Directed Forest Fire Models

The mean field forest fire model is an abstract mathematical model introduced by Rath and Toth (EJP, 2009) to study the phenomenon of self-organized criticality (SOC) by a rigorous mathematical approach. It is an example of a random fragmentation-coalescence process out of equilibrium. It is an evolving random subgraph of the complete graph Kn, in which each possible undirected edge is added to the graph rate n-1, independently, as in the dynamical Erdos-Renyi graph, but connected components are randomly deleted at a rate proportional to their number of vertices. The deletions, called fires, are triggered by a lightning strikes, hitting each vertex independently at rate n-1/2. The limiting behaviour for large n is known to exhibit SOC, in the sense that the limiting distribution of connected component sizes becomes critical in a finite time and remains critical thereafter: it has a polynomially decaying tail, with the same exponent as one sees in the critical Erdos-Renyi graph and in critical branching processes.

This PhD research project aims to extend known theorems about the above model to a variant in which the edges are directed. This means that fire can only propagate in a specified direction along each edge. When lightning strikes a vertex v, then v burns and so does every vertex w that may be reached at that moment by a directed path from v. Each fire causes all edges incident on any burned vertex to be deleted instantaneously. To study the directed forest fire dynamics, one must study the overlapping structures of in-graphs and out-graphs of the n vertices.

The scientific intention is to extend the qualitative applicability of the model to real-life SOC systems which have directional or causal relationships between their individual elements, such as the neuronal cascades in the brain and financial networks.

The first goal of the project is to prove that this directed model displays SOC in the above sense, when looking at in-graph and out-graph sizes rather than connected component sizes. It is possible to prove this for a restricted class of initial conditions using a coupling argument that relates vertex survival times in the directed and undirected processes. The first technical task is to extend this to more general initial conditions, using the martingale approximation arguments of Rath and Toth to prove that the empirical distribution of in-graph sizes in the directed mean field forest fire model has a limit that solves the critical forest fire equations.

We expect the evolution of the empirical distribution of out-graph sizes to be much more complicated than the in-graph evolution. We expect that its limiting dynamics cannot be described autonomously, but instead one must work with a distribution on some larger space of labelled rooted trees. The second goal of the project will be to prove a limit theorem describing the dynamics of the out-graphs as n tends to infinity.

The third goal will be to study the dynamics of the out-graph of a tagged vertex. In the special case where the initial condition is an age-driven inhomogeneous random graph, the out-graph will be approximated by a multitype Galton-Watson tree, which experiences both growth and pruning events. This process will be studied in the time-stationary setting where the distribution of types is the unique fixed point of the age distribution PDE. A challenging question will be to decide whether the local limit process of the out-graph of a tagged vertex is explosive or almost surely remains finite for all time. It is planned to begin by studying some simpler dynamic random branching trees that have both growth and pruning events.

This project falls within the EPSRC Mathematical Sciences research area, specifically the following themes: applied probability and statis