Ultimate "SIR" in Autonomous Linear Networks with Symmetric Weight Matrices, and Its Use to Stabilize the Network - A Hopfield-like network

In this paper, we present and analyse two Hopfield-like nonlinear networks, in continuous-time and discrete-time respectively. The proposed network is based on an autonomous linear system with a symmetric weight matrix, which is designed to be unstable, and a nonlinear function stabilizing the whole network thanks to a manipulated state variable calledultimate SIR''. This variable is observed to be equal to the traditional Signal-to-Interference Ratio (SIR) definition in telecommunications engineering. The underlying linear system of the proposed continuous-time network is $\dot{{\mathbf x}} = {\mathbf B} {\mathbf x}$ where {\bf B} is a real symmetric matrix whose diagonal elements are fixed to a constant. The nonlinear function, on the other hand, is based on the defined system variables called SIR’’s. We also show that the SIR''s of all the states converge to a constant value, called system-specific Ultimate SIR’’; which is equal to $\frac{r}{\lambda_{max}}$ where $r$ is the diagonal element of matrix ${\bf B}$ and $\lambda_{max}$ is the maximum (positive) eigenvalue of diagonally-zero matrix $({\bf B} - r{\bf I})$, where ${\bf I}$ denotes the identity matrix. The same result is obtained in its discrete-time version as well. Computer simulations for binary associative memory design problem show the effectiveness of the proposed network as compared to the traditional Hopfield Networks.

💡 Research Summary

The paper introduces two Hopfield‑style recurrent networks—one continuous‑time and one discrete‑time—built on an autonomous linear subsystem with a symmetric weight matrix B. Unlike conventional Hopfield networks, B is deliberately chosen to be unstable: its diagonal entries are fixed to a constant r, while the off‑diagonal part B – rI possesses a positive maximum eigenvalue λ_max. The linear dynamics alone would cause exponential divergence of the state vector x.
Stability is achieved by embedding a nonlinear feedback that depends on a state‑dependent “Signal‑to‑Interference Ratio” (SIR) defined for each neuron i as

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