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Human Mobility Modelling: Exploration and Preferential Return Meet the Gravity Model

Authors:

Abstract

Modeling the properties of individual human mobility is a challenging task that has received increasing attention in the last decade. Since mobility is a complex system, when modeling individual human mobility one should take into account that human movements at a collective level influence, and are influenced by, human movement at an individual level. In this paper we propose the $d$-EPR model, which exploits collective information and the gravity model to drive the movements of an individual and the exploration of new places on the mobility space. We implement our model to simulate the mobility of thousands synthetic individuals, and compare the synthetic movements with real trajectories of mobile phone users and synthetic trajectories produced by a prominent individual mobility model. We show that the distributions of global mobility measures computed on the trajectories produced by the $d$-EPR model are much closer to empirical data, highlighting the importance of considering collective information when simulating individual human mobility.
Available online at www.sciencedirect.com
Procedia Computer Science 00 (2016) 000–000
www.elsevier.com/locate/procedia
The 5th International Workshop on Agent-based Mobility, Trac and Transportation Models,
Methodologies and Applications (ABMTRANS)
Human mobility modelling:
exploration and preferential return meet the gravity model
Luca Pappalardoa,b,, Salvatore Rinzivillob, Filippo Siminic
aDepartment of Computer Science, University of Pisa, 56127 Pisa, Italy
bInstitute of Information Science and Technologies (ISTI), National Research Council (CNR), 56124 Pisa, Italy
cDepartment of Engineering Mathematics, University of Bristol, Merchant Venturers Building, Woodland Road, BS8 1UB Bristol, UK
Abstract
Modeling the properties of individual human mobility is a challenging task that has received increasing attention in the last decade.
Since mobility is a complex system, when modeling individual human mobility one should take into account that human movements
at a collective level influence, and are influenced by, human movement at an individual level. In this paper we propose the d-EPR
model, which exploits collective information and the gravity model to drive the movements of an individual and the exploration
of new places on the mobility space. We implement our model to simulate the mobility of thousands synthetic individuals, and
compare the synthetic movements with real trajectories of mobile phone users and synthetic trajectories produced by a prominent
individual mobility model. We show that the distributions of global mobility measures computed on the trajectories produced
by the d-EPR model are much closer to empirical data, highlighting the importance of considering collective information when
simulating individual human mobility.
c
2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Conference Program Chairs.
Keywords: Human Mobility, Data Science, Mobility Modeling
1. Introduction
The analysis of the patterns of human mobility has received increasing attention in the last decade, given the
availability of massive digital traces of human movements and its importance in domains such as urban planning,
sustainability, transportation engineering, public health, and economic forecasting. Particular interest has been put
on modeling the properties of individual human mobility, with the purpose of reproducing the movements of an
individual in a realistic manner1. For example in the prominent Exploration and Preferential Return (EPR) model an
individual can choose either to return to previously visited locations (preferential return) or to explore new locations
at a given distance from the current location (exploration), according to well-known distributions of standard mobility
measures such as the waiting time and the jump length2. In the EPR model and its recent improvements3,4 no
Corresponding author
E-mail address: lpappalardo@di.unipi.it
1877-0509 c
2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the Conference Program Chairs.
2Author name /Procedia Computer Science 00 (2016) 000–000
collective information about the movements of other moving individuals is taken into account when deciding the
new location an individual explores. Notwithstanding, human mobility is a complex system and the movements of an
individual influence, and are influenced by, the collective mobility behavior of other individuals on the mobility space.
Omitting this information can produce simulations that are unable to capture accurately human mobility patterns at a
global level, e.g., the distribution of the radius of gyration, the distribution of location relevance, or the distribution of
population density on the space5,6. Here, we advocate that the movements of individuals on a space are also driven by
apreferential exploration force, which depends on the collective relevance of locations on the mobility space.
In this paper, we propose the d-EPR model, which improves the EPR model by using collective information and
the gravity model to drive the movements of a synthetic individual. In particular, the model exploits information about
the relevance of locations on the space: when an individual explores a new location, she is attracted to new places with
a force that depends on the relevance of such places at a collective level (preferential exploration). We implement the
d-EPR model to simulate the mobility of 50,000 synthetic individuals and compare the synthetic movements with real
trajectories of mobile phone users and synthetic trajectories produced by a spatial version of the EPR model where
individuals are constrained to move in a confined geographical space. We observe that the distributions of global
mobility measures computed on the trajectories produced by the d-EPR model are much closer to empirical data
than those produced by the EPR model. Our results highlight the importance of considering collective information
when simulating individual mobility, enforcing the intuition that individual movements are strongly influenced by the
collective mobility behavior of other people. In other words, individuals express individual preference when returning
the previously visited places and collective preference when exploring new places on the mobility space.
The paper is organized as follows. Section 2 introduces the EPR model which is the base of our model. Section 3
describes in detail the d-EPR model and introduces the algorithm to reproduce it. In Section 4 we compare the results
of our model with real trajectories of mobile phone users and the synthetic trajectories produced by the EPR model.
Finally, Section 5 concludes the paper and discuss some possible extensions and improvements of the proposed model.
2. Related Work
All the main studies in human mobility document a stunning heterogeneity of human travel patterns that coexists
with a high degree of predictability: individuals exhibit a broad spectrum of mobility ranges while repeating daily
schedules dictated by routine5,7. How to combine such ingredients to create a realistic model which captures the
salient aspects of individual human mobility is a challenging task. Many individual human mobility models have been
proposed so far, the majority of which do not use spatio-temporal realism about population densities thus producing
unrealistic mobility patterns1. The model proposed by Isaacman et al.8, for example, exploits several distributions
sampled from mobile phone data or census data to simulate the movements of individuals between a predefined
number of locations on a given territory. Although this model produces realistic population density distributions, it is
not able to produce realistic distributions of standard mobility measures, such as the radius of gyration.
Among the many proposed models, the Exploration and Preferential Return (EPR) model is one of the most used
ones, especially because it does not fix in advance the number of visited locations but let them emerge spontaneously2.
The model exploits two basic mechanisms that together describe human mobility: exploration and preferential return.
Exploration is a random walk process with a truncated power-law jump size distribution 2. Preferential return repro-
duces the propensity of humans to return to the locations they visited frequently before5. An agent in the model
selects between these two mechanisms: with a given probability the individual returns to one of the Spreviously
visited places, with the preference for a location proportional to the frequency of the individual’s previous visits. With
complementary probability the individual moves to a new location, whose distance from the current one is chosen
from the truncated power-law distribution of displacements as measured on empirical data 5. The probability to ex-
plore decreases as the number of visited locations Sincreases and, as a result, the model has a warmup period of
greedy exploration, while in the long run individuals mainly move around a set of previously visited places. Recently
the EPR model has been improved in dierent directions, such as by adding information about the recency of loca-
tion visits during the preferential return step3or adding information about moving from home or other places4. It is
worth noting that in the EPR model both exploration and preferential return mechanisms depend on individual forces.
During a preferential return the individual returns to one of her previously visited locations, during an exploration
the individual explores a new location randomly chosen at a given distance: none of the two mechanisms take into
Author name /Procedia Computer Science 00 (2016) 000–000 3
account the relevance of locations on the space or its population density. In this paper, we advocate the need of con-
sidering such information during the exploration phase, relying on the intuition that individuals move preferably to
dense places, where the variety and the number of locations available are large. For this reason we propose the d-EPR
model, which improves the EPR model in two directions: first, it works on a finite mobility space using a predefined
tessellation of the space into locations; second, it considers the relevance of locations on the space when choosing a
new location to explore, hence defining a preferential exploration step.
3. The d-EPR model
The d-EPR model incorporates two competing mechanisms, one driven by an individual force (preferential return)
and the other driven by a collective force (preferential exploration). The intuition underlying the model can be easily
understood: when an individual returns, she is attracted to previously visited places with a force that depends on the
relevance of such places at an individual level. In contrast, when an individual explores she is attracted to new places
with a force that depends on the relevance of such places at a collective level. In the preferential exploration phase,
an individual selects a new location to visit depending on both its distance from the current position, as well as its
relevance measured as the total number of visits of all users. In the model, hence, the synthetic individual follows
a personal preference when returning and a collective preference when exploring new locations. We use the gravity
model9,10 to assign the probability of a trip between any two locations, which automatically constrains individuals
within a territory’s boundaries. The usage of the gravity model is justified by the accuracy of the gravity model to
estimate origin-destination matrices even at the country level 11,12,13,14 .
Algorithm 1 describes the d-EPR model in detail. The model takes in input several variables: (i) a list Lof
tuples each representing a location on the space; (ii) an integer M axT ime, the length (in hours) of the time period
during which the individual moves on the space; (iii) the parameters of the waiting time distribution βand τ; (iv)
the parameters for defining the probability of returning ρand γ. Every tuple in Lcontains information about the
geographical coordinates of the location and its relevance.
Given the input list, the algorithm computes for every pair of locations i,jthe probability of moving from ito j
(Algorithm 1, line 2). Every probability is computed as pi j =1
N
didj
r2
ij
, where di(j)is the relevance of location i(j), rij
is the geographic distance between iand j, and N=Pi,j,ipi j is a normalisation constant (see Algorithm 1, function
computeProbabilityMatrix). Starting from a location chosen randomly according to its relevance (Algorithm 1,
line 3), until time <Ma xT ime the algorithm iterates four basic steps: (i) waiting time choice, (ii) action selection,
(iii) movement, (iv) variable updates.
In the waiting time choice step, the model extracts a waiting time tfrom the distribution P(t)t1βexp(t)
(Algorithm 1, line 7) 2. In the action selection phase, with probability Pnew =ρSγwhere Sis the number of dis-
tinct locations previously visited2, the individual chooses to explore a new location (Algorithm 1, line 10), oth-
erwise she returns to a previously visited location (Algorithm 1, line 16). If the individual explores and is in lo-
cation i, the new location j,iis selected according to the precomputed probability pi j (Algorithm 1, function
PreferentialExploration) and the number of distinct locations visited, S, is increased by one. If the individual
returns to a previously visited location, it is chosen with probability proportional to the number of her previous visits
to that location (Algorithm 1, function preferentialReturn). After the movement step, the time elapsed (Algo-
rithm 1, line 20) and the current location (line 21) are updated. When the maximum time expires (time M axT ime),
the algorithm terminates and returns in output the sequence Vof locations visited by the individual.
For a comparison with the EPR model we design the s-EPR model, a spatial version of the original EPR model
where individuals are constrained to move in a confined geographical space. The s-EPR model diers from the
original EPR model in the exploration phase: when an individual explores a new location a distance ris extracted
from the distribution P(r)= ∆r(1+α), and an angle θbetween 0 and 2πis extracted with uniform probability; if the
location at distance rand angle θfrom the current location is not in space’s boundaries a new distance and a new
angle are extracted until this condition is satisfied. It is worth highlighting an important dierence between the s-EPR
model and the d-EPR model. In the former, the exploration phase depends on the individual, i.e., when exploring
the individual does not take into account the location relevance on the mobility space. In contrast, in the d-EPR the
individual does take into account location relevance and is more likely to explore relevant locations.
4Author name /Procedia Computer Science 00 (2016) 000–000
input :Ma xT ime: the period of time the individual moves on the space
L: a list of tuples [t1,t2,...,tn] where ti=(xi,yi,di) describes a location
β,τ,ρ,γ: parameters of distributions
output:V: the sequence of locations visited by the synthetic individual
1S=1, time =0// Sis the number of visited locations
2M=computeProbabilityMatrix (L)// computes for every pair i,jthe probability of moving from ito j
3i=weightedRandom (L)// choose randomly a location according to its relevance
4vi=(xi,yi,1)
5V.append(vi)
6while time Ma xT ime do
7t=getWaitingTime () // Extract a waiting time from the distribution P(t)t1βexp(t)
8Pnew =getReturnProbability () // Choose a probability to return or to explore Pnew =ρSγ
9if Pnew ρSγthen
10 j=PreferentialExploration (i,M)// Explore a new location
11 vj=(xj,yj,1)
12 V.append(vj)
13 S=S+1
14 end
15 else
16 j=PreferentialReturn () // Return to a previously visited location
17 vj=(xj,yj,countj+1)
18 V.update(vj)
19 end
20 time =time + ∆t
21 i=j
22 end
1Function computeProbabilityMatrix(L)
2foreach tiLdo
3foreach tjL,j,ido
4pi j =didj
dist(i,j)2// compute probability according to locations’ density and gravity model
5M[i,j]=pi j
6end
7end
8N=Pi,j,iM[i,j]// Nis a normalization factor to ensure pi j [0,1]
9foreach tiLdo
10 foreach tjL,j,ido
11 M[i,j]=M[i,j]/N
12 end
13 end
1515 return M
1Function PreferentialExploration(i)
2j=weightedRandom (M[i]) // choose randomly a location jaccording to its probability in list M[i]
44 return j
1Function PreferentialReturn()
2j=weightedRandom (V)// choose randomly a location jaccording to countiin list V
44 return
Algorithm 1: The algorithm describing how the d-EPR model works.
Author name /Procedia Computer Science 00 (2016) 000–000 5
4. Model validation
We implement the d-EPR model to simulate the mobility of 50,000 synthetic individuals. Each individual moves
for a period of three months (2,160 hours) between a set Lof locations consisting in GSM towers dislocated on a
European country. We estimate the relevance of each location in Las the number of calls from that location made
during three months by 50,000 anonymized mobile phone users. We set the input parameters to β=0.8, τ=17
hours, ρ=0.6 and γ=0.21, which are the parameters’ values for the waiting time distribution and the probability of
returning estimated by Song et al. on GSM data2.
We compare the results of the d-EPR model with two other mobility datasets. The first one is an anonymized
GSM dataset collected by a European carrier for billing and operational purposes5,2,6. The dataset consists of Call
Detail Records (CDR) describing each phone call performed by 50,000 users in a period of three months. Each call is
characterized by timestamp, caller and callee identifiers, duration of the call and the geographical coordinates of the
tower serving the call. The time ordered list of towers from which a user made her calls forms a trajectory, capturing
her movements during the period of observation. The other dataset consists of the mobility trajectories produced by
50,000 synthetic individuals obtained by running the s-EPR model 6, where agents are constrained within a country
boundary (the same country as GSM data and d-EPR model). We set the exponent for the distribution of distance
lengths to α=0.55, as estimated by Gonzalez et al. on GSM data5.
(a) (b) (c)
Fig. 1. A comparison of d-EPR model, s-EPR model and empirical GSM data.(a) The distribution of the radius of gyration rgof individuals
computed on the three datasets. We observe that rgfor d-EPR and GSM data (blue and black solid curves) are similar and well approximated by a
power-law with exponential cut-o, while for s-EPR model (dashed curve) we observe a peaked distribution. (b) The distribution of overall number
of visits per location for the three datasets. Also in this case the distribution for the s-EPR model diers from the other two distributions. (c) The
distribution of nL1, the number of individuals for which a location is the most frequent location. We observe that nL1for d-EPR is more similar to
GSM data than the distribution for s-EPR.
Figure 1 compares the three datasets on: (i) the distribution of radius of gyration rg, a measure of the characteristic
distance traveled by a given individual during the period of observation defined as rg=q1
NPiLdi(rircm)2where
Nis the total number of visits to any location by the individual, Lis the set of locations visited, diis the relevance of
location i,riare the coordinates of location i,rcm the coordinates of the center of mass of the individual5,15; (ii) the
distribution of overall visits per location, i.e., the total number of visits by all the individuals in that location during
the period of observation; (iii) the distribution of nL1per location, where nL1is the number of individuals for which
that location is the most frequent location L1, i.e. the phone tower where the user performs the highest number of calls
during the period of observation. We observe that the distribution of the radius of gyration for GSM data and d-EPR
data are similar (a power-law distribution with exponential cuto), while for s-EPR data it is a peaked distribution
(Figure 1(a), green dashed curve). Similarly, the distribution of the number of visits per location of s-EPR data diers
from the other two distributions, which are similar to each other (Figure 1(b)). In Figure 1(c) we plot the distributions
of nL1, the number of individuals for whom a given location is the most frequent locations (L1), an estimate of the
number of individuals living in a given location. We observe that all the three distributions are heavy-tailed, reflecting
an uneven distribution of population density on the space. However the curves for GSM data and d-EPR data are more
The original EPR model works in an infinite mobility space, we implement the s-EPR model (which works on a finite mobility space) to make
the results of the model comparable with the GSM and the d-EPR datasets.
6Author name /Procedia Computer Science 00 (2016) 000–000
similar to each other than the curve for s-EPR data, which starts to dier from the others for low values of nL110
(Figure 1(c)). These results show that the s-EPR model fails in capturing some global human mobility patterns, and
that we can overcome this shortcoming by implementing a preferential exploration phase.
5. Conclusion
In this paper we proposed the d-EPR, a generative model to simulate individual human mobility. In contrast with
the EPR, our model exploits collective information about location relevance and implements a preferential exploration
phase, producing results that are much in better agreement with empirical data. Our results show that the patterns of
individual mobility are driven by two competing forces: an individual force during the preferential return phase, and
a collective force during the exploration phase where the movements of an individual are influenced by the relevance
of locations on the mobility space. In the approach we proposed, the distribution of visitation relevance of locations is
given as input variable to the d-EPR model. Although such a distribution can be easily computed from mobile phone
data or census data, information about location relevance on a space are not always available. As future work, we
plan to turn the individual model into a collective model, making the relevance of locations to emerge naturally during
the running of the model: in the preferential exploration phase the probability of an individual to visit a new location
will be proportional to the number of visits to that location made by other synthetic agents moving at the same time
on the space. It will be interesting to investigate whether the empirical distribution of visitation relevance emerges
spontaneously from the collective model.
Acknowledgements
This work has been partially funded by the following European projects: Cimplex (grant agreement 641191),
PETRA (grant agreement 609042), SoBigData RI (grant agreement 654024).
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... The third strand of literature focuses on designing generative algorithms, i.e., models that can generate synthetic trajectories able to reproduce, realistically, the laws of human mobility. A class of algorithms aim to reproduce spatial properties of mobility (Song, Koren, Wang, and Barabási 2010a;Pappalardo, Rinzivillo, and Simini 2016a); another one focuses on the accurate representation of the time-varying behavior of individuals (Barbosa, de Lima-Neto, Evsukoff, and Menezes 2015;Alessandretti et al. 2018). More recently, some approaches rely on machine learning to propose generative algorithms that are realistic with respect to both spatial and temporal properties of human mobility (Pappalardo and Simini 2018;Jiang, Yang, Gupta, Veneziano, Athavale, and González 2016;Luca et al. 2021). ...
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... Density exploration and preferential return model (Pappalardo et al. 2016a;Barbosa et al. 2018). name = 'Density EPR model' rho = 0.6, gamma = 0.21, beta = 0.8 tau = 17, min_wait_time_minutes = 20 SpatialEPR Spatial exploration and preferential return model (Song et al. 2010b;Barbosa et al. 2015;Pappalardo et al. 2016a). ...
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People flow data are utilized in diverse fields such as urban and commercial planning and disaster management. However, people flow data collected from mobile phones, such as using global positioning system and call detail records data, are difficult to obtain because of privacy issues. Even if the data were obtained, they would be difficult to handle. This study developed pseudo-people-flow data covering all of Japan by combining public statistical and travel survey data from limited urban areas. This dataset is not a representation of actual travel movements but of typical weekday movements of people. Therefore it is expected to be useful for various purposes. Additionally, the dataset represents the seamless movement of people throughout Japan, with no restrictions on coverage, unlike the travel surveys. In this paper, we propose a method for generating pseudo-people-flow and describe the development of a "Pseudo-PFLOW" dataset covering the entire population of approximately 130 million people. We then evaluated the accuracy of the dataset using mobile phone and trip survey data from multiple metropolitan areas. The results showed that a coefficient of determination of more than 0.5 was confirmed for comparisons regarding population distribution and trip volume.
... The model successfully reproduces several of the statistical characteristics of human mobility, such as the number of visited locations, mean square displacement, etc. Based on their research, many studies have been conducted that improve the model in various directions, adding other factors such as the collective relevance of new locations (d-EPR) [13], the recency of location visits (recency-EPR) [14], limited-memory of exploration and preferential return (memory-EPR) [15], social contacts (STSEPR) [16], etc. Jiang et al., proposed a time-varying Markov Chain model, namely TimeGeo, which captures not only the circadian rhythm of daily trips with a global variable, but the likelihood of conducting activity with three individual-specific parameters [28]. Pappalardo and Simini proposed a method of generating trajectories in a framework named DITRAS in two steps: the generation of mobility diary, and the generation of trajectory [29]. ...
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... The exploration and preferential return (EPR) model incorporates the random walk process of exploration and the human propensity to revisit places (preferential return). In EPR, one of these two competing mechanisms is chosen probabilistically each time [45]. Exploration and exploitation are two essential components of any optimization algorithm [46]. ...
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... Recently, various approaches have been proposed to model human mobility [1,5,13,14,16,17,23] and specific aspects such as community and motif formation [15,26] and the movement behavior of each individual [7,18]. Mobile devices are increasingly used to measure movement [8][9][10]17,27]. ...
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We introduce a Markov Modulated Process (MMP) to describe human mobility. We represent the mobility process as a time-varying graph, where a link specifies a connection between two nodes (humans) at any discrete time step. Each state of the Markov chain encodes a certain modification to the original graph. We show that our MMP model successfully captures the main features of a random mobility simulator, in which nodes moves in a square region. We apply our MMP model to human mobility, measured in a library.
... Logarithmic growth is a key feature of human movements, characterizing the anomalous ultra-slow diffusion and home range effect 28 . More recently, the d-EPR model generalizes the IMM by introducing a background field where an individual explores a new location with a probability proportional to its population density 29,30 . Although IMM and d-EPR successfully capture individual movements on a daily basis, individuals move independently and do not interact with each other. ...
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