Nonlocal pagerank

Cipolla, Stefano; Durastante, Fabio; Tudisco, Francesco

doi:10.1051/m2an/2020071

Cited by 14 publications

(9 citation statements)

References 60 publications

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“…we retrieve the Mellin transformed and the Laplace transformed path graph Laplacians, respectively, introduced by E. Estrada. See [18,20,21,34], and [13,19] for the most recent developments. We will call ∆ α a path graph Laplacian.…”

Section: Path and Fractional Graph Laplaciansmentioning

confidence: 99%

Compatibility, embedding and regularization of non-local random walks on graphs

Bianchi,

Donatelli,

Durastante

et al. 2021

Preprint

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Several variants of the graph Laplacian have been introduced to model non-local diffusion processes, which allow a random walker to "jump" to non-neighborhood nodes, most notably the transformed path graph Laplacians and the fractional graph Laplacian. From a rigorous point of view, this new dynamics is made possible by having replaced the original graph G with a weighted complete graph G on the same node-set, that depends on G and wherein the presence of new edges allows a direct passage between nodes that were not neighbors in G.We show that, in general, the graph G is not compatible with the dynamics characterizing the original model graph G: the random walks on G subjected to move on the edges of G are not stochastically equivalent, in the wide sense, to the random walks on G. From a purely analytical point of view, the incompatibility of G with G means that the normalized graph Ĝ can not be embedded into the normalized graph Ĝ . Eventually, we provide a regularization method to guarantee such compatibility and preserving at the same time all the nice properties granted by G .

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Section: Path and Fractional Graph Laplaciansmentioning

confidence: 99%

Compatibility, embedding and regularization of non-local random walks on graphs

Bianchi,

Donatelli,

Durastante

et al. 2021

Preprint

Self Cite

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“…For the remaining part of the manuscript, we will focus on the properties of the underlying fractional extension. We refer to [15] for some comparison of the two approaches, and to [12] for a discussion on using different type of series and distances on the graph G.…”

Section: Non-local Navigation Strategiesmentioning

confidence: 99%

Nonlocal diffusion of variable order on graphs

Bertaccini,

Durastante

2021

Preprint

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Some aspects of nonlocal dynamics on directed and undirected networks for an initial value problem whose Jacobian matrix is a variable-order fractional power of a Laplacian matrix are discussed here. Both directed and undirected graphs are considered. Under appropriate assumptions, the existence, uniqueness, and uniform asymptotic stability of the solutions of the underlying initial value problem are proved. Some examples giving a sample of the behavior of the dynamics are also included.

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“…On the other hand, since there are some well-connected city centre stations that intersect many underground lines, we may expect traditional eigenvector centrality to display localization, where most of the centrality mass is placed on a small subset of the nodes. With this network, we also have the benefit of independent passenger usage data from [9], which gives a criterion on which to quantify success.…”

Section: Computational Examplementioning

confidence: 99%

“…We see from the figure that the measure dramatically increases in localization in the regime where backtracking is not suppressed (θ ≈ 1) and we are close to an eigenvector measure (α ≈ α ). For Figure 5 we made use of independent data from [9] that records the annual passenger usage at each station. We took data for the most recent available year, 2017.…”

Section: Computational Examplementioning

confidence: 99%

A Theory for Backtrack-Downweighted Walks

Arrigo¹,

Higham²,

Noferini³

2021

SIAM J. Matrix Anal. Appl.

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We develop a comprehensive theory for the combinatorics of walk-counting on a directed graph in the case where each backtracking step is downweighted by a given factor. By deriving expressions for the associated generating functions, we also obtain linear systems for computing centrality measures in this setting. In particular, we show that backtrack-downweighted Katz-style network centrality can be computed at the same cost as standard Katz. Studying the limit of this centrality measure at its radius of convergence also leads to a new expression for backtrack-downweighted eigenvector centrality that generalizes previous work to the case where directed edges are present. The new theory allows us to combine advantages of standard and nonbacktracking cases, avoiding certain types of localization while accounting for tree-like structures. We illustrate the behaviour of the backtrack-downweighted centrality measure on both synthetic and real networks.

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Nonlocal pagerank

Cited by 14 publications

References 60 publications

Compatibility, embedding and regularization of non-local random walks on graphs

Compatibility, embedding and regularization of non-local random walks on graphs

Nonlocal diffusion of variable order on graphs

A Theory for Backtrack-Downweighted Walks

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