Field Theoretical Approach to the Conservation of Identity of a Complex Network System

The concept of “identity” of a complex network system is proposed based on the method of the gauge field theory. The system is assumed to consist of many elements interacting with each other. The interaction weight is treated as a gauge potential. Change in the external environment surrounding the system is assumed to induce a unitary transformation of the state vector of the system, which is regarded as a gauge transformation. The identity is defined by the fact that the system has some invariant quantities under gauge transformation. Here the total hamiltonian of the whole system is assumed to be that of the invariant quantities representing the identity. The invariance of the identity is conserved by changing the gauge potential according to the given external environment. Some invariant functionals are obtained by introducing the noncommutative gauge field derived from the gauge potential of mutual interaction. Based on the concept of identity, new learning dynamics written in a covariant form are presented as a field equation of the gauge field, which is utilized to realize the state requisite for adaptation to a given new external environment. The learning dynamics are extended to the case of nonlinear interaction among the elements.