A review of the structure of street networks

We review measures of street network structure proposed in the recent literature, establish their relevance to practice, and identify open challenges facing researchers. These measures' empirical values vary substantially across world regions and development eras, indicating street networks' geometric and topological heterogeneity.

Due to strong physical constraints, many quan==es studied in complex networks (Latora, et al., 2017;Menczer, et al., 2020) are not relevant for street networks (Lämmer, et al., 2006;Mossa, et al., 2002).For example, street networks have a narrow degree distribu=on and a very high clustering coefficient (Barthelemy, 2022).Also, many indicators introduced in transporta=on geography, such as alpha, beta, and gamma indices (Kansky, 1963), mainly depend on the average degree and are redundant.To effec=vely characterize street networks, it is also crucial to consider spa=al proper=es such as planarity, road segment length distribu=on, betweenness centrality spa=al distribu=on, block shape factor, and street angle distribu=on.We will present here results for what could cons=tute a minimal set of measures that characterizes a street network.
The most basic indicator, the number of nodes, increases with the popula=on -which makes theore=cal sense as residents can share public infrastructure -at most linearly (Strano, et al., 2012;Barthelemy, et al., 2013), and in general slightly sublinearly (Boeing, 2022): a 1% increase in urban area popula=on is associated with a 0.95% (±0.03%) increase in intersec=on count.The total length scales accordingly to a simple argument (Barthelemy & Flammini, 2008) as √ where  is the area size.Basic measures (see Supp.Info.for the defini=on of all quan==es discussed) which convey important informa=on about these networks are summarized in table 1 1.Basic network sta=s=cs computed for the set of 100 world ci=es discussed in (Boeing, 2019): average degree , propor=on of dead-ends  !, propor=on  " of  = 4 intersec=ons, average length  !(in meters) of edges.
We observe for all measures a large diversity of values and illustrate the possible networks according to their average degree value in Figure 1.These networks are not completely planar due to the presence of tunnels and bridges.A measure, the Spa=al Planarity Ra=o,  was introduced in (Boeing, 2020): a spa=ally planar network with no overpasses or underpasses will have  = 1.0, while lower values indicate the extent to which the network is planar.Among drivable street networks for 50 world ci=es, only 20% are formally planar and on average  = 0.88, and ranging from 100% in six of these ci=es to a low of 54% for Moscow.
T he betweenness centrality (BC) defined in (Freeman, 1977) measures the importance of a node (or edge) for flows on the network.In this sense it could serve as a simple proxy for traffic on the network (although it assumes in general a flat OD matrix), but also as an interes=ng structural probe of the network.The distribu=on of betweenness centrality (BC) is invariant for street networks, despite the existence of structural differences between them (Kirkley, et al., 2018).For a regular network, the BC decreases with the distance to the gravity center of nodes, but when disorder is present, we observe the emergence of different paFerns.In par=cular, we observe the presence of loops with large BC (Lion & Barthelemy, 2017;Barthelemy, et al., 2013) signaling the importance of these structures for large ci=es (see figure 2).The geometry of street networks is also fundamental.Using a large global database comprising all major roads on the Earth, (Strano, et al., 2017) showed that the road length distribu=on within croplands is indis=nguishable from urban ones, once rescaled by the average road length.The area  of blocks is another important feature of these spa=al networks, and it was shown that its distribu=on is universal of the form ()~ #$ (Lämmer, et al., 2006;Louf & Barthelemy, 2014).The organiza=on and overall geometry of the street network can also be characterized by the distribu=on of street angles.There are very regular networks (almost lajce like) such as in Chicago and very disordered ones such as in Rome (Figure 3).The orienta=on order can then be characterized by the entropy of street compass bearings (Boeing, 2019).
Finally, several authors proposed to construct a typology of these street networks: Marshall (Marshall, 2004) proposed a first approach based on general considera=ons, a typology based on the block size distribu=on and shape was proposed in (Louf & Barthelemy, 2014), and more recently machine learning approaches were proposed.In (Thompson, et al., 2020) a greater propor=on of railed public transport networks combined with dense road networks characterised by smaller blocks is correlated with the lowest rates of road traffic injury, and in (Boeing, et al., 2024) it was shown that straighter, more-connected, and less-overbuilt street networks are associated with lower transport emissions, all else equal.This review covers a minimal set of topological and spa=al measures of street network structure: node count, average degree, frac=on of dead ends and intersec=ons, average edge length, planarity index, BC spa=al distribu=on, street angle distribu=on.Despite the universality of street networks, these measures show the diversity of structural paFerns of streets reflec=ng local culture, poli=cs, era, and transport technologies.Important challenges remain open.First, we need a beFer understanding of the spa=o-temporal evolu=on of these networks and their co-evolu=on with ci=es (Capel-Timms, et al., 2024).More efforts in the digi=za=on of historical maps are needed to advance our theore=cal understanding and modeling of this phenomenon.Second, merely iden=fying empirical values is not enough for the professional disciplines of city-making.A key future challenge is to link the "what" (descrip=ve measures) to the "how" to build new and improve exis=ng networks to meet broader societal goals (sustainability, resilience, public health, economic health, etc).

Degree
The street network is described by a network  = (, ) where  is a set of  nodes and  the set of links between these nodes.The nodes represent the intersec=ons and the links segments of roads between these intersec=ons.The degree  of a node is the number of streets converging to it.A node of degree  = 1 is a dead-end, nodes of degree  = 2 are generally removed and nodes of degree 3, 4. (or more) represent typical intersec=ons.The average degree is then simply given by where  ! is the degree of node .In general, the number of nodes of degree  is denoted by () and the propor=on of dead-ends reads then and of  = 4 intersec=ons:

Detour index
The detour index (or stretch factor) for a pair of nodes  and  is defined as (Aldous & Shun, 2010;Barthelemy, 2022) where  ' is the Euclidean distance between  and , and  & is the route distance computed on the network.We then have  ()* = max !,,

𝑄(𝑖, 𝑗)
We can also average over pairs of nodes at a given distance  and construct the detour profile where () is the number of pairs of nodes at distance .

Total and average length
The total length of the network is defined as (Barthelemy, 2022) total length () is the length of edge .The average edge length is then

Spa9al planarity ra9o
The spa=al Planarity Ra=o,  (Boeing, 2020) represents the ra=o of the number  6 of nonplanar intersec=ons (i.e., non-dead-end nodes in the nonplanar, three-dimensional, spa=ally-embedded graph) to the number  7 of planar intersec=ons (i.e., edge crossings in the planar, two-dimensional, spa=ally-embedded graph): The (posi=ve) quan=ty  7 −  6 is then equal to the number of nonplanar edge crossings such as overpasses and underpasses in the network.

Frac9on of one-way streets
The frac=on  of one-way streets is defined as (Verbavatz & Barthelemy, 2021) where  $ is the total length of one-way streets and  the total length of the network.
where (, ) is the number of shortest paths from  to , and  !(, ) is the number of such shortest path that go through the node  (and a similar defini=on for the BC of edges).The normaliza=on (here chosen as the number of pairs of nodes different from ) can be slightly different according to different authors.

Figure 1 .
Figure 1.Street networks with (a) small average degree (Helsinki  = 2.35), (b) typical value (Singapore  = 3.0), and (c) a large value (Buenos Aires  = 3.55).The overall organiza=on of street networks is diverse, reflec=ng a wide range of paFerns and structures depending on urban planning, geography, and cultural factors (see the Supp.Info.for details about the degree  calcula=on).

Figure 2 .
Figure 2. Spa=al distribu=on of node betweenness centrality (weighted by edge length) for Beijing, China (yellow nodes are more central).We observe the emergence of non-trivial paFerns of large BC nodes (Data from OSM).

Figure 3 .
Figure 3. Polar histograms (bar direc=ons represent streets' compass bearings and bar lengths represent rela=ve frequency of streets) and the corresponding street maps illustra=ng different types of organiza=on: low entropy (Chicago), typical (New Orleans), and high entropy (Rome).