Spatial age-dependent random graphs
Final Report Abstract
In this project we have investigated qualitative mathematical models, which include two essential features of many large networks arising in nature, science or technology: On the one hand the occasional occurrence of powerful nodes with an extremely large number of adjacent links, and on the other hand the increased tendency of similar nodes to establish a link. The models are based on embedding nodes into space and making the existence of links dependent on features of the nodes themselves, such as their age and weight, as well as their distance in space. We have given sharp conditions when such a network is robust under random attack and which qualitative network features lead to robustness. We have analysed how many links are typically needed to connect two spatially remote nodes of the network. Finally, we have investigated in several different models how fast infections, or indeed information, can spread on such a network and how many nodes will be infected in case of an outbreak.
Publications
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Percolation phase transition in weight-dependent random connection models. Advances in Applied Probability, 53(4), 1090-1114.
Gracar, Peter; Lüchtrath, Lukas & Mörters, Peter
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The contact process on random hyperbolic graphs: Metastability and critical exponents. The Annals of Probability, 49(3).
Linker, Amitai; Mitsche, Dieter; Schapira, Bruno & Valesin, Daniel
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Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution. Communications in Mathematical Physics, 395(2), 859-906.
Gracar, Peter; Grauer, Arne & Mörters, Peter
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Percolation in weight-dependent random connection models. PhD Thesis, Uni Köln
Lukas Lüchtrath
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The occurrence and effects of short paths in scalefree geometric random graphs. PhD Thesis, Uni Köln
Arne Grauer
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The contact process on scale-free geometric random graphs. Stochastic Processes and their Applications, 173, 104360.
Gracar, Peter & Grauer, Arne
