Computation of Balanced Equivalence Relations and Their Lattice for a Coupled Cell Network

A coupled cell network describes interacting (coupled) individual systems (cells). As in networks from real applications, coupled cell networks can represent inhomogeneous networks where different types of cells interact with each other in different ways, which can be represented graphically by different symbols, or abstractly by equivalence relations. Various synchronous behaviors, from full synchrony to partial synchrony, can be observed for a given network. Patterns of synchrony, which do not depend on specific dynamics of the network, but only on the network structure, are associated with a special type of partition of cells, termed balanced equivalence relations. Algorithms in Aldis [J. W. Aldis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 407–427] and Belykh and Hasler [I. Belykh and M. Hasler, Chaos, 21 (2011), 016106] find the unique pattern of synchrony with the fewest clusters. In this paper, we compute the set of all possible patterns of synchrony and show their hierarchy structure as a complete lattice. We represent the network structure of a given coupled cell network by a symbolic adjacency matrix encoding the different coupling types. We show that balanced equivalence relations can be determined by a matrix computation on the adjacency matrix which forms a block structure for each balanced equivalence relation. This leads to a computer algorithm to search for all possible balanced equivalence relations. Our computer program outputs the balanced equivalence relations, quotient matrices, and a complete lattice for user specified coupled cell networks. Finding the balanced equivalence relations of any network of up to 15 nodes is tractable, but for larger networks this depends on the pattern of synchrony with the fewest clusters.