Interaction graphs are a useful tool to investigate graphically the influence of feebacks in dynamical systems modelling complex systems with many interacting components in a non-spatial context. In  [ 14 ] and [ 15 ] the authors showed  with the help of a multi-scale anal- ysis that stochastic systems with infinite and finite degrees of freedom can be approximated by dynamical systems whose leading order time evolution is given by a combination of ordi- nary differential equations 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 such systemswith discrete and continuous state spaces. There are many complex systems that can be conveniently described inside such a state space setting, for example bio-molecular interactions. Both ODEs and Markov chains and their mutual interaction can be better understood and analysed through the  introduction of the so called Interaction Graph, a con- cept originally invented for time-continuous dynamical systems  [ 4 ]. In this paper the average dynamics  is introduced as an heuristic tool to construct models of reaction networks with their qualitative properties explored in terms of connectivity properties of the Interaction Graphs for the ODEs, the Markov chain and the resulting average dynamics. In a second paper the question of stability, robustness and a possible modular structure of the complex system defined by the average dynamics is addressed .
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 .
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Interaction graphs are a useful tool to investigate graphically the influence of feebacks in dynamical systems modelling
Time-continuous dynamical systems defined on graphs are often used to model
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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
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14
25
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that
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anal- ysis that
analysis
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infinite and finite degrees of freedom
both continuous variables and finite state spaces
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ordi- nary differential equations
ordinary differential equations (ODEs)
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such systemswith discrete and continuous state spaces. There are many complex systems that can be conveniently described inside such a state space setting, for example bio-molecular interactions. Both
a class of simplified hybrid systems. Once the dynamics is defifined the mutual interaction of both
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and their mutual interaction can be better understood and
can be
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(reverse)
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con- cept
concept
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, see
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4
5
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In this paper
Here we transfer this graph concept to
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, which itself
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networks with their qualitative properties explored
or contact networks. The graphical concepts introduced form the basis for any subsequent study of the qualitative properties of hybrid models
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properties of the Interaction Graphs for the ODEs, the Markov chain and the resulting average dynamics. In a second paper the question of stability, robustness and a possible modular structure of the complex system defined by the average dynamics is addressed
and (feedback) loop formation