Self-organised criticality in random graph processes

• Main Author: Yeo, Dominic
• Other Authors: Goldschmidt, Christina ; Martin, James
• Published: University of Oxford 2016
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.735872

In the first half of this thesis, we study the random forest obtained by conditioning the Erdös-Rényi random graph G(N,p) to include no cycles. We focus on the critical window, in which <style> td.upper_line { border-top:solid 1px black; } table.fraction { text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em; } </style> <table class="fraction" align="center" cellpadding="0" cellspacing="0"> <tr> <td rowspan="2" nowrap="nowrap"> <i>p(N)</i> = </td><td nowrap="nowrap"> 1+&lambda;N^-1/3 </td> </tr><tr> <td class="upper_line"> N </td> </tr> </table> , as studied by Aldous for G(N,p). We describe a scaling limit for the sizes of the largest trees in this critical random forest, in terms of the excursions above zero of a particular reflected diffusion. We proceed by showing convergence of the reflected exploration process associated to the critical random forests, using careful enumeration of classes of forests, and the asymptotic properties of uniform trees. In the second half of this thesis, we study a random graph process where vertices have one of k types. An inhomogeneous random graph represents the initial connections between vertices, and over time new edges are added homogeneously, as in the classical random graph process. Each vertex is frozen at some rate, resulting in the removal of its entire component. This is a version of the frozen percolation model introduced by R\'ath, which (under mild conditions) exhibits self-organised criticality: the dynamics first drive the system to a critical state, and from then on maintain it in criticality. We prove a convergence result for the proportion of vertices of each type which survive until time t, and describe the local limit in terms of a multitype branching process whose parameters are critical and given by the solution to an unusual differential equation driven by Perron--Frobenius eigenvectors. The argument relies on a novel multitype exploration process, leading to a concentration result for the proportion of types in all large components of a near-critical inhomogeneous random graph; and on a stronger convergence result for mean-field frozen percolation, when the initial graphs may be random.