Title: Dynamics of networks: The mean field approach to probabilistic cellular automata on random and small-world graphs
Authors: Lourens Waldorp - University of Amsterdam (Netherlands) [presenting]
Jolanda Kossakowski - University of Amsterdam (Netherlands)
Abstract: The dynamics of networks are described using one-dimensional discrete time dynamical systems theory obtained from a mean field approach to(elementary) probabilistic cellular automata (PCA). Often the mean field approach is used on a regular graph (a grid or torus) where each node has the same number of edges and the same probability of becoming active. We consider elementary PCA where each node has two states (two-letter alphabet): ``active'' or ``inactive'' (0/1). We then use the mean field approach to describe the dynamics of a random graph and a small-world graph. The mean field can now be viewed as a weighted average of the behaviour of the nodes in the graph, since the behaviour of the nodes is determined by a different number of edges. The mean field predicts (pitchfork) bifurcations. The application we have in mind is that of psychopathology. A mental disorder can be viewed as a network of symptoms, each symptom influencing other symptoms. For instance, lack of sleep during the night could lead to poor concentration during the day, which in turn could lead to lack of sleep again by worrying that your job may be on the line. The symptom graph is more likely to be a small-world than a grid. The mean field approach then allows possible explanations of ``jumping'' behaviour in depression, for instance.