From trees to graphs: collapsing continuous-time branching processes

Garavaglia, Alessandro; Hofstad, Remco van der

doi:10.1017/jpr.2018.57

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…This construction is already used in [5,6,32,48]. The embedding holds for any m ≥ 2, but the topological description of the graph as a CTBP is used only in [32].…”

Section: Garavaglia Van Der Hofstad Litvakmentioning

confidence: 99%

Local weak convergence for PageRank

Garavaglia

Hofstad

Litvak³

2020

Ann. Appl. Probab.

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PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called 'power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees. Up to date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in-and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as preferential attachment models.

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“…This construction is already used in [5,6,32,48]. The embedding holds for any m ≥ 2, but the topological description of the graph as a CTBP is used only in [32].…”

Section: Garavaglia Van Der Hofstad Litvakmentioning

confidence: 99%

Local weak convergence for PageRank

Garavaglia

Hofstad

Litvak³

2020

Ann. Appl. Probab.

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“…The above construction is reminiscent of the collapsed continuous-time branching processes defined in [GvdH18]. However, the two constructions are different as the newly born vertices in (τ l , τ l+1 ) are not allowed to reproduce in the above construction, while they are in [GvdH18].…”

Section: Branching Process Embeddingmentioning

confidence: 99%

Degree centrality and root finding in growing random networks

Banerjee¹,

Huang²

2021

Preprint

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We consider growing random networks {Gn} n≥1 where, at each time, a new vertex attaches itself to a collection of existing vertices via a fixed number m ≥ 1 of edges, with probability proportional to a function f (called attachment function) of their degrees. It was shown in [BB21] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding to ∞ i=1 f (i) −2 < ∞, where the top K maximal degree vertices fixate over time for any given K, and (ii) the non-persistent regime, with ∞ i=1 f (i) −2 = ∞, where the identities of these vertices keep changing infinitely often over time. In this article, we develop root finding algorithms based on the empirical degree structure of a snapshot of such a network at some large time. In the persistent regime, the algorithm is purely based on degree centrality, that is, for a given error tolerance ε ∈ (0, 1), there exists Kε ∈ N such that for any n ≥ 1, the confidence set for the root in Gn, which contains the root with probability at least 1 − ε, consists of the top Kε maximal degree vertices. In particular, the size of the confidence set is stable in the network size. Upper and lower bounds on Kε are explicitly characterized in terms of the error tolerance ε and the attachment function f . In the non-persistent regime, analogous algorithms are developed based on centrality measures where one assigns to each vertex v in Gn the maximal degree among vertices in a neighborhood of v of radius rn, where rn is much smaller than the diameter of the network. A bound on the size of the associated confidence set is also obtained, and it is shown that, when f (k) = k α , k ≥ 1, for any α ∈ (0, 1/2], this size grows at a smaller rate than any positive power of the network size.

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“…The downside of this approach is that applications are restricted to trees, something that is not always realistic. However, in a recent paper [6] the authors shows a way of collapsing a branching process resulting in a (specific) preferential attachment network, while still being able to apply powerful branching process results. This is done for the single type case with affine rate functions, and extensions to the multi-type case seems possible.…”

Section: Introductionmentioning

confidence: 99%

A Multi-type Preferential Attachment Tree

Rosengren¹

2018

Internet Mathematics

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A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the p-type case we derive a framework for studying the tree where a type i vertex generates new type j vertices with rate w ij (n 1 , n 2 , . . . , np) where n k is the number of type k vertices previously generated by the type i vertex, and w ij is a non-negative function from N p to R. The framework is then used to derive results for trees with more specific attachment rates.In the case with linear preferential attachment-where type i vertices generate new type j vertices with rate w ij (n 1 , n 2 , . . . , np) = γ ij (n 1 + n 2 + · · · + np) + β ij , where γ ij and β ij are positive constants-we show that under mild regularity conditions on the parameters {γ ij }, {β ij } the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.Multi-type preferential attachment; preferential attachment tree; multi-type general branching process; power law degree distribution; asymptotic composition.

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From trees to graphs: collapsing continuous-time branching processes

Cited by 6 publications

References 27 publications

Local weak convergence for PageRank

Local weak convergence for PageRank

Degree centrality and root finding in growing random networks

A Multi-type Preferential Attachment Tree

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