A dictionary of coupled maps

Most often we do not know the number of agents in a network. Without knowing the number of agents, or how they are coupled, predcting long-term behavior may seem challenging.

One approach, Noelle Thew's senior thesis project, is to build a dictionary of coupled maps. Different numbers of maps, different logistic parameters, different coupling strengths. In order to keep the dictionary size manageable, some care was needed in selecting representative system parameters.

For each network in the dictionary, the address populations, the number of points in each address ij, in each address ijk, and so on, are stored. Call these Dij, Dijk, and so on.

To compare an experimental data time series X with a dictionary entry, first adjust the bin boundaries on X so X1 = D1, X2 = D2, X3 = D3, and X4 = D4. Then the 2-address correlation is

κ2(X,D) = (N - (1/2)(|X11 - D11| + ... + |X44 - D44|))/N

Closest matches are those dictionary entries that maximize κ2(X,D). κ3(X,D), and κ4(X,D).

For example, comparing the dictionary entries to a system of three maps not in the dictionary gave these matches. The black line is κ2, the dark gray is κ3, and the light gray is κ4.

Here are the driven IFS for the original network (left), dictionary entry 33 (middle), and dictionary entry 2908 (right). Dictionary entry 33 is a single logistic map, entry 2908 is two coupled maps. At least as far as address occupancies are concerned, systems with substantially different parameters can exhibit similar behavior. This result suggests that the networks have synchronized and are acting like a single logistic map.

When the time series of a complex system matches the dictionary entry of a small number of maps, we look for that system to organize itself into a small number of internally synchronized subsystems. This is a way to start to identify trees in a forest.