Cities as Complex Systems
Mathematical simulation of urban dynamics — from scaling laws and growth PDEs through cellular automata, agent-based models, and stochastic dynamics to traffic lattice gases, mean field games, optimal control, topological data analysis, pollution modeling, and SUMO microsimulation.
Course Overview
Cities are the most complex structures built by humanity — self-organizing systems where millions of individual decisions produce emergent order at scales from street canyons to metropolitan regions. This course develops the mathematical machinery to model, simulate, and understand urban dynamics using tools from statistical physics, PDEs, stochastic processes, network theory, algebraic topology, game theory, and optimal control.
What You'll Learn
- • Urban scaling laws and Kleiber's exponent derivation
- • Fisher-KPP traveling waves and instanton rare events
- • SLEUTH and FLUS cellular automata models
- • Schelling segregation and Helbing social forces
- • Langevin, Fokker-Planck, and Schrödinger Bridges
- • Graph Laplacian spectral analysis and percolation
- • TASEP, ASEP+Langmuir, and KPZ universality
- • Mean Field Games and Hamilton-Jacobi-Bellman
- • Kuramoto synchronization and topological data analysis
- • SUMO microsimulation and street canyon pollution
Prerequisites
- • Calculus and Linear Algebra
- • Probability and Statistics
- • Statistical Mechanics (helpful)
- • Ordinary and partial differential equations
- • Python programming; Fortran exposure helpful
- • Basic graph theory and network concepts
Course Structure
18 Modules covering 54 chapters • From scaling laws to integrated multi-physics simulation • Python simulations and Fortran high-performance solvers throughout • Bridges statistical physics, PDEs, stochastic processes, game theory, topology, and traffic engineering • Suitable for graduate students in physics, applied mathematics, urban science, and transportation
Course Modules
Module 1: Scaling Laws & Infrastructure
Derives the $\beta = 3/4$ infrastructure exponent from hierarchical network optimization, the $\beta = 5/4$ social superlinearity from interaction integrals, and entropy production using Glansdorff-Prigogine theory. Python fits real scaling exponents; Fortran computes via log-log regression.
Module 2: Urban Growth PDEs
Full derivations of logistic ODE (analytical solution + stability), Fisher-KPP traveling wave with minimum speed $c_{\min} = 2\sqrt{Dr}$, and instanton formalism for rare urban transitions. Fortran solver uses explicit upwind scheme with CFL check.
Module 3: Cellular Automata
Stochastic CA with formal neighborhood influence function $\Psi_{ij}$, SLEUTH model components, Figure of Merit (Jaccard index), and FLUS adaptive inertia. High-performance Fortran CA core.
Module 4: Agent-Based Models
Schelling model with entropy-based segregation index, Alonso-Muth-Mills bid-rent gradient ($p(r) \propto e^{-tr/q^*}$), logit choice model, and Helbing Social Force pedestrian dynamics.
Module 5: Stochastic Dynamics
Langevin equation from microscopic jump processes, Itô vs Stratonovich discussion, Fokker-Planck with Chang-Cooper scheme, and full Schrödinger Bridge / Sinkhorn algorithm with explicit connection to quantum mechanics via Nelson stochastic mechanics.
Module 6: Network Analysis
OSMnx street network analysis (real city data), graph Laplacian spectral analysis, Fiedler value, heat diffusion, effective resistance as accessibility metric, and percolation threshold via Molloy-Reed criterion.
Module 7: Calibration & KPZ
Bayesian MCMC calibration with Metropolis-Hastings, Fortran brute-force parameter search, and KPZ equation with roughness exponent measurement.
Module 8: Fractals & Quantum
DLA with box-counting fractal dimension, fractional Fokker-Planck for urban memory effects, and quantum game theory for urban coordination.
Module 9: Multi-Scale Methods
Homogenization (CA → PDE) proving $D = \text{spread}/4$, two-scale expansion deriving the cell problem, renormalization group deriving $\beta = 3/4$ from RG fixed points, equation-free methods (Kevrekidis lift-evolve-restrict), and Fisher-Rao / Schrödinger Bridge geometry.
Module 10: Traffic as Lattice Gas
Langmuir → fundamental diagram mapping, Langmuir-Hinshelwood for intersections, TASEP exact phase diagram (LD, HD, MC) via Matrix Ansatz $DE - ED = D + E$, LWR PDE from conservation, Rankine-Hugoniot shocks, and Cahn-Hilliard spinodal decomposition of traffic jams.
Module 11: ASEP + Langmuir
General ASEP with forward/backward hopping, Langmuir coupling with $\Omega = (\omega_A + \omega_D)L$, five-phase diagram, Bethe ansatz and $U_q(\mathfrak{sl}_2)$ quantum group symmetry, domain wall theory, and Tracy-Widom GUE distribution for KPZ universality in traffic.
Module 12: Mean Field Games
Single-agent HJB with Cole-Hopf transform to Schrödinger equation, coupled HJB+FP MFG system from $N \to \infty$ limit, stationary MFG recovering Alonso-Muth-Mills endogenously, MFG on networks, Achdou-Capuzzo Dolcetta numerics, and CCD equilibrium formalization.
Module 13: Optimal Control & Congestion Pricing
Pontryagin Maximum Principle for store-and-forward, bang-bang signal timing and Green Wave as singular arc, LQR via algebraic Riccati equation, Pigouvian toll $\tau^* = f \cdot dt/df$, and model predictive control.
Module 14: Kuramoto Synchronization
Kuramoto model from averaging theorem, critical coupling $K_c = 2/(\pi g(0))$, network generalization connecting to Fiedler value from Module 6, Green Wave as locked Kuramoto state, and morning commute as synchronization phase transition.
Module 15: Topological Data Analysis
Simplicial homology ($H_k = \ker\partial_k / \text{im}\,\partial_{k+1}$), Betti numbers for five urban forms, persistent homology via Vietoris-Rips filtration with stability theorem, and Mapper algorithm for topological skeletons of cities. Fortran union-find implementation.
Module 16: Street Canyon Pollution
Canyon flow regimes classified by aspect ratio $\alpha = H/W$, advection-diffusion with OSPM ($C = C_D + C_R + C_B$), k-ε turbulence model, WHO health metrics, and MFG extension with pollution-aware Wardrop equilibrium.
Module 17: SUMO Microsimulation
SUMO architecture with Krauss and IDM car-following models, OSMnx → SUMO pipeline via netconvert, SUMO-canyon coupling for emission fields, and multi-physics feedback loop connecting all previous modules.
Module 18: Integration Architecture
TraCI binary protocol specification, four bridge architectures (REST, ZeroMQ, gRPC, subprocess) with latency benchmarks, Protocol Buffer schema for urban simulation, and complete startup orchestration connecting Java SUMO controller to Python physics engines.