By Stephen Zhang, The University of Melbourne
Random processes are present everywhere in our world, from the motion of molecules and particles suspended in air, to closing prices on the stock market. Many of these processes can be formulated as random walks, a class of mathematical processes in which a moving agent takes a continual sequence of random steps. In the simple random walk, a single agent is equally likely to move in each of four directions, and its properties are well understood in the mathematical literature.
However, suppose we are interested in the motion of individuals moving within a dense crowd. Two weaknesses of the simple random walk become apparent:
- The simple random walk doesn’t consider interactions (individuals in the crowd bumping into each other). Instead, under the simple random walk, agents are able to move through each other
- In a crowd, individuals will tend to persist in their direction of motion, that is, an individual moving right will tend to continue moving right.
We formulated a mathematical model for this scenario, called a persistent exclusion processes, in which many agents move on a grid, and the motion is (1) random, but with an element of persistence, and (2) restricted so that two agents can never occupy the same place (called simple exclusion). This model was implemented as a computer program, and through running simulations we were able to observe the behaviour of each individual moving agent, as well as the overall behaviour of the crowd.
Running simulations can be slow, and it is often difficult or costly to extract insights from the results. Thus, a large part of our work was to derive certain sets of equations (called partial differential equations or PDEs) that could be used to describe the average behaviour of this system. We found that these equations matched very well with simulation data. Although these equations still require a computer in order to be solved, solutions can be computed much more quickly. Furthermore, inspection of the equations themselves can often give insight as to the properties of the solution and how that may change with different parameter values.
Finally, we were able to extend our equations to cover the multi-species case of the process, in which there is more than one type of individual — for instance, one might consider a situation in which there are both fast-moving, persistent individuals and slow-moving, non-persistent individuals in the same crowd. Whilst the single-species equations matched well, the multi-species equations were often unable to accurately reproduce behaviour seen in the simulations, indicating that a key approximation (called a mean-field approximation) was failing for the multi-species case.
Stephen Zhang was a recipient of a 2018/19 AMSI Vacation Research Scholarship.