Title: Asymptotic behavior of memristive circuits and combinatorial optimization,
Abstract: The interest in memristors has risen due to their possible application both
as memory units and as computational devices in combination with CMOS. This is
in part due to their nonlinear dynamics and a strong dependence on the circuit
topology. We provide evidence that also purely memristive circuits can be
employed for computational purposes. We show that a Lyapunov function,
polynomial in the internal memory parameters, exists for the case of DC
controlled memristors. Such Lyapunov function can be asymptotically mapped to
quadratic combinatorial optimization problems. This shows a direct parallel
between memristive circuits and the Hopfield-Little model. In the case of
Erdos-Renyi random circuits, we provide numerical evidence that the
distribution of the matrix elements of the couplings can be roughly
approximated by a Gaussian distribution, and that they scale with the inverse
square root of the number of elements. This provides an approximated but direct
connection to the physics of disordered system and, in particular, of mean
field spin glasses. Using this and the fact that the interaction is controlled
by a projector operator on the loop space of the circuit, we estimate the
number of stationary points of the Lyapunov function, and provide a scaling
formula as an upper bound in terms of the circuit topology only. In order to
put these ideas into practice, we provide an instance of optimization of the
Nikkei 225 dataset in the Markowitz framework, and show that it is competitive
compared to exponential annealing.