Modeling the Social, Spatial, and Temporal dimensions of Human Mobility in a unifying framework

Dmytro Karamshuk

IMT - Institutions Markets TechnologiesInstitute for Advanced Studies, Lucca

January 2013

Why do we study human mobilityWhy 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

Opportunistic NetworksOpportunistic 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

Properties of Human MobilityProperties of Human Mobility

● in human mobility we study in human mobility we study howhow people visitpeople visit different different placesplaces● we are interested in we are interested in socialsocial, , spatialspatial, and , and temporaltemporal characteristics of the characteristics of the visitsvisits

Mobility Properties - SpatialMobility Properties - Spatial

How far we travel from place to place?How far we travel from place to place?

M. Gonzalez, C. Hidalgo, A. Barabasi, Understanding individual human mobility patterns, Nature

Mobility Properties – TemporalMobility Properties – Temporal● returning time probability ● visits of top k-th location

How frequently we visit different places?How frequently we visit different places?C. Song, T. Koren, P. Wang, A. Barabasi, Modelling the scaling propertiesof human mobility, Nature Physics

Mobility Properties - SocialMobility Properties - Social

How our social ties influence the choice of the places we visit?How our social ties influence the choice of the places we visit?

● 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?

Mobility Properties – Social (another view)Mobility Properties – Social (another view)● inter-contact time

i.e. time between two consecutive contacts of two persons (mobile devices)

● this this inte r-c o nta c t t im e sinte r-c o nta c t t im e s characteristic is crucial for studying mobile social characteristic is crucial for studying mobile social networks, particularly opportunistic networks based on p2p communications networks, particularly opportunistic networks based on p2p communications

● usually this is the usually this is the o utput o f the m o b ility o utput o f the m o b ility m o de lingm o de ling T. Karagiannis, J. Le Boudec, M. Vojnovic, Power law and exponentialdecay of intercontact times between mobile devices, Mobile Computing

Mobility Models Mobility Models

D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. Human mobility models for opportunistic networks. IEEE Commun. Mag, 2011

● existing models does not combine all directions ● existing models are neither flexible nor controllable

A survey of existing models:

Arrival Based Mobility FrameworkArrival 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

Adding Spatial Dimension to Social GraphsAdding 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 We develop an algorithmalgorithm that: that:● takes a social graph as inputtakes a social graph as input● partitions the graph into a set of overlapping and hierarchically organized cliquespartitions the graph into a set of overlapping and hierarchically organized cliques● generates arrival network by assigning each clique a separate meeting placegenerates arrival network by assigning each clique a separate meeting place

Adding Spatial Dimension to Social GraphsAdding Spatial Dimension to Social GraphsThe The clique partitioningclique partitioning algorithm consists of two main parts: algorithm consists of two main parts:

● finding the cover of the finding the cover of the maximum overlapping cliquesmaximum overlapping cliques in the input social graph (we in the input social graph (we use BronKerbosch algorithm) use BronKerbosch algorithm)

● reproducing reproducing hierarchical cliqueshierarchical cliques structure by randomly splitting the cliques structure by randomly splitting the cliques

ExampleExample

step N1step N1 step N2step N2

step N3step N3 resultresult

Adding Temporal DimensionAdding Temporal Dimension

To To characterizecharacterize the temporal dimension of the temporal dimension of human mobility we model time sequences of human mobility we model time sequences of users' arrivals to places with stochastic point users' arrivals to places with stochastic point processes. 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.

Customizing the modelCustomizing the model

Input: ● social graph

● link removal probability

● arrival processes

Output:● statistics of contact sequences

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

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

Flexibility of the FrameworkOutput:

● statistics of contact sequences

Input: ● social graph and link removal

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

Analytical analysis - PrerequisitesAnalytical analysis - Prerequisites

● A. Passarella and M. Conti. Characterizing aggregate inter-contact times in heterogeneous opportunistic networks. NETWORKING 2011

A: Does aggregate power-law imply power-law for individual

components?

Q: Not necessarily

Analytical analysis - IdeaAnalytical analysis - Idea

In the same network with the same arrival processes

we can obtain very different inter-contact times

distributions.

Analytical Analysis – Contact ProcessAnalytical Analysis – Contact ProcessContacts between two users in a

single meeting place.Contacts between two users in all

shared meeting places.

The rate of the resulting contact process depends on arrival rates as:The rate of the resulting contact process depends on arrival rates as:

Analytical Analysis – SchemeAnalytical 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)

Case study N1 – long-tail ICTCase study N1 – long-tail ICT

Output:● long-tail distribution of inter-contact

times

Input:● random graph with number of nodes n and probability of link χ

● removal probability α

● Bernoulli arrival processes with rates where Y is a standard normal random variable

Case study N2 – exponential ICTCase study N2 – exponential ICT

Input:● similar as in the first case but the

Bernoulli arrival processes with identical rates

Output:● inter-contact times distribution with

exponential shape

ConclusionConclusion● 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 D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. An arrival based framework for human mobility modeling. WoWMoM, 2012framework for human mobility modeling. WoWMoM, 2012

D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. SPoT: Representing D. Karamshuk, C. Boldrini, M. Conti, and A. Passarella. SPoT: Representing the Social, Spatial, and Temporal Dimensions of Human Mobility with a the Social, Spatial, and Temporal Dimensions of Human Mobility with a Unifying Framework. Under submission.Unifying Framework. Under submission.

Thank you for attention!

Dmytro KaramshukPhD student @ IMT Lucca

Research Associate @ IIT CNR di Pisaemail: karamshuk@gmail.com

follow me on Twitter: @karamshuk