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).

Urban planners and engineers have long been interested in measuring street connectivity and typically do so with relatively simple measures such as the link to node ratio (called the Beta Index in the Transport Geography field), which divides the total number of links (i.e. road segments between intersections) by the total number of nodes (i.e. intersections including dead ends).  In the above image, the connected network has link to node ratio of 1.6 while the dendritic network’s link to node ratio is 1.0 (a link to node ration of 1.4 is typically considered a well-connected street network).

The connected node ratio divides the number of connected nodes (i.e. nodes that are not dead ends) by total number of nodes (Handy, Paterson, & Butler, 2003).  The networks above have a connected node ratio of 1.0 and 0.6, respectively.  The underlying intent is distinguish between well-connected or gridded street networks and dendritic, treelike networks  – as highlighted in the figure above – in researching relevant issues such as travel behavior, road safety, VMT, and public health outcomes.

Topology takes a slightly different approach to understanding this issue.  The Meshedness Coefficient, for instance, measures connectivity by looking at the number of cycles in the network with respect to the maximum number of cycles (a cycle is a closed path that begins and ends at the same node with no fewer than three links).  A Meshedness Coefficient of 0 represents full tree structure (i.e. no cycles), and 1 represents complete connectivity (i.e. every node is directly connected to every other node, which is not feasible in a large surface transportation network) (Buhl et al., 2006).  In non-planar networks, this measure is also known in Transport Geography as the Alpha Index.  The Alpha for the connected network above is 0.4 and for the dendritic network, it is just 0.03.  For large networks, Beta and Alpha are highly correlated.

Xie and Levinson (2007) developed another useful metric called Treeness.  Instead of counting the number of cycles, Treeness is instead calculated by dividing the length of street segments not within a cycle by the total length of street segments.  The Treeness measure also provides a value between 0 and 1, but in this case, the higher number represents a more treelike or dendritic network (Xie & Levinson, 2007).

Networks with good overall connectivity are called integrated networks.  Networks with low connectivity are called fractured networks (although fractured networks can still be comprised of connected components).  Again, these measures relate to issues of resilience.  When a single node failure can significantly erode network functionality, the system is fragile.  The image below shows a fallen tree in Lake Oswego, OR that cut off more than 50 families from the outside world (or more specifically, the cars of more than 50 households were trapped) (Florip, 2010).  If only that network had a little less Treeness.

When we have nodes or links with high Betweenness values, it is often because our network is split into various sub-groups that can be called clusters.  Clusters tend to have their own unique set of properties, so it is useful to be able to identify clusters quantitatively.

While there are a growing number of clustering algorithms, the basic idea behind them is to capture the degree to which nodes cluster.  The Clustering coefficient, for instance, represents how likely is it that two connected nodes are part of a larger group of highly connected nodes.  It can be calculated by dividing number of actual connections between the neighbors of a node (i.e. the nodes directly connected to the node in question) by the number of possible connections between these same neighboring nodes.  For instance in the image above, the red node is the node of interest, and it has a Degree of 4.  Those 4 neighboring nodes make 4 actual connections (i.e. the black lines in the figure on the right) but have 6 possible connections (i.e. the black lines plus the red dashed lines).  Thus, the Clustering coefficient for the red node is 4 divided by 6 or 0.67.

The value represented by the Clustering coefficient ranges from 0 (i.e. no clustering) to 1 (i.e. complete clustering).  If we are interested in the amount of clustering for an entire network, we average the Clustering coefficients for all of the nodes.  Clustering tends to be higher in real-world networks than in random networks.  So when a network becomes more centralized (i.e. a small percentage of nodes have high connectivity), the overall topology becomes more differentiated and clusters begin to emerge.

Other related terms include component and clique.  When a given sub-group of nodes is also highly connected, that is called a component.  When the nodes in a component have few connections to other nodes outside of the component, that is a clique.  Understanding clusters, components, and cliques in networks can be useful because they can hold more influence over behavior than overall network structure (Neal, 2013).  Imagine, for instance, a New Urbanist neighborhood with great street connectivity set into a city with poor overall street connectivity.  Analyzing network structure for the overall city might lead us to one conclusion; yet, we could find very different outcomes in the New Urbanist neighborhood.  While factors such as land use, street design, and demographics influence transportation-related outcomes as well, the concept of clustering holds value for those interested in truly understanding transportation networks.

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.