EIGENTIME IDENTITIES OF FRACTAL FLOWER NETWORKS

The eigentime identity for random walks on networks is the expected time for a walker going from a node to another node. In this paper, our purpose is to calculate the eigentime identities of flower networks by using the characteristic polynomials of normalized Laplacian and recurrent structure of Markov spectrum.

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Eigentime identity of the weighted (m,n)-flower networks

International Journal of Modern Physics B ◽

10.1142/s0217979220501593 ◽

2020 ◽

Vol 34 (18) ◽

pp. 2050159

Author(s):

Changxi Dai ◽

Meifeng Dai ◽

Tingting Ju ◽

Xiangmei Song ◽

Yu Sun ◽

...

Keyword(s):

Random Walks ◽

Characteristic Polynomial ◽

Network Size ◽

Weight Factor ◽

Laplacian Eigenvalues ◽

Weighted Networks ◽

Expected Time ◽

Leading Term ◽

Analytic Expression ◽

Markov Spectrum

The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted [Formula: see text]-flower networks with the weight factor [Formula: see text]. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.

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Many Random Walks Are Faster Than One

Combinatorics Probability Computing ◽

10.1017/s0963548311000125 ◽

2011 ◽

Vol 20 (4) ◽

pp. 481-502 ◽

Cited By ~ 42

Author(s):

NOGA ALON ◽

CHEN AVIN ◽

MICHAL KOUCKÝ ◽

GADY KOZMA ◽

ZVI LOTKER ◽

...

Keyword(s):

Distributed Computing ◽

Random Walks ◽

Efficient Algorithms ◽

Large Collection ◽

Cover Time ◽

Expected Time ◽

Probabilistic Distribution ◽

Random Walks On Graphs ◽

Speed Up ◽

Time Required

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

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Intelligibility and first passage times in complex urban networks

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.0329 ◽

2008 ◽

Vol 464 (2096) ◽

pp. 2153-2167 ◽

Cited By ~ 17

Author(s):

Philippe Blanchard ◽

Dimitri Volchenkov

Keyword(s):

Random Walks ◽

First Passage Time ◽

Local Property ◽

Passage Time ◽

Urban Environments ◽

Recurrence Time ◽

First Passage ◽

Graph Representations ◽

Expected Time ◽

Urban Patterns

Topology of urban environments can be represented by means of graphs. We explore the graph representations of several compact urban patterns by random walks. The expected time of recurrence and the expected first passage time to a node scales apparently linearly in all urban patterns we have studied. In space syntax theory, a positive relation between the local property of a node (qualified by connectivity or by the recurrence time) and the global property of the node (estimated in our approach by the first passage time to it) is known as intelligibility. Our approach, based on random walks, allows us to extend the notion of intelligibility onto the entire domain of complex networks and graph theory.

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