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Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
Diversity of graphs with highly variable connectivity
A popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large-scale connectivity. While it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent, the relationship between competing descriptions is not entirely understood. This paper lends perspective on this problem by showing how the degree sequence and other constraints (e.g., connectedness, no self-loops or parallel edges) on a particular graph play a primary role in dictating many features, including its correlation structure. Building on recent work, we show how a simple structural metric characterizes key differences between graphs having the same degree sequence. More broadly, we show how the (often implicit) choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions
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