In binary networks, the focus is on whether or not a connection between two nodes exists.  However, not all links (or nodes) are created equal, particularly when it comes to transportation networks.  When we know about the presence of a link as well as the strength of that link, it is called a valued network.  For instance when traveling from A to B in a street network, there is usually discontinuity in street type.  In other words, one might move from a local street to a collector road to an arterial road and then back to a collector before reaching their destination.  While engineers know this sort of differentiation as functional classification, it is also referred to as hierarchy.

Hierarchy, which is embedded in many natural and societal systems such as biologic cells and the Internet, is a common transportation complexity that requires a more complicated topological analysis (Tomko, Winter, & Claramunt, 2008).  Typical topological measures such as Degree or Betweenness can be useful in helping understand network hierarchy, particularly in tree-like networks; however, such measures would fail to properly distinguish between streets in a gridded street network.  In the above version of Metropolis’ street network, the major streets are represented by thicker lines and easily discerned, even in a gridded network (Fleischer, 1941).   Using the basic set of topological metrics, we would have no idea that 8^th Street is functionally different from 7^th Street or F Street from D Street.  These metrics fail to consider attributes – such as urban design, number of lanes, active transportation infrastructure, adjacent land uses, and speed – beyond network structure and would not necessarily be able to distinguish such streets.

Working with hierarchical networks often involves dividing networks in multiple layers or tiers.  Measurements of heterogeneity have also become common proxies for characterizing hierarchy.  To identify heterogeneity among street segments, researchers have used entropy measures as well as discontinuity measures (Xie, 2005).  Discontinuity, for example, does not necessarily denote a disconnected network; rather, the reference is to the discontinuity in moving from one street type to another.  If we sum the number of times a traveler goes from one type of street to another while traveling along a shortest path route, we find the trip discontinuity.  Dividing that number by the length of the trip gives us the relative discontinuity (Parthasarathi, 2011).  Other simplistic hierarchy measures calculate the relative percentage of a particular type of road.  For instance, we might divide the number or length of arterials by the total number or length of roads to find the relative percent arterials (Parthasarathi, 2011).

Interestingly, it is not uncommon for large-scale transportation models to delete streets on the lower end of the hierarchical spectrum (i.e. local streets) for the sake of computational efficiency.  Yet, removing such streets creates a bias against more connected networks because less connected networks typically need to be supported by major streets with more capacity than would be needed in more connected networks (Bern & Marshall, 2012).  Some topological researchers – where the focus should be on understanding the full network – unfortunately reach the same conclusion: “urban streets demonstrate a hierarchical structure in the sense that a majority is trivial, while a minority is vital” (Jiang, 2009).  If we only care about vehicle traffic flow, such statements may be true.  However, my previous street network research confirms that understanding the full network holds the key to pushing toward improved safety, increased active transportation, and better environmental and health outcomes (Bern & Marshall, 2013; Marshall & Garrick, 2009, 2010a, 2010b, 2012).