A Phase Transition for the Diameter of the Configuration Model

Hofstad, RW Remco van der; Hooghiemstra, Gerard; Znamenski, Dmitri

doi:10.1080/15427951.2007.10129138

Cited by 9 publications

(1 citation statement)

References 33 publications

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“…in the configuration model with i.i.d. degrees [22] and Barabási-Albert preferential attachment model [11]. This suggests the time complexity of the algorithm in applications to be O(N 3 log N ), deeming it practical for shape comparison for graphs of up to a moderate order.…”

Section: Algorithm For Estimating D Gh (X Y )mentioning

confidence: 99%

Efficient estimation of a Gromov--Hausdorff distance between unweighted graphs

Oles¹,

Lemons²,

Panchenko³

2019

Preprint

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Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-hard optimization problem, deeming the notion impractical for applications. In this paper we propose a polynomial algorithm for estimating the so-called modified Gromov-Hausdorff (mGH) distance, whose topological equivalence with the standard Gromov-Hausdorff (GH) distance was established in [36] (Mémoli, F, Discrete & Computational Geometry, 48 (2) 2012). We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library scikit-tda, and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world social and computer networks.

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Section: Algorithm For Estimating D Gh (X Y )mentioning

confidence: 99%

Efficient estimation of a Gromov--Hausdorff distance between unweighted graphs

Oles¹,

Lemons²,

Panchenko³

2019

Preprint

View full text Add to dashboard Cite

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Diameter in ultra‐small scale‐free random graphs

Caravenna

Garavaglia

Hofstad

2018

Random Struct Algorithms

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It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k −( τ − 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to and , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

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Diameters in Preferential Attachment Models

2010

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In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent τ > 2.We prove that the diameter of the PA-model is bounded above by a constant times log t, where t is the size of the graph. When the power-law exponent τ exceeds 3, then we prove that log t is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for τ > 3, distances are of the order log t. For τ ∈ (2, 3), we improve the upper bound to a constant times log log t, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order log log t.These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order log log t when τ ∈ (2, 3), and of order log t when τ > 3.

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A Phase Transition for the Diameter of the Configuration Model

Cited by 9 publications

References 33 publications

Efficient estimation of a Gromov--Hausdorff distance between unweighted graphs

Efficient estimation of a Gromov--Hausdorff distance between unweighted graphs

Diameter in ultra‐small scale‐free random graphs

Diameters in Preferential Attachment Models

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