### Connectivity in spatial networks

Probability Seminar

31st March 2017, 3:40 pm – 4:40 pm

Main Maths Building, SM3

In many networks, the nodes are physically located in space, and links are more likely between close pairs of nodes. The most studied model is that of a random geometric graph, in which nodes are distributed randomly in a finite domain using a Poisson point process, and links formed for mutual distances up to a threshold. As the threshold is increased, the value at which the whole network is connected asymptotically almost surely coincides with the linking of the last isolated node. Recent work has incorporated a second source of randomness, forming links independently with a probability depending on mutual distance through a “connection function,” with specific choices arising from wireless ad-hoc network applications. Calculations show that the connection probability can be approximated using just a few moments of the connection function for a wide variety of domain geometries. Considering a number of generalisations including k-connectivity, anisotropy, inhomogeneity and fractal geometry leads to the surprising observation that more complicated models are in many respects better behaved.

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