  Time-continuous dynamical systems defined on graphs are often used to model
complex systems with many interacting components in a non-spatial context. In
the reverse sense attaching meaningful dynamics to given 'interaction diagrams'
is a central bottleneck problem in many application areas, especially in cell
biology where various such diagrams with different conventions describing
molecular regulation are presently in use. In most situations these diagrams
can only be interpreted by the use of both discrete and continuous variables
during the modelling process, corresponding to both deterministic and
stochastic hybrid dynamics. The conventions in genetics are well-known, and
therefore we use this field for illustration purposes. In [25] and [26] the
authors showed that with the help of a multi-scale analysis stochastic systems
with both continuous variables and finite state spaces can be approximated by
dynamical systems whose leading order time evolution is given by a combination
of ordinary differential equations (ODEs) and Markov chains. The leading order
term in these dynamical systems is called average dynamics and turns out to be
an adequate concept to analyse a class of simplified hybrid systems. Once the
dynamics is defifined the mutual interaction of both ODEs and Markov chains can
be analysed through the (reverse) introduction of the so called Interaction
Graph, a concept originally invented for time-continuous dynamical systems, see
[5]. Here we transfer this graph concept to the average dynamics, which itself
is introduced as an heuristic tool to construct models of reaction or contact
networks. The graphical concepts introduced form the basis for any subsequent
study of the qualitative properties of hybrid models in terms of connectivity
and (feedback) loop formation.
