Optimal networks revealed by global mean first return time

The first-return time (FRT) is an effective measurement of random walks. Presently, it has attracted considerable attention with a focus on its scalings with regard to network size. In this paper, we propose a family of generalized and weighted transfractal networks and obtain the scalings of the FRT for a prescribed initial hub node. By employing the self-similarity of our networks, we calculate the first and second moments of FRT by the probability generating function and obtain the scalings of the mean and variance of FRT with regard to network size. For a large network, the mean FRT scales with the network size at the sublinear rate. Further, the efficiency of random walks relates strongly with the weight factor. The smaller the weight, the better the efficiency bears. Finally, we show that the variance of FRT decreases with more number of initial nodes, implying that our method is more effective for large-scale network size and the estimation of the mean FRT is more reliable.

Distribution of the first return time in fractional Brownian motion and its application to the study of on-off intermittency

Physical Review E ◽

10.1103/physreve.52.207 ◽

1995 ◽

Vol 52 (1) ◽

pp. 207-213 ◽

Cited By ~ 111

Author(s):

Mingzhou Ding ◽

Weiming Yang

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Return Time ◽

A return time invariant for finitary isomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700002406 ◽

1984 ◽

Vol 4 (2) ◽

pp. 225-231 ◽

Author(s):

U. R. Fiebig

Keyword(s):

Finite Measure ◽

Measurable Subset ◽

Return Time ◽

Time Function ◽

Markov Shifts ◽

Measure Preserving Transformation ◽

Recurrence Theorem ◽

First Return Time ◽

Time Invariant ◽

Finitary Isomorphisms

AbstractPoincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.58 ◽

2018 ◽

Vol 40 (3) ◽

pp. 663-698 ◽

Author(s):

HENK BRUIN ◽

DALIA TERHESIU

Keyword(s):

Regular Variation ◽

Return Time ◽

Main Task ◽

Two Dimensional ◽

Dimensional Torus ◽

Anosov Diffeomorphisms ◽

Infinite Measure ◽

Mixing Rates ◽

First Return Time ◽

Almost Anosov

The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.

Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

Random Matrices Theory and Application ◽

10.1142/s2010326315500112 ◽

2015 ◽

Vol 04 (03) ◽

pp. 1550011 ◽

Cited By ~ 1

Author(s):

O. Marchal

Keyword(s):

Return Time ◽

Unitary Matrix ◽

Unitary Matrices ◽

Toeplitz Determinants ◽

Recurrence Times ◽

Toeplitz Determinant ◽

Large N ◽

Single Arc ◽

Random Unitary Matrices ◽

The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

SCALING PROPERTIES OF FIRST RETURN TIME ON WEIGHTED TRANSFRACTALS (1,3)-FLOWERS

Fractals ◽

10.1142/s0218348x18500950 ◽

2018 ◽

Vol 26 (06) ◽

pp. 1850095 ◽

Cited By ~ 6

Author(s):

MEIFENG DAI ◽

HUIJIA CHI ◽

XIANBIN WU ◽

YUE ZONG ◽

WENJING FENG ◽

...

Keyword(s):

Complex Networks ◽

Real Life ◽

Probability Generating Function ◽

Return Time ◽

Exact Probability ◽

Scaling Properties ◽

Mean And Variance ◽

The Mean ◽

First Return Time ◽

The Impact

Complex networks are omnipresent in science and in our real life, and have been the focus of intense interest. It is vital to research the impact of their characters on the dynamic progress occurring on complex networks for weight-dependent walk. In this paper, we first consider the weight-dependent walk on one kind of transfractal (or fractal) which is named the weighted transfractal [Formula: see text]-flowers. And we pay attention to the first return time (FRT). We mainly calculate the mean and variance of FRT for a prescribed hub (i.e. the most concerned nodes) in virtue of exact probability generating function and its properties. Then, we obtain the mean and the secondary moment of the first return time. Finally, using the relationship among the variance, mean and the secondary moment, we obtain the variance of FRT and the scaling properties of the mean and variance of FRT on weighted transfractals [Formula: see text]-flowers.

Mean first return time for random walks on weighted networks

International Journal of Modern Physics C ◽

10.1142/s0129183115500680 ◽

2015 ◽

Vol 26 (06) ◽

pp. 1550068 ◽

Cited By ~ 1

Author(s):

Xing-Li Jing ◽

Xiang Ling ◽

Jiancheng Long ◽

Qing Shi ◽

Mao-Bin Hu

Keyword(s):

Random Walk ◽

Complex Networks ◽

Random Walks ◽

Stationary Distribution ◽

Real World ◽

Return Time ◽

Edge Weight ◽

Weighted Networks ◽

Strength Based ◽

Random walks on complex networks are of great importance to understand various types of phenomena in real world. In this paper, two types of biased random walks on nonassortative weighted networks are studied: edge-weight-based random walks and node-strength-based random walks, both of which are extended from the normal random walk model. Exact expressions for stationary distribution and mean first return time (MFRT) are derived and examined by simulation. The results will be helpful for understanding the influences of weights on the behavior of random walks.

Average convergence rate of the first return time

Colloquium Mathematicum ◽

10.4064/cm-84/85-1-159-171 ◽

2000 ◽

Vol 84 (1) ◽

pp. 159-171 ◽

Author(s):

Geon Choe ◽

Dong Kim

Keyword(s):

Convergence Rate ◽

Return Time ◽

A vague memory can affect first-return time

Journal of Physics Communications ◽

10.1088/2399-6528/ab9801 ◽

2020 ◽

Vol 4 (6) ◽

pp. 065005

Author(s):

Tomoko Sakiyama

Keyword(s):

Return Time ◽

AVALANCHES OF BAK–SNEPPEN COEVOLUTION MODEL ON DIRECTED SCALE-FREE NETWORK

Fractals ◽

10.1142/s0218348x09004259 ◽

2009 ◽

Vol 17 (02) ◽

pp. 233-237 ◽

Author(s):

KYOUNG EUN LEE ◽

JAE WOO LEE

Keyword(s):

Probability Distribution ◽

Power Law ◽

Critical Exponents ◽

Return Time ◽

Scale Free Network ◽

Avalanche Size ◽

Scale Free ◽

Free Network ◽

First Return Time ◽

Physical Quantities

We study the critical properties of the Bak–Sneppen coevolution model on scale-free networks by Monte Carlo method. We report the distribution of the avalanche size and fractal activity through the branching process. We observe that the critical fitness fc(N) depends on the number of the node such as fc(N) ~ 1/ log (N) for both the scale-free network and the directed scale-free network. Near the critical fitness many physical quantities show power-law behaviors. The probability distribution P(s) of the avalanche size at the critical fitness shows a power-law like P(s) ~ s-τ with τ = 1.53(5) regardless of the scale-free network and the directed scale free network. The probability distribution Pf(t) of the first return time also shows a power-law such as Pf(t) ~ t-τf. The critical exponents τf are equivalent for both the scale-free network and the directed scale-free network. We obtain the critical exponents as τf = 1.776(5) at the scalinge regime. The directionality of the network does not change the universality on the network.