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Article overview
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Analytic formulae for random walks on stochastic uniform growth trees | | Fei Ma
; Ping Wang
; | | Date: |
28 Oct 2020 | | Abstract: | Random walk, as a representation depicting discrete-time unbiased Markov
process, has attracted increasing attention in the past. The most important in
studying random walk on networks is to measure a structural parameter called
mean first-passage time, denote by $overline{mathcal{F}}$. As known, the
commonly-utilized methods for determining $overline{mathcal{F}}$ are mainly
based on Laplacian spectra on networks under consideration. On the other hand,
methods of this type can become prohibitively complicated and even fail to work
in the worst case where the corresponding Laplacian matrix is difficult to
describe in the first place.
In this paper, we will propose an effective approach to addressing this kind
of issues on some tree networks, such as, Vicsek fractal, with intriguing
structural properties, for instance, fractal feature. As opposed to most of
previous work focusing on estimating $overline{mathcal{F}}$ on growth trees
that share deterministic structure, our goal is to consider stochastic cases
where probability is introduced into the process of growing trees. To this end,
we first build up a general formula between Wiener index, denoted by
$mathcal{W}$, and $overline{mathcal{F}}$ on a tree. This enables us to
convert issues to answer into calculation of $mathcal{W}$ on networks in
question, which helps us gain what we are seeking for. Additionally, it is
straightforward to obtain Kirchhoff index on our tree networks using the
approach proposed instead of spectral technique. As an immediate consequence,
the previously published results in deterministic cases are easily covered by
formulae established in this paper. Most importantly, our approach is more
manageable than many other methods including spectral technique in situations
considered herein. | | Source: | arXiv, 2010.14815 | | Services: | Forum | Review | PDF | Favorites | |
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