Weighted distances in scale‐free preferential attachment models

Jorritsma, Joost; Komjáthy, Júlia

doi:10.1002/rsa.20947

Cited by 8 publications

(17 citation statements)

References 95 publications

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“…First-passage percolation. In view of the above discussion, we recall the static counterpart of Theorem 1.1 by the authors in [37] that generalizes earlier results on graph distances in PAMs [15,21,25]. In [37,Theorem 2.8] it is shown that, for weight distributions satisfying…”

Section: Introductionmentioning

confidence: 65%

“…Now we prove the lower bound of Theorem 1.1, i.e., we show that with probability close to one there is no too short path between u t and v t for any t ≥ t. The main contribution of this section versus existing literature, e.g. [15,21,37], is the following proposition concerning the graph distance. In its proof we develop a path-decomposition technique that uses the dynamical construction of PA t in a refined way to get strong error bounds that are summable over t ≥ t. After the notational and conceptual set-up of the argument, we state and prove some technical lemmas.…”

Section: Proof Of the Lower Boundmentioning

confidence: 74%

“…In its proof we develop a path-decomposition technique that uses the dynamical construction of PA t in a refined way to get strong error bounds that are summable over t ≥ t. After the notational and conceptual set-up of the argument, we state and prove some technical lemmas. In the end of the section, we extend Proposition 3.1 to the edge-weighted setting, using refinements of the error bounds in [37]. We abbreviate u := u t and v := v t , respectively.…”

Section: Proof Of the Lower Boundmentioning

confidence: 99%

“…while for β ∈ (0, 1) it holds that I 2 (L) < ∞ We are mostly interested in distributions that satisfy I 1 (L) = ∞, as by [37,Theorem 2.8] the typical weighted distance is already of constant order if I 1 (L) < ∞, making Theorem 1.1 a trivial statement in this case.…”

Section: Introductionmentioning

confidence: 99%

“…These PAMs are mathematically defined by Bollobás and Riordan [13], and Dereich and Mörters [22]. For an overview of rigorous results and references we refer to [32], but also to recent works on these models [8,15,21,22,23,25,37]. Since the original PAM, many variants with more involved dynamics and connection functions have been introduced.…”

Section: Introductionmentioning

confidence: 99%

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Distance evolutions in growing preferential attachment graphs

Jorritsma¹,

Komjáthy²

2020

Preprint

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We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent τ ∈ (2, 3), sample two vertices ut, vt uniformly at random when the graph has t vertices, and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in t ≥ t, called the distance evolution. We show that there is a tight strip around the function 4 log log(t)−log(1∨log(t /t))∨ 4 that the distance evolution never leaves with high probability as t tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a non-negative random variable L.

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Section: Introductionmentioning

confidence: 65%

Section: Proof Of the Lower Boundmentioning

confidence: 74%

Section: Proof Of the Lower Boundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Distance evolutions in growing preferential attachment graphs

Jorritsma¹,

Komjáthy²

2020

Preprint

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Long-Range First-Passage Percolation on the Torus

van der Hofstad,

Lodewijks

2024

J Stat Phys

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We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph $$\mathcal {K}_n$$ K n are embedded in the d-dimensional torus $$\mathbb T_n^d$$ T n d , and each edge e is assigned an independent transmission time $$T_e=\Vert e\Vert _{\mathbb T_n^d}^\alpha E_e$$ T e = ‖ e ‖ T n d α E e , where $$E_e$$ E e is a rate-one exponential random variable associated with the edge e, $$\Vert \cdot \Vert _{\mathbb T_n^d}$$ ‖ · ‖ T n d denotes the torus-norm, and $$\alpha \ge 0$$ α ≥ 0 is a parameter. We are interested in the case $$\alpha \in [0,d)$$ α ∈ [ 0 , d ) , which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $$\mathbb {Z}^d$$ Z d studied by Chatterjee and Dey [14], and which extends first-passage percolation on the complete graph (the $$\alpha =0$$ α = 0 case) studied by Janson [24]. We consider the typical distance, flooding time, and diameter of the model. Our results show a 1, 2, 3-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on $$\mathbb {Z}^d$$ Z d .

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A network community detection method with integration of data from multiple layers and node attributes

Reittu

Leskelä

Räty

2023

Net Sci

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Multilayer networks are in the focus of the current complex network study. In such networks, multiple types of links may exist as well as many attributes for nodes. To fully use multilayer—and other types of complex networks in applications, the merging of various data with topological information renders a powerful analysis. First, we suggest a simple way of representing network data in a data matrix where rows correspond to the nodes and columns correspond to the data items. The number of columns is allowed to be arbitrary, so that the data matrix can be easily expanded by adding columns. The data matrix can be chosen according to targets of the analysis and may vary a lot from case to case. Next, we partition the rows of the data matrix into communities using a method which allows maximal compression of the data matrix. For compressing a data matrix, we suggest to extend so-called regular decomposition method for non-square matrices. We illustrate our method for several types of data matrices, in particular, distance matrices, and matrices obtained by augmenting a distance matrix by a column of node degrees, or by concatenating several distance matrices corresponding to layers of a multilayer network. We illustrate our method with synthetic power-law graphs and two real networks: an Internet autonomous systems graph and a world airline graph. We compare the outputs of different community recovery methods on these graphs and discuss how incorporating node degrees as a separate column to the data matrix leads our method to identify community structures well-aligned with tiered hierarchical structures commonly encountered in complex scale-free networks.

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Weighted distances in scale‐free preferential attachment models

Cited by 8 publications

References 95 publications

Distance evolutions in growing preferential attachment graphs

Distance evolutions in growing preferential attachment graphs

Long-Range First-Passage Percolation on the Torus

A network community detection method with integration of data from multiple layers and node attributes

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