sde.py

import numpy as np
from scipy.integrate import solve_ivp
from geomstats.geometry.riemannian_metric import RiemannianMetric
from geomstats.geometry.hyperbolic import Hyperbolic
from geomstats.learning.frechet_mean import FrechetMean

class SocietalDynamicsEngine:
    def __init__(self, manifold_dim=100, num_variables=50):
        self.manifold = Hyperbolic(dim=manifold_dim)
        self.metric = self.manifold.metric
        self.num_variables = num_variables
        self.current_state = self.manifold.random_point()
        self.drift_coefficient = 0.1
        self.diffusion_coefficient = 0.05
        self.temperature = 1.0
        self.interaction_strength = 0.1

    def set_initial_state(self, initial_state):
        assert len(initial_state) == self.manifold.dim
        self.current_state = self.manifold.from_vector(initial_state)

    def dynamics(self, t, state, noise):
        # Implement the drift and diffusion terms of the SDE
        drift = self.calculate_drift(state)
        diffusion = self.calculate_diffusion(state)
        return drift + diffusion * noise

    def calculate_drift(self, state):
        # Calculate the drift term based on the gradient of the potential energy
        gradient = self.metric.grad(self.calculate_energy, state)
        return -self.drift_coefficient * gradient

    def calculate_diffusion(self, state):
        # Calculate the state-dependent diffusion term
        return self.diffusion_coefficient * np.sqrt(self.calculate_energy(state))

    def evolve_society(self, time_span, time_step):
        t_eval = np.arange(0, time_span, time_step)
        noise = np.random.normal(0, 1, (len(t_eval), self.manifold.dim))
        
        def ode_func(t, y):
            return self.dynamics(t, y, noise[int(t/time_step)])
        
        solution = solve_ivp(ode_func, (0, time_span), self.current_state, t_eval=t_eval, method='RK45')
        self.current_state = solution.y[:, -1]
        return solution.y.T

    def calculate_energy(self, state):
        # Calculate the energy of a given societal state using a simplified Ising-like model
        energy = 0
        for i in range(self.manifold.dim):
            for j in range(i+1, self.manifold.dim):
                energy -= self.interaction_strength * state[i] * state[j]
        return energy

    def calculate_probability(self, state):
        # Calculate the probability of a state using the Boltzmann distribution
        energy = self.calculate_energy(state)
        return np.exp(-energy / self.temperature) / self.calculate_partition_function()

    def calculate_partition_function(self):
        # Approximate the partition function using Monte Carlo integration
        num_samples = 10000
        samples = [self.manifold.random_point() for _ in range(num_samples)]
        energies = [self.calculate_energy(sample) for sample in samples]
        return np.mean(np.exp(-np.array(energies) / self.temperature))

    def parallel_transport(self, vector, curve):
        # Perform parallel transport of a vector along a curve on the manifold
        return self.metric.parallel_transport(vector, curve[0], curve[1:], n_steps=100)

    def geodesic(self, initial_point, end_point, n_steps=100):
        # Compute the geodesic between two points on the manifold
        return self.metric.geodesic(initial_point=initial_point, end_point=end_point)(np.linspace(0, 1, n_steps))

    def frechet_mean(self, points, weights=None):
        # Calculate the Fréchet mean of a set of points on the manifold
        mean = FrechetMean(metric=self.metric, point_type='vector')
        mean.fit(points, weights=weights)
        return mean.estimate_

    def curvature(self, point):
        # Compute the sectional curvature at a point
        return self.manifold.sectional_curvature(point)

    def stability_analysis(self, state):
        # Perform stability analysis by computing the Jacobian of the drift term
        epsilon = 1e-6
        jacobian = np.zeros((self.manifold.dim, self.manifold.dim))
        for i in range(self.manifold.dim):
            perturbed_state = state.copy()
            perturbed_state[i] += epsilon
            jacobian[:, i] = (self.calculate_drift(perturbed_state) - self.calculate_drift(state)) / epsilon
        eigenvalues = np.linalg.eigvals(jacobian)
        return eigenvalues

    def lyapunov_exponent(self, initial_state, time_span, time_step):
        # Estimate the largest Lyapunov exponent
        trajectory = self.evolve_society(time_span, time_step)
        perturbation = np.random.normal(0, 1e-10, self.manifold.dim)
        perturbed_trajectory = self.evolve_society(time_span, time_step, initial_state + perturbation)
        distances = np.linalg.norm(trajectory - perturbed_trajectory, axis=1)
        return np.mean(np.log(distances[1:] / distances[:-1])) / time_step

    def bifurcation_analysis(self, parameter_range, parameter_name):
        # Perform bifurcation analysis by varying a parameter
        results = []
        original_value = getattr(self, parameter_name)
        for param_value in parameter_range:
            setattr(self, parameter_name, param_value)
            steady_state = self.evolve_society(time_span=100, time_step=0.1)[-1]
            stability = self.stability_analysis(steady_state)
            results.append((param_value, steady_state, stability))
        setattr(self, parameter_name, original_value)
        return results