Probability Seminar: Julia Komjathy

Joint work with Enrico Baroni and Remco van der Hofstad.

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following power law distribution with exponent \tau \in (2, 3). In this model two colors spread with fixed but not necessarily equal speeds on the unweighted random graph. We show that almost all vertices ultimately get the "faster" color, while only a random subpolynomial fraction of the vertices gets the "slower" color. We also show that even if the speeds are equal, there is no coexistence with high probability, and further the "loser" color paints a polynomial fraction of the vertices with a random exponent.
 
This work is the counterpart of Deijfen and van der Hofstad, where there are exponential edge weights on the same graph model. Changing the edge weights significantly changes the picture: in the exponential case, even when the speeds are unequal, the "winner" can be either of the two colors, and the "loser" can only paint a set of vertices of constant order size.