Dynamics on Networks

  • Networks consist of a certain number of vertices (or nodes) that are connected by edges (or links). The structural properties of such networks have long been studied in the context of mathematical graph theory. Using graph theoretical methods, one can characterize the structure of single networks as well as the average properties of network ensembles.

  • Networks provide a general framework for the representation of many different systems. First, any system that attains a discrete number of states can be viewed as a network, in which the vertices represent the different states of the system and the edges correspond to transitions between the states. One example is provided by network representations of molecular motors. [1] These networks describe the dynamical motor properties as observed in single molecule experiments [2] and lead to fundamental thermodynamic relations in the form of energy balance conditions [3].

  • Another type of network representation arises for composite systems that consist of many different but similiar components, which are represented by the network's vertices, and "interactions" between the different components, which are described by the network's edges. [4] The different components or vertices may be different structural components such as different proteins within a certain biological cell, different functional components such as the different genes of an organism, or different cells such as neurons that interact with each other via chemical signals.

  • In general, the different components of the system and, thus, the different vertices of the network can attain several internal states that evolve with time. The simplest case is provided by binary variables that can be active or inactive (on or off, up or down, etc). [5] [6] [7] At any given time, the state of the composite system is then characterized by an activity pattern of the network, and the system's dynamics is described by the time evolution of this pattern. Depending on the inital pattern, the system will reach one of the dynamical attractors that govern the system's long-time behavior.

  • A relatively simple dynamics is provided by local majority rules. For scale-free networks without degree-degree correlations, the activity patterns then evolve either towards the all-active pattern or towards the all-inactive pattern. [5] [6] These two patterns correspond to two fixed points with two basins of attraction. The separatrix between these two basins can be smooth or spiky depending on the structure of the scale-free networks. For scale-free networks with degree-degree correlations, on the other hand, majority rule dynamics is governed by a huge number of attractors. [7, 8] Furthermore, for dissortative degree-degree correlatoins, this attractor number attains even a maximum at intermediate network sizes.

  • For more information, see publications on dynamic networks.