Human mobility: Models and applications
Introduction
While the term mobility has multiple connotations, in the context of this review it refers to the movement of human beings (individuals as well as groups) in space and time and thus implicitly refers to human mobility. Indeed, from the migration of Homo sapiens out of Africa around 70,000 years ago, through the European discovery of the “New World”, to the existence of expatriate populations in contemporary times, the existence of human beings has always been inextricably linked with their movement. While earlier movement patterns were primarily driven by factors such as climate change, inhospitable landscapes, conflict and food scarcity, in modern times, socio-economic factors such as wage imbalance, differences in welfare and living conditions and globalization play an increasing role.
From hunters and gatherers of prehistoric times to present-day commuters of large metropolitan areas, humans are bound to move on a daily basis in order to earn their livelihood. However, daily trips are also undertaken to perform social and leisure activities. The temporal and spatial scales of these trips are much shorter than those of migratory flows, and they are often characterized by the regularities and periodicities that mark human lives. The daily movement of a large and growing population has important impact on the lives of the individuals and the environmental conditions. Studies conducted in Europe and the United States found that the average household spending on transportation is between 15 and 25 percent of the total expenditures, making transportation the second largest expenditure category after housing. Transportation is also the second source of greenhouse gas emissions to the atmosphere. From these few examples it should be clear that mobility has an enormous impact on human societies and an accurate quantitative description of human mobility is of fundamental importance to understand the processes related to human movement and their impact on the community and the environment. A quantitative theory of human mobility ought to be able to provide answers to relevant questions, such as, what determines the decision to start a trip? which factors determine the choice of the destination? to what extent is human movement predictable, and what is the intrinsic degree of randomness? is it possible to find general rules or laws to explain empirical patterns and regularities exhibited by travels in many diverse countries, such as the distribution of commuting times and distances, and the degree of predictability of future whereabouts?
While the study of human mobility currently spans several disciplines, arguably, geography was the first discipline to analyze mobility data and put forward corresponding theories to describe travel patterns. Indeed, the pioneers of quantitative and theoretical geography in the 50’s defined “geography as (the scientific discipline of) spatial interaction ” [1]. Early quantitative studies of the movements of people and vehicles were held in large US metropolitan areas [2] (see the first chapters of [3] for details), and initial studies of human travel were of scales ranging from international migrations [[4], [5], [6], [7]] to journey-to-work commuting [[8], [9], [10], [11], [12]]. Indeed, the elucidation and understanding of these patterns was motivated by its relation to several real-world applications such as traffic forecasting [[3], [13], [14]], urban planning [[15], [10]], internal security [[16], [17]] and epidemic modeling [[18], [19], [20]] to name but a few.
The first systematic analysis of the concept of distance as a constraint to movement was proposed in the 19th century: in his 1965 review [21] Gunnar Olsson cites Henry C. Carey’s Principles of Social Science (1858) as the first work to explicitly make the observation about the amount of interaction between two cities being proportional to their population size and inversely proportional to the intervening distance. Few decades later, the geographer Ernst Ravenstein further developed and popularized the idea in a seminal work where he formulated his laws of migration [22]. Further refinements on this theme were made in the 1940’s by the American sociologist Samuel Stouffer [[5], [23]] in his law of intervening opportunities, and by the American linguist George Kingsley Zipf [6]. Zipf’s formulation led to what is now conventionally referred to as the gravity law. The increasing availability of datasets on population movements at various levels of granularity, coupled with the quantitative revolution in geography [[24], [25], [26]], led to the introduction of more sophisticated mathematical methods such as hidden Markov models and diffusion processes. Gender and socio-economic factors behind population movement were further analyzed thanks to richer datasets resulting from surveys and interviews [[27], [28], [29]], and through theories of labor economics focusing on wage differential between locations [30]. Thus through the 20th century, contributions to the theories of human movement were made chiefly in geography, sociology, and economics, while the scale at which this was primarily studied was at the population level.
To provide context for what is to follow, we briefly describe a selection of influential historical contributions, keeping in mind “Stigler’s law of eponymy” which states: “No scientific discovery is named after its original discoverer” [31]. However, if we restrict ourselves to the recognition of distance as a primary factor in determining movement and interactions between places, then it is reasonable to start with the work of Ernst Georg Ravenstein [22].
Laws of Migration. Ravenstein was a German–English geographer who made important contributions to cartography as well as providing one of the first rough estimates of a “maximal” global population based on resource consumption. He was also one of the first to attempt an explanation and prediction of migration patterns within and between countries. Considering the effect of distance as well as the type of migrant (male or female, old or young) as primary factors, he posited the following seven laws:
- 1.
Most migrants only travel short distances, and “currents of migrations” are in the direction of the great centers of commerce and industry given that these can absorb the migrants.
- 2.
The process of absorption occurs in the following manner: inhabitants of the areas immediately surrounding a rapidly growing town flock to it, thus leaving gaps in the rural areas that are filled by migrants of more remote districts, creating migration flows that reach to “the most remote corner of the kingdom”.
- 3.
The process of dispersion is inverse to that of absorption, and exhibits similar features.
- 4.
Each main current of migration produces a compensating counter-current.
- 5.
Migrants traveling long distances generally go by preference to one of the great centers of commerce or industry.
- 6.
The natives of towns are less migratory than those of the rural parts of the country.
- 7.
Females are more migratory than males.
Ravenstein added another two laws in 1889 [32]:
- 8.
Towns grow more by immigration than by natural increase.
- 9.
The volume of migration increases as transport improves and industry grows.
While the laws are non-quantitative and observational in character, Ravenstein correctly identified socio-economic factors as well as distance-constraints to be the essential ingredients behind modern population movement. Consequently, his laws stimulated an enormous volume of work, and although they have been refined and adjusted over the years, the essential ingredients of his formulation remain relevant even today.
Law of Intervening Opportunities. One of the most important refinements was made in the 1940’s by the American sociologist Samuel Stouffer [5]. Roughly speaking, Stouffer was looking to expand upon Ravenstein’s observations regarding migrants moving shorter distances and flocking to commercial centers. To account for this, he proposed that the number of people going a given distance is directly proportional to the number of opportunities at that distance and inversely proportional to the number of intervening opportunities. In other words, trips between two locations are driven primarily by relative accessibility of socio-economic opportunities that lie between those two locations. In this context, opportunity is defined as a potential destination for the termination of a traveler’s journey, whereas an intervening opportunity is one that the traveler rejects in favor of continuing on. In Stouffer’s original formulation, this can be mathematically expressed as where is the cumulative number of migrants that move a distance from their original location, and is the cumulative number of intervening opportunities. Assuming that itself is a continuous function of distance, then the expression above can be integrated to yield where is some constant denoting the number of opportunities at the origin location. Thus, the relationship between mobility and distance is indirect and is established only through an auxiliary dependence via the intervening opportunities: the higher the number of intervening opportunities between two locations at distance , the smaller the number of migrants that would travel that distance. This may explain why rural migrants may flock to urban centers over large distances, whereas those already in commercial centers are comparatively more stationary.
Distance-Decay and the Gravity Law. Around the same time as Stouffer, the Harvard philologist, George Kingsley Zipf, was expanding upon his famous observation of the rank-frequency dependence in linguistics; the eponymous Zipf’s law, where the frequency of a word ranked – in terms of usage – has the statistical dependence [33]. Zipf found that this relation was expandable to other realms of society, specifically the size of cities [34], where the occurrence of a city with population and consequent rank also follows the relation Broadly speaking, Zipf’s argument for this relation was due to the tension between two competing factors. The first, which he called Force of Diversification relates to the likelihood of populations living close to the source of raw materials (commodities) in order to minimize the cost of transportation to production centers. The second effect, referred to as the Force of Unification, is the tendency of populations to aggregate in urban centers due to the minimization of work required to transport finished products to consumers. While the former leads to the formation of multiple urban centers (given that the commodity sources are not localized in one part of a country) with smaller populations, the latter has the competing effect of urban agglomeration in a few centers of large population. Assuming some kind of equilibrium between these quasi-forces, the rank-frequency relation of Eq. (3) with naturally follows [35]. Deviations from equilibrium, where one force dominates over the other, then leads to a change in the exponent.
Carrying the argument further, under somewhat unrealistic assumptions of equitable share of national income as well as urban centers being self-sufficient (i.e. production and consumption at equal levels), the share of any center in the total flow of goods is proportional to its population . Therefore, the flow of goods between two centers is proportional to the product of their populations. Finally, if one would like to minimize the cost and work associated with transportation of goods, this flow must be inversely proportional to the distance between centers. Putting all this together we arrive at the relation where represents the flow of goods between two population centers and is the distance between these two centers. Zipf tested his theory on both freight and population movement data and got good qualitative agreement. This formulation is of course quite different form Stouffer’s explanation as the effect of distance is quite explicit in Eq. (4). Indeed, the form of Eq. (4) is such that it leads to a distance-decay effect suppressing long-range movement. Furthermore, the flow of populations and goods is seen to be as a result of some dynamic equilibrium between the cost of transportation, manufacture, and distribution of goods and services. The references to “forces” as well as its functional dependence on distance eventually led to Zipf’s formulation being dubbed the Gravity law in analogy to Newtonian mechanics.
The common theme connecting Ravenstein, Stouffer and Zipf of course is the geographical distance, though its functional effect on movement is quite different in the Intervening Opportunities and Gravity Law models. Nevertheless, both these models were quite influential on subsequent work, setting off two major strains of parallel research as well as attempts at unification.
Time geography and ICT Data. Measuring, understanding and forecasting the displacements of individuals in space and time has long been part of the program of quantitative and theoretical geography, a branch of geography “born” academically in the 1950’s [[25], [26]]. While the first efforts in capturing human displacements focused on the aggregated levels of flows between spatial units, some also focused on individual trajectories. Torsten Hägerstrand, a Swedish geographer, laid down in the early 1950’s the basis of time geography, and brought a number of conceptual and graphical tools to formalize the trajectories of individuals through space and time. His seminal work [36] remains famous for its proposal to represent individual trajectories in a cube (also known as the “space–time aquarium”), in which the horizontal plane represents the geographical space, while the vertical axis represents time (as depicted in Fig. 1). Hägerstrand proposed a number of graphical conventions (a “notation system” in his own words) to picture the constraints imposed by social life on individual daily trajectories. He also provided means to represent the co-presence and synchronization of several individuals in space, and more generally a set of (essentially graphical) conventions useful to represent the structure and behavior of individual human mobility.
Time geography was naturally invoked (and somewhat rediscovered) in the 1990’s, when the modeling of human mobility shifted towards individual-based simulation (micro/multi-agent/agent-based simulation) [37]. This can be understood in the twin contexts of increased computing power and the development of more expressive programming frameworks, allowing for semantically richer and more ambitious models of human dynamics. However, while the models increased in complexity, they were somewhat artificial; relevant data of comparable complexity and resolution was not available for their calibration. Indeed, while the models progressed, the data lagged behind and the best one had to work with was longitudinal survey data collected since the 1970’s.
The beginning of the 21 century saw the introduction and subsequent widespread adoption of mobile phones, as well as the pervasive usage of Global Positioning System (GPS). This led to an exponential increase in data-generation on human movement. Coupled with further progress in computing power and sophisticated data-mining methods, it enabled to capture the movement not just of populations at finer levels of spatial granularity, but potentially of individuals. In particular, the large volume and frequency of call detail records (CDRs) from mobile phones (see Section 2.3) enabled the analysis of human movement at very fine temporal scales. Thus statistical information on mobility became available, on the scale of hours to decades; from the individual, through communities to the level of country-wide populations.
This review predominantly focuses on these later developments. As we will see throughout the text, these new sources of data have rejuvenated the scientific interest for human mobility, opening the door to new questions and measures, as well as enhancing and validating the insight gained from studying traditional data sources. For reasons which are probably technical, historical, and societal, research teams in physics and computer science have broadly “invested” in these new georeferenced (meta)data resulting from human activity [38]. This is evidenced by a quick search on the online website arXiv1 for physics papers that include the terms “Human Mobility” or “Mobility Patterns”. As seen in Fig. 2, since about 2004–2005, the number of such papers displays an almost exponential increase. One must keep in mind that this is probably an incomplete sample, and is underestimating the volume of research in this field. It is likely that one would see an even more dramatic trend if a more comprehensive list of publications were to be included.
While this growing interest of the Physics community in human mobility has multiple reasons, it is somewhat to be expected given the traditional interest of statistical physicists in studying emergent collective properties of systems of many interacting particles and in describing the dynamics of randomly moving particles undergoing (anomalous) diffusion. The availability of data at multiple spatio-temporal scales is helping uncover robust statistical patterns as well as aiding the development of phenomenological theories of human mobility, that are well suited to the tools, methods and paradigms of statistical physics.
The recent pace of developments and the volume of work published are such that a comprehensive review of the findings appeared necessary. Moreover, the study of human mobility is a highly interdisciplinary endeavor and progress has occurred in parallel across different academic communities, sometimes with one unaware of the others’ works. This is reflected in different terminology of common metrics and similar results being cast in different and seemingly unrelated contexts. Consequently with this review, we also aim at (i) bringing disparate communities together by unifying the findings in a common context, and (ii) providing new researchers in the field with an accessible starting point and a minimal set of tools, metrics, concepts and models.
One must note that humans, of course, do not move in a vacuum. Just as the diffusion of molecules in materials is mediated by the structural and thermodynamic properties of the material, similarly, spatio-temporal patterns of human movement are necessarily shaped by spatial constraints and limitations of geography. Examples of this are the topography of urban centers or the pattern of roads and transportation infrastructure, the properties of which are studied under the aegis of spatial networks [39]. Yet, this survey is not about Spatial Networks, nor do we give details on the so-called “science of cities” (e.g., city forms, properties of urban transport networks, urban scaling laws, etc. [40]); these two aspects will be considered only when made necessary for the reader to understand the concepts related to human mobility.
We begin in Section 2 with a comprehensive (but by no means exhaustive) list of data sources used for empirical studies. In Section 3 we introduce key metrics, measures, spatio-temporal scales, as well as an overview of the fundamental physics behind the study of human mobility. In Section 4 we describe the state-of-the-art families of models (both generative and phenomenological) that best describe the empirical observations of human mobility. The models are categorized according to scale, starting from the level of individuals, through to the collective level of populations, and finally a mixture of the two incorporating the concept of (inter) modality. Continuing with the theme of scales, in Section 5 we describe some selected applications of the framework and models ranging from intra-urban movement to flows between countries and continents including the case of epidemic spreading, transportation systems, and a brief digression on new results on virtual mobility (web browsing). Finally, we conclude in Section 6 with challenges and future directions for the field. We also added Appendix where we provide descriptions of some basic tools and algorithms (agent-based modeling, random walkers, etc.) that may be of use to researchers making initial forays into the field.
Section snippets
Data sources
A natural starting point is to describe the nature of empirical data which has been used in mobility research. Indeed, empirical data has been vital to both aggregate and individual mobility investigation, by providing means of parameter calibration as well as model validation. In this section, we outline the main sources available for mobility research and the relevant information that can be extracted from them. The data sources are presented in (rough) chronological order of their historical
Metrics, physics and scales
In this section, we will discuss some of the fundamental metrics used to characterize mobility as well as the associated spatio-temporal scales at which they are relevant. We will then move onto a discussion of some of the physics associated with mobility, including the relations between distance, time and velocity. We end the section with a discussion of energy arguments and interpolation of spatio-temporal scales through the lens of multimodality.
General mobility models
Models of human mobility can be aimed at reproducing individual mobility patterns or general population flows. In both cases one must necessarily take into account the characteristic spatial and temporal scales of the mobility process, which can vary from hundreds of meters to thousands of kilometers and from hours to years. For this reason, each of these cases have been tackled with distinct modeling frameworks. Individual mobility is subject to a certain level of uncertainty associated with
Selected domains of application
Here we discuss some selected applications of the frameworks, concepts, models and datasets introduced thus far. We organize this section by scale, ranging from single scale (mostly dealing with transportation modes), multi-scale applications with a focus on movement in cities and epidemic spreading, and finally end with some new developments related to virtual scales, i.e mobility patterns seen in online activity.
Conclusions
In this work we discussed the state-of-the art in the field of human mobility. After a short historical description of the field, we focused on current work starting with the type of data utilized in most research projects, followed by the description of metrics and models of individual mobility and population mobility. We illustrated a list of selected applications of human mobility modeling, demonstrating the insights the field can bring to the solution of real-world problems.
It should be
Acknowledgments
HB, GG, RM and MT were partially supported by the US Army Research Office under Agreement Number W911NF-17-1-0127. FS is supported by EPSRC First Grant EP/P012906/1. JJR received funding from the Spanish Ministry of Economy, Industry and Competitiveness (MINEICO) and FEDER (EU) under the project ESOTECOS (FIS2015-63628-C2-2-R), and also from the European Commission H2020 project BigData4ATM (699260) under the SESAR Joint Undertaking.
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