Growing Networks Through Random Walks Without Restarts

Amorim, Bernardo Nascimento de; Figueiredo, Daniel R.; Iacobelli, Giulio; Neglia, Giovanni

doi:10.1007/978-3-319-30569-1_15

Cited by 5 publications

(25 citation statements)

References 10 publications

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“…We have seen in Corollary 1 that for s = 1 the degree distribution of non-leaf vertices (i.e., vertices having degree greater than one) is bounded from above by a geometric distribution. In sharp contrast, we show that for every s even the degree distribution of non-leaf vertices is bounded from below by a power-law distribution whose exponent depends on s. The dichotomy between the exponential tail and the heavy tail for the degree distribution in NRRW for s odd and s even was already empirically observed in simulations [1]. Proposition 1.…”

Section: Degree Distribution Of a Vertex With S Evensupporting

confidence: 53%

“…Last, various properties of networks generated by NRRW were first observed by means of extensive numerical simulations [1], such as the dichotomy in the degree distribution and distance distribution as a function of the parity of s. In this paper we provide rigorous theoretical treatment for some of the results observed empirically.…”

Section: Introductionmentioning

confidence: 92%

“…The graphs shown in Figure 1 (top plots) indicate that in this scenario the trees grow in depth as the number of vertices increases. A more substantial empirical evidence of this phenomenon is provided in [1]. This suggests that the random walk is extending the tree to lower depths just never to return to its origins.…”

Section: Nrrw With Step Parameter S=1mentioning

confidence: 97%

“…A promising approach to capture such principles in a network growth model is to leverage random walks [1,5,9,11,13]. In such models, the network grows as the walker moves with both processes being mutually dependent.…”

Section: Introductionmentioning

confidence: 99%

“…In the Random Walk Model [5,11], the walker takes s steps, a new node is added and connected to the current walker position, and the random walk restarts (choosing a new node uniformly at random). The "No Restart Random Walk" (NRRW) Model [1] builds on this model by not requiring the walker to restart. In particular, NRRW can be algorithmically described as follows:…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Transient and slim versus recurrent and fat: Random walks and the trees they grow

2019

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Network growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In this paper we consider the No Restart Random Walk (NRRW) model where a walker builds its graph (tree) while moving around. In particular, the walker takes s steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another s steps, and the process repeats. We analyze this process from the perspective of the walker and the network, showing a fundamental dichotomy between transience and recurrence for the walker as well as power law and exponential degree distribution for the network. More precisely, we prove the following results: s = 1: the random walk is transient and the degree of every node is bounded from above by a geometric distribution. s even: the random walk is recurrent and the degree of non-leaf nodes is bounded from below by a power law distribution with exponent decreasing in s. We also provide a lower bound for the fraction of leaves in the graph, and for s = 2 our bound implies that the fraction of leaves goes to one as the graph size goes to infinity. NRRW exhibits an interesting mutual dependency between graph building and random walking that is fundamentally influenced by the parity of s. Understanding this kind of coupled dynamics is an important step towards modeling more realistic network growth processes.

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Section: Degree Distribution Of a Vertex With S Evensupporting

confidence: 53%

Section: Introductionmentioning

confidence: 92%

Section: Nrrw With Step Parameter S=1mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transient and slim versus recurrent and fat: Random walks and the trees they grow

2019

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Theory of chain walking catalysis: From disordered dendrimers to dendritic bottle-brushes

Dockhorn

Sommer

2022

The Journal of Chemical Physics

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The chain walking (CW) polymerization technique has the unique property of a movable catalyst synthesizing its own path by creating branch-on-branch structures. By successive attachment of monomers, the resulting architecture ranges from dendritic to linear growth depending on the walking rate, which is defined by the ratio of walking steps and reaction events of the catalyst. The transition regime is characterized by local dendritic sub-structures (dendritic blobs) and a global linear chain feature forming a dendritic bottle-brush. A scaling model for structures obtained by CW catalysis is presented and validated by computer simulation relating the extensions of CW structures to the catalyst’s walking ability. The limiting case of linear (low walking rate) and dendritic growth (high walking rate) is recovered, and the latter is shown to bear analogies to the Barabási–Albert graph and Bernoulli growth random walk. We could quantify the size of the dendritic blob as a function of the walking rate by using spectral properties of the connectivity matrix of the simulated macromolecules. This allows us to fit the numerical constants in the scaling approach. We predict that independent of the underlying chemical process, all CW polymerization syntheses involving a highly mobile catalyst ultimately result in bottle-brush structures whose properties depend on a unique parameter: the walking rate.

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On a random walk that grows its own tree

Figueiredo¹,

Iacobelli²,

Oliveira³

et al. 2021

Electron. J. Probab.

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Random walks on dynamic graphs have received increasingly more attention from different academic communities over the last decade. Despite the relatively large literature, little is known about random walks that construct the graph where they walk while moving around. In this paper we study one of the simplest conceivable discrete time models of this kind, which works as follows: before every walker step, with probability p a new leaf is added to the vertex currently occupied by the walker.The model grows trees and we call it the Bernoulli Growth Random Walk (BGRW). We show that the BGRW walker is transient and has a well-defined linear speed c(p) > 0 for any 0 < p ≤ 1. Moreover, we show that the tree as seen by the walker converges (in a suitable sense) to a random tree that is one-ended. Some natural open problems about this tree and variants of our model are collected at the end of the paper.

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Growing Networks Through Random Walks Without Restarts

Cited by 5 publications

References 10 publications

Transient and slim versus recurrent and fat: Random walks and the trees they grow

Transient and slim versus recurrent and fat: Random walks and the trees they grow

Theory of chain walking catalysis: From disordered dendrimers to dendritic bottle-brushes

On a random walk that grows its own tree

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