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We apply Game Theory to a mathematical representation of two competing teams of agents connected within a complex network, where the ability of each side to manoeuvre their resource and degrade that of the other depends on their ability to internally synchronise decision-making while out-pacing the other. Such a representation of an adversarial socio-physical system has application in a range of business, sporting, and military contexts. Specifically, we unite here two physics-based models, that of Kuramoto to represent decision-making cycles, and an adaptation of a multi-species Lotka-Volterra system for the resource competition. For complex networks we employ variations of the Barab\'asi-Alberts scale-free graph, varying how resources are initially distributed between graph hub and periphery. We adapt as equilibrium solution Nash Dominant Game Pruning as a means of efficiently exploring the dynamical decision tree. Across various scenarios we find Nash solutions where the side initially concentrating resources in the periphery can sustain competition to achieve victory except when asymmetries exist between the two. When structural advantage is limited we find that agility in how the victor stays ahead of decision-state of the other becomes critical., Comment: 11 pages, 27 figures

We apply computational Game Theory to a unification of physics-based models that represent decision-making across a number of agents within both cooperative and competitive processes. Here the competitors try to both positively influence their own returns, while negatively affecting those of their competitors. Modelling these interactions with the so-called Boyd-Kuramoto-Lanchester (BKL) complex dynamical system model yields results that can be applied to business, gaming and security contexts. This paper studies a class of decision problems on the BKL model, where a large set of coupled, switching dynamical systems are analysed using game-theoretic methods. Due to their size, the computational cost of solving these BKL games becomes the dominant factor in the solution process. To resolve this, we introduce a novel Nash Dominant solver, which is both numerically efficient and exact. The performance of this new solution technique is compared to traditional exact solvers, which traverse the entire game tree, as well as to approximate solvers such as Myopic and Monte Carlo Tree Search (MCTS). These techniques are assessed, and used to gain insights into both nonlinear dynamical systems and strategic decision making in adversarial environments.

We propose new mathematical optimization models for generating sparse dynamical graphs, or networks, that can achieve synchronization. The synchronization phenomenon is studied using the Kuramoto model, defined in terms of the adjacency matrix of the graph and the coupling strength of the network, modelling the so-called coupled oscillators. Besides sparsity, we aim to obtain graphs which have good connectivity properties, resulting in small coupling strength for synchronization. We formulate three mathematical optimization models for this purpose. Our first model is a mixed integer optimization problem, subject to ODE constraints, reminiscent of an optimal control problem. As expected, this problem is computationally very challenging, if not impossible, to solve, not only because it involves binary variables but also some of its variables are functions. The second model is a continuous relaxation of the first one, and the third is a discretization of the second, which is computationally tractable by employing standard optimization software. We design dynamical graphs that synchronize, by solving the relaxed problem and applying a practical algorithm for various graph sizes, with randomly generated intrinsic natural frequencies and initial phase variables. We test robustness of these graphs by carrying out numerical simulations with random data and constructing the expected value of the network's order parameter and its variance under this random data, as a guide for assessment., Comment: The adjacency matrices of the 10 graphs studied in this manuscript can be found in the ancillary folder

We consider the influence of correlated noise on the stability of synchronisation of oscillators on a general network using the Kuramoto model for coupled phases $\theta_i$. Near the fixed point $\theta_i \approx \theta_j \ \forall i,j$ the impact of the noise is analysed through the Fokker-Planck equation. We deem the stochastic system to be `weakly unstable' if the Mean First Passage Time for the system to drift outside the fixed point basin of attraction is less than the time for which the noise is sustained. We argue that a Mean First Passage Time, computed near the phase synchronised fixed point, gives a useful lower bound on the tolerance of the system to noise. Applying the saddle point approximation, we analytically derive general thresholds for the noise parameters for weak stochastic stability. We illustrate this by numerically solving the full Kuramoto model in the presence of noise for an example complex network., Comment: Accepted by Physica D, 18 Figures, main body 33 pages with 3 Appendices for technical aspects; paper significantly different to v.1. after referee reviews

We study the synchronisation properties of the Kuramoto model of coupled phase oscillators on a general network. Here we distinguish the ability of such a system to self-synchronise from the stability of this behaviour. While self-synchronisation is a consequence of genuine non-perturbative dynamics, the stability in dynamical systems is usually accessible by fluctuations about a fixed point, here taken to be the synchronised solution. We examine this problem in terms of modes of the graph Laplacian, by which the absolute Lyapunov stability of the synchronised fixed point is readily demonstrated. Departures from stability are seen to arise at the next order in fluctuations where the dynamical equations resemble those for species population models, the logistic and Lotka-Volterra equations. Methods from these systems are exploited to analytically derive new critical couplings signalling deviation from classical stability. We observe in some cases an intermediate regime of behaviour, between incoherence and synchronisation, where system wide periodic behaviours are exhibited. We discuss these results in light of simulations., Comment: 11 pages, 4 figures

We study the weak decay of the neutral pion to a neutrino-antineutrino pair, $\pi^0\to \nu \bar{\nu}$, in the Skyrme model. In baryon free-space the process is forbidden by helicity while in a dense baryonic medium, the process becomes possible already to leading order in $G_F$ due to the break-down of Lorentz symmetry in the background medium., Comment: 7 pages, RevTeX, 4 figures. Expanded discussion in light of referee comments

We study the density dependence of the decay $\pi^0\to \gamma \gamma$ using the Skyrme Lagrangian to describe simultaneously both the matter background and mesonic fluctuations. Pion properties such as mass and decay constant are modified by the medium. This leads to large suppression at high density of both photo-production from the neutral pion and the reverse process. The in-medium effective charge of $\pi^{\pm}$ are also discussed in the same framework., Comment: 8 pages, 4 figures. Corrections in light of referee comments