The Annals of Applied Probability

Spatial Gibbs random graphs

Jean-Christophe Mourrat and Daniel Valesin

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Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with small average graph distance between vertices, but adding an edge comes at a cost measured according to the geometry of the ambient physical space. In most cases, we identify the order of magnitude of the average graph distance as a function of the parameters of the model. As the proofs reveal, hierarchical structures naturally emerge from our simple modeling assumptions. Moreover, a critical regime exhibits an infinite number of discontinuous phase transitions.

Article information

Ann. Appl. Probab., Volume 28, Number 2 (2018), 751-789.

Received: June 2016
Revised: February 2017
First available in Project Euclid: 11 April 2018

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Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35] 05C80: Random graphs [See also 60B20]

Spatial random graph Gibbs measure phase transition


Mourrat, Jean-Christophe; Valesin, Daniel. Spatial Gibbs random graphs. Ann. Appl. Probab. 28 (2018), no. 2, 751--789. doi:10.1214/17-AAP1316.

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