Maximizing Activity in Ising Networks via the TAP Approximation

A wide array of complex biological, social, and physical systems have recently been shown to be quantitatively described by Ising models, which lie at the intersection of statistical physics and machine learning. Here, we study the fundamental question of how to optimize the state of a networked Ising system given a budget of external influence. In the continuous setting where one can tune the influence applied to each node, we propose a series of approximate gradient ascent algorithms based on the Plefka expansion, which generalizes the na\"{i}ve mean field and TAP approximations. In the discrete setting where one chooses a small set of influential nodes, the problem is equivalent to the famous influence maximization problem in social networks with an additional stochastic noise term. In this case, we provide sufficient conditions for when the objective is submodular, allowing a greedy algorithm to achieve an approximation ratio of $1-1/e$. Additionally, we compare the Ising-based algorithms with traditional influence maximization algorithms, demonstrating the practical importance of accurately modeling stochastic fluctuations in the system

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Nuclides as a liquid phase of SU(2)L×SU(2)RSU(2)_L \times SU(2)_R chiral perturbation theory I: emergence of pion-less SU(2) χ\chi PT

The Standard Model of particle physics, augmented with neutrino mixing, is at least very nearly the complete theory of interactions of known particles at energies accessible to Nature on Earth. Candidate effective theories of nuclear structure must therefore reflect SM symmetries, especially the chiral global $SU(2)_L \times SU(2)_R$ symmetry of two-massless-quark QCD. For ground-state nuclei, SU(2) chiral perturbation theory (XPT) enables perturbation in inverse powers of $\Lambda_{XSB}\simeq 1 GeV$, with analytic operators renormalized to all loop orders. We show that pion-less "Static Chiral Nucleon Liquids" (SXNL) emerge as a liquid phase of SU(2) XPT of protons, neutrons and 3 Nambu-Goldstone boson pions. Far-IR pions decouple from SXNL, simplifying the derivation of saturated nuclear matter and microscopic liquid drops (ground-state nuclides). We trace to the global symmetries of two-massless-quark QCD the power of pion-less SU(2) XPT to capture experimental ground-state properties of certain nuclides with even parity, spin zero, even proton number Z, and neutron number N. We derive the SXNL effective SU(2) XPT Lagrangian, including all order $\Lambda_{XSB},\Lambda^0_{XSB}$ operators. These include: all 4-nucleon operators that survive Fierz rearrangement in the non-relativistic limit, and effective Lorentz-vector iso-vector neutral "$\rho$-exchange" operators. SXNL motivate nuclear matter as non-topological solitons at zero pressure: the Nuclear Liquid Drop Model and Bethe-Weizsacker Semi-Empirical Mass Formula emerge in an explicit Thomas-Fermi construction provided in the companion paper. For chosen nuclides, nuclear Density Functional and Skyrme models are justified to order $\Lambda_{\chi SB}^0$. We conjecture that inclusion of higher order operators will result in accurate "natural" Skyrme, No-Core-Shell, and neutron star models