Analysis of Reaction Network Systems Using Tropical Geometry

We discuss a novel analysis method for reaction network systems with polynomial or rational rate functions. This method is based on computing tropical equilibrations defined by the equality of at least two dominant monomials of opposite signs in the differential equations of each dynamic variable. In algebraic geometry, the tropical equilibration problem is tantamount to finding tropical prevarieties, that are finite intersections of tropical hypersurfaces. Tropical equilibrations with the same set of dominant monomials define a branch or equivalence class. Minimal branches are particularly interesting as they describe the simplest states of the reaction network. We provide a method to compute the number of minimal branches and to find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201

Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks

The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be trained to accurately mimic the Boltzmann distribution using unsupervised learning from system configurations generated using standard MCMC algorithms. The trained NADE is then employed as smart proposal distribution for the Metropolis-Hastings algorithm. This allows us to perform efficient MCMC simulations, which provide unbiased results even if the expectation value corresponding to the probability distribution learned by the NADE is not exact. Notably, we implement a sequential tempering procedure, whereby a NADE trained at a higher temperature is iteratively employed as proposal distribution in a MCMC simulation run at a slightly lower temperature. This allows one to efficiently simulate the spin-glass model even in the low-temperature regime, avoiding the divergent correlation times that plague MCMC simulations driven by local-update algorithms. Furthermore, we show that the NADE-driven simulations quickly sample ground-state configurations, paving the way to their future utilization to tackle binary optimization problems.Comment: 13 pages, 14 figure

Quantum Optimization of Fully-Connected Spin Glasses

The Sherrington-Kirkpatrick model with random $\pm1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph. The performance of the optimizer compares and correlates to simulated annealing. When considering the effect of the static noise, which degrades the performance of the annealer, one can estimate an improvement on the comparative scaling of the two methods in favor of the D-Wave machine. The optimal choice of parameters of the embedding on the Chimera graph is shown to be associated to the emergence of the spin-glass critical temperature of the embedded problem.Comment: includes supplemental materia

Equilibration through local information exchange in networks

We study the equilibrium states of energy functions involving a large set of real variables, defined on the links of sparsely connected networks, and interacting at the network nodes, using the cavity and replica methods. When applied to the representative problem of network resource allocation, an efficient distributed algorithm is devised, with simulations showing full agreement with theory. Scaling properties with the network connectivity and the resource availability are found.Comment: v1: 7 pages, 1 figure, v2: 4 pages, 2 figures, simplified analysis and more organized results, v3: minor change

Tropicalization and tropical equilibration of chemical reactions

Systems biology uses large networks of biochemical reactions to model the functioning of biological cells from the molecular to the cellular scale. The dynamics of dissipative reaction networks with many well separated time scales can be described as a sequence of successive equilibrations of different subsets of variables of the system. Polynomial systems with separation are equilibrated when at least two monomials, of opposite signs, have the same order of magnitude and dominate the others. These equilibrations and the corresponding truncated dynamics, obtained by eliminating the dominated terms, find a natural formulation in tropical analysis and can be used for model reduction.Comment: 13 pages, 1 figure, workshop Tropical-12, Moskow, August 26-31, 2012; in press Contemporary Mathematic