Average Distance in Growing Trees

We show that the structure of a growing tree preserves an information on the shape of an initial graph. For the exponential trees, evidence of this kind of memory is provided by means of the iterative equations, derived for the moments of the node-node distance distribution. Numerical calculations confirm the result and allow to extend the conclusion to the Barabasi--Albert scale-free trees. The memory effect almost disappears, if subsequent nodes are connected to the network with more than one link.Comment: 9 pages, 9 figure

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“…2. An example of the construction S for trees (m = 1) is given in [10]. for exponential trees and for scale-free trees, respectively.…”

Section: Numerical Algorithmmentioning

confidence: 99%

Memory effect in growing trees

Malarz

Kułakowski

2005

Physica A: Statistical Mechanics and its Applications

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“…The site coordination number z varies for each site with a distribution which depends mainly on the network growth rules. For example, when subsequent sites are attached randomly to already existing ones the site degree becomes exponential [12,14]. When the attachment is preferential [12,15] the distribution follows a power law, and for so-called classical random graphs it is given by a Poisson distribution [12,16].…”

mentioning

confidence: 99%

Restoring site percolation on damaged square lattices

Malarz

2005

Phys. Rev. E

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Restoring site percolation on a damaged square lattice with the nearest neighbor (N2) is investigated using two different strategies. In the first one, a density y of new sites are created on the empty sites with longer range links, either next-nearest neighbor (N3) or next-next-nearest neighbor (N4), but without N2. In the second one, new longer range links N3 or N4 are added to N2 but only for a fraction v of the remaining non-destroyed sites. Starting at p(c)(N2), with a density x of randomly destroyed sites, the values of y(c) and v(c), which restore site percolation, are calculated for both strategies with, respectively, N3 and N4 using Monte Carlo simulations. Results are obtained for the whole range 0 < or = x < or = p(c)(N2).

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“…also asymptotically grows as d n ∼ log n for growing trees [8,38,39] and as d n ∼ √ n for aged trees [26]. The sum in the last equation is over all pairs i, j of vertices of the tree.…”

Section: Tree Ageingmentioning

confidence: 99%

Ageing of complex networks

2020

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Many real-world complex networks arise as a result of a competition between growth and rewiring processes. Usually the initial part of the evolution is dominated by growth while the later one rather by rewiring. The initial growth allows the network to reach a certain size while rewiring to optimise its function and topology. As a model example we consider tree networks which first grow in a stochastic process of node attachment and then age in a stochastic process of local topology changes. The ageing is implemented as a Markov process that preserves the node-degree distribution. We quantify differences between the initial and aged network topologies and study the dynamics of the evolution. We implement two versions of the ageing dynamics. One is based on reshuffling of leaves and the other on reshuffling of branches. The latter one generates much faster ageing due to non-local nature of changes.

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Average Distance in Growing Trees

Cited by 9 publications

References 6 publications

Memory effect in growing trees

Memory effect in growing trees

Restoring site percolation on damaged square lattices

Ageing of complex networks

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