scholarly journals Transient and slim versus recurrent and fat: Random walks and the trees they grow

2019 ◽  
Vol 56 (3) ◽  
pp. 769-786
Author(s):  
Giulio Iacobelli ◽  
Daniel R. Figueiredo ◽  
Giovanni Neglia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Exponential Distribution ◽  
Power Law ◽  
Random Network ◽  
Power Law Distribution ◽  
Current Position ◽  
Network Growth ◽  
Model Driven ◽  
Bounded Below

AbstractThe no restart random walk (NRRW) is a random network growth model driven by a random walk that builds the graph while moving on it, adding and connecting a new leaf node to the current position of the walker every s steps. We show a fundamental dichotomy in NRRW with respect to the parity of s: for ${s}=1$ we prove that the random walk is transient and non-leaf nodes have degrees bounded above by an exponential distribution; for s even we prove that the random walk is recurrent and non-leaf nodes have degrees bounded below by a power law distribution. These theoretical findings highlight and confirm the diverse and rich behaviour of NRRW observed empirically.

scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (01) ◽  
pp. 189-220
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

scholarly journals Type Transition of Simple Random Walks on Randomly Directed Regular Lattices

2014 ◽  
Vol 51 (4) ◽  
pp. 1065-1080 ◽  
Author(s):  
Massimo Campanino ◽  
Dimitri Petritis
Keyword(s):  
Random Walk ◽  
Periodic Function ◽  
Random Walks ◽  
Power Law ◽  
Random Perturbation ◽  
Simple Random Walk ◽  
Critical Value ◽  
Absolute Value ◽  
Horizontal Orientation ◽  
Regular Lattices

Simple random walks on a partially directed version ofZ2are considered. More precisely, vertical edges between neighbouring vertices ofZ2can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function; the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of simple random walk that is recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient.

scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (1) ◽  
pp. 189-220 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

HOW LONG CAN YOU ENJOY BLACKJACK WITH 100 CHIPS?

2011 ◽  
Vol 22 (10) ◽  
pp. 1161-1171
Author(s):  
TAO YANG ◽  
QIANQIAN LI ◽  
XINGANG XIA ◽  
ERBO ZHAO ◽  
GUO LIU ◽  
...  
Keyword(s):  
Decision Making ◽  
Random Walk ◽  
Power Law ◽  
Risk Taking ◽  
Real World ◽  
Power Law Distribution ◽  
Related Research ◽  
Fat Tail ◽  
Playing Time ◽  
Pathologic Gambling

Gambling-related research has implications in financial area understandings and applications. Researches in this area usually focus on pathology, risk-taking, decision-making and addiction. Few works have been done to demonstrate the distribution of the playing time before players go bankrupt. One problem is that it is difficult to get statistics in real world gambling. In this paper, we do simulations in a Blackjack game with a selected strategy. We find the distribution of playing time before players lose a certain amount of money as a power law distribution, indicating the existence of very long playing time players. We also find that double is the most important factor that causes the fat tail. Comparison shows that when removing double, split and three to two payoff, Blackjack goes back to a random walk. The increase of the number of decks somewhat decreases the average playing time. Our results may have pathologic gambling intervention implications.

scholarly journals An Agent-Based Model of a Pricing Process with Power Law, Volatility Clustering, and Jumps

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yu Shi ◽  
Qixuan Luo ◽  
Handong Li
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Asset Prices ◽  
Structural Changes ◽  
Power Law Distribution ◽  
Volatility Clustering ◽  
Agent Based ◽  
Tail Distribution ◽  
Pricing Process ◽  
Security Price

In this paper, we propose a new model of security price dynamics in order to explain the stylized facts of the pricing process such as power law distribution, volatility clustering, jumps, and structural changes. We assume that there are two types of agents in the financial market: speculators and fundamental investors. Speculators use past prices to predict future prices and only buy assets whose prices are expected to rise. Fundamental investors attach a certain value to each asset and buy when the asset is undervalued by the market. When the expectations of agents are exogenously driven, that is, entirely shaped by exogenous news, then they can be modeled as following a random walk. We assume that the information related to the two types of agents in the model will arrive randomly with a certain probability distribution and change the viewpoint of the agents according to a certain percentage. Our simulated results show that this model can simulate well the random walk of asset prices and explain the power-law tail distribution of returns, volatility clustering, jumps, and structural changes of asset prices.

scholarly journals Anomalous recurrence properties of many-dimensional zero-drift random walks

2016 ◽  
Vol 48 (A) ◽  
pp. 99-118 ◽  
Author(s):  
Nicholas Georgiou ◽  
Mikhail V. Menshikov ◽  
Aleksandar Mijatović ◽  
Andrew R. Wade
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Markov Processes ◽  
Zero Drift ◽  
Spherical Case ◽  
Current Position ◽  
Spatial Homogeneity ◽  
Second Moments ◽  
Spatially Homogeneous ◽  
Recurrence Classification

AbstractFamously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

Information spreading on metapopulation networks with heterogeneous contacting

2021 ◽  
Author(s):  
Yanyi Nie ◽  
Liming Pan ◽  
Tao Lin ◽  
Wei Wang
Keyword(s):  
Normal Distribution ◽  
Exponential Distribution ◽  
Power Law ◽  
Social Systems ◽  
Real Data ◽  
Power Law Distribution ◽  
Specific Distribution ◽  
Mean And Variance ◽  
Information Spreading ◽  
Individual Contact

Extensive real-data reveals that individuals exhibit heterogeneous contacting frequency in social systems. We propose a mathematical model to investigate the effects of heterogeneous contacting for information spreading in metapopulation networks. In the proposed model, we assume the number of contacting (NOC) distribution follows a specific distribution, including the normal, exponential, and power-law distributions. We utilize the Markov chain method to study the information spreading dynamics and find that mean and variance display no significant effect on the outbreak threshold for all the considered distributions. Under the same values of NOC distribution’s mean and variance, the information prevalence is largest when the distribution of NOC follows the normal distribution and second-largest for the exponential distribution, the smallest for the power-law distribution. When the distribution of NOC obeys the normal distribution, experimental results show that the information prevalence will decrease with individual contact ability heterogeneity. We observe similar phenomena when the distribution of NOC follows a power-law and exponential distribution. Furthermore, a larger mean of individual contact capacity distribution will result in higher information prevalence.

scholarly journals Effect of Hub Radius on Rotational Stability of Functionally Graded Timoshenko Beams

2019 ◽  
Vol 9 (1) ◽  
pp. 4246-4251
Keyword(s):  
Exponential Distribution ◽  
Power Law ◽  
Functionally Graded ◽  
Rotational Stability ◽  
Timoshenko Beams ◽  
Power Law Distribution ◽  
Exponential Law ◽  
Radius Parameter ◽  
Graded Material ◽  
The Stability

This work is concerned to examine the rotational stability of functionally graded cantilever Timoshenko beams. Power law with various indices as well as exponential law were used to find out the effect of hub radius parameter on the stability of both functionally graded ordinary (FGO) beam. Floquet’s theory was used to establish the stability boundaries. The governing equation of motion was followed by Hamilton’s principle and solved by Finite element method. Dependence of Bulk modulus on thickness of beam was studied using both power law and exponential distribution. The influence of hub radius parameter was found to be enhancing the stability of FGO beams. It has further been confirmed that the effect of hub radius with exponential distribution of constituent phases renders better stability compared to power law distribution of the phases in the functionally graded material(FGM).

scholarly journals Type Transition of Simple Random Walks on Randomly Directed Regular Lattices

2014 ◽  
Vol 51 (04) ◽  
pp. 1065-1080 ◽  
Author(s):  
Massimo Campanino ◽  
Dimitri Petritis
Keyword(s):  
Random Walk ◽  
Periodic Function ◽  
Random Walks ◽  
Power Law ◽  
Random Perturbation ◽  
Simple Random Walk ◽  
Critical Value ◽  
Absolute Value ◽  
Horizontal Orientation ◽  
Regular Lattices

Simple random walks on a partially directed version ofZ2are considered. More precisely, vertical edges between neighbouring vertices ofZ2can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function; the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of simple random walk that is recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient.

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