Lecture 10
Network models in
CUS

Esteban Moro

Network Science Institute | Northeastern University

NETS 7983 Computational Urban Science

2025-03-31

Welcome!

This week:

Urban Network Models

Mobility (A) and social (B) networks in urban areas, from [1]

You are here

Data -> Methods -> Models -> Applications.

Aims

• Understand the role of network models in urban science
• Learn about the main properties of urban networks
• Understand the impact of network structure on urban dynamics

Contents

• Introduction
• Properties of urban networks
• Impact of network structure on urban dynamics
• Conclusions

Introduction

Introduction

Cities are networks of people, goods, information, and resources.

• Transportation networks: roads, subways, buses, etc.
• Mobility networks: flows, trips, etc.
• Social networks: people, organizations, etc.
• Consumption networks: goods, services, etc.

Representing these interconnections as networks helps us to:

• Map out and analyze how different parts of urban areas relate.
• Characterize the dynamics of urban systems better by understanding complex interdependences between urban areas, social groups, etc.
• Understand better dynamical processes as they spread or evolve through and on these networks.

Introduction

Example: Road network

• Not only allows us to understand how people move from one place to another,
• but also how traffic jams propagate in the city

Introduction

Example: Social network

• Not only allows us to understand how people interact with each other
• But also how information spreads in the city, segregation patterns, etc.

Central Places in France, Spain, Portugal social networks. From [2]

Introduction

We know there is a strong relationship between the structure/dynamics of networks and processes happening on those networks. The same applies to urban networks:

• Connectivity (shortest paths, centrality, etc.) can help us identify key areas/places in the city.
• Assortativity can help us identify areas where people with similar characteristics interact more.
• Community structure can help us identify areas where different social groups interact, or epidemic spreading happens.
• Network resilience can help us identify areas/infrastructure where the system is more/less robust to perturbations.

Thus, investigating the properties of urban networks can help us understand better the dynamics of urban systems.

Properties of urban networks

Properties of urban networks

• Space: Urban networks are embedded in space. This means that the structure of urban networks is shaped by the physical constraints of the city.

• Distance: geographical proximity shapes connectivity people are more likely to interact with people close to them or to shop in places close to them.

• Wiring/Infrastructure costs: the cost of building infrastructure (e.g., roads, cables, etc.) makes redundant connections expensive.

• City layout: the layout of the population, resources, roads in the city shapes the structure of urban network.

Properties of urban networks

• Heterogeneity: cities are heterogeneous, with different areas having different characteristics. This heterogeneity is reflected in the structure of urban networks.

• Temporal dynamics: urban networks are not static, they evolve over time. This evolution can be due to changes in the city layout, changes in the population, external shocks

Change in the mobility networks between counties due to COVID-19 lockdown, from [3]

Properties of urban networks

• Multilayer: urban networks are not isolated, they are interconnected with other networks. For example, the different transportation networks in the city are interconnected.

Multilayer structure of transportation networks, from [4]

Spatial and planar networks

Urban networks can be classified in different ways. One of the most important classifications is based on the spatial embedding of the network:

• Spatial networks: networks where the nodes are embedded in space. For example, social connections in the city. In spatial networks, most edges happen with nodes that are close to each other due to some cost (e.g., travel time, wiring cost, etc.).

• Planar networks: spatial networks that can be embedded in a plane without edges crossing. For example, the road network, the subway network, etc. Note that some transportation networks can have edges crossing, but they are a minority.

In both cases, the probability of connecting two nodes decreases with the distance between them, similar to the gravity model.

Spatial and planar networks

• However, the characteristics of the network depend critically on the number of long-distance connections present. For example, in online social networks or airport networks, long-range connections are frequent, making networks ” small worlds.” However, planar or quasi-planar networks tend not to have many of those connections and are not “small worlds.”

Spatial graph with long-range connections

Road networks are almost planar graphs

Local and global properties

Because of their nature, spatial networks have different properties to other type of networks:

• Degree distribution: in spatial networks, the degree distribution \(P(k)\) is usually not heavy-tailed and has an exponential cutoff.

  • This is because of many reasons: the number of neighbors is bounded, the probability of connecting to a node decreases with distance, wiring costs, etc.

  • Thus, in spatial or planar graphs, \(P(k)\) is generally peaked around a certain value (number of neighbors) fixed number of neighbors. Thus \(P(k)\) is of less interest than in other networks.

Local and global properties

• Clustering coefficient: in spatial networks, the clustering coefficient is usually high (compared to random models). This is because, in planar networks, nodes are usually connected to neighbors that are also connected to each other. There is, of course, a difference if the network is infrastructure (e.g., roads, mid clustering), mobility (e.g., O-D matrices, mid-high clustering), or social (e.g., friendship, high clustering). see [5]

Clustering of mobility networks, from [6]

Local and global properties

• Shortest paths: in spatial networks, the shortest path between nodes is long because we have to transverse the space. The extreme case is planar networks, where the shortest path is usually very large. This means that the diameter of the network is usually large. Instead of scaling like \(\log N\) (small world) it scales like \(N^{1/2}\) (as in a 2-dimensional network).

Centrality

• Centrality: in non-spatial networks, centrality of nodes (e.g., betweenness) depends on the degree of the node \(g(k) \sim k^\eta\).

  • However, in spatial networks, \(k\) is bounded.
  • Typically, in spatial networks, betweenness centrality has large fluctuations for equal \(k\).
  • This is because dependence of centrality with \(k\) competes with spatial position.
  • Since betweenness centrality is a proxy for traffic in those networks, that means that there is a small number of nodes that carry most of the traffic and thus are points of congestion or failure.

Weighted networks

• Strength: urban networks are typically weighted, meaning that each edge/link is characterized by a capacity, weight or intensity \(w_{ij}\).

• This weight can represent the number of people traveling between two locations, the number of goods exchanged, the number of friendships between areas, etc. Nodes are then characterized by the weight strength

\[ s_i = \sum_{j} w_{ij} \]

• The strength of the nodes is typically correlated with the weight

\[ s_i \propto k_i^\alpha \]

where \(\alpha\) is a constant that depends on the network. For example, Barrat et al [7] found that while \(\alpha \approx 1\) for social networks, \(\alpha \approx 1.5\) for transportation networks.

Assortativity

• Assortativity: it depends on the nature of the network. For example,
  • in social networks, (demographic and degree) assortativity is usually high because people tend to connect with people who are similar to them, creating patterns of segregation.
  • In transportation networks, (degree) assortativity is usually low or even negative because transportation networks are designed to efficiently connect different parts of the city, creating transportation hubs and hierarchies.

Assortativity in mobility (A) and social connectivity (B) in urban areas, from [1]

Community structure

• Community structure: In spatial networks, the community structure is usually related to the network’s spatial layout.

  • For example, in social networks, communities are usually related to geographical areas.
  • In transportation networks, communities are usually related to areas of the city connected by a specific type of transportation (e.g., subway lines).

(Patchy) Community structure in road networks, from [8]

Impact of network structure on urban dynamics

Impact of network structure on urban dynamics

Because of their properties, the structure of urban networks has a strong impact on the dynamics of urban systems. For example:

• Traffic congestion:

  • Shortest paths in road networks are the routes that people take to navigate the city
  • The variability of the centrality of transportation networks makes some streets or transportation hubs more prone to congestion.
  • Also, the removal of some nodes can have a strong impact on the traffic flow.
  • Thus, in general, transportation networks are very sensitive to perturbations.

Impact of network structure on urban dynamics

Example: in a recent paper [9], Boeing and Ha found that road networks with less variability in centrality (less chokepoint score) are more robust towards centrality-based attacks.

Impact of network structure on urban dynamics

• Epidemic spreading: the existence of hubs and local flows in the mobility network can make some activities or areas of the city more prone to epidemic spreading than others.

Mobility network and contact network in urban areas, from [10]

Mobility network and contact network in urban areas, from [10]

Impact of network structure on urban dynamics

• Urban segregation: the assortativity of social networks can make some areas/activities/places in the city more segregated than others. This can have a strong impact on the dynamics of the city, for example, in terms of economic development, social capital, disaster resilience, etc.

Mobility networks create segregated places in urban areas, from [11]

Conclusions

• Urban networks are networks of people, goods, information, and resources in urban areas.

• They are embedded in space and have specific properties that make them different from other types of networks.

• The structure of urban networks has a strong impact on the dynamics of urban systems, from traffic congestion to epidemic spreading, urban segregation, etc.

• Understanding the properties of urban networks can help us understand better the dynamics of urban systems and design better urban policies.

Further reading

• Spatial Networks by Barthelemy [5]
• A review of the Structure of Street Networks by Barthelemy and Boeing
• The anatomy of urban social networks and its implications in the searchability problem Herrera-Yagüe et al. [2]
• A survey of results on mobile phone datasets analysis by Blondel et al. [12]
• Social connectedness in urban areas by Bailey et al. [13]

References

[1]

X. Dong et al., “Segregated interactions in urban and online space,” EPJ Data Science, vol. 9, no. 1, pp. 1–22, Dec. 2020, doi: 10.1140/epjds/s13688-020-00238-7.

[2]

C. Herrera-Yagüe et al., “The anatomy of urban social networks and its implications in the searchability problem,” Scientific Reports, vol. 5, no. 1, p. 10265, Jun. 2015, doi: 10.1038/srep10265.

[3]

Y. Kang, S. Gao, Y. Liang, M. Li, J. Rao, and J. Kruse, “Multiscale dynamic human mobility flow dataset in the U.S. During the COVID-19 epidemic,” Scientific Data, vol. 7, no. 1, p. 390, Nov. 2020, doi: 10.1038/s41597-020-00734-5.

[4]

F. Asgari, A. Sultan, H. Xiong, V. Gauthier, and M. A. El-Yacoubi, “CT-Mapper: Mapping sparse multimodal cellular trajectories using a multilayer transportation network,” Computer Communications, vol. 95, pp. 69–81, Dec. 2016, doi: 10.1016/j.comcom.2016.04.014.

[5]

M. Barthélemy, “Spatial networks,” Physics reports, vol. 499, no. 1–3, pp. 1–101, 2011.

[6]

H. Gibbs, R. M. Eggo, and J. Cheshire, “Clustering reveals key behaviours driving human movement network structure.” medRxiv, Nov. 2023. doi: 10.1101/2023.11.06.23298140.

[7]

A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, “The architecture of complex weighted networks,” Proceedings of the National Academy of Sciences, vol. 101, no. 11, pp. 3747–3752, Mar. 2004, doi: 10.1073/pnas.0400087101.

[8]

W. Guo, G. M. Donate, S. Law, S. Johnson, M. Liakata, and A. Wilson, “Urban Analytics: Multiplexed and Dynamic Community Networks.” arXiv, Dec. 2018. doi: 10.48550/arXiv.1706.05535.

[9]

G. Boeing and J. Ha, “Resilient by design: Simulating street network disruptions across every urban area in the world,” Transportation Research Part A: Policy and Practice, vol. 182, p. 104016, Apr. 2024, doi: 10.1016/j.tra.2024.104016.

[10]

A. Aleta et al., “Quantifying the importance and location of SARS-CoV-2 transmission events in large metropolitan areas,” Proceedings of the National Academy of Sciences, vol. 119, no. 26, p. e2112182119, Jun. 2022, doi: 10.1073/pnas.2112182119.

[11]

E. Moro, D. Calacci, X. Dong, and A. Pentland, “Mobility patterns are associated with experienced income segregation in large US cities,” Nature Communications, vol. 12, no. 1, pp. 1–10, Jul. 2021, doi: 10.1038/s41467-021-24899-8.

[12]

V. D. Blondel, A. Decuyper, and G. Krings, “A survey of results on mobile phone datasets analysis,” EPJ Data Science, vol. 4, no. 1, p. 10, Dec. 2015, doi: 10.1140/epjds/s13688-015-0046-0.

[13]

M. Bailey, R. Cao, T. Kuchler, J. Stroebel, and A. Wong, “Social Connectedness: Measurement, Determinants, and Effects,” Journal of Economic Perspectives, vol. 32, no. 3, pp. 259–280, Aug. 2018, doi: 10.1257/jep.32.3.259.