
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 longtime behavior.

A relatively simple dynamics is provided by
local majority rules.
For scalefree networks without degreedegree correlations, the
activity patterns then evolve either towards the
allactive pattern or towards the allinactive 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 scalefree
networks. For scalefree networks with degreedegree
correlations, on the other hand, majority rule dynamics is governed by a huge number of
attractors. [7,
8] Furthermore,
for dissortative degreedegree correlatoins, this attractor number attains even a maximum
at intermediate network sizes.

For more information, see
publications
on dynamic networks.

