Architectures for arbitrarily connected synchronization networks

In a synchronization (sync) network
¹
containing N nodes, it is shown (Theorem 1c) that an arbitrarily connected sync network D is the union of a countable set of isolated connecting sync networks {D
i
, i = 1, 2, …, L}, i.e., D = U
^L
i=1
D
i
. It is shown (Theorem 2e) that a connecting sync network is the union of a set of disjoint irreducible subnetworks having one or more nodes. It is further shown (Theorem 3a) that there exists at least one closed irreducible subnetwork in D
i
. It is further demonstrated that a connecting sync network is the union of both a mastergroup and a slave group of nodes. The master group is the union of closed irreducible subnetworks in D
i
. The slave group is the union of non-closed irreducible subnetworks in D
i
. The relationships between master-slave (MS), mutual synchronous (MUS) and hierarchical MS/MUS networks are clearly manifested [1]. Additionally, Theorem 5 shows that each node in the slave group is accessible by at least one node in the master group. This allows one to conclude that the synchronization information available in the master group can be reliably transported to each node in the slave group. Counting and combinatorial arguments are used to develop a recursive algorithm which counts the number A
n
of arbitrarily connected sync network architectures in an N-nodal sync network and the number C
N
of isolated connecting sync networks in D. Examples for N=2, 3, 4, 5 and 6 are provided. Finally, network examples are presented which illustrate the results offered by the theorems. The notation used and symbol definitions are listed in Appendix A.