Question

In a rapidly developing city, house prices change over time based on their neighborhood’s influence. The city can be represented as an N×N grid, where each cell represents a property with a certain price.

The price evolution of each house is influenced by:

1. Its neighboring house prices (spatial component)
2. A constant appreciation rate α (temporal component)
3. A development factor β that represents local infrastructure improvements

The price P(x, y, t) at position (x, y) at time t follows standard diffusion.

Given the following inputs,

• N: grid size
• Initial price configuration P₀(x, y)
• Number of years T to simulate
• α (price diffusion coefficient)
• β (development factor)

Simulates the price evolution over 10 years and output

• the average price across the city
• the maximum and minimum prices
• [extension] a heatmap showing the price distribution

Input

N = 50  # 50x50 grid
T = 5   # Simulate for 5 years
alpha = 0.2  # Diffusion coefficient
beta = 0.05  # Development factor

# Initial condition: central high-price district
P0 = np.zeros((N, N))
P0[20:30, 20:30] = 1000000  # Million-dollar properties in center
P0[0:10, 0:10] = 500000     # Another wealthy area

Solution

Here’s the code for reference and some notes on the solution below.

This simulation models how house prices spread and evolve across a city using principles similar to heat diffusion in physics. The key ideas are:

• Prices tend to influence nearby areas (diffusion effect)
• There’s natural price growth over time (development factor)
• Different initial conditions represent different city features

The code uses what’s called a reaction-diffusion equation:

dP/dt = α∇²P + βP

where:

• P is the price at each location
• α (alpha) is the diffusion coefficient - how quickly prices influence nearby areas
• β (beta) is the development factor - natural price growth rate
• ∇²P (Laplacian) measures how different a price is from its neighbors

initial city layout:

P0[20:30, 20:30] = 1000000  # A central expensive district
P0[0:10, 0:10] = 500000     # A secondary wealthy area
P0 += 200000 * X            # Prices increase from left to right

This creates:

• A high-price downtown area ($1M)
• A secondary wealthy district ($500K)
• A general price gradient where east is more expensive than west

So, to see how the prices evolve over time, we divide time into small steps (dt = 0.01 years) and for each step:

• Calculates how different each location’s price is from its neighbors (Laplacian)
• Updates prices based on both diffusion (spreading) and development (growth)
• Stores the state at the end of each year

Post computing everything we output the required stats and visualize the price distribution.