Elements of Access: Betweenness

Degree is often good for measuring local circumstance, but adequately characterizing centrality is a bit more complicated.  When trying to figure out centrality in terms of how connected and influential a node or link is, it is useful to get a sense of relative network flow through a particular node or link.


Betweenness measures attempt to capture this relative flow by quantifying the number of times a node or link is on a shortest path between two other nodes.  The first step would be to calculate the shortest path between every origin and every destination.  Next, we count the number of times that a particular node or link shows up on a shortest path.  The resulting number represents the relative role of a node or link as a connector between clusters of nodes or links.  In the above street network, the intersection highlighted in red must be included in over half of the shortest paths.  We call this count Betweenness, which is essentially an attempt to quantify how necessary a node or link is to get from one side of the network to the other.  The Panama Canal, for instance, is a key maritime link connecting the Atlantic and Pacific Oceans.  Without it, ships would have to route around Cape Horn at the southernmost tip of Chile or through the Straits of Magellan.  For a ship traveling from New York to San Francisco, the Panama Canal – due to its high Betweenness value – cuts more than 7,500 miles from the journey.  In terms of other transportation issues, Betweenness usually relates to metrics such as accessibility and traffic congestion.


In addition to revealing relative importance, Betweenness also indicates how irreplaceable a node or link may be to a network.  In other words, what happens if we remove a certain node or link from the network?  Very high betweenness values can indicate a critical connection between various groups of nodes or links.  In some cases, this represents a vulnerability where we would want to add redundancies to the network.


In transportation networks, if we assume all travelers take the shortest path and treat each traveler as having a unique origin and destination, Betweenness is the same as the flow (number of travelers) on the link.  We call this Flow-weighted Betweenness.

Elements of Access: Degree

Centrality measures help gauge the overall importance of a node.  In other words, how connected and how influential is a node within the overall network?



One of the simplest measures of centrality is Degree, which measures the number of connections between a node and all other nodes.  For instance if we are considering a street network with intersections as nodes, a nodal Degree of 4 would indicate a typical 4-way intersection.  The image above depicts a rendition of the Metropolis street network with a Degree value shown at each intersection and a 4-way intersection highlighted in red (Fleischer, 1941).  When we focus on what is happening at one particular node, it is called the ego network (in that we are looking at the network from the perspective of a single node while ignoring all nodes not directly connected, which can be deemed a bit narcissistic).  The entire network can be called the complete, whole, or global network.  So if we want an overall Degree measure, we can calculate Average Degree, which is the average number of connections for all the nodes within the overall network.  When the Average Degree exceeds 1, every node has at least one connection, on average.  When the Average Degree approaches log(n), where n equals the number of nodes in the network, every node starts to become accessible from every other node (Neal, 2013).  For the Metropolis network, there are 78 nodes with an Average Degree of 3.4.

Degree Distribution Graph for Metropolis
Degree Distribution Graph for Metropolis

Analyzing Degree measures for a complete network also entails generating a Degree Distribution, which literally equates to the plotting the frequency of each Degree for all the nodes as shown in the image below for the Metropolis street network.  The idea is to try to capture the relative differences in connectivity between the nodes in order to gain a sense of network structure.  For instance, every node in a homogenous network would have the exact same number of connections and not much of a distribution.  A more centralized network might have one node with a high Degree value and all other nodes with low Degree values.

Elements of Access: Introduction to Topology

Networks play a role in nearly all facets of our daily lives, particularly when it comes to transportation. Even within the transportation realm lays a relatively broad range of different network types such as air networks, freight networks, bus networks, and train networks (not to mention the accompanying power and communications networks). We also have the ubiquitous street network, which not only defines how you get around a city, but it provides the form upon which our cities are built and experienced. Cities around the world that are praised for having good street networks come in many different configurations ranging from the medieval patterns of cities like Prague and Florence, to the organic networks of Boston and London, and the planned grids of Washington D.C. and Savannah, Georgia. But how do researchers begin to understand and quantify the differences in such networks?


The primary scientific field involved with the study of shapes and networks is called topology. Based in mathematics, topology is a subfield of geometry that allows one to transform a network via stretching or bending. Under a topological view, a network that has been stretched like a clock in a Salvador Dali painting would be congruent with the original, unstretched network. This would not be the case in conventional Euclidean geometry where differences in size or angle cannot be ignored. The transportation sector typically models networks as a series of nodes and links (Levinson & Krizek, 2008). The node (or vertex) is the fundamental building block of the model; links (or edges) are not independent entities but rather are represented as connections between two nodes. Connectivity – and the overall structure of the network that emerges from that connectivity – is what topology is all about. In other words, topology cares less about the properties of the objects themselves and more about how they come together.

For instance if we look at the topology of a light rail network, the stations would typically be considered the nodes and the rail lines would be the links. In this case, the stations are the actors in the network, and the rail lines represent the relationships between the actors (Neal, 2013). Those relationships – and more specifically, those connections – embody what is important. Taking a similar approach with a street network, we might identify the intersections as the nodes and the street segments as the links as shown in the network based on an early version of Metropolis above (Fleischer, 1941). For most street networks, however, the street segments are just as important as the intersections, if not more so. The ‘space syntax’ approach takes the opposite (or ‘dual’) approach with street networks: the nodes represent the streets, and the lines between the nodes (i.e. the links or edges) are present when two streets are connected at an intersection, as shown below using the same Metropolis network (Jiang, 2007).


Initial theories related to topology trace back to 1736 with Leonhard Euler and his paper on the Seven Bridges of Königsberg. Graph theory based topological measures first debuted in the late 1940s (Bavelas, 1948) and were initially developed in papers analyzing social networks (Freeman, 1979) and the political landscape (Krackhardt, 1990). Since then, topological analyses have been widely adopted in attempting to uncover patterns in biology (Jeong, Tombor, Albert, Oltvai, & Barabasi, 2000), ecology (Montoya & Sole, 2002), linguistics (Cancho & Sole, 2001), and transportation (Carvalho & Penn, 2004; Jiang & Claramunt, 2004; Salingaros & West, 1999). Topology represents an effort to uncover structure and pattern in these often complex networks (Buhl et al., 2006). The topological approach to measuring street networks, for instance, is primarily based upon the idea that some streets are more important because they are more accessible, or in the topological vernacular, more central (Porta, Crucitti, & Latora, 2006). Related to connectivity, centrality is another important topological consideration. A typical Union Station, for example, is a highly central (and important) node because it acts as a hub for connecting several different rail lines. Some common topological measures of centrality include Degree and Betweenness, which we will discuss in more detail in subsequent sub-sections.

There are also some peculiarities worth remembering when it comes to topology.

When thinking about the Size of a network, our first inclination might be measures that provide length or area. In topological terms, however, Size refers to the number of nodes in network. Other relevant size-related measures include: Geodesic Distance (the fewest number of links between two nodes); Diameter (the highest geodesic distance in a network); and Characteristic Path Length (the average geodesic distance of a network).

Density is another tricky term in the topological vernacular. When talking about the density of a city, we usually seek out measures such as population density, intersection density, or land use intensity. In most cases, these metrics are calculated in terms of area (e.g. per km2). In topology, however, Density refers to the density of connections. In other words, the density of a network can be calculated by dividing the number of links by the number of possible links. Topologically, the fully-gridded street networks of Portland, OR and Salt Lake City, UT are essentially the same in terms of Density; with respect transportation and urbanism, however, there remain drastic functionality differences between the 200’ (~60m) Portland blocks and the 660’ (~200 m) Salt Lake City blocks.

As illustrated with the Portland/Salt Lake City example, one limitation of topology is that it ignores scale. However, this can also be an advantage. For instance, Denver might be much closer to Springfield, IL than Washington, DC as the crow flies, but a combination of several inexpensive options for direct flights to DC and relatively few direct flight options for Springfield mean that DC is essentially closer in terms of network connectivity. Topology captures such distinctions by focusing on connectedness rather than length.

While topological analyses such as the above are scale-free, we also need to be careful about use of this term because scale-free networks are not equivalent to scale-free analyses. In topological thinking, scale-free networks are highly centralized. More specifically, if we plot the number of connections for each node, the resulting distribution for what is known in topology as a scale-free network would resemble a Power law distribution with some nodes having many connections but most having very few. A hub-and-spoke light-rail system, for instance, tends to exhibit scale-free network qualities with relatively few stations connecting many lines. The nodes in a random network, on the other hand, tend to have approximately the same number of connections. For instance when we define the intersections of a street network as the nodes and the segments as the links, the results tends towards a random network. If we flip the definition again, so that the streets are the nodes and the intersections the links, we trend back towards a scale-free network (Jiang, 2007; Jiang & Claramunt, 2004).

One reason to look at connectivity in these terms has to do with the critical issues of resilience and vulnerability. In general, robustness is associated with connectivity. When we have good connectivity, removing one node or link does not make much of a difference in terms of overall network performance. In contrast, scale-free networks are more susceptible to strategic attacks, failures, or catastrophes. However, as shown in a recent paper about urban street network topology during a Zombie apocalypse, good connectivity could actually be a double-edged sword (Ball, Rao, Haussman, & Robinson, 2013).


Access Across America: Auto 2015

CTS Catalyst September 2016 just came out, and announces our Access Across America: Auto 2015 study: Study estimates accessibility to jobs by auto in U.S. cities. The article is reprinted below:

Map of Accessibility to jobs by auto in U.S.
Accessibility to jobs by auto

A new report from the University’s Accessibility Observatory estimates the accessibility to jobs by auto for each of the 11 million U.S. census blocks and analyzes these data in the 50 largest (by population) metropolitan areas.

“Accessibility is the ease and feasibility of reaching valuable destinations,” says Andrew Owen, director of the Observatory. “Job accessibility is an important consideration in the attractiveness and usefulness of a place or area.”

Travel times are calculated using a detailed road network and speed data that reflect typical conditions for an  8 a.m. Wednesday morning departure. Additionally, the accessibility results for 8 a.m. are compared with accessibility results for 4 a.m. to estimate the impact of road and highway congestion on job accessibility.

Map of U.S. showing reduced job accessibility due to congestion
Reduced job accessibility due to congestion

Rankings are determined by a weighted average of accessibility, with a higher weight given to closer, easier-to-access jobs. Jobs reachable within 10 minutes are weighted most heavily, and jobs are given decreasing weights as travel time increases up to 60 minutes.

Based on this measure, the research team calculated the 10 metropolitan areas with the greatest accessibility to jobs by auto (see sidebar).

A similar weighting approach was applied to calculate an average congestion impact for each metropolitan area. Based on this measure, the team calculated the 10 metropolitan areas where workers experience, on average, the greatest reduction in job access due to congestion (see sidebar).

Areas with the greatest loss in job accessibility due to congestion

  1. Los Angeles
  2. Boston
  3. Chicago
  4. New York
  5. Phoenix
  6. Houston
  7. Riverside
  8. Seattle
  9. Pittsburgh
  10. San Francisco

Metropolitan areas with the greatest job accessibility by auto

  1. New York
  2. Los Angeles
  3. Chicago
  4. Dallas
  5. San Jose
  6. San Francisco
  7. Washington, DC
  8. Houston
  9. Boston
  10. Philadelphia

“Rather than focusing on how congestion affects individual travelers, our approach quantifies the overall impact that congestion has on the potential for interaction within urban areas,” Owen explains.

“For example, the Minneapolis–St. Paul metro area ranked 12th in terms of job accessibility but 23rd in the reduction in job access due to congestion,” he says. “This suggests that job accessibility is influenced less by congestion here than in other cities.”

The report—Access Across America: Auto 2015—presents detailed accessibility and congestion impact values for each metropolitan area as well as block-level maps that illustrate the spatial patterns of accessibility within each area. It also includes a census tract-level map that shows accessibility patterns at a national scale.

The research was sponsored by the National Accessibility Evaluation Pooled-Fund Study, a multi-year effort led by the Minnesota Department of Transportation and supported by partners including the Federal Highway Administration and 10 state DOTs.

Related Links

“Transit Makes you Short”: On Health Impact Assessment of Transportation and the Built Environment

How the p-value and coefficient changes with observation size
How the p-value and coefficient changes with observation size

Recently published:


This study conducts an in-depth analysis to alert policymakers and practitioners to erroneous results in the positive impacts of transit use on health measures. We explore the correlation of transit use and accessibility by transit and walking with self-reported general health, Body Mass Index (BMI), and height. We develop a series of linear regression and binary logit models. We also depict the coefficient-p-value-sample-size chart, and conduct the effect size analysis to scrutinize the practically significant impacts of transit use and accessibility on health measures. The results indicate transit use and accessibility by transit and walking are significantly associated with general health and BMI. However, they are practically insignificant, and the power of the large sample in our particular case causes the statistically insignificant variable to become significant. At a deeper level, a 1% increase in transit use at the county level diminishes the BMI by only 0.0037% on average. The elasticity of transit use also demonstrates that every 1% increase in transit use would escalate the chance of having excellent or very good general health by 0.0003%. We show there is a thin line between false positive and true negative results. We alert both researchers and practitioners to the dangerous pitfalls deriving from the power of large samples and the weakness of p-values. Building the results on just statistical significance and sign of the parameter of interest is worthless, unless the magnitude of effect size is carefully quantified post analysis.

A random walk down Main Street

Recently published:Figure_6


US suburbs have often been characterized by their relatively low walk accessibility compared to more urban environments, and US urban environments have been characterized by low walk accessibility compared to cities in other countries. Lower overall density in the suburbs implies that activities, if spread out, would have a greater distance between them. But why should activities be spread out instead of developed contiguously? This brief research note builds a positive model for the emergence of contiguous development along “Main Street” to illustrate the trade-offs that result in the built environment we observe. It then suggests some policy interventions to place a “thumb on the scale” to choose which parcels will develop in which sequence to achieve socially preferred outcomes.

The City is flatter: Changing patterns of job and labor access

Recently published:

Standard deviational ellipse: Accessibility to 750,000 jobs or more within 20 minutes in the AM peak: 1995-2005
Standard deviational ellipse: Accessibility to 750,000 jobs or more within 20 minutes in the AM peak: 1995-2005


This study measures accessibility by automobile for the Minneapolis – Saint Paul (Twin Cities) region from 1995 to 2005. In contrast to most previous analyses of accessibility, this study uses travel time estimates derived, to the extent possible, from actual observations of network performance by time of day. A set of cumulative opportunity measures are computed with transport analysis zones (TAZs) as the unit of analysis for 1995 and 2005. Analysis of the changes in accessibility by location over the period of study reveals that, for the majority of locations in the region, accessibility increased over this period, though the increases were not uniform. A “flattening” or convergence of levels of accessibility across locations was observed over time, with faster-growing suburban locations gaining the most in terms of employment accessibility. An effort to decompose the causes of changes in accessibility into components related to transport network structure and land use (opportunity location) reveals that both causes make a contribution to increasing accessibility, though the effects of changes to the transportation network tend to be more location-specific. Overall, the results of the study demonstrate the feasibility and relevance of using accessibility as a key performance measure to describe the regional transport system.


Accessibility Evaluation of the Metro Transit A-Line

Recently Published:
Change in number of jobs accessible within 30 minutes by transit available with Metro Transit A-Line
Change in number of jobs accessible within 30 minutes by transit available with Metro Transit A-Line
Abstract: This report evaluates the Accessibility of the Metro Transit A-Line arterial bus rapid transit system serving St. Paul and Minneapolis, Minnesota. It is found that overall the A-Line increases accessibility compared to the previous service provided by the local bus, though some areas lose accessibility if they are not near a new A-Line stop.
Keywords: Accessibility, Bus Rapid Transit, Public Transport, Evaluation
Persistent link to this item http://hdl.handle.net/11299/180900
Funding information
United States Department of Transportation, Federal Highway Administration
Suggested Citation
Palmateer, Chelsey; Owen, Andrew; Levinson, David M. (2016). Accessibility Evaluation of the Metro Transit A-Line.Accessibility Observatory, University of Minnesota. Retrieved from the University of Minnesota Digital Conservancy, http://hdl.handle.net/11299/180900.

Improving Urban Access: New Approaches to Funding Transport Investment

A new book is out, I had the opportunity to preview and blurb it:

Improving Urban Access: New Approaches to Funding Transport Investment

Edited by Elliott D. SclarMåns LönnrothChristian Wolmar. 2016 – Routledge



“Improving Urban Access” provides a wide-ranging introduction to the issues of funding and financing urban transport, ranging from how we got into the current predicament to the prospects for a variety of solutions that might make transport more inclusive, efficiently funded, and soundly managed. The ideas discussed here should be deeply understood by everyone concerned with transport policy and planning.” – David M. Levinson, Professor, Department of Civil Engineering, Richard P. Braun/CTS Chair in Transportation Engineering

Improving Urban Access is a must-read for the 21st century generation of transport and urban planners. Lessons learned have called for a bold rethinking of planning and implementation of a highway-centered landscape. With an emphasis on access – where access addresses quality of life and place, old models of mobility give way to rethinking the institutions that serve our growing urban areas, the ways in which citizens can finance new transport modes and how – we can achieve a more equitable social structure.” – Robert E. Paaswell, Distinguished Professor City University, City College of New York, former CEO of the Chicago Transit Authority

“Many public servants are so desperate to “find” additional revenues for urban transportation that they may lose sight of what they are trying to accomplish and why. This book does an excellent job of reminding us that how something is funded directly impacts the societal outcomes we are wishing to achieve, pointing out that careful consideration of funding mechanisms is absolutely critical to success.” – Joshua Schank, Chief Innovation Officer, Los Angeles County Metro, former President of the Eno Center for Transportation

“Transportation policy scholarship is changing slowly but dramatically, and this second stimulating milestone book by these editors charts that transition. Contributors forcefully address the most important unresolved questions as transportation thinking moves from forecasting demand and providing facilities to a new emphasis on access, social and economic equity, and environmental sustainability.” – Martin Wachs, Distinguished Professor Emeritus, Urban Planning and Civil Engineering University of California Los Angles and University of California, Berkeley