### How to stop explosion by penalising transmission to hubs

Probability Seminar

25th October 2019, 3:30 pm – 4:30 pm

Fry Building, LG.22

In this talk we study the spread of information on infinite inhomogeneous spatial random graphs.

We take a scale-free spatial random graph, where the degree of a vertex follows a power law with exponent tau >1. Then we equip each edge with a random and iid transmission delay L, and study the ball-growth of the first-passage infected cluster around the source vertex as a function of time. For a second, more realistic spreading model, the iid random transmission delay L through an edge with expected degrees W and Z is multiplied by a factor that is a polynomial of W,Z, (the penalty factor).

We call the model outwards (inwards) explosive if it is possible to reach infinitely many vertices within finite time (if infinitely many vertices can reach a target vertex within finite time).

We will discuss the criterion for explosion in the original model (no penalty factor) and in the penalised model. In particular, we will discuss that asymmetric penalty functions can lead to `outwards' explosion but no `inwards’ explosion or the other way round.

Joint work with John Lapinskas and Johannes Lengler.

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