This document proposes a framework for modeling human mobility that unifies the social, spatial, and temporal dimensions. It summarizes existing models and their limitations in capturing all dimensions. The proposed framework models visits as sequences rather than trajectories. It takes a social graph and arrival processes as input and outputs contact statistics. Case studies show it can generate different inter-contact time distributions by varying the arrival rates. The framework is customizable and allows analytical analysis of temporal dependencies. It was tested on a real mobility dataset.

1. Modeling the Social, Spatial, and Temporal dimensions of
Human Mobility in a unifying framework
Dmytro Karamshuk
IMT - Institutions Markets Technologies
Institute for Advanced Studies, Lucca
January 2013

2. Why do we study human mobility
● modeling ad-hoc wireless networks
● modeling information propagation, disease
spreading etc.
● developing new mobile services, e.g., location
recommendation systems
● security systems in location based social networks
● transportation, urban infrastructure

3. Opportunistic Networks
● Motivation: 5,3 billion mobile devices, 10 billion ARM
processors in embedded systems of vehicles, street
cameras etc.
● Approach: based on 'stare, carry and forward' principle
● Main challenge: forwarding (routing) protocols and more
generally information dissemination

4. Properties of Human Mobility
● in human mobility we study how people visit different places
● we are interested in social, spatial, and temporal characteristics of the
visits

5. Mobility Properties - Spatial
How far we travel from place to place?
M. Gonzalez, C. Hidalgo, A. Barabasi, Understanding individual human mobility
patterns, Nature

6. Mobility Properties – Temporal
● returning time probability ● visits of top k-th location
How frequently we visit different places?
C. Song, T. Koren, P. Wang, A. Barabasi, Modelling the scaling properties
of human mobility, Nature Physics

7. Mobility Properties - Social
● To what extend our
movements depend
on our social ties?
● How the influence of
our social ties depend
on time?
● How the places
associated with
different social
communities are
spatially distributed?
How our social ties influence the choice of the places we visit?

8. Mobility Properties – Social (another view)
● inter-contact time
i.e. time between two consecutive contacts
of two persons (mobile devices)
●
this in t e r-c o n ta c t tim e s characteristic is crucial for studying mobile social
networks, particularly opportunistic networks based on p2p communications
●
usually this is the o u tpu t o f th e m o b ilit y m o de lin g
T. Karagiannis, J. Le Boudec, M. Vojnovic, Power law and exponential
decay of intercontact times between mobile devices, Mobile Computing

9. Mobility Models
● existing models does not combine all directions
● existing models are neither flexible nor controllable
A survey of existing models:
D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. Human mobility models for
opportunistic networks. IEEE Commun. Mag, 2011

10. Arrival Based Mobility Framework
● defines mobility in terms of visits sequences not trajectories
● customizable for any temporal patterns of visits
● provides a framework for analytical analysis of the temporal
dependencies between visits and contacts

11. Adding Spatial Dimension to Social Graphs
● cliques (i.e., fully connected
sub-graphs) of users share
common meeting places
● cliques are overlapping and
hierarchically organized
● example: a company has
meeting rooms shared by all
employees, while each
subdivision of the company
has their own offices, shared
only by the members of the
subdivision. The subdivisions
might share common
members.
We develop an algorithm that:
● takes a social graph as input
● partitions the graph into a set of overlapping and hierarchically organized cliques
● generates arrival network by assigning each clique a separate meeting place

12. Adding Spatial Dimension to Social Graphs
The clique partitioning algorithm consists of two main parts:
● finding the cover of the maximum overlapping cliques in the input social graph (we
use BronKerbosch algorithm)
● reproducing hierarchical cliques structure by randomly splitting the cliques

13. Example
step N1 step N2
step N3 result

14. Adding Temporal Dimension
To characterize the temporal dimension of
human mobility we model time sequences of
users' arrivals to places with stochastic point
processes.
For simplicity we consider that arrival processes are:
● discrete (e.g., with the time unit equal to one day)
● the contact between persons happen if they both arrive in
the same place in the same time slot
Although, the framework could be extended to other cases.

15. Customizing the model
Input: Output:
● social graph ● statistics of contact sequences
● link removal probability
● arrival processes

16. Data Analysis
● 27M check-in records
● 619K users
● 2.4M venues
● 15M user-place pairs and 94K of them
with at least 20 repeats
● 1.3K user pairs with at least 20
contacts
● time period from 21.01.09 to 07.08.11
T. Hossmann, T. Spyropoulos, F. Legendre, Putting contacts into context: Mobility
modeling beyond inter-contact times

17. Individual arrival sequences
● fitting geometric distribution with Maximum Likelihood Estimation
● Pearson's chi-squared test to attest the quality of approximation
● 70% of individual inter-arrivals sequences follows a geometric distribution
● arrival sequences can be potentially approximated by a simple Bernoulli process

18. Flexibility of the Framework
Input: Output:
● social graph and link removal ● statistics of contact sequences
probability measured from the
Gowalla data
● homogenous Bernoulli arrival
processes with the distribution of
rates measured from the Gowalla
data
model is in agreement with data

19. Analytical analysis - Prerequisites
A: Does aggregate power-law
imply power-law for individual
components?
Q: Not necessarily
●A. Passarella and M. Conti. Characterizing aggregate inter-contact times in
heterogeneous opportunistic networks. NETWORKING 2011

20. Analytical analysis - Idea
In the same network with the
same arrival processes
we can obtain very
different inter-contact times
distributions.

21. Analytical Analysis – Contact Process
Contacts between two users in a Contacts between two users in all
single meeting place. shared meeting places.
The rate of the resulting contact process depends on arrival rates as:

22. Analytical Analysis – Scheme
where
● different shapes of the inter-contact times distribution can be obtained by tuning
the distribution of arrival rates
● although we cannot derive a closed-form expression for a general case, we can
do for specific cases, e.g., for exponential or long-tail F(τ)

23. Case study N1 – long-tail ICT
Input: Output:
● random graph with number of nodes ● long-tail distribution of inter-contact
n and probability of link χ times
●
removal probability α
● Bernoulli arrival processes with
rates where Y is a
standard normal random variable

24. Case study N2 – exponential ICT
Input: Output:
● similar as in the first case but the ● inter-contact times distribution with
Bernoulli arrival processes with exponential shape
identical rates

25. Conclusion
● The framework allows us to model the way users visit different
places and contact each other in those places
● The framework is customizable for any social environment by
taking social graph as an input parameter
● The framework is customizable for any temporal patterns of
users' visits to places by taking arrival stochastic processes as an
input parameter
● Temporal characteristics of the contact sequences can be analyzed
analytically
D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. An arrival based
framework for human mobility modeling. WoWMoM, 2012
D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. SPoT: Representing
the Social, Spatial, and Temporal Dimensions of Human Mobility with a
Unifying Framework. Under submission.

26. Thank you for attention!
Dmytro Karamshuk
PhD student @ IMT Lucca
Research Associate @ IIT CNR di Pisa
email: karamshuk@gmail.com
follow me on Twitter: @karamshuk