Regulation of the evolution of the architecture of a network by tensors operating on coalitions of actors

Socio-economic networks, neural networks and genetic networks describe collective phenomena through constraints relating actions of several actors, coalitions of these actors and multilinear connectionist operators acting on the set of actions of each coalition. We provide a class of control systems governing the evolution of actions, coalitions and multilinear connectionist operators under which the architecture of the network remains viable. The controls are the “viability multipliers” of the “resource space” in which the constraints are defined. They are involved as “tensor products” of the actions of the coalitions and the viability multiplier, allowing us to encapsulate in this dynamical and multilinear framework the concept of Hebbian learning rules in neural networks in the form of “multi-Hebbian” dynamics in the evolution of connectionist operators. They are also involved in the evolution of coalitions through the “cost” of the constraints under the viability multiplier regarded as a price. Copyright Springer-Verlag Berlin Heidelberg 2003