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# Emergence of Criticality in the Transportation Passenger Flow: Scaling and Renormalization in the Seoul Bus System

PLoS ONE

Public Library of Science

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### Abstract

Social systems have recently attracted much attention, with attempts to understand social behavior with the aid of statistical mechanics applied to complex systems. Collective properties of such systems emerge from couplings between components, for example, individual persons, transportation nodes such as airports or subway stations, and administrative districts. Among various collective properties, criticality is known as a characteristic property of a complex system, which helps the systems to respond flexibly to external perturbations. This work considers the criticality of the urban transportation system entailed in the massive smart card data on the Seoul transportation network. Analyzing the passenger flow on the Seoul bus system during one week, we find explicit power-law correlations in the system, that is, power-law behavior of the strength correlation function of bus stops and verify scale invariance of the strength fluctuations. Such criticality is probed by means of the scaling and renormalization analysis of the modified gravity model applied to the system. Here a group of nearby (bare) bus stops are transformed into a (renormalized) “block stop” and the scaling relations of the network density turn out to be closely related to the fractal dimensions of the system, revealing the underlying structure. Specifically, the resulting renormalized values of the gravity exponent and of the Hill coefficient give a good description of the Seoul bus system: The former measures the characteristic dimensionality of the network whereas the latter reflects the coupling between distinct transportation modes. It is thus demonstrated that such ideas of physics as scaling and renormalization can be applied successfully to social phenomena exemplified by the passenger flow.

### Most cited references17

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### Understanding individual human mobility patterns

(2008)
Despite their importance for urban planning, traffic forecasting, and the spread of biological and mobile viruses, our understanding of the basic laws governing human motion remains limited thanks to the lack of tools to monitor the time resolved location of individuals. Here we study the trajectory of 100,000 anonymized mobile phone users whose position is tracked for a six month period. We find that in contrast with the random trajectories predicted by the prevailing Levy flight and random walk models, human trajectories show a high degree of temporal and spatial regularity, each individual being characterized by a time independent characteristic length scale and a significant probability to return to a few highly frequented locations. After correcting for differences in travel distances and the inherent anisotropy of each trajectory, the individual travel patterns collapse into a single spatial probability distribution, indicating that despite the diversity of their travel history, humans follow simple reproducible patterns. This inherent similarity in travel patterns could impact all phenomena driven by human mobility, from epidemic prevention to emergency response, urban planning and agent based modeling.
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### The scaling laws of human travel

(2006)
The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronise populations of interacting species, stabilise them, and diversify gene pools [1-3]. Human travelling, e.g. is responsible for the geographical spread of human infectious disease [4-9]. In the light of increasing international trade, intensified human mobility and an imminent influenza A epidemic [10] the knowledge of dynamical and statistical properties of human travel is thus of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Based on a comprehensive dataset of over a million individual displacements we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale free random walks known as Levy flights. Secondly, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails which attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two parameter continuous time random walk model to a surprising accuracy and conclude that human travel on geographical scales is an ambivalent effectively superdiffusive process.
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### Self-similarity of complex networks

(2005)
Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks \cite{ab-review,mendes,vespignani,newman,amaral}. A large number of real networks are called scale-free'' because they show a power-law distribution of the number of links per node \cite{ab-review,barabasi1999,faloutsos}. However, it is widely believed that complex networks are not {\it length-scale} invariant or self-similar. This conclusion originates from the small-world'' property of these networks, which implies that the number of nodes increases exponentially with the diameter'' of the network \cite{erdos,bollobas,milgram,watts}, rather than the power-law relation expected for a self-similar structure. Nevertheless, here we present a novel approach to the analysis of such networks, revealing that their structure is indeed self-similar. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given "size". Concurrently, we identify a power-law relation between the number of boxes needed to cover the network and the size of the box defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, social, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena.
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### Author and article information

###### Affiliations
[1 ]Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul, Korea
[2 ]Department of Geography, Sungshin Women's University, Seoul, Korea
[3 ]CNRS, Institut Jean Lamour, Département de Physique de la Matière et des Matériaux, UMR 7198, Vandoeuvre-les-Nancy, France
National Research & Technology Council, Argentina
###### Author notes

Competing Interests: The authors have declared that no competing interests exist.

Conceived and designed the experiments: KL MYC. Performed the experiments: SG KL. Analyzed the data: SG. Contributed reagents/materials/analysis tools: JYF. Wrote the paper: SG MYC JYF.

###### Contributors
Role: Editor
###### Journal
PLoS One
PLoS ONE
plos
plosone
PLoS ONE
Public Library of Science (San Francisco, USA )
1932-6203
2014
5 March 2014
: 9
: 3
24599221
3943856
PONE-D-13-46945
10.1371/journal.pone.0089980
(Editor)

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

###### Counts
Pages: 10
###### Funding
This work was supported by the National Research Foundation of Korea (Grant Nos. NRF-2013017764/1, 2012R1A2A4A01004419, and 2011-0012331). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
###### Categories
Research Article
Engineering
Mechanical Engineering
Transportation
Mathematics
Applied Mathematics
Complex Systems
Statistics
Statistical Theories
Equilibrium Statistical Theory
Scaling Theory
Physics
Interdisciplinary Physics
Statistical Mechanics
Social and Behavioral Sciences
Geography
Economic Geography
Spatial Economic Analysis
Human Geography
Distance-Decay Models
Spatial Analysis

Uncategorized