On the second eigenvalue and random walks in randomd-regular graphs

COMBINATORICA ◽  
1991 ◽  
Vol 11 (4) ◽  
pp. 331-362 ◽  
Author(s):  
Joel Friedman
Keyword(s):  
Random Walks ◽  
Regular Graphs ◽  
Second Eigenvalue

scholarly journals Cutoff phenomena for random walks on random regular graphs

2010 ◽  
Vol 153 (3) ◽  
pp. 475-510 ◽  
Author(s):  
Eyal Lubetzky ◽  
Allan Sly
Keyword(s):  
Random Walks ◽  
Regular Graphs

Potential Theory on Distance-Regular Graphs

1993 ◽  
Vol 2 (3) ◽  
pp. 243-255 ◽  
Author(s):  
Norman L. Biggs
Keyword(s):  
Random Walks ◽  
Potential Theory ◽  
Effective Resistance ◽  
Electrical Network ◽  
Regular Graphs ◽  
Regular Case ◽  
Explicit Formulae ◽  
Distance Regular Graphs ◽  
A Current

A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case. A well-known link with random walks motivates a conjecture about the maximum effective resistance. Arguments are given that point to the truth of the conjecture for all known distance-regular graphs.

scholarly journals Analytical results for the distribution of cover times of random walks on random regular graphs

2021 ◽  
Author(s):  
Ido Tishby ◽  
Ofer Biham ◽  
Eytan Katzav
Keyword(s):  
Random Walks ◽  
Cumulative Distribution ◽  
Large Network ◽  
Regular Graphs ◽  
Time Step ◽  
Single Node ◽  
Initial Node ◽  
Analytical Results ◽  
Partial Cover ◽  
Cover Times

Abstract We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of N nodes of degree c (c ≥ 3). Starting from a random initial node at time t = 1, at each time step t ≥ 2 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time TCis the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution Pt(S = s) of the number of distinct nodes s visited by an RW up to time t and solve it analytically. Inserting s = N we obtain the cumulative distribution of cover times, namely the probability P (TC ≤ t) = Pt(S = N) that up to time t an RW will visit all the N nodes in the network. Taking the large network limit, we show that P (TC ≤ t) converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times P (TPC,k = t), which is the probability that at time t an RW will complete visiting k distinct nodes. We also calculate the distribution of random cover (RC) times P (TRC,k = t), which is the probability that at time t an RW will complete visiting all the nodes in a subgraph of k randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

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