scholarly journals The Diameter and Laplacian Eigenvalues of Directed Graphs

10.37236/1142 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Fan Chung
Keyword(s):  
Random Walk ◽  
Directed Graphs ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Undirected Graphs ◽  
Laplacian Eigenvalues ◽  
Strongly Connected ◽  
Perron Vector

For undirected graphs it has been known for some time that one can bound the diameter using the eigenvalues. In this note we give a similar result for the diameter of strongly connected directed graphs $G$, namely $$ D(G) \leq \bigg \lfloor {2\min_x \log (1/\phi(x))\over \log{2\over 2-\lambda}} \bigg\rfloor +1 $$ where $\lambda$ is the first non-trivial eigenvalue of the Laplacian and $\phi$ is the Perron vector of the transition probability matrix of a random walk on $G$.

Ideal flow of Markov Chain

2018 ◽  
Vol 10 (06) ◽  
pp. 1850073
Author(s):  
Kardi Teknomo
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Scaling Factor ◽  
Matrix Perturbation ◽  
Flow Network ◽  
Ideal Flow ◽  
The Matrix ◽  
Strongly Connected ◽  
Flow Equilibrium ◽  
Matrix Perturbation Analysis

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

scholarly journals Stochastic LU factorizations, Darboux transformations and urn models

2018 ◽  
Vol 55 (3) ◽  
pp. 862-886 ◽  
Author(s):  
F. Alberto Grünbaum ◽  
Manuel D. de la Iglesia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Parameter ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Urn Models ◽  
Triangular Matrices ◽  
New Family ◽  
Step Transition

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

scholarly journals A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Iwao Sato
Keyword(s):  
Random Walk ◽  
Zeta Function ◽  
Characteristic Polynomial ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Regular Graphs ◽  
Correlated Random Walk ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph. 

scholarly journals A Random Walk-Based Energy-Aware Compressive Data Collection for Wireless Sensor Networks

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Keming Dong ◽  
Chao Chen ◽  
Xiaohan Yu
Keyword(s):  
Energy Efficiency ◽  
Wireless Sensor Networks ◽  
Markov Chain ◽  
Random Walk ◽  
Sensor Networks ◽  
Data Collection ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Wireless Sensor ◽  
Expected Cost

The energy efficiency for data collection is one of the most important research topics in wireless sensor networks (WSNs). As a popular data collection scheme, the compressive sensing- (CS-) based data collection schemes own many advantages from the perspectives of energy efficiency and load balance. Compared to the dense sensing matrices, applications of the sparse random matrices are able to further improve the performance of CS-based data collection schemes. In this paper, we proposed a compressive data collection scheme based on random walks, which exploits the compressibility of data vectors in the network. Each measurement was collected along a random walk that is modeled as a Markov chain. The Minimum Expected Cost Data Collection (MECDC) scheme was proposed to iteratively find the optimal transition probability of the Markov chain such that the expected cost of a random walk could be minimized. In the MECDC scheme, a nonuniform sparse random matrix, which is equivalent to the optimal transition probability matrix, was adopted to accurately recover the original data vector by using the nonuniform sparse random projection (NSRP) estimator. Simulation results showed that the proposed scheme was able to reduce the energy consumption and balance the network load.

scholarly journals Correlated random walks with stay

1988 ◽  
Vol 1 (3) ◽  
pp. 197-222
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Equilibrium Solution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Finite State ◽  
Direction Of Movement ◽  
One Step ◽  
Bivariate Markov Chain

A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability distribution. In a correlated random walk (CRW) the movement follows a Markov chain and induces correlation in the state of the walk at various epochs. Then, the walk can be modelled as a bivariate Markov chain with the location of the particle and the direction of movement as the two variables. In such random walks, normally, the particle is not allowed to stay at one location from one step to the next. In this paper we derive explicit results for the following characteristics of the CRW when it is allowed to stay at the same location, directly from its transition probability matrix: (i) equilibrium solution and the fast passage probabilities for the CRW restricted on one side, and (ii) equilibrium solution and first passage characteristics for the CRW restricted on bath sides (i.e., with finite state space).

STRING EDIT DISTANCE, RANDOM WALKS AND GRAPH MATCHING

2004 ◽  
Vol 18 (03) ◽  
pp. 315-327 ◽  
Author(s):  
ANTONIO ROBLES-KELLY ◽  
EDWIN R. HANCOCK
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Steady State ◽  
Adjacency Matrix ◽  
Transition Matrix ◽  
Edit Distance ◽  
Graph Matching ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
String Edit Distance

This paper shows how the eigenstructure of the adjacency matrix can be used for the purposes of robust graph matching. We commence from the observation that the leading eigenvector of a transition probability matrix is the steady state of the associated Markov chain. When the transition matrix is the normalized adjacency matrix of a graph, then the leading eigenvector gives the sequence of nodes of the steady state random walk on the graph. We use this property to convert the nodes in a graph into a string where the node-order is given by the sequence of nodes visited in the random walk. We match graphs represented in this way, by finding the sequence of string edit operations which minimize edit distance.

scholarly journals Incorporating Reverse Search for Friend Recommendation with Random Walk

2020 ◽  
Vol 17 (3) ◽  
pp. 291-298
Author(s):  
Qing Yang ◽  
Haiyang Wang ◽  
Mengyang Bian ◽  
Yuming Lin ◽  
Jingwei Zhang
Keyword(s):  
Social Networks ◽  
Random Walk ◽  
Social Relationships ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Forward Search ◽  
Search Mode ◽  
Friend Recommendation ◽  
Proposed Model ◽  
Better Than

Recommending friends is an important mechanism for social networks to enhance their vitality and attractions to users. The huge user base as well as the sparse user relationships give great challenges to propose friends on social networks. Random walk is a classic strategy for recommendations, which provides a feasible solution for the above challenges. However, most of the existing recommendation methods based on random walk are only weighing the forward search, which ignore the significance of reverse social relationships. In this paper, we proposed a method to recommend friends by integrating reverse search into random walk. First, we introduced the FP-Growth algorithm to construct both web graphs of social networks and their corresponding transition probability matrix. Second, we defined the reverse search strategy to include the reverse social influences and to collaborate with random walk for recommending friends. The proposed model both optimized the transition probability matrix and improved the search mode to provide better recommendation performance. Experimental results on real datasets showed that the proposed method performs better than the naive random walk method which considered the forward search mode only.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

Further enhancement on H∞ sliding mode control design for singular Markovian jump systems with partly known transition probabilities

2021 ◽  
pp. 107754632198920
Author(s):  
Zeinab Fallah ◽  
Mahdi Baradarannia ◽  
Hamed Kharrati ◽  
Farzad Hashemzadeh
Keyword(s):  
Transition Probabilities ◽  
Sliding Mode ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Linear Matrix ◽  
Sliding Mode Controller ◽  
Sliding Surface ◽  
Markovian Jump ◽  
Markovian Jump System ◽  
Singular Markovian Jump Systems

This study considers the designing of the H ∞ sliding mode controller for a singular Markovian jump system described by discrete-time state-space realization. The system under investigation is subject to both matched and mismatched external disturbances, and the transition probability matrix of the underlying Markov chain is considered to be partly available. A new sufficient condition is developed in terms of linear matrix inequalities to determine the mode-dependent parameter of the proposed quasi-sliding surface such that the stochastic admissibility with a prescribed H ∞ performance of the sliding mode dynamics is guaranteed. Furthermore, the sliding mode controller is designed to assure that the state trajectories of the system will be driven onto the quasi-sliding surface and remain in there afterward. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design algorithms.

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