Small-sample quantization effects on the equilibrium behavior of combined fuzzy cognitive maps

Fuzzy cognitive maps (FCMs) allow experts to express their knowledge by drawing weighted causal digraphs. Experts can pool or fuse their knowledge by adding the underlying FCM causal matrices. This naturally extends the ordered-weighted-averaging technique to averaging dynamical systems and can create complex dynamical systems from several simpler ones. Edge quantization allows experts to state their real-valued knowledge in the simpler terms of causal increase (1), decrease (-1), or absence (0). We model the expert FCMs as a sequence of random fields to study the small-sample effects of quantizing both the causal edges and fuzzy-set concept nodes. The averaged quantized random matrices exhibit large-sample convergence to the population means of the unquantized matrices in accordance with the strong law of large numbers, but the small-sample averages can show substantial diversity of equilibrium attractors. We use statistical tests and a novel measure based on the fuzzy equality of limit-cycle histograms to show that this small-sample equilibrium diversity increases as the node multivalence or fuzzy-set quantization increases. More realistic probabilistic assumptions can more accurately model the equilibrium behavior of large-scale knowledge fusion.