Advances in Applied Probability, Vol. 50, No. 1 (2018), pp. 272-301 (30 pages) We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the ...

Graph limit theory provides a rigorous framework for analysing sequences of large graphs by representing them as continuous objects known as graphons – symmetric measurable functions on the unit ...

We study the uniform random graph Cn with n vertices drawn from a subcriticai class of connected graphs. Our main result is that the rescaled graph On ${C_n}/\sqrt n $ converges to the Brownian ...

This lecture course is devoted to the study of random geometrical objects and structures. Among the most prominent models are random polytopes, random tessellations, particle processes and random ...

When the mathematicians Jeff Kahn and Gil Kalai first posed their “expectation threshold” conjecture in 2006, they didn’t believe it themselves. Their claim — a broad assertion about mathematical ...

Quantum graphs—networks composed of vertices connected by edges on which quantum wave dynamics are defined—have emerged as a versatile model for exploring the interplay between geometry, topology, and ...

According to mathematical legend, Peter Sarnak and Noga Alon made a bet about optimal graphs in the late 1980s. They’ve now both been proved wrong. It started with a bet. In the late 1980s, at a ...