Interaction graphs of isomorphic automata networks I: complete digraph and minimum in-degree
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An automata network with n components over a finite alphabet Q of size q is a discrete dynamical system described by the successive iterations of a function f:Q^nā Q^n. In most applications, the main parameter is the interaction graph of f: the digraph with vertex set [n] that contains an arc from j to i if f_i depends on input j. What can be said on the set š¾(f) of the interaction graphs of the automata networks isomorphic to f? It seems that this simple question has never been studied. Here, we report some basic facts. First, we prove that if nā„ 5 or qā„ 3 and f is neither the identity nor constant, then š¾(f) always contains the complete digraph K_n, with n^2 arcs. Then, we prove that š¾(f) always contains a digraph whose minimum in-degree is bounded as a function of q. Hence, if n is large with respect to q, then š¾(f) cannot only contain K_n. However, we prove that š¾(f) can contain only dense digraphs, with at least ā n^2/4 ā arcs.
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