Discrete dynamical network models have proved to be useful in a variety of applications, such as logistics, social sciences and systems biology.  Consequently, efficient analysis methods for structural characteristics and dynamical properties of such models are in high demand. The network structure describes dependencies and character of interactions between components and can be represented by signed digraphs. For discrete models, corresponding network dynamics can also be captured by a directed graph, the so-called state transition graph. Its vertex set is exponential in the number of network components, and its topology can be very complex when dealing with realistic representations of real-world systems, in particular in biological applications. Comprehensive analysis of the state transition graph is no easy task, efficient methods need to avoid explicit calculation of the graph in its entirety.

In this talk, we will give a number of results relating structural and dynamical characteristics of a discrete regulatory network using methods from iteration, Boolean spectral and graph theory. As we will see, positive and negative cycles in the graph representing the network structure are closely linked to the existence and shape of attractors of the system. Results of this nature allow us to focus on the comparably small network structure to obtain information on the system's dynamics without having to generate the state transition graph.

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