modelling Level 4

Spatial Diffusion of Innovation

A new technology emerges in one city and spreads to others. A disease outbreak starts locally and radiates outward. Both follow spatial diffusion processes — they spread through contact, and contact depends on proximity. This model models diffusion as a discrete-time process on a spatial network.

Prerequisites: discrete time models, logistic spread, network adjacency, contagion

26 Feb 2026 Updated 12 Mar 2026 13 min read

1. The Question

How do innovations spread through a population distributed across space?

A farmer in Iowa adopts hybrid corn in 1935. By 1940, neighboring counties have adopted it. By 1950, it’s widespread across the Midwest. By 1960, it’s standard practice nationwide.

Why this pattern?

Hierarchical diffusion: Large cities adopt first, then spread to smaller towns (down the urban hierarchy)
Contagious diffusion: Adoption spreads to nearby locations through social contact and observation

The mathematical question: Can we model both mechanisms — proximity-based contagion and size-based hierarchy — in a single framework?

2. The Conceptual Model

Diffusion as a Two-Stage Process

Stage 1: Who can adopt?
Only locations that have heard about the innovation can adopt it. Information spreads through:

Contact with adopters (neighbors, trade partners)
Media coverage (proportional to city size or existing adoption)

Stage 2: Who actually adopts?
Among those aware, adoption depends on:

Benefits (economic advantage, social status)
Costs (upfront investment, risk, learning curve)
Social pressure (more adopters → stronger influence)

This creates an S-curve (logistic) at each location, but the timing varies spatially.

Network Structure

Represent locations as nodes in a network:

Edges connect locations that can transmit the innovation
Edge weights represent transmission probability (function of distance, trade volume, etc.)

Adjacency matrix $A$:

$A_{ij} = 1$ if locations $i$ and $j$ are connected (neighbors)
$A_{ij} = 0$ otherwise
For weighted networks: $A_{ij} = w_{ij}$ (transmission strength)

3. Building the Mathematical Model

Simple Contagion Model (SIS-style)

Divide each location’s population into:

S: Susceptible (not yet adopted)
I: Adopters (have adopted the innovation)

At each time step $t$:

\[I_{i}(t+1) = I_{i}(t) + \beta \cdot S_{i}(t) \cdot \sum_{j} A_{ij} \frac{I_j(t)}{N_j}\]

Where:

$I_i(t)$ is the number of adopters in location $i$ at time $t$
$S_i(t) = N_i - I_i(t)$ is the number of susceptibles
$N_i$ is the total population of location $i$
$\beta$ is the transmission rate (probability of adoption upon contact)
$A_{ij}$ is the adjacency weight (contact frequency between $i$ and $j$)

Interpretation:
The change in adopters depends on:

How many susceptibles remain ($S_i(t)$)
The fraction of adopters in neighboring locations ($I_j(t) / N_j$)
How strongly connected the locations are ($A_{ij}$)

Logistic Growth at Each Location

Once the innovation arrives, adoption within a location often follows a logistic curve:

\[\frac{dI_i}{dt} = r_i I_i \left(1 - \frac{I_i}{N_i}\right)\]

But the start time of the curve varies — late-arriving locations lag behind.

Combined model:
The innovation spreads between locations via the network (contagion), and within each location via the logistic S-curve (internal diffusion).

Discrete-Time Update Rule

For computational implementation:

\[I_i(t+1) = I_i(t) + \Delta I_{\text{internal}} + \Delta I_{\text{external}}\]

Internal diffusion (logistic within the location):

\[\Delta I_{\text{internal}} = r I_i(t) \left(1 - \frac{I_i(t)}{N_i}\right) \Delta t\]

External diffusion (contagion from neighbors):

\[\Delta I_{\text{external}} = \beta (N_i - I_i(t)) \sum_{j} A_{ij} \frac{I_j(t)}{N_j} \Delta t\]

Constraint: $I_i(t+1) \leq N_i$ (can’t exceed total population)

4. Worked Example by Hand

Problem: Three cities with populations and adjacency:

City	Population	Initial adopters
A	1000	100
B	500	0
C	800	0

Adjacency matrix (binary, undirected):

\[A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}\]

A is connected to B. B is connected to both A and C. C is connected to B.

Parameters: $\beta = 0.2$ (transmission rate), $r = 0.1$ (internal growth rate), $\Delta t = 1$ year.

Calculate $I_A(1)$, $I_B(1)$, $I_C(1)$ after one time step.

Solution

City A (t=1):

Internal diffusion:

\[\Delta I_{\text{int}, A} = 0.1 \times 100 \times \left(1 - \frac{100}{1000}\right) = 10 \times 0.9 = 9\]

External diffusion (only neighbor is B, which has 0 adopters):

\[\Delta I_{\text{ext}, A} = 0.2 \times 900 \times \left(1 \times \frac{0}{500}\right) = 0\] \[I_A(1) = 100 + 9 + 0 = 109\]

City B (t=1):

Internal diffusion (starts at 0):

\[\Delta I_{\text{int}, B} = 0.1 \times 0 \times \left(1 - \frac{0}{500}\right) = 0\]

External diffusion (neighbors: A with 100/1000, C with 0/800):

\[\Delta I_{\text{ext}, B} = 0.2 \times 500 \times \left(1 \times \frac{100}{1000} + 1 \times \frac{0}{800}\right)\] \[= 100 \times (0.1 + 0) = 10\] \[I_B(1) = 0 + 0 + 10 = 10\]

City C (t=1):

Internal diffusion:

\[\Delta I_{\text{int}, C} = 0\]

External diffusion (only neighbor is B with 0/500):

\[\Delta I_{\text{ext}, C} = 0.2 \times 800 \times \left(1 \times \frac{0}{500}\right) = 0\] \[I_C(1) = 0\]

Summary after 1 year:

City A: 109 adopters
City B: 10 adopters (innovation has jumped to B)
City C: 0 adopters (still waiting)

5. Computational Implementation

Below is an interactive spatial diffusion simulation.

Network type: Transmission rate (β): 0.20 Internal growth rate (r): 0.10

Try this:

Grid network: Diffusion spreads outward in waves from the center
Hub network: Central city infects all spokes rapidly, then they spread internally
Random network: Irregular diffusion pattern, depends on network structure
Distance-weighted: Closer neighbors exert stronger influence
Increase β: Faster spatial spread
Increase r: Faster internal adoption once innovation arrives

Key insight: Network structure determines diffusion speed and pattern. A well-connected hub accelerates spread; isolated nodes lag behind.

6. Interpretation

Diffusion Curves at Different Locations

Early adopters (large cities, well-connected nodes):

Innovation arrives early
S-curve starts sooner
High adoption by time T

Late adopters (small towns, peripheral nodes):

Innovation arrives late
S-curve starts later
Low adoption by time T

Result: At any snapshot in time, locations are at different stages of the adoption curve.

Barriers to Diffusion

Physical barriers:

Mountains, oceans → low adjacency weights
Poor roads → slow transmission

Social barriers:

Language differences
Cultural resistance
Incompatible infrastructure (e.g., technology standards)

Economic barriers:

High cost → only wealthy locations adopt
Network externalities → need critical mass

Policy Implications

To accelerate diffusion:

Seed innovation in hubs (large, central cities)
Subsidize early adoption (reduce cost barrier)
Improve connectivity (roads, communication networks)
Demonstrate benefits (pilot programs in visible locations)

7. What Could Go Wrong?

Assuming Homogeneous Populations

The model treats all individuals within a location as identical. In reality:

Some people are innovators (adopt early regardless of neighbors)
Others are laggards (resist even with high social pressure)

Solution: Divide population into categories with different adoption thresholds.

Ignoring Hierarchical Effects

The model emphasizes contagious diffusion (neighbor-to-neighbor). Real diffusion often includes:

Hierarchical jumps (New York → Los Angeles, skipping intermediate towns)
Media influence (independent of spatial proximity)

Solution: Add long-range connections weighted by city size.

Forgetting that Innovations Can Fail

Not all innovations succeed. Some spread initially but then reverse (CB radios, fax machines, MySpace).

The logistic model assumes monotonic growth. Reversals require a decline phase after saturation.

Assuming Static Networks

Networks evolve:

New roads are built
Cities grow or shrink
Trade patterns shift

Dynamic networks require updating adjacency $A(t)$ over time.

8. Extension: Epidemic Models (SIR)

The same framework models disease spread:

S → I → R

S: Susceptible (not infected)
I: Infected (can transmit disease)
R: Recovered (immune, removed from transmission)

Equations:

\[\frac{dS_i}{dt} = -\beta S_i \sum_j A_{ij} \frac{I_j}{N_j}\] \[\frac{dI_i}{dt} = \beta S_i \sum_j A_{ij} \frac{I_j}{N_j} - \gamma I_i\] \[\frac{dR_i}{dt} = \gamma I_i\]

Where $\gamma$ is the recovery rate.

Key parameter: Basic reproduction number $R_0 = \beta / \gamma$

$R_0 > 1$: epidemic spreads
$R_0 < 1$: epidemic dies out

Spatial structure affects $R_0$ — clustered networks slow spread; well-mixed networks accelerate it.

9. Math Refresher: Discrete-Time Dynamics

Difference Equations

A discrete-time model updates state at fixed intervals:

\[x(t+1) = f(x(t))\]

Example: $x(t+1) = 2x(t)$ (doubling each step)

Solution: $x(t) = x(0) \cdot 2^t$ (exponential growth)

Stability

An equilibrium $x^*$ is stable if small perturbations decay:

\[|f(x^* + \epsilon) - x^*| < |\epsilon|\]

Graphical method: Plot $x(t+1)$ vs. $x(t)$. Equilibria are where the curve crosses the diagonal. Stable if slope $< 1$ at the crossing.

Coupled Systems

When multiple locations interact:

\[\mathbf{x}(t+1) = \mathbf{A} \mathbf{x}(t) + \mathbf{b}\]

Where $\mathbf{x}$ is a vector of states, $\mathbf{A}$ is the adjacency/interaction matrix.

Eigenvalues of $\mathbf{A}$ determine long-term behavior (growth, decay, oscillation).

Summary

Spatial diffusion: Innovations spread through networks via contact
Two mechanisms: Internal (logistic within a location), external (contagion between locations)
Network structure determines diffusion speed: hubs accelerate, periphery lags
Transmission rate β and growth rate r control dynamics
Adjacency matrix $A_{ij}$ encodes who can transmit to whom
Applications: technology adoption, disease spread, cultural diffusion, information flow
Real diffusion involves hierarchical jumps, barriers, and heterogeneous populations