Science and Culture: Can the principles of topology help improve the world’s slums?
Stephen Ornes
According to the United Nations (UN), nearly 1 billion of the 8 billion people in the world live in slums (1). These dense and disordered communities form in urban areas without a formal plan; their residents frequently lack addresses. Often they don’t have immediate access to water, power, or emergency services.
Slums stem from all sorts of complex social and economic ills that don’t have easy solutions. But these chaotic communities share a common trait: a lack of connectivity. “Many parcels aren’t connected to a broader urban infrastructure — roads, power, telecommunications, sewer lines,” explains Christa Brelsford, a fellow at the Oak Ridge National Laboratory in Oak Ridge, TN. Brelsford’s research examines ways that people interact with urban environments.
That lack of connectivity, she says, is essentially a math problem. For the last 5 years, she has been working with interdisciplinary researchers to analyze data, collected firsthand by residents, to define mathematical ways in which slums differ from more functional neighborhoods. The researchers are seeking efficient and nondisruptive ways to improve lives.
But scholars who’ve studied slums and their multifaceted origins are skeptical that applying such mathematical approaches will help. Some fear they could even make the problem worse.
The Bridges of Konigsberg
Topologists study surfaces and spaces; they want to know what properties remain after transformations such as stretching, squashing, or twisting. If two surfaces can be deformed into one another without tearing, ripping, or stitching, then they’re topologically equivalent. Topologists, according to an old math joke, don’t know the difference between a coffee cup and a donut. But the field’s roots can be traced back to geography. “There’s a long history of using mathematics to describe connectivity, and to describe urban problems,” says Brelsford.
Topology traces back to the Konigsberg Bridge problem, probably the most famous connectivity problem in math history. Konigsberg, a city near the Baltic Coast, was founded in the 13th century on the Pregolya River in Prussia. The city spanned both banks of the river and islands in-between; in total, it had four land masses, connected by seven bridges.
The famous problem: Is it possible for a person to walk through the city, crossing each bridge once and only once? People who tried quickly realized that they either had to cross at least one bridge twice — or skip one altogether.
The conundrum caught the attention of prolific Swiss mathematician Leonard Euler, who in 1727 had moved to St. Petersburg, Russia. “This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it,” he wrote to Italian mathematician Giovanni Marinoni, who was Court Astronomer to Kaiser Leopold I, in Vienna (2). In his solution to the problem, published in 1736, Euler noted that one could list all possible routes, but such a brute-force solution would require a considerable amount of time. Plus, it wouldn’t be applicable to other configurations of bridges and islands.
Euler proved that no such route existed for Konigsberg and at the same time generalized his results to describe what kinds of networks would have a path with no double-crosses (3). In doing so, he was the first to use the language of the emerging fields of graph theory — the rigorous study of networks — and topology. Along with Luis Bettencourt at the University of Chicago in Illinois, the project’s leader, Brelsford and her collaborators are applying that same language to urban settlements.
Teasing Apart Topology
Brelsford’s work has already turned up some surprises. In an article published in Science Advances in August (4), she and her team reported that the spatial character of a city is best understood not by using the tools of geometry but by turning to topology. Slums, they found, have a different topological structure than organized cities.
The difference between geometry and topology offers some insight as to why the latter could be useful. A geometric approach considers the shapes and sizes of blocks, as well as the lengths, positions, and orientations of roads. It would note curves and straightaways. But topologists study surfaces and connections. The topological approach described by Brelsford and her collaborators instead focuses on connections to probe at the network-like nature of a neighborhood.
Brelsford has helped develop new tools that use topology to improve slums. This method could determine, for example, the fewest streets that would need to be added to provide street access to everyone, and at minimal cost and with minimal disruption to the residents. The tools are now being put to use by residents, urban planners, and local governments to test and implement mathematically derived strategies in large slums in Cape Town, South Africa; Mumbai, India; and Nairobi, Kenya.
Evolving the City
Bettencourt’s research focuses on using big data and quantitative methods to understand how cities grow and evolve. Even Konigsberg, in 1875, built an additional bridge that reduced the famous problem to a triviality. With that additional connection in place, visiting the city by crossing each bridge only once became simple. Bettencourt says that over time, urban areas naturally tend toward becoming more connected. “In a curvy or weird way, all cities that become formal do this,” he says. “But slums don’t.”
What the mathematical method does, in effect, is speed up this natural evolution, bringing connectivity into the slums. Bettencourt sees their work as a bridge between the kind of formal solution that an urban planner might propose and the natural evolution of the “fabric of the city.”
The idea to turn to topology came from slum residents themselves, says Brelsford. While she was a postdoctoral researcher at the Santa Fe Institute in Santa Fe, NM, where Bettencourt was a faculty member at the time, she studied firsthand reports from Slum Dwellers International in Palo Alto, CA, a nonprofit group. “The folks who live in these communities were literally saying they were disconnected,” she says. That word, “disconnected,” was a tipoff.
Her group’s topological view of a city begins by observing that all the streets and paths in an urban area together can be seen as a connected 2D space. That means a traveler can move from one location to another without leaving the space or making any jumps or breaks. With that in mind, blocks and buildings become “holes” in the surface — you have to go around them as you travel from one place to another.
Just as a donut and coffee cup are equivalent because they each have one hole, dense, urban areas with the same number of city blocks are also said to be topologically equivalent. A video that accompanies the 2018 Science Advances article (4) shows how developed sections of Mumbai can be deformed, step by step, to take on the shape of city blocks in Manhattan or suburban developments in Las Vegas. Similarly, the researchers note, parts of Baghdad can be reshaped to look like Beijing.
“The folks who live in these communities were literally saying they were disconnected.”
— Christa Brelsford
Changing the topological structure of slums to look more like cities, the researchers suggest, will open up access. In one case, the team applied their method to data collected by residents of one section of a slum near Harare, Zimbabwe. Currently, the section includes seven sections that are completely blocked in the interior with no access to roads. Using their model, the researchers determined that a minimum of six paths — totaling 119 meters — was needed to ensure that every parcel had access to a path or road.
Asking the Neighbors
Slum upgrading has long been a goal of cities around the world. In a 2014 paper (5), The United Nations-Habitat, the UN’s agency for sustainable development, cautioned against one-size-fits-all strategies and recommends that neighborhoods pursue “reblocking,” which means reconfiguring an area so that every building can be reached with a road and services such as running water, gas, and sanitation are available to everyone.
Topology is one way to find reblocking strategies, says Brelsford. But it doesn’t guarantee success. Two problems need to be solved simultaneously to manage a slum-upgrading program, she says. “You have to rearrange space to make space for roads. And you have to get a lot of people to agree on something,” Brelsford says. “It’s not my place to be telling people who live in a developing country what they should be doing with their own communities and neighborhood.” The math can provide guidance, she says, but it doesn’t account for the day-to-day reality of living in a slum.
Plus, the optimal solution might be impossible on the ground — the most efficient proposed new connections might require cutting through existing buildings. The test cities offer dramatically different experimental cases. Mumbai’s slums are denser than Cape Town’s, for example, and they include more concrete buildings. “It’s hard to go into a city like Mumbai and find ways to create these accesses,” says Bettencourt. Thus far, he and his team have taken the math-driven ideas for new streets to residents, hoping they’ll use those proposals as a way to find changes that both improve life in the slums and don’t cause major disruptions.
Solutions or More Problems?
Some urban researchers argue such a mathematical approach won’t make a dent in the existing problems and could raise new ones. “I think it’s naive,” says Alain Bertaud at New York University’s Marron Institute of Urban Management and author of the 2018 book Order Without Design: How Markets Shape Cities (6). Bertaud, who has studied slums for more than 4 decades, disagrees that the central problem of slums is layout; he says the most successful strategies he’s seen involved upgrading the existing infrastructure rather than redesigning it. “They’re trying to solve a nonexistent problem,” he says. “The main problem isn’t geometric. Or topological. And my experience with slum upgrading is that any change in existing boundaries within a slum [is] always extremely disruptive.” He points to the kampong neighborhoods of Indonesia where people in even the smallest dwellings have access to clean water and sanitation because of upgraded infrastructure driven by necessity — and not through top-down interventions. A large-scale redesign there, he says, would be disastrous and disrupt the welfare of inhabitants. Changes in boundaries might work for small parcels but not for entire neighborhoods.
Bertaud says the real-world complexity is the very thing that could derail the topological strategy. New roads need space, and that space will have to be carved from the living space of people who already live in the slums, he says. Disruption is unavoidable. “You have to push some people out.”
Bettencourt thinks the method can be scaled up from the block level to entire slums, but Bertaud has doubts. “They should shift scales,” he says. The topological test cases describe solutions for blocks only a few hundred meters on a side, but in large cities, he says, blocks can sprawl over kilometers. He thinks such large swaths of land require broader solutions. Path networks in slums, he says, have been established by the people who live there, and for any large-scale changes in layout, parcel boundaries and topography would likely thwart any benefit derived from using topology. Plus, analytical and big-data approaches to solving problems in slums may fail to recognize the forces that gave rise to the disordered neighborhood in the first place, says city planner Jason Corburn at the University of California, Berkeley.
“As we focus on measurement, methods and modeling, we tend to ignore the local knowledge and expertise that already exists in these communities,” says Corburn, who co-edited the 2016 interdisciplinary book Slum Health: From the Cell to the Street (7). “We ignore the institutions and actors that use discriminatory process, like racial or ethnic segregation, to create and re-create these living conditions.” Other political scientists, similarly, have argued that using algorithms to fight poverty or homelessness can actually exacerbate the problem.
Nevertheless, Bettencourt remains optimistic. He says the slum residents are deeply involved in his projects, from studying maps to commenting on solutions to finding ways to reach consensus.
“This is all driven by local communities,” Bettencourt says. “We try to find tools that are congenial to the kinds of knowledge that the people already have. Part of the challenge today is to find balance between rich new technologies that allow us to see so much data and deep social processes.”
Published under the PNAS license.
