Pattern formation on networks with reactions: A continuous-time random-walk approach

C. N. Angstmann, I. C. Donnelly, B. I. Henry

We derive the generalized master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). The nontrivial incorporation of the reaction process into the CTRW is achieved by splitting the derivation into two stages. The reactions are treated as birth-death processes and the first stage of the derivation is at the single particle level, taking into account the death process, while the second stage considers an ensemble of these particles including the birth process. Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non-ad hoc manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases: one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in nonsymmetric and symmetric CTRW Laplacians, respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in nonsymmetric Laplacians, pattern formation may be possible with or without a Turing instability.

http://link.aps.org/doi/10.1103/PhysRevE.87.032804

Continuous Time Random Walks with Reactions, Forcing and Trapping

 C. N. Angstmann, I. C. Donnelly, B. I. Henry

One of the central results in Einstein’s theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

Accepted to appear in Mathematical Modelling of Natural Phenonmena 2013

Selfish routing and the price of anarchy on complex systems

I. C. Donnelly

Superivisor: Prof. Bruce Henry

This thesis investigates inefficiencies of Selfish Routing on networks. We present a review of Game Theory for networks, then introduce and investigate a measure of the inefficiency of selfish behaviour called the Price of Anarchy. We will consider methods of optimality control, most notably Braess’ Paradox. Finally we present our own results regarding selfish behaviour on the internet routing network and consider their applicability in the face of reality.

This was my honours thesis, submitted in 2011