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Abstract
We consider a model for complex networks that was introduced by Krioukov et al.
In this model, N points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance.
The N points are distributed according to a quasi-uniform distribution, which is a distorted version of the uniform distribution.
The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges.
Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering.
The model is controlled by two parameters \alpha and \nu where, roughly speaking, \alpha controls the exponent of the power-law and \nu controls the average degree. The present paper focuses on the evolution of the component structure of the random graph.
We show that (a) for \alpha > 1 and \nu arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for \alpha < 1 and \nu arbitrary with high probability there is
a "giant'' component of linear order,
and (c) when \alpha=1 then there is a non-trivial phase transition for the existence of a linear-sized component in terms of \nu.
Original language | English |
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Article number | P3.24 |
Number of pages | 46 |
Journal | The Electronic Journal of Combinatorics |
Volume | 22 |
Issue number | 3 |
Early online date | 14 Aug 2015 |
Publication status | Published - 14 Aug 2015 |
Keywords
- giant component
- random graphs on the hyperbolic plane
- component structure
- Complex networks
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Dive into the research topics of 'On the largest component of a hyperbolic model of complex networks'. Together they form a unique fingerprint.Projects
- 1 Finished
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FP7 MC CIG - HYPERBOLIC GRAPHS: Hyperbolic random graphs
Fountoulakis, N. (Principal Investigator)
1/09/11 → 31/08/15
Project: EU