Thermodynamic properties of fully connected Q-Ising neural networks

Abstract

Using a probabilistic approach we study the parallel dynamics of fully connected Q-Ising neural networks for arbitrary Q. A Lyapunov function is shown to exist at zero temperature. A recursive scheme is set up to determine the time evolution of the order parameters through the evolution of the distribution of the local field. As an illustrative example, an explicit analysis is carried out for the first three time steps. For the case of the Q = 3 model these theoretical results are compared with extensive numerical simulations. Finally, equilibrium fixed-point equations are derived and compared with the thermodynamic approach based upon the replica-symmetric mean-field approximation.

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“…Using a probabilistic approach (see, e.g., [4], [8]) we calculate the distribution of the local field for a general time step for Q ≥ 2 systems analogously to the fully connected case studied very recently [9]. This allows us to obtain recursion relations determining the full time evolution of the relevant order parameters.…”

“…From the work on fully connected networks [9] we know that due to the correlations we have to study carefully the influence of the non-condensed patterns in the time evolution of the system, expressed by the variance of the residual overlaps. The latter is defined as…”

“…the central limit theorem (CLT) can not be applied directly to this residual overlap. Therefore we follow a procedure similar to that used for the fully connected model [9] by taking out of the local field precisely the contributions arising from these dependences. In order to do so we apply the dynamics writing the residual overlap (17) as…”

“…At this point we remark that instead of using explicitly the set of indices I k where the term 1 √ C ξ µ j r µ T j (t) becomes relevant in the gain function g b (·) (see [9] for more details), we replace it, more conveniently, by the sum over j of the Heaviside function. We then consider the limit C, N → ∞.…”

“…Recently, a complete solution for the parallel dynamics of fully connected Q-Ising networks at zero-temperature has been obtained in this way [9]. Similar to the fully connected architecture, and in contrast with the extremely diluted asymmetric and layered network architectures, the local field contains both a discrete and a normally distributed part.…”