Structure and Dynamics of Discrete Networks

  1. 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.
  2. 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.