon goes to infinity. In this paper, we prove the same result for the equilateral random polygons in R³. More precisely, if EP_n is an equilateral random polygon of n steps, then we have

The geometric features of the square and triadic Koch snowflake drums are compared using a position entropy defined on the grid points of the discretizations (prefractals) of the two domains. Weighted graphs using the geometric quantities are created and random walks on the two prefractals are performed. The aim is to understand if the existence of narrow channels in the domain may cause the "localization" of eigenfunctions.

In this paper, we present two deterministic weighted scale-free networks controlled by a weight parameter $r(0<r\le 1)$. One is fractal network, the other one is non-fractal network, while they have the same weight distribution when the parameter $r$ is identical. Based on their special network structure, we study random walks on network with a trap located at a fixed node. For each network, we calculate exact solutions for average trapping time (ATT). Analyzing and comparing the obtained solutions, we find that their ATT all grow asymptotically as a power-law function of network order (number of nodes) with the exponent $f(r)$ dependent on the weight parameter, but their exponent $f(r)$ are obviously different, one is an increasing function of $r$, while the other is opposite. Collectively, all the obtained results show that the efficiency of trapping on weighted Scale-free networks has close relation to the weight distribution, but there is no stable positive or negative correlation between the weight distribution and the trapping time on different networks. We hope these results given in this paper could help us get deeper understanding about the weight distribution on the property and dynamics of scale-free networks.

In this paper, we study two kinds of random walks with a trap in a class of scale-free fractal hierarchical lattices. One is standard random walks belonging to unbiased random walks, while the other one named mixed random walks is biased. The structural properties of these hierarchical lattices are controlled by a parameter $q$. We derive exact solutions of the average trapping time (ATT) for the two trapping issue, respectively. The results show that in large networks, both of the ATT grow asymptotically as a power-law function of network size with the exponent related to the parameter $q$. It indicates that network structure has a substantial effect on the efficiency of trapping processes performed in scale-free networks. Comparing the results obtained for the two different random walks, we find that changes of the walking rule have no effect on the leading exponent of the ATT, but could modify the coefficient of the formula for the ATT. The findings are helpful for better understanding the influence factor of random walks in complex systems.

In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. In this paper I further test the truncated Levy paradigm using high frequency data from the Australian All Ordinaries share market index. I then consider an optimal option hedging strategy which is appropriate for the early Levy dominated regime. This is compared with the usual delta hedging approach and found to differ significantly.

We investigate the probabilities for a return to the origin at step n of a random walker on a finite lattice. As a consistent measure only the first returns to the origin appear to be of relevance; these include paths with self-intersections and self-avoiding polygons. Their return probabilities are power-law distributed giving rise to 1/f^b noise. Most striking is the behavior of the self-avoiding polygons exhibiting a slope b=0.83 for d=2 and b=0.93 for d=3 independent on lattice structure.

For a multi-agent spatial Parrondo's model that it is composed of games A and B, we use the discrete time Markov chains to derive the probability transition matrix. Then, we respectively deduce the stationary distribution for games A and B played individually and the randomized combination of game A + B. We notice that under a specific set of parameters, two absorbing states instead of a fixed stationary distribution exist in game B. However, the randomized game A + B can jump out of the absorbing states of game B and has a fixed stationary distribution because of the "agitating" role of game A. Moreover, starting at different initial states, we deduce the probabilities of absorption at the two absorbing barriers.

Parrondo’s paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec’s theorem. The regime switching mechanism gives our models a richer and more complex asymptotic behavior than the simple random walks in random environments appearing in the literature, in terms of transience and recurrence.

Let be a dynamical system where is a probability space and T an invertible transformation preserving the measure μ. Let (S_k)_k≥0 be a transient ℤ-random walk. Let f ∈ L²(μ) and H ∈ ]0,1[, we study the convergence in distribution of the sequence

In this paper, we study dynamical properties such as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreducible Markov chains. As particular cases, we consider simple random walks on $\mathbb{Z}$ and ${\mathbb{Z}}_{+}$.

Random walks are one of the best investigated dynamical processes on graphs. A particularly fascinating phenomenon is the scaling relationship of fluctuations σ with the average flux 〈f〉. Here we analyze how network topology and nodes with finite capacity lead to deviations from a simple scaling law σ ~ 〈f〉^α. Sources of randomness are the random walk itself (internal noise) and the fluctuation of the number of walkers (external noise). We obtained exact results for the extreme case of a star network which are indicative of the behavior of large scale systems with a broad degree distribution. The latter are subsequently studied using Monte Carlo simulations. We find that the network heterogeneity amplifies the effects of external noise. By computing the "effective" scaling of each node we show that multiple scaling relationships can coexist in a graph with a heterogeneous degree distribution at an intermediate level of external noise. Finally, we analyze the effect of a finite capacity of nodes for random walkers and find that this also can lead to a heterogeneous scaling of fluctuations.

In this paper, we present a method based on local density and random walks (LDRW) for core-attachment complexes detection in protein-protein interaction (PPI) networks whether they are weighted or not. Our LDRW method consists of two stages. Firstly, it finds all the protein-complex cores based on local density of subnetwork. Then it uses random walks with restarts for finding the attachment proteins of each detected core to form complexes. We evaluate the effectiveness of our method using two different yeast PPI networks and validate the biological significance of the predicted protein complexes using known complexes in the Munich Information Center for Protein Sequence (MIPS) and Gene Ontology (GO) databases. We also perform a comprehensive comparison between our method and other existing methods. The results show that our method can find more protein complexes with high biological significance and obtains a significant improvement. Furthermore, our method is able to identify biologically significant overlapped protein complexes.

The question of quantum computation gates and quantum dissipation maps generated by means of classical random walks is addressed here in operations involving qubit states. Classical variables determining the manifold of qubit state vectors and density matrices are left to perform a classical random walk formulated algebraically by means of Hopf algebras. It is shown that this induces quantum operations on the qubits and density matrices which are further identified with e.g. known unitary transformations and completely positive trace preserving maps, depending upon the kind of random walk chosen.

It is shown that the Schrödinger equation can be written in the form of the diffusion equation for classical particles moving in a continuous space. A class of classical random processes described by the Schrödinger equation is considered. It is shown that such classical random processes can be used as a tool for the numerical solution of the Schrödinger equation.

By measured graphs, we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe algebras.

Community detection is a highly active research area that aims to identify groups of vertices with similar properties or interests within complex real-world networks. Over the years, a large number of papers have been published, resulting in the development of various community detection algorithms that consider factors such as the type of networks, the nature of communities, and applications. Despite numerous relevant review and survey papers, the literature lacks a comprehensive analysis of the computational complexity of existing community detection algorithms. This review aims to address this gap by providing a more detailed analysis and evaluation of the computational complexity of hierarchical clustering algorithms for community detection, including two main categories: agglomerative and divisive algorithms. We also highlight the main theoretical concepts, emphasizing the benefits and drawbacks of each approach, both in theory and in practical applications. This review helps researchers and practitioners in this field better understand valuable information on the differences and unique features of community detection algorithms with hierarchical community structure.

We present a distributed algorithm for constructing a random spanning tree, making use of random walks as network traversal scheme. Our approach is novel and make use of distributed random walks, each one represented by a token annexing a territory over the underlying graph. These multiple random walks collapse into a final one, that defines the final territory and provides the random spanning tree. The scheme is parallel and make use of waves to merge very efficiently the spanning forest computed by the random walks into one final random spanning tree.

Optimal stopping has been one of Y.S. Chow's major research areas in probability. This paper reviews the seminal work of Chow and Robbins on the optimal stopping problem for S_n/n and subsequent developments and intercrosses with other areas of probability theory. It also uses certain ideas and techniques from these developments to devise relatively simple but accurate approximations to the optimal stopping boundary.

In this paper, we consider knotting of Gaussian random polygons in 3-space . A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kesten's Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity . We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. This formula allows us to treat the CTRW as a discrete-space discrete-time random walk that in the continuum limit tends towards a generalized diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Plots of simulations for some case-studies are given in order to display the sample paths for the fractional diffusion processes, generally non Markovian, that are obtained by the composition of two Markovian processes.