
How do cities grow? Why do some organisms thrive while others struggle?
In Scale, Geoffrey West reveals the mathematical patterns that govern everything from urban development to metabolism. Drawing on his groundbreaking research at the Santa Fe Institute, West shows how all complex systems—from organisms to organizations—follow predictable scaling laws that determine their success and survival.
Read below for a brief overview of West’s book Scale.
Overview of Scale by Geoffrey West
In Scale, Geoffrey West reveals a hidden mathematical order that governs how systems (from organisms to companies to cities) grow and change. While we might assume that a doubling in size simply means a doubling in all other characteristics, West shows that reality is more surprising: Many complex systems in the manmade and natural worlds follow precise mathematical patterns as they scale up or down—patterns that transcend the apparent differences between biological and social systems.
Consider how cities change as they grow: When a city doubles in population, it doesn’t just need twice as many roads or generate twice as much economic activity. Instead, it consistently needs about 85% more infrastructure but produces about 115% more wealth, patents, and innovation. Similar mathematical regularities appear in everything from the way animals’ metabolic rates change with body size to how companies’ productivity shifts as they expand. West contends these aren’t just interesting mathematical coincidences: They reflect basic principles about how networks distribute energy, resources, and information within complex systems.
West is a theoretical physicist whose work has been motivated by a search for unifying principles,—core principles that connect and explain diverse phenomena and build a foundation for further scientific inquiry. Just as Newton’s laws help us see the mathematical principles governing motion, West’s work on scaling reveals the mathematical architecture underlying the complex systems in the world around us.
What Is Scaling?
West explains that scaling describes the mathematical relationships between different characteristics of a system as its size changes. These relationships often follow precise mathematical patterns called power laws, where one quantity varies as a power of another. For example, when examining how cities change with size, we might ask: If a city’s population doubles, how do its other characteristics—like road surface area, crime rates, or patent production—change? Surprisingly, these characteristics don’t simply double; instead, they change according to consistent mathematical ratios that can be described by power laws.
These patterns appear consistently across a wide range of systems, from biological organisms to cities to social networks. While these systems might seem fundamentally different from one another, they share common principles in how their characteristics change with size. In this section, we’ll examine three crucial mathematical relationships that West identifies: economy of scale (where systems become more efficient as they grow), increasing returns to scale (where growth leads to proportionally greater outputs), and inverse relationships (where an increase in one characteristic leads to a predictable decrease in another).
Economy of Scale
First, West explains that a foundational pattern we observe in scaling is economy of scale: systems become more efficient at larger sizes than at smaller sizes. This pattern demonstrates what mathematicians call sublinear scaling—one of several ways that variables can relate to each other as systems change in size.
To understand sublinear scaling, it helps to first understand linear scaling, where two variables increase proportionally. In linear relationships, as one variable increases, the other increases at exactly the same rate. For example, if you earn $20 per hour, your pay scales linearly with the hours you work: Two hours equals $40, four hours equals $80, and so on. If you plotted this relationship on a graph, with hours worked on the X axis and dollars earned on the Y axis, you’d get a straight line moving upward at a constant angle.
West explains that not all relationships in nature are linear, and sublinear scaling follows a different pattern. When something scales sublinearly, the output increases more slowly than the input. If you plotted a sublinear relationship on the same graph as a linear one, its line would curve below the straight line, showing that it advances more quickly along the X-axis than up the Y-axis.
West explains that we can see sublinear scaling at work in biological organisms through their metabolic rates: the energy required for an organism to stay alive. An animal twice the size of another needs only about 75% more energy, not 100% more, to function. This efficiency explains how giant animals like blue whales can exist: Even though they must solve the same basic biological problems as smaller animals—like circulating blood and breathing—they can do so more efficiently per unit of body mass.
Another way to explain this mathematical pattern is to say that the sublinear relationship between size and metabolic rate follows a power law with an exponent of about 0.75 across many species. Consider a 120-pound woman and her 60-pound dog. While the woman’s mass is twice that of her dog’s, her metabolic rate (around 1,300 calories per day) is not twice her dog’s (around 880 calories per day). This demonstrates that larger organisms require less energy per unit of mass than smaller ones.
This sublinear scaling of metabolic rate has implications for how organisms live and grow, and it creates natural limits on size. While larger organisms become more efficient at using energy as they grow, they eventually reach a point where they can’t generate enough extra energy to grow further. This also affects the pace of life: Larger animals tend to have slower metabolic rates, longer lifespans, and slower life rhythms compared to smaller ones. This pattern helps explain why mice live fast and die young, while elephants live slow and die old.
Increasing Returns to Scale
Another important pattern we observe in scaling is increasing returns to scale: when systems become more productive (not just more efficient) at larger sizes. This pattern demonstrates what mathematicians call superlinear scaling, where outputs grow faster than inputs. Again, it helps to compare this to linear scaling. If a restaurant chain’s revenue scales linearly with its number of locations, opening 10 new restaurants would mean generating exactly 10 times more revenue. But some systems show a different pattern: They produce proportionally more output as they get bigger.
West explains that we can see superlinear scaling most clearly in social systems like cities. When a city doubles in population, it typically more than doubles in measures like economic output, innovation, and even crime rates. If you plotted these relationships on a graph, the line would curve above the straight line of linear growth, showing that outputs are increasing faster than inputs.
Social networks provide a compelling example of why this happens. Consider a college campus that grows from 2,000 to 4,000 undergraduate students. While the student population doubles, its measures of social and intellectual output more than double. The number of student organizations might increase from 100 to roughly 250 (rather than the 200 we’d expect with linear scaling), and the number of research collaborations between students might jump from 50 per semester to 140 (rather than 100).
West explains that the superlinear growth in a university’s innovation occurs because each new person added to a network doesn’t just bring their individual contribution—they create new possibilities for interaction with everyone already in the network. When a campus doubles in size, it doesn’t just have twice as many people; it has more than twice as many possible connections between people. Each new student can form relationships, share ideas, and collaborate with all existing students, creating an amplified effect on creative and social output.
West points out that superlinear scaling helps explain many phenomena we observe in the real world: why larger cities generate disproportionately more patents and startup companies than smaller ones, why cultural innovations tend to emerge from urban centers rather than rural areas, and why universities with larger student populations often produce dramatically more research breakthroughs than smaller institutions.
Inverse Relationships
A third crucial pattern in scaling involves inverse relationships: when two characteristics of a system move in opposite directions in a precise, predictable way. This pattern is different from both sublinear and superlinear scaling because it describes how two variables relate to each other, rather than how a system changes with size.
To understand inverse relationships, consider how a river’s speed changes with its width. When a river channel narrows, the water must flow faster to maintain the same volume flow rate. When the channel widens, the water slows down. This creates an inverse relationship between width and speed: As one increases, the other must decrease proportionally. The product of width and speed stays constant because the same volume of water must pass through each point of the river in the same amount of time.
West explains that we can see inverse relationships at work in both natural and human-made systems. In biology, there’s often an inverse relationship between the number of offspring an organism produces and the energy it invests in each one. Fish might lay thousands of eggs but invest little energy in each, while elephants typically have one calf at a time but invest enormous resources in raising them. The product of offspring number and energy per offspring tends to remain constant within similar types of organisms.
Why Do Natural and Manmade Systems Scale the Same Way?
It may seem counterintuitive that the same mathematical principles apply to both naturally occurring systems and human-made systems. But West explains that systems as different as organisms and cities scale in similar ways because, at their core, they’re all built on networks. Whether we’re looking at blood vessels in a body or roads in a city, these networks share fundamental characteristics that determine how they grow and function. In this section, we’ll take a closer look at universal scaling laws as a consequence of networks’ basic features, including their tendency to create specific repeating patterns.
The Power of Networks
West explains that most complex systems are built around “hierarchical branching networks.” Think of these like a tree with a thick trunk that splits into major branches, which split into smaller branches, and so on. This basic pattern appears across nature and human design: In biology, blood vessels branch from the aorta down to tiny capillaries. In cities, highways branch into main roads, then local streets. In social networks, close relationships branch out into wider circles of acquaintances.
West explains that the same kind of networks occur not only in nature but also in the complex systems that we build. For instance, engineers and designers lay out roads, power lines, and water pipes in hierarchical branching networks that distribute resources from central supply points to individual buildings. At each level of these hierarchies, the capacity—to carry cars, electricity, or water—scales down geometrically as the branches grow smaller. For example, a few major highways feed into larger arterial roads, which further branch into smaller residential streets leading to individual homes.
The space-filling architecture of hierarchical branching networks creates economies of scale as a city’s size increases. Increasing an infrastructure network’s capacity requires extending the network, but not building new systems from scratch. For example, adding a new neighborhood only requires extending existing roads, power lines, and water lines from the nearest branches, rather than building new systems.
West explains that even human social networks show a similar hierarchical, modular structure: We make strong connections with a small core group (about five people) and have larger but more weakly tied groups of about 15, 50, and 150 people. The flow of information and the strength of bonds diminish through these levels.
The Fractal Pattern
West explains that hierarchical branching networks often mimic a pattern mathematicians call “fractals.” To understand fractals, imagine looking at a tree in winter: First, you might look at the whole tree and see a pattern of branches splitting into smaller branches. Then, if you zoom in on just one branch, you’ll see the same pattern repeated: branches splitting into smaller branches. Then, if you zoom in on an even smaller branch, the pattern appears yet again.
This self-repeating pattern is called “self-similarity,” and it’s a defining feature of fractals. While perfect fractals (like those created by mathematicians) repeat infinitely, natural systems show this pattern across multiple, but not infinite, scales. It’s from these self-repeating patterns that scaling laws governing metabolic rate, growth, lifespan, and other biological traits emerge.
The Result: Universal Scaling Laws
West explains that because networks as diverse as biological circulatory systems, urban transportation networks, and corporate communication networks share such fundamental properties, they lead to predictable scaling patterns. This explains why we can predict an animal’s metabolic rate just from its size, regardless of species; why cities show consistent patterns in everything from road surface area to innovation rates, regardless of location or culture; and why social networks develop similar hierarchical structures, whether in small towns or major metropolises.
Since universal scaling laws are a consequence of network characteristics, they give us a mathematical framework to see the simplicity underlying complex organisms and networks. Using this mathematical framework, scientists can make precise, testable predictions about biological quantities (like metabolic rate, growth curves, and lifespans) based solely on an organism’s size, transcending the specifics of anatomy or taxonomy. They can also apply this knowledge on larger scales, using it to learn about the fundamental constraints and tradeoffs that shaped the evolution of life.
The same idea applies to experts who study and work to improve human systems, like cities. Cities as diverse as New York, Tokyo, and São Paulo, with vastly different histories and cultures, all exhibit similar scaling patterns in metrics like infrastructure, wages, innovation, and crime rates. This highlights how universal scaling laws originate from the common network dynamics underlying all cities, rather than being influenced by specific local contexts.
How Can We Use This Network Perspective to Improve Our World?
West explains that understanding how networks shape our cities gives us powerful tools for improving them. His research suggests four key strategies:
1. Design Smarter Infrastructure Networks
First, West explains that cities can save enormous resources and make more efficient planning decisions by optimizing their physical networks. Because these networks follow sublinear scaling patterns, simply adding more infrastructure linearly (like doubling roads when the population doubles) is inefficient and unnecessary. Instead, cities should use their understanding of network scaling to make strategic improvements.
Transportation: Traditional approaches to traffic congestion often focus on adding capacity everywhere—widening roads or adding lanes across the system. However, West explains that this linear solution ignores how traffic flows through the hierarchical network of a city’s roads. Instead, planners can analyze traffic patterns to identify critical bottlenecks where the hierarchical network breaks down. For example, a city might reduce congestion dramatically by adding just a few strategic connector roads between major arteries, allowing traffic to flow more efficiently through the entire system.
Utilities: The same principle applies to water, power, and other utility systems. When cities need to expand these services, West recommends using network analysis to identify the most efficient points for expansion. For example, rather than laying new pipes along every street in a growing area, a city might be able to accommodate 20% more residents while adding only 5% more pipelines by carefully analyzing where new main lines would have the greatest impact on the system’s overall capacity.
Public Services: West explains that emergency services also function as networks, and their effectiveness depends more on their structure than their size. Instead of simply adding more facilities, cities can use network modeling to optimize their existing resources. For instance, rather than building new fire stations, a city might significantly reduce emergency response times by repositioning existing stations based on population density and traffic patterns, creating more efficient routes between stations and high-risk areas, and coordinating responses between multiple stations to provide better coverage.
2. Create Spaces That Spark Connection
Second, West explains that since cities exhibit superlinear scaling in innovation and economic growth—driven by human interactions and information exchange—cities should deliberately design spaces that catalyze these connections. This isn’t just about creating pleasant environments; it’s about leveraging the mathematical advantage that comes from increasing social interactions in urban spaces.
Mixed-Use Development: Traditional urban planning often separates residential, commercial, and business areas from each other, which limits spontaneous interactions between people. In contrast, West points out that when cities convert several downtown streets into pedestrian zones that mix shops, cafes, and housing, they create multiple opportunities for unplanned encounters and collaborations. This increased social connectivity leads to measurable economic benefits.
Innovation Districts: Some cities have taken this concept further by deliberately combining research facilities, startups, and housing into “innovation districts.” West explains that this approach creates a powerful feedback loop: When researchers, entrepreneurs, and creative professionals live and work in proximity, the number of potential collaborative connections grows superlinearly. The result is often dramatic: Cities typically see a huge increase in patent applications from companies in these districts, far beyond what would be expected from simply gathering the same number of people in separate locations.
Public Spaces: West emphasizes that even simple infrastructure changes can trigger superlinear growth in social and economic activity if they increase connectivity between people. When cities create unbroken walking networks connecting neighborhoods, they’re not just building pleasant walkways: They create a continuous space for social interaction that links previously separated communities. This enhanced connectivity typically leads to a surge in local business formation along walking paths, as the increased foot traffic and social interactions create new economic opportunities.
3. Balance Growth’s Costs and Benefits
Third, West explains that cities face a fundamental challenge: The same superlinear scaling that produces increased innovation and wealth also amplifies negative outcomes like crime and disease transmission. Understanding this mathematical reality means cities must actively manage both the advantages and disadvantages of urban scaling, rather than assuming growth will automatically solve problems.
Crime Prevention: West points out that larger cities have disproportionately more crime—not just more total crime, but more crime per capita than smaller cities. This is a predictable result of superlinear scaling: The increased social connectivity that drives innovation also creates more opportunities for criminal activity. Understanding network principles helps cities respond more effectively. Instead of adding more police officers in proportion to population growth (a linear response to a superlinear problem), cities can use network analysis to identify critical points where criminal networks intersect and focus on these “hot spots.”
Public Health: The Covid-19 pandemic demonstrated how disease transmission also scales superlinearly with city size. Larger cities experienced faster spread rates per capita than smaller ones. West explains that cities that understand how networks affect public health are better equipped to manage an infectious disease crisis.
For example, Seoul, South Korea used network analysis to identify and respond to transmission clusters, tracking how infections spread through different locations and age groups. Finding that certain venues (like nightclubs and churches) and age groups (people in their 20s) created disproportionately more connections, Seoul implemented targeted interventions rather than blanket policies. This network-based approach helped the city maintain relatively low disease transmission rates while keeping other parts of the city functioning.
Economic Opportunity: While cities generate more economic opportunity per capita as they grow, West explains that these benefits are often distributed unevenly. This happens because economic opportunities cluster in certain areas (like business districts or innovation hubs) while other neighborhoods become structurally isolated due to physical barriers (like highways or rivers), inadequate public transit connections, or historical patterns of disinvestment.
The network features that make cities efficient can inadvertently create “opportunity deserts” in areas that lack good connections to these hubs. However, cities can use network analysis to identify isolated neighborhoods and strategically add connections to their transportation network, effectively connecting more people to job opportunities without building entirely new transportation systems.
4. Build for Sustainable Growth
Lastly, West explains that cities face a critical challenge rooted in their scaling patterns: While infrastructure becomes more efficient as cities grow (scaling sublinearly), resource consumption and waste production increase faster than population growth (scaling superlinearly). This means that bigger cities, despite their per-capita efficiency advantages, still use resources at an accelerating rate. Understanding this mathematical reality is crucial for designing sustainable urban systems.
Energy Networks: Traditional city energy systems are often designed linearly, with each building or block operating independently. However, West explains that cities can significantly reduce energy consumption by creating “superblocks”—groups of buildings that share resources and infrastructure in a networked way. This approach takes advantage of sublinear scaling: Just as larger organisms use proportionally less energy per unit of mass, these superblocks can achieve better efficiency than individual buildings operating alone—for example, by using excess heat from one building’s cooling system to heat another building’s water.
Circular Economy: West points out that cities typically consume resources and produce waste faster than their population growth would predict. City planners can counter this trend by using network analysis to map material flows and create closed-loop systems.
For example, some cities have mapped their construction material flows and created networks connecting demolition sites with new construction projects, allowing building materials to be reused rather than discarded. Others have created networks where one facility’s waste becomes another’s raw material, like using excess heat from data centers to warm greenhouses or converting food waste into compost for urban farms. By tracking how resources move through these networks, cities can identify opportunities to keep materials circulating in the economy rather than ending up in landfills.
Innovation Cycles: While increased resource consumption is a challenge of urban scaling, West explains that cities can leverage another scaling pattern—the superlinear growth of innovation—to address sustainability challenges. When cities put out calls for sustainability solutions, they tap into their dense networks of creative and technical professionals. For example, some cities have developed smart grid systems that reduce power consumption through collaborations between multiple groups: Utilities contribute their infrastructure expertise, tech companies develop the software, universities research optimal patterns of energy use, and residents provide real-time feedback through smart meters.
Practical Implementation Steps
Systems and networks are complicated, and it can be difficult to imagine where to start if you want to improve your city or town. West suggests communities start by:
- Mapping their current networks (physical, social, and economic)
- Measuring how different variables scale as the city grows
- Identifying where network breakdowns create inefficiencies
- Testing small-scale interventions before scaling up
- Continuously monitoring and adjusting based on network metrics
West emphasizes that we don’t have to choose between growth and sustainability. By understanding and working with network principles, we can help our communities grow more efficiently by optimizing existing networks, fostering innovation while managing responsible resource use, creating connected communities while reducing environmental impact, and building resilience through redundancy in critical systems.
The key is to use network science not just to understand cities, but to shape their development in ways that enhance both human flourishing and environmental sustainability. By recognizing the universal dynamics at play, cities can better navigate the tensions between efficiency, connectivity, and sustainability, paving the way for more livable, resilient, and equitable urban environments.