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Percolation phase transition in weight-dependent random connection models

Part of: Graph theory Special processes

Published online by Cambridge University Press:  22 November 2021

Peter Gracar Open the ORCID record for Peter Gracar [Opens in a new window] ,
Lukas Lüchtrath Open the ORCID record for Lukas Lüchtrath [Opens in a new window]  and
Peter Mörters Open the ORCID record for Peter Mörters [Opens in a new window]
Peter Gracar*
Affiliation:
University of Cologne
Lukas Lüchtrath*
Affiliation:
University of Cologne
Peter Mörters*
Affiliation:
University of Cologne
*
*Postal address: Weyertal 86-90, 50931 Köln, Germany.
*Postal address: Weyertal 86-90, 50931 Köln, Germany.
*Postal address: Weyertal 86-90, 50931 Köln, Germany.
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Abstract

We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

Keywords

Subcritical regimerandom geometric graphBoolean modelscale-free percolationlong-range percolationspatial networkrobustnessage-based spatial preferential attachment

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1090 - 1114
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Prob. 6, paper no. 23, 13 pp.CrossRefGoogle Scholar
Van den Berg, J. (1996). A note on disjoint-occurrence inequalities for marked Poisson point processes. J. Appl. Prob. 33, 420–426.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular variation. Encyclopaedia Math. Appl. Cambridge University Press.Google Scholar
Deijfen, M., van der Hofstad, R. and Hooghiemstra, G. (2013). Scale-free percolation. Ann. Inst. H. Poincaré Prob. Statist. 49, 817–838.CrossRefGoogle Scholar
Deprez, P., Subhra Hazra, R. and Wüthrich, M. V. (2015). Inhomogeneous long-range percolation for real-life network modeling. Risks 3, 1–23.CrossRefGoogle Scholar
Deprez, P. and Wüthrich, M. V. (2019). Scale-free percolation in continuum space. Commun. Math. Statist. 7, 269–308.CrossRefGoogle Scholar
Gouéré, J.-B. (2008). Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Prob. 36, 1209–1220.CrossRefGoogle Scholar
Gouéré, J.-B. (2009). Subcritical regimes in some models of continuum percolation. Ann. Appl. Prob. 19, 1292–1318.CrossRefGoogle Scholar
Gracar, P., Grauer, A., Lüchtrath, L. and Mörters, P. (2019). The age-dependent random connection model. Queueing Systems 93, 309–331.CrossRefGoogle Scholar
Gracar, P., Heydenreich, M., Mönch, C. and Mörters, P. (2019). Recurrence versus transience for weight-dependent random connection models. Preprint. Available at https://arxiv.org/abs/1911.04350.Google Scholar
Heydenreich, M., van der Hofstad, R., Last, G. and Matzke, K. (2019). Lace expansion and mean-field behavior for the random connection model. Preprint. Available at https://arxiv.org/abs/1908.11356.Google Scholar
Heydenreich, M., Hulshof, T. and Jorritsma, J. (2017). Structures in supercritical scalefree percolation. Ann. Appl. Prob. 27, 2569–2604.CrossRefGoogle Scholar
Jacob, E. and Mörters, P. (2015). Spatial preferential attachment networks: power laws and clustering coefficients. Ann. Appl. Prob. 25, 632–662.CrossRefGoogle Scholar
Jacob, E. and Mörters, P. (2017). Robustness of scale-free spatial networks. Ann. Prob. 45, 1680–1722.CrossRefGoogle Scholar
Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge University Press.CrossRefGoogle Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.CrossRefGoogle Scholar
Newman, C. M. and Schulman, L. S. (1986). One-dimensional $1/\vert j-i\vert^s$ percolation models: the existence of a transition for $s\leq 2$. Commun. Math. Phys. 104, 547–571.CrossRefGoogle Scholar
Penrose, M. D. (1991). On a continuum percolation model. Adv. Appl. Prob. 23, 536–556.CrossRefGoogle Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar