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The scaling of human interactions with city size

Markus Schläpfer, Luis M. A. Bettencourt, Sebastian Grauwin, Mathias Raschke, Rob Claxton, Zbigniew Smoreda, Geoffrey B. West, Carlo Ratti

arXiv:1210.5215v3physics.soc-phcs.SIphysics.data-an

TL;DR

The paper asks how city size relates to the structure of human interaction networks, a relationship not previously studied empirically in detail. It maps society-wide communication networks from two European countries onto urban areas and finds that contacts and communication activity rise superlinearly with population, while local clustering remains largely unchanged. These patterns support a scale-invariant acceleration of interaction-based spreading as cities grow.

  • Problem

    The relationship between city size and the structure of underlying human interaction networks lacked detailed empirical investigation.

  • Method

    The study maps society-wide communication networks from Portugal and the UK to urban areas and analyzes their scaling with city population.

  • Results

    Total contacts and communication activity obey superlinear scaling with city size, affecting most citizens, while average local clustering does not change with city size.

  • Takeaways & Limitations

    The findings provide a microscopic basis for linking increasing social connectivity to superlinear socioeconomic quantities and faster interaction-based spreading in larger cities.

Abstract

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The size of cities is known to play a fundamental role in social and economic life. Yet, its relation to the structure of the underlying network of human interactions has not been investigated empirically in detail. In this paper, we map society-wide communication networks to the urban areas of two European countries. We show that both the total number of contacts and the total communication activity grow superlinearly with city population size, according to well-defined scaling relations and resulting from a multiplicative increase that affects most citizens. Perhaps surprisingly, however, the probability that an individual's contacts are also connected with each other remains largely unaffected. These empirical results predict a systematic and scale-invariant acceleration of interaction-based spreading phenomena as cities get bigger, which is numerically confirmed by applying epidemiological models to the studied networks. Our findings should provide a microscopic basis towards understanding the superlinear increase of different socioeconomic quantities with city size, that applies to almost all urban systems and includes, for instance, the creation of new inventions or the prevalence of certain contagious diseases.

I. INTRODUCTION

The paper addresses the lack of detailed empirical evidence linking city size to the structure of human interaction networks. Using nationwide communication records from Portugal and the UK, it finds that contacts and communication activity increase superlinearly with population, while local clustering remains largely unchanged.

  • I. INTRODUCTION: Empirical evidence on how city size relates to human interaction-network structure has been limited, despite the importance of interactions for social and economic phenomena.Traditional surveys are time-consuming, limited in scope, and vulnerable to sampling bias.
  • I. INTRODUCTION: Superlinear urban scaling is commonly expressed as Y ∝ N^β, with β ≈ 1.15 > 1 for many socioeconomic quantities.Reported examples include economic output, wages, patents, violent crime, and contagious-disease prevalence.
  • I. INTRODUCTION: The study tests whether superlinear socioeconomic scaling originates in a scale-invariant increase of social connectivity per capita.The proposed link is motivated by the role of human interactions in wealth generation, innovation, crime, and disease spread.
  • I. INTRODUCTION: Nationwide communication records from Portugal and the UK are mapped to urban areas to examine city size and interaction-network structure.The Portugal dataset contains 15 months of mobile calls, with 1.6 × 10^6 nodes and 6.8 × 10^6 reciprocated links; the UK dataset contains 24 × 10^6 landline phones and 119 × 10^6 links.
  • I. INTRODUCTION: In Portugal, contacts and communication activity grow superlinearly with city population through a continuous shift in individual-level distributions.The UK data suggest that superlinear total social connectivity also holds across communication means and national urban systems.
  • I. INTRODUCTION: The probability that an individual’s contacts are mutually connected remains largely constant, while denser networks facilitate interaction-based spreading in larger cities.This combination is presented as a microscopic basis for the superlinear scaling of certain socioeconomic quantities.

A. Superlinear scaling of social connectivity

Social connectivity increases superlinearly with city population in Portugal and shows comparable scaling in the UK. The increase reflects more contacts and communication activity per person, while sampling and device-level assumptions constrain interpretation.

  • A. Superlinear scaling of social connectivity: Rescaled Portuguese social connectivity follows Kr ∝ N^β with β = 1.12 > 1 (95% CI [1.11,1.14]) across several orders of magnitude.Kr rescales cumulative degree by observed coverage to extrapolate average nodal degree to the full city population.
  • A. Superlinear scaling of social connectivity: 12% more mobile phone contacts per person accompany every doubling of city population on average.This follows from ⟨k⟩ ∝ N^(β−1) with β − 1 ≈ 0.12.
  • A. Superlinear scaling of social connectivity: An average resident of Lisbon accumulated about twice as many reciprocated contacts as an average resident of Lixa during the 15-month observation period.Lisbon had N = 5 × 10^5 and Lixa had N = 4 × 10^3.
  • A. Superlinear scaling of social connectivity: Non-reciprocal networks have β = 1.13−1.24 (95% CI [1.05,1.25]), indicating social solicitations grow faster with city size than reciprocated contacts.The comparison concerns non-reciprocal versus reciprocated interaction networks.
  • A. Superlinear scaling of social connectivity: Portugal-wide predictions are limited by observing only about 20% of the overall mobile network, although the qualitative scaling persists in better-sampled cities.The authors expect the observed qualitative behaviour to apply to the full network because coverage has no clear city-size trend.
  • A. Superlinear scaling of social connectivity: UK reciprocal connectivity has exponents β = 1.08−1.14 (95% CI [1.05,1.17]), supporting superlinear scaling across another communication system.Landline phones may be shared, so their average degree does not necessarily represent an individual-based network.

B. Probability distributions for individual social connectivity

The study examines how city-size scaling emerges from individual connectivity distributions. These distributions shift toward higher interaction levels in larger cities, indicating that superlinear scaling reflects broader increases across callers rather than a few exceptionally connected individuals.

  • B. Probability distributions for individual social connectivity: Individual-level network data are used to investigate how urban scaling relations emerge from distributions of connectivity properties.This extends earlier city-wide analyses to the underlying distributions of nodal degree, call volume, and call count.
  • B. Probability distributions for individual social connectivity: The means of all logarithmic interaction variables consistently increase with city size, producing a shift toward higher individual connectivity values.Standard-deviation trends are inconsistent across city definitions and indicators.
  • B. Probability distributions for individual social connectivity: Nodal degree follows a skewed lognormal distribution, while call volume and call count are approximated by conventional lognormal distributions.The transformations k* = ln k, v* = ln v, and w* = ln w are used in the distributional descriptions.
  • B. Probability distributions for individual social connectivity: Superlinear scaling is not simply caused by a few individuals with extreme connectivity; it reflects increased connectivity among most callers.The authors distinguish this pattern from one dominated by a power-law tail.
  • B. Probability distributions for individual social connectivity: The lognormal results suggest that new acquaintances may arise through stochastic cascades of social encounters in space and time facilitated by larger cities.This is presented as a hypothesis about the process generating new social connections.
  • B. Probability distributions for individual social connectivity: Interpretation of the complete communication network remains constrained by average mobile-phone coverage of approximately 20%, despite preserved distribution shapes in highly covered cities.The authors hypothesize that the qualitative behaviour also holds at approximately 100% coverage.

C. Invariance of the average clustering coefficient

The average clustering coefficient remains approximately constant as city size grows, indicating that local social-network structure is retained even as connections reach larger populations.

  • Definition: C_i measures how many links exist among node i’s contacts relative to all possible links between them.C_i is close to one when most contacts know one another and equals zero when they are mutual strangers.
  • Expectation: Larger cities would be expected to have lower contact clustering if contacts were selected randomly from a larger population.The probability that two contacts are mutually connected would then decrease rapidly with city size.
  • Result: ⟨C⟩≈0.25 in Portugal’s individual-based network, and its average remains largely unaffected by city size.This invariance also holds when link weights based on call volume and number of calls are considered.
  • Interpretation and scope: Under the assumption that the mobile-phone data reliably proxy social-relation strength, constant ⟨C⟩ indicates that urban networks retain local structures while extending into larger populations.The authors note that the observed Portuguese network is only a sample, which may affect the absolute value of ⟨C⟩.

D. Acceleration of spreading processes

The paper tests whether greater urban connectivity accelerates spreading by simulating an epidemiological model on Portugal’s mobile-phone network. Spreading speed increases systematically with city size, following a weak power-law scaling.

  • Motivation: Connectivity increases imply similar, scale-invariant gains in spreading potential for residents of larger cities.This interpretation combines increasing connectivity with invariant link clustering and continuously shifting distributions.
  • Caveat: Community structure and assortative degree mixing may also influence the resulting spreading dynamics.These network effects are identified as additional factors beyond the measured connectivity and clustering patterns.
  • Model: The model represents each node as susceptible or infected and transmits information across links with probability P_ij = xν_ij.ν_ij is accumulated call-volume weight, while x controls overall spreading speed.
  • Results: R ∝ N^δ, with δ = 0.11−0.15 (95% CI [0.02, 0.26]), showing that spreading speed increases with city size.The increase is also found in simulations on the unweighted network.
  • Conclusion: The simulations confirm an acceleration of spreading processes as cities become larger.The spreading paths can involve the entire nationwide network rather than remaining within city boundaries.

III. DISCUSSION

Mapping communication networks across two European countries reveals that city size is associated with superlinear growth in contacts and communication activity, while local clustering remains stable. These patterns suggest scale-invariant acceleration of interaction-based spreading and provide a basis for studying socioeconomic differences across cities.

  • Society-wide communication networks were mapped to urban areas in two European countries to test how human interaction scales with city size.
  • The total number of contacts and communication activity follow superlinear power-law scaling, reflecting a multiplicative increase affecting most citizens.
  • The average local clustering coefficient does not change with city size, indicating that larger-city groups remain as tightly knit as those in smaller towns.
  • Larger cities may facilitate diffusion of information, ideas, and other interaction-based spreading processes.
  • The findings support the hypothesis that social-network structure underlies the superlinear scaling of socioeconomic quantities with population size.
  • Generality remains to be tested with other individual-based communication datasets, ideally covering the population completely; causal links to socioeconomic characteristics also remain unresolved.

A. Data sets

The study combines large communication datasets from Portugal and the UK, with anonymized records and different degrees of geographic and network coverage. These datasets provide nationwide interaction observations but do not fully represent all mobile-phone connections or the complete population.

  • The Portugal dataset contains 440 million Call Detail Records from 2006–2007 and covers approximately 2 million mobile-phone users.
  • The Portugal observations cover approximately 20% of the country’s population, despite near-universal mobile-phone penetration.
  • Portugal records include caller identities, call duration, timestamps, and the cell towers routing each call, with anonymized phone identifiers.
  • The UK dataset contains 7.6 billion calls from one month in 2005 and includes 44 million landline and 56 million mobile-phone numbers.
  • Because UK mobile-to-mobile access and spatial information were limited, the analysis included mobile numbers connected to landlines and assigned exchange areas to urban units.

B. City definitions

Because city boundaries lack an unambiguous definition, the study compares multiple urban units in Portugal and uses Urban Audit Cities in the UK. Population data from 2001 support direct comparison across countries.

  • Portugal was analyzed using Statistical Cities, Municipalities, and Larger Urban Zones as alternative city definitions.
  • Statistical Cities and Municipalities use Portuguese national definitions and 2001 population data, while Larger Urban Zones represent extended urban regions defined by Eurostat.
  • The Portuguese inventory contains 156 Statistical Cities, 308 Municipalities, and 9 Larger Urban Zones.
  • Municipalities cover the entire national territory, although interpreting them as urban units can be flawed in some cases.
  • The UK analysis uses 30 Urban Audit Cities equivalent to Local Administrative Units, Level 1, enabling direct comparison with Portuguese Municipalities.

C. Spatial interaction networks

The researchers construct reciprocal and non-reciprocal communication networks, assign users and exchange areas geographically, and filter potentially business-related activity. The resulting city-level node counts closely track population size.

  • The reciprocal network connects mobile-phone users when each initiated at least one call to the other, whereas the non-reciprocal network includes any one-way call.
  • Reciprocal links reduce potential bias from business usage, call centers, and accidental calls, while non-reciprocal links capture more superficial interactions.
  • The largest connected cluster was extracted from both network types, and additional filtering removed likely business hubs and extreme links in the UK data.
  • Users were assigned to cities by mapping their predominant call-routing cell tower to city polygons.
  • The analysis retained 140 Statistical Cities, 9 Larger Urban Zones, and 293 Municipalities after excluding units without suitable spatial data or assigned towers.
  • The number of assigned network nodes strongly correlates with city population size, with r=0.95, 0.97, and 0.92 for Statistical Cities, Larger Urban Zones, and Municipalities, respectively.
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