Gradial: Or the Unreasonable Network

The reasonable network adapts itself to the world; the unreasonable one persists in trying to adapt the world to itself. Therefore all progress depends on the unreasonable network.1

The physical location of network infrastructure is one of the most permanent decisions cities make. The Cardo Maximus in the old city of Jerusalem is still a main north-south shopping street, constructed when Emporer Hadrian rebuilt the city in the 130s CE.

A street right-of-way, once created is seldom destroyed. A segment of that infrastructure is designed to be optimal at a moment of time, with a particular land use (either the realized development of today or an imagined place of tomorrow), enmeshed within a particular network context of all the other nodes and links, compatible with a particular technology. That it functions at all when land use, networks, and technologies change radically, as they do over centuries, is testament to the general flexibility inherent in networks. But the implication is that if it is optimal for the world in which it was designed, it is unlikely to be optimal as that world changes.

Some adaptations do occur. Streets designed for horses were adapted for streetcars (trams) and bicycles and cars and buses and pedestrians.

Still, it may be the best that can be done. Embedded infrastructure, the dictionary example of sunk costs,2 cannot adapt much to the world around them. Instead we expect the world to adapt to the infrastructure.

Following Shaw, we might say such infrastructures are `unreasonable’, in that they cannot be reasoned with.

Many, if not most, planned cities have been laid out with a network of streets “with the sombre sadness of right-angles,” as Jules Verne, quoting Victor Hugo, described the American grid in Salt Lake City, of streets at 90-degree angles to each other, in his classic road trip story: Around the World in 80 Days. Street grids don’t plan themselves, so while all street grids were planned, not all plans result in street grids.

Organically developed3 cities are often more naturalistic, radial cities, with streets feeding the city from the hinterlands, allowing more than 4-directions of entry. All roads lead to Rome, as the saying goes. The Romans themselves were a bit adverse to this organic radial system once they got their own growth machine going, laying out encampments and new settlements on the grid system. The radial system leading to and from the town would bend once it reached the town gates. But as cities themselves were generally not conceived of as whole, but rather themselves emerged, often as conurbations of smaller settlements, towns, and villages, there are often radial webs centered on town A overlapping radial webs centered on town B. Rome was famously built on seven hills, which can be read as meaning Rome is a conurbation of seven earlier villages. (See Elements of Access, Chapter 3.3)

Each of these networks typologies has its advantages and disadvantages.

DCMetro
Washington DC Metro. The center is a space, not a point. A `triangle’ is formed by L’Enfant Plaza (Yellow/Green with Orange/Blue/Silver), Metro Center (Red with Orange/Blue/Silver), and Gallery Place (Red with Yellow/Green)

 

We observe that radial networks are optimal to maximize access for many-to-one types of movements (suburbs to central city). So rail transit networks, which serve the high loads demanded by, and making possible, high density city centers tend toward being radial. But when they are large they are usually not so radial that all the branches meet at one junction. From a network design perspective, intersecting more than two lines at a station can lead to other types of conflicts, and many systems are designed with triangular center to avoid overloading a single transfer station. Washington DC’s largely radial Metrorail system, shown in  the first figure, illustrates this design. Cities are spaces, not points.

In contrast, the 90-degree grid is reasonably well-suited to maximize access for scattered trips, what network analysts would call a many-to-many pattern. We see this especially in dispersed point-to-point (suburb to suburb, within city to within city) flows that are enabled by and reinforce the grid. This is the network for the automobile. The Los Angeles freeway grid, the famous Milton Keynes arterial grid, and numerous other  late twentieth century cities have been designed in a grid-like way (though not so orthogonal that Victor Hugo would object). Even though the topology is not as efficient from a distance perspective as say a 60-degree mesh, by remaining out of the city core it can keep speeds higher.

But in response to the landscape that emerged with the automobile, transit planners like Jarrett Walker (2012) have called for more grid-like transit networks, so people can move, via public transport, from suburb to suburb without going through the city centre. This is relatively easy to reconfigure for buses, the very definition of  mobile capital, while very difficult for the more capital intensive rail networks with their physically embedded infrastructure.

Still, core radial lines will always be the backbone of transit systems so long as at least one important center justifies a disproportionate amount of service.

So how can we grid the radial, or square the circle, so to speak?

A better network topology might be the 60-degree, hexagonal pattern. (Ben Joseph 2000) But remaking street grids for existing cities is tough-going, as property rights are well established, and requires efforts like those of Haussmann in 19th century Paris. (Willms 1997).

daganzo
Possible system layouts: (a) hub-and-spoke; (b) grid; (c) hybrid. Source: Figure 1 in Daganzo (2010)

Instead, we have overlapping network topologies, ideally which are grade-separated in some fashion, so trains are radial and don’t intersect streets or motorways, and bus services can be more grid-like, and rapid or express bus networks serve the market niche in-between.

Thus the original street level networks are still topologically grids, but the services running on that grid, while still largely parallel and perpendicular, are compressed near the center, so the bus lines, for instance, bend towards the center, as illustrated in the second figure. The regulatory layer of through streets for automobiles may be constructed to defer to the orientation of bus services.

There are no optimal network configurations independent of the enveloping land use pattern or the technological regime. Similarly there are no optimal land use allocations independent of the network pattern or technology. Finally, there is no optimal mode independent of the land use or network. All three of these systems are interlocking. Moving one requires adapting the others.

The unreasonable network forces the land use pattern to adapt to it, such that relocating network elements is more costly than keeping them in place. Similarly, in many ways the network, designed for a given technology, is very hard to adapt to a different technology. That doesn’t stop people and cities from trying, the misfit we see with the automobile in the urban core is the product of failing to acknowledge this unreasonableness. But as the number of European cities restricting cars in the city center are showing, the unreasonable network wins out over technology too.

The Grid/Radial Gradial network is also Gradual. These systems seldom change all-at-once, instead they gradually evolve over decades, centuries, and millenia.


Notes:

1. This is an adaptation of a famous George Bernard Shaw quote.

The reasonable man adapts himself to the world; the  unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.

2. The economist’s adage that “sunk costs are sunk” means that once something has been built, and that money spent, it no longer factors into benefit-cost analysis about how prospective decisions should be made, except to the extent it changes the costs of various options. Logically, you shouldn’t go to a concert just because you bought tickets if you don’t want to go, though if you are considering going to a concert or a bookstore after you bought the tickets, you don’t need to account for paying for the tickets again. You might also consider the `opportunity cost’ of going as the loss from not scalping the tickets. You shouldn’t throw good money after bad. But the sunk infrastructure cannot be unbuilt.

3. Organic development is often largely systematically unplanned, though obviously some degree of planning often goes into laying out a street, even if it is disjoint from any other decisions. When we think of `planning,’ we are generally referring to longer-term more strategic type spatial plans, that consider interactions between prospective decisions, rather than short-term tactical plans that optimize a single decision alone decontextualized from the rest of the city.

On Hypo- and Hyper-connectivity in Transport

Connectivity is good. Is more connectivity better?

During the early stages of a useful technology like roads or transit, adding links generally adds more benefits than costs. However there are limits. A four way intersection is good does not mean a five way intersection (or six or seven) is necessarily better. The more complex intersection adds to the friction of travel and cost of construction over its simpler alternatives.

Muller's Hexagonal Network
Muller’s Hexagonal Network

A grid network, with streets at 90-degree angles to each other might not be as good as a network with streets at 60-degree angles, which reduces travel costs and increases directness (reduces circuity), but it is most assuredly better than a fine mesh with streets at 10-degrees or 1-degree, where almost all is pavement and little is actually buildable land. While 1-degree network would reduce surface travel distance, it does so at many other costs, including a reduction in accessibility because of fewer development opportunities.

Consider the circuity additions based on network angle. If all places are connected via a 90-degree square grid, the circuity at worst is SQRT(2), but on average 1.21.  So travel distance increases by 21% over a straight-line path. With a 60-degree grid, the circuity is lower, at worst 1.22, on average nearer 1.11. (Bus transit networks, which tend not to follow the shortest path, have much worse circuity.)

The optimal level of connectivity depends on what you are trying to optimize.

Hypo and Hyper are antonyms. Wiktionary says:

I would maintain that most developed countries are pretty close to optimal in terms of road connectivity, that there are few missing links whose costs outweigh their benefits. If subsidies for modes were to be eliminated, some large cities might be under-developed in terms of transit connectivity because of a bias towards coverage (and circuity) aims rather than frequency.

Let’s think of this in the context of induced demand. More connectivity in one sense means a faster network, which users exploit by traveling longer distances in the same amount of time. They gain utility by being in a house they prefer. However they use up the capacity gains of the network. But more connectivity increases the friction of connections (junctions, interchanges, transfers) which slows down the network. Induced demand due to connectivity is thus self-limiting.

Braess Paradox is the most famous supply side example of hyper-connectivity. In this situation, removing a link improves travel for road users at large because the additional network link induces travelers to use a link with a lower average cost but higher social marginal cost.

A key point is that whether a network is over or under-connected depends on the technology of travel, as well as the amount. A network which is overconnected for cars may be underconnected for pedestrians who don’t congest so easily. A network which is overconnected for 2000 cars may be underconnected for 1000. This is the challenge in building cities. Networks last for seemingly forever, but technologies that use them change more frequently. How can you design a permanent infrastructure flexible enough to serve future technology?

 

Evolution of the Sydney Trains Network

Some work we have done at TransportLab at the University of Sydney.

Evolution of the Sydney Trams Network

Some work we have done at the University of Sydney’s TransportLab on Network Growth in Sydney:

Accessibility and the choice of network investments in the London Underground

Recent working paper:

Levinson, D., Giacomin, D., and Badsey-Ellis, A. (2014) Accessibility and the choice of network investments in the London Underground. Presented at the World Symposium on Transport and Land Use Research, June 2014, at Delft.

Accessibility in London in 1881
Accessibility in London in 1881

 

 

  • Abstract: In 1863, the Metropolitan Railway of what came to be known as the London Underground successfully opened as the world’s first subway. Its high ridership spawned interest in additional links. Entrepreneurs secured funding and then proposed new lines to Parliament for approval, though only a portion were actually approved. While putative rail barons may have conducted some economic analysis, the final decision lay with Parliament, which did not have available modern transportation economic or geographic analysis tools. How good were the decisions that Parliament made in approving Underground Lines? This paper explores the role accessibility played on the decision to approve or reject proposed early London Tube Schemes. It finds that maximizing accessibility to population (highly correlated with revenue and ridership) largely explains Parliamentary approvals and rejections.
    Keywords: Accessibility, Network Growth, Subways, Public Transport, Travel Behavior, Networks

Modeling the Minneapolis Skyway Network

Recent working paper

Adopting an agent-based approach, this paper explores the topological evolution of the Minneapolis Skyway System from a microscopic perspective. Under a decentralized decision-making mechanism, skyway segments are built by self-interested building owners. We measure the accessibility for the blocks from 1962 to 2002 using the size of office space in each block as an indicator of business opportunities. By building skyway segments, building owners desire to increase their buildings’ value of accessibility, and thus potential business revenue. The skyway network in equilibrium generated from the agent model displays similarity to the actual skyway system. The network topology is evaluated by multiple centrality measures (e.g., degree centrality, closeness centrality, and betweenness centrality) and a measure of road contiguity, roadness. Sensitivity tests such parameters as distance decay parameter and construction cost per unit length of segments are performed. Our results disclose that the accessibility- based agent model can provide unique insights for the dynamics of the skyway network growth.

Special Issue on the Evolution of Transportation Network Infrastructure in Networks and Spatial Economics: Volume 9, Issue 3 (2009)

A special issue of Networks and Spatial Economics on the Evolution of Transportation Network Infrastructure, for which I was the editor, is now out. Many thanks to my co-authors and the journal for making this happen. (I cannot however see the final version, as it is behind a pay-wall and my university does not yet subscribe to the journal. I have read all of the articles though, and it is well worth reading if you do have access).
Introduction to the Special Issue on the Evolution of Transportation Network Infrastructure
David Levinson
289-290
Modeling the Growth of Transportation Networks: A Comprehensive Review
Feng Xie and David Levinson
291-307
Inter-Modal Network Externalities and Transport Development: Evidence from Roads, Canals, and Ports During the English Industrial Revolution
Dan Bogart
309-338
The Efficiency of the Victorian British Railway Network: A Counterfactual Analysis
Mark Casson
339-378
Graph-Theoretical Analysis of the Swiss Road and Railway Networks Over Time
Alexander Erath, Michael Löchl and Kay W. Axhausen
379-400
Co-evolution of Density and Topology in a Simple Model of City Formation
Marc Barthélemy and Alessandro Flammini
401-425
The Topology of Transportation Networks: A Comparison Between Different Economies
Efrat Blumenfeld-Lieberthal
427-458
Jurisdictional Control and Network Growth
Feng Xie and David Levinson
459-483

The Co-Evolution of London’s Land Use and Transport

updated August 25, 2009:
For those of you who doubt I am doing work over in London, I have completed two other papers (in addition to “Too Expensive to Meter” based on my research over here):

  • Levinson, David (2008) The Orderliness Hypothesis: Does Population Density Explain the Sequence of Rail Station Opening in London? Journal of Transport History 29(1) March 2008 pp.98-114.[download]
  • Network growth is a complex phenomenon. Some have suggested that it occurs in an orderly or rational way, based on the size of the places that are connected. David Levinson examines the order in which stations were added to the London surface rail and Underground rail networks in the nineteenth and twentieth centuries, testing the extent to which order correlates with population density. While population density is an important factor in explaining order, he shows that other factors were at work. The network itself helps to reshape land uses, and a network that may have been well ordered at one time may drift away from order as activities relocate.

  • Levinson, David (2008) Density and Dispersion: The Co-Development of Land use and Rail in London. Journal of Economic Geography 8(1) 55-57.
    JEG: [doi]
  • This article examines the changes that occurred in the rail network and density of population in London during the 19th and 20th centuries. It aims to disentangle the ‘chicken and egg’ problem of which came first, network or land development, through a set of statistical analyses clearly distinguishing events by order. Using panel data representing the 33 boroughs of London over each decade from 1871 to 2001, the research finds that there is a positive feedback effect between population density and network density. Additional rail stations (either Underground or surface) are positive factors leading to subsequent increases in population in the suburbs of London, while additional population density is a factor in subsequently deploying more rail. These effects differ in central London, where the additional accessibility produced by rail led to commercial development and concomitant depopulation. There are also differences in the effects associated with surface rail stations and Underground stations, as the Underground was able to get into central London in a way that surface rail could not. However, the two networks were weak (and statistically insignificant) substitutes for each other in the suburbs, while the density of surface rail stations was a complement to the Underground in the center, though not vice versa.

Perhaps more interesting for the non-academic, we (Ahmed El-Geneidy, Feng Xie, and myself of the Nexus group) have put together three quicktime movies

  • 1.The co-evolution of London population density and surface (National) rail
  • 2.The co-evolution of London population density and the Underground
  • 3.The co-evolution of London population density and surface (National) rail and the Underground

These can be accessed from here.