Random graphs were introduced as a convenient example for demonstrating the impossibility of ‘complete disorder’ by Erdos, who also thought that these objects will never become useful in the applied areas outside of pure mathematics. In this talk, I will view random graphs as objects in the field of applied mathematics and discus how the application-driven objectives have set new directions for studying random graphs. I will focus on characterising the sizes of connected components in graphs with a given degree distribution, on the percolation-like processes on such structures, and on generalisations to the coloured graphs. These theoretical questions have interesting implications for studying resilience of networks with nontrivial structures, and for materials science where they explain kinetics-driven phase transitions. Even more surprisingly, the results reveal intricate connections between random graphs and non-linear partial differential equations indicating new possibilities for their analysis.
