scholarly journals A New Framework for Analysis of Coevolutionary Systems—Directed Graph Representation and Random Walks

2019 ◽  
Vol 27 (2) ◽  
pp. 195-228 ◽  
Author(s):  
Siang Yew Chong ◽  
Peter Tiňo ◽  
Jun He ◽  
Xin Yao
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Directed Graph ◽  
Pairwise Comparison ◽  
Complete Characterization ◽  
Graph Representation ◽  
The Rich ◽  
Cycle Structures ◽  
Expected Hitting Time ◽  
New Framework

Studying coevolutionary systems in the context of simplified models (i.e., games with pairwise interactions between coevolving solutions modeled as self plays) remains an open challenge since the rich underlying structures associated with pairwise-comparison-based fitness measures are often not taken fully into account. Although cyclic dynamics have been demonstrated in several contexts (such as intransitivity in coevolutionary problems), there is no complete characterization of cycle structures and their effects on coevolutionary search. We develop a new framework to address this issue. At the core of our approach is the directed graph (digraph) representation of coevolutionary problems that fully captures structures in the relations between candidate solutions. Coevolutionary processes are modeled as a specific type of Markov chains—random walks on digraphs. Using this framework, we show that coevolutionary problems admit a qualitative characterization: a coevolutionary problem is either solvable (there is a subset of solutions that dominates the remaining candidate solutions) or not. This has an implication on coevolutionary search. We further develop our framework that provides the means to construct quantitative tools for analysis of coevolutionary processes and demonstrate their applications through case studies. We show that coevolution of solvable problems corresponds to an absorbing Markov chain for which we can compute the expected hitting time of the absorbing class. Otherwise, coevolution will cycle indefinitely and the quantity of interest will be the limiting invariant distribution of the Markov chain. We also provide an index for characterizing complexity in coevolutionary problems and show how they can be generated in a controlled manner.

A directed graph representation for computer simulation of belief systems

1968 ◽  
Vol 2 (1-2) ◽  
pp. 19-40 ◽  
Author(s):  
Lawrence Tesler ◽  
Horace Enea ◽  
Kenneth Mark Colby
Keyword(s):  
Computer Simulation ◽  
Directed Graph ◽  
Belief Systems ◽  
Graph Representation

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

scholarly journals One dimensional quantum walks with memory

2010 ◽  
Vol 10 (5&6) ◽  
pp. 509-524
Author(s):  
M. Mc Gettrick
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Walks ◽  
One Dimensional ◽  
With Memory ◽  
Previous Step

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

The Study on Convergence and Convergence Rate of Genetic Algorithm Based on an Absorbing Markov Chain

2012 ◽  
Vol 239-240 ◽  
pp. 1511-1515 ◽  
Author(s):  
Jing Jiang ◽  
Li Dong Meng ◽  
Xiu Mei Xu
Keyword(s):  
Genetic Algorithm ◽  
Markov Chain ◽  
Convergence Rate ◽  
Hitting Time ◽  
Sufficient Condition ◽  
First Hitting Time ◽  
Absorbing Markov Chain ◽  
Expected Hitting Time ◽  
Optimization Efficiency ◽  
Theoretical Issues

The study on convergence of GA is always one of the most important theoretical issues. This paper analyses the sufficient condition which guarantees the convergence of GA. Via analyzing the convergence rate of GA, the average computational complexity can be implied and the optimization efficiency of GA can be judged. This paper proposes the approach to calculating the first expected hitting time and analyzes the bounds of the first hitting time of concrete GA using the proposed approach.

Determining Equivalent Resistance and Nodal Potentials of a Resistive Circuit Using a Reduced Transition Matrix

2020 ◽  
Vol 02 (01) ◽  
pp. 2050004
Author(s):  
Je-Young Choi
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Transition Matrix ◽  
Present Approach ◽  
Voltage Source ◽  
Equivalent Resistance ◽  
Electrical Circuits ◽  
Electric Potentials ◽  
Dominant Eigenvector

Several methods have been developed in order to solve electrical circuits consisting of resistors and an ideal voltage source. A correspondence with random walks avoids difficulties caused by choosing directions of currents and signs in potential differences. Starting from the random-walk method, we introduce a reduced transition matrix of the associated Markov chain whose dominant eigenvector alone determines the electric potentials at all nodes of the circuit and the equivalent resistance between the nodes connected to the terminals of the voltage source. Various means to find the eigenvector are developed from its definition. A few example circuits are solved in order to show the usefulness of the present approach.

Random walks on a dodecahedron

1980 ◽  
Vol 17 (02) ◽  
pp. 373-384 ◽  
Author(s):  
G. Letac ◽  
L. Takács
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Regular Dodecahedron

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn +1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X 0 = A] and P[In = A | I 0 = A]. The methods used are applicable to other solids.

scholarly journals Composition Markov chains of multinomial type

2009 ◽  
Vol 41 (01) ◽  
pp. 270-291 ◽  
Author(s):  
Hua Zhou ◽  
Kenneth Lange
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Continuous Time ◽  
Spectral Decomposition ◽  
Equilibrium Distribution ◽  
Convergence To Equilibrium ◽  
Light Bulb ◽  
Krawtchouk Polynomials ◽  
Particle Counts ◽  
Coalescence Times

Suppose that n identical particles evolve according to the same marginal Markov chain. In this setting we study chains such as the Ehrenfest chain that move a prescribed number of randomly chosen particles at each epoch. The product chain constructed by this device inherits its eigenstructure from the marginal chain. There is a further chain derived from the product chain called the composition chain that ignores particle labels and tracks the numbers of particles in the various states. The composition chain in turn inherits its eigenstructure and various properties such as reversibility from the product chain. The equilibrium distribution of the composition chain is multinomial. The current paper proves these facts in the well-known framework of state lumping and identifies the column eigenvectors of the composition chain with the multivariate Krawtchouk polynomials of Griffiths. The advantages of knowing the full spectral decomposition of the composition chain include (a) detailed estimates of the rate of convergence to equilibrium, (b) construction of martingales that allow calculation of the moments of the particle counts, and (c) explicit expressions for mean coalescence times in multi-person random walks. These possibilities are illustrated by applications to Ehrenfest chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA evolution, a light bulb chain, and random walks on some specific graphs.

scholarly journals A PATH GUESSING GAME WITH WAGERING

2010 ◽  
Vol 24 (3) ◽  
pp. 375-396 ◽  
Author(s):  
Marcus Pendergrass
Keyword(s):  
Markov Chain ◽  
Directed Graph ◽  
Optimal Strategies ◽  
Chain Dynamics ◽  
Coin Toss ◽  
Player Game

We consider a two-player game in which the first player (the Guesser) tries to guess, edge-by-edge, the path that second player (the Chooser) takes through a directed graph. At each step, the Guesser makes a wager as to the correctness of her guess and receives a payoff proportional to her wager if she is correct. We derive optimal strategies for both players for various classes of graphs, and we describe the Markov-chain dynamics of the game under optimal play. These results are applied to the infinite-duration Lying Oracle Game, in which the Guesser must use information provided by an unreliable Oracle to predict the outcome of a coin toss.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

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