Spatial Diffusion: Relocation and Expansion Models

Spatial diffusion is the process by which an idea, innovation, disease or practice spreads from its place of origin to other places over time; relocation diffusion happens when people physically move and carry the idea with them, while expansion diffusion happens when the idea spreads through a population and remains strong in the origin area. In brief: expansion diffusion has three types: contagious (spreading by direct contact, like a flu), hierarchical (spreading from large places or influential people down to smaller ones) and stimulus (the idea itself is not adopted, but it inspires a related innovation). Adoption over time usually follows an S-shaped curve.

Diffusion explains how languages, religions, technologies, fashions and diseases reach new places and why some places adopt them earlier than others. This guide explains the main types of diffusion, the factors that help or hinder it, the S-curve of adoption, compares the types in a table, works through calculations, outlines the history of diffusion research and corrects common mistakes.

What is relocation diffusion?

In relocation diffusion, the people who carry the idea move from one place to another, and the idea leaves the origin or at least does not spread there in the same way. Examples include the spread of the Spanish and Portuguese languages to Latin America through colonization, the arrival of Chinese and Indian communities in many parts of the world with their languages, foods and religions, the movement of Mormon settlers to Utah in the 19th century, and the transport of enslaved Africans to the Americas, who brought music, religion and food traditions that shaped new cultures. Migration is the main mechanism, so the pattern follows the flows of people, and new cultural centers may appear far from the origin, such as Hindu temples in London or Toronto.

What is expansion diffusion?

In expansion diffusion, the idea spreads outward from its source, and the origin remains a center. There are three forms.

• Contagious diffusion: spreading from person to person through contact, so that nearby places are affected first and the effect falls with distance, like a wave. Examples are the spread of influenza, rumors and early adoption of a local innovation by neighbors.
• Hierarchical diffusion: spreading from nodes of authority or large places to others, skipping intermediate areas, according to rank rather than distance. Examples are fashions and new music moving from major cities to smaller cities and then to towns, or new ideas passed from leaders to followers. The spread of the Covid-19 virus first followed the network of international airports and large cities, before local spread filled in the gaps.
• Stimulus diffusion: the underlying idea spreads, but the specific form is changed. For example, the example of Japan’s industrial success inspired other Asian economies to follow their own variants, and the California roll adapted sushi to American tastes by using avocado and putting the rice on the outside.

What affects the spread?

The friction of distance reduces the spread with distance, because of the cost and time of contact. Barriers include physical (mountains, oceans), political (borders, censorship, trade restrictions), cultural (language, religion, values, taboos) and economic ones (cost, lack of infrastructure). Channels help the spread: roads, railways, air routes, media, the internet and social networks. The properties of the innovation also matter: its advantage over existing options, compatibility with local culture, ease of use and visibility. Adopters differ: Everett Rogers described groups of people who adopt an innovation at different times, including innovators, early adopters, an early majority, a late majority and laggards.

What is the S-curve of adoption?

When the share of adopters is plotted against time, it usually forms an S-shaped (logistic) curve. At the start, adoption is slow, because few people know the innovation. Then it accelerates as more people adopt and influence others, reaching the fastest growth when about half have adopted. Finally, it slows as the market is saturated and only the last holdouts remain. The Swedish geographer Torsten Hägerstrand studied this pattern for farm innovations in Sweden in the 1950s, and found that adoption spread with distance from early adopters and with the frequency of personal contact. His simulations of “neighborhood effect” diffusion became a model for studies of the spread of everything from agricultural practices to technology.

Diffusion types compared

Worked example: adoption and spread

1. Adopter categories: Rogers’s groups make up 2.5% innovators, 13.5% early adopters, 34% early majority, 34% late majority and 16% laggards, which total 100%. By the end of the early majority, 2.5 + 13.5 + 34 = 50% have adopted.
2. Logistic curve: the share adopting by time t (in years) is P(t) = 100 ÷ (1 + e^{−r(t − t₀)}). With r = 0.5 per year and t₀ = 6 (the year when half have adopted), P(6) = 100 ÷ (1 + 1) = 50%.
3. Earlier year: at t = 4, P = 100 ÷ (1 + e¹) = 100 ÷ (1 + 2.718) = 26.9%. At t = 8, P = 100 ÷ (1 + e⁻¹) = 100 ÷ (1 + 0.368) = 73.1%.
4. Fastest growth: between t = 4 and t = 8, the share grows by 73.1 − 26.9 = 46.2 percentage points in 4 years, which is 11.6 points a year, much faster than at the start (at t = 0, P = 100 ÷ (1 + e³) = 100 ÷ 21.09 = 4.7%).
5. Contagious spread speed: a disease front advances 15 km a day. A town 300 km from the origin is reached after 300 ÷ 15 = 20 days.
6. Hierarchical spread: a new product reaches a city of 5 million in month 1, a city of 1 million in month 3, a city of 200,000 in month 6 and a town of 20,000 in month 12, so the timing follows the size, not the distance.

How did diffusion theory develop?

Early geographers and anthropologists, such as Carl Sauer, studied the diffusion of crops, animals and cultural traits from “hearths.” The Swedish geographer Torsten Hägerstrand published “Innovation Diffusion as a Spatial Process” in 1953 (translated into English in 1967), which introduced quantitative modeling. Everett Rogers’s book Diffusion of Innovations (1962) described the adopter categories. Since the 1960s, models of the diffusion of epidemics and information have been developed, and in the 21st century, digital networks have allowed ideas and trends to spread across the world within hours, so that distance matters less, but other factors, such as language, platform access and censorship, still shape the geography of diffusion.

Which mistakes are common?

Mistake 1: Assuming that diffusion always depends on distance

Distance matters in contagious diffusion, but hierarchical diffusion follows networks and rank, and the internet reduces the role of distance.

Mistake 2: Confusing relocation and expansion diffusion

In relocation diffusion, people move with the idea. In expansion diffusion, the idea spreads through the population, and the origin stays a source.

Mistake 3: Expecting that everyone adopts at the same speed

Adopters differ widely, and adoption begins slowly, speeds up and then levels off, along an S-shaped curve.

Mistake 4: Seeing diffusion as a one-way flow from a single center

Ideas move in many directions, change as they spread, and often arise independently in several places.