Structure and Dynamics of Discrete Networks
- Structure and dynamics of acyclic networks. Acyclic networks are dynamical systems whose dependency graph (or wiring diagram) is an acyclic graph. It is known that such systems will have a unique steady state and that it will be globally asymptotically stable. Such result is independent of the mathematical framework used. More precisely, this result is true for discrete-time finite-space, discrete-time discrete-space, discrete-time continuous-space and continuous-time continuous-space dynamical systems; however, the proof of this result is dependent on the framework used. In this paper we present a novel and simple argument that works for all of these frameworks. Our arguments support the importance of the connection between structure and dynamics.
- Bifurcations in Boolean Networks. This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions 0 to 1 (up-threshold) and 1 to 0 (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference Delta of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for Delta >= 2 they may have long periodic orbits. The limiting case of Delta=2 is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.