Dynamics of interactions in a large communication network

Márton Karsai, Mikko Kivelä, Raj Pan, Jari Saramäki, Kimmo Kaski, Albert-László Barabási

János KertészBudapest University of Technology and EconomicsandAalto University, Helsinki

Support: EU FP7 FET-Open 238597, FiDiPro, OTKA

In collaboration with

Temporal aspects of social behavior from mobile phone data

Transmission on a linear chain

VERY slow

May 29, 2003: calls to the Hungarian Central Office for Combatting Catastrophiesthat people obtained sms messages like:„Large nuclear accident in PaksStay home, close doors-windows, don’t eat lettuce”

In reality nothing happened.

Police investigation revealed that a nurse had heared children talking in the kindergarden about such things (who heared them from their parents as a rumor); she called her relatives, the „information” reached a journalist, who started an sms campaign...

Was spreading fast or slow?

Outline- Spreading phenomena in complex networks,

small world- Mobile phone call network: A proxy for the social

network. Nodes, links and weights- The Granovetterian structure of the society- Event list and modeling spreading - Different sources of correlations:

- topology- weight-topology- daily (weekly) patterns- burstiness- link-link

- Differentiating between contributions to spreading- Burst statistics- Summary

Spreading phenomena in networks

- epidemics (bio- and computer)- rumors, information, opinion- innovations- etc.

Nodes of a network can be: - Susceptible- Infected- Recovered (immune)

Corresponding models: SI, SIR, SIS...

Important: speed of spreading (SLOW)

Spreading curve (SI)

Early

Late

Intermediate

m(t)=Ninf /Ntot

Spreading in the society

Small world property; “Six Degrees of Separation”, Erdős number, WWW,

collaboration network, Kevin Bacon game etc.

Not only social nw-s: Internet, genetic transcription, etc.

In many networks the average distance btw two

arbitrary nodes is small (grows at most log with system size).

Distance: length of shortest path btw two nodes

Impossible to know – use a proxy:

The usage of mobile phones in the adult population is close to 100%

All interactions are recorded – use call network as a proxy of the network at the societal level

Small world: fast spreading?There are short, efficient paths. Are they used?

Needed information: - Structure of the society: Network at the societal level - Local transmission dynamics: Detailed description, how information (rumor, opinions etc) is transmitted

• Over 7 million Over 7 million private mobile phoneprivate mobile phone subscriptions subscriptions• Focus: voice calls within the home operator Focus: voice calls within the home operator

• Data aggregated from a period of 18 weeksData aggregated from a period of 18 weeks• Require reciprocity (Require reciprocity (XXY AND YY AND YXX) for a link) for a link

• Customers are anonymous (hash codes)Customers are anonymous (hash codes)• Data from Data from anan European mobile operator (20% market share) European mobile operator (20% market share)• Weights:Weights: either call duration or number of calls either call duration or number of calls

Constructing Constructing social nsocial networetwork from k from mobilemobile

phone data phone data

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Huge network: proxy for network at societal level

Small world

The strength of weak ties (M.Granovetter, 1973)

Hypothesis about the small scale (micro-) structure of the society:

1. “The strength of a tie is a (probably linear) combination of the amount of time, the emotional intensity, the intimacy (mutual confiding), and the reciprocal services which characterize the tie.”2. “The stronger the tie between A and B, the larger the proportion of individuals S to whom both are tied.”

Consequences on large (macro-) scale:Society consists of strongly wired communities linked by weak ties. The latter hold the society together.

Granovetter, Mark S. (May 1973), "The Strength of Weak Ties", American Journal of Sociology 78 (6): 1360–1380

Overlap

• Definition: relative neighborhood overlap (topological)

where the number of triangles around edge (vi, vj) is nij

• Illustration of the concept:

ijji

ijij nkk

nO

)1()1(

Empirical Verification

• Let <O>w denote Oij averaged over a bin of w-values

• Use cumulative link weight distribution: (the fraction of links with weights less than w’)

´

cum )(´)(ww

wPwP

• Relative neighbourhood overlap increases as a function of link weight Verifies Granovetter’s hypothesis (~95%) (Exception: Top 5% of weights)

Blue curve: empirical network

Red curve: weight randomised network

High Weight Links?

• Weak links: Strengh of both adjacent nodes (min & max) considerably higher than link weight

• Strong links: Strength of both adjacent nodes (min & max) about as high as the link weight

• Indication: High weight relationships clearly dominate on-air time of both, others negligible

• Time ratio spent communicating with one other person converges to 1 at roughly w ≈ 104

• Consequence: Less time to interact with others

• Explaining onset of decreasing trend for <O>w

ijji wss /),min(

ijji wss /),max(

wijsi sj

si=Σjwij

Possible to ask unprecedented questions and even find the answers to them

Study revealed the structure of the network, the interplay btw weigths and communities, the relations btw local, mesoscopic and global structure

Spreading of informationKnowledge of information diffusion based on unweighted networksUse the present network to study diffusion on a weighted network: Does

the local relationship between topology and tie strength have an effect? Spreading simulation: infect one node with new information

(1) Granovetterian: pij wij

(2) Reference: pij <w>

Spreading significantly faster on the reference (average weight) networkInformation gets trapped in communities in the real network

Reference

Granovetterian

Small but slow worldWe have data about- who called whom, voice, SMS, MMS - when- how long they talked(+ metadata – gender, age, postal code+ mostly used tower,…)306 million mobile call records of 4.9 millionindividuals during 4 months with 1s resolution

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voice calls SMS

Time sequence is made periodic

Is this fast or slow? What to compare with?

The problem of null models

More accurate study of spreading is possible: Infect (info, gossip, etc.) a node at time t0=0.

Transmission, whenever call with uninfected takes place (SI model). Watch m(t)=<I(t)/N>, the ratio of infected nodes with an average over initiators.

Correlations influence spreading speed-Topology (community structure)- Weight-topology (Granovetter-structure)- Bursty dynamics- Daily pattern- Link-link dynamic correlations

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Bursty dynamics: inhomogeneous activity patterns

Poissonian

Bursty

Average user

Busy user

Note the different scales

Bursty call patterns for individual users

Daily pattern of call density.

Weekly pattern too (here disregarded)

Calls are non-Poissonian

Scaled inter-event time distr.Binned according to weights (here: number of calls)

Inset: time shuffled

Dynamic link-link correlationstriggered calls, cascades, etc.

How to identify the effect of the different correlations on the spreading?

Introduce different null models by appropriate shuffling of the data.

Time shuffling

Link1 Link2 Link3... LinkN

t11 t21 t31... tN1t12 t22 t32... tN2. . . .

. . t3n_3.... .

t1n_1 . .

t2n_2 .

tNn_N

Destroyes burstiness (and link-link correlations) but keeps weight and daily pattern

Link1 Link2 Link3... LinkN

t11 t21 t31... tN1t12 t22 t32... tN2. . . .

. . t3n_3... .

t1n_1 . .

t2n_2 .

tNn_N

Link sequence shuffling

Select random pairs of links sequences and exchangeDestroys topology-weight and link-link correlation, keeps burstiness

Link1w Link2w Link3w.. LinkNw

t1’1 t2’1 t3’1... tN’1t1’2 t2’2 t3’2... tN’2. . . .

.t1’n_1’

.t2’n_2’

.t3’n_3’...

.tN’n_N’

Equal weight link sequence shuffling

Destroyes link-link correlationsKeeps weight-topology correlations and bursty dynamics

Long time behavior

The role of the daily patternModel calculation: Take the empirical topology, with weightsCompare homogeneous amd imhomogeneous Poissonians

Little effect

Slowing down mainly due to Granovetterian structure and bursty character of human activity

A closer look to burstinessDefine a bursty period (BP)In a series of signals a BP(Δt) is a sequence of

signals with an empty period of length Δt both at the beginning and at the end of the sequence

{

Δt

The end is measured from the end of the talk.

Bursty dynamics: a closer look

What is a burst?Define it relative to a window Δt:A bursty period (BP) is a sequence of events

separated from the rest by empty periods of at least Δt lengths.

ΔtΔt

Statistics of bursts:

length of BP

Frequency

of

len

gth

of

BP

Δt

Δt=10 is toosmall

Distribution of number of events E in bursts

-4.2

Autocorrelation of events

Modeling:

1. Independent events:

Whatever the distribution of inter-event times is, we get for P(E = n) ~ exp(-An) in contrast to the observed power law

Queuing model (Barabási, 2005)

Qualitatively good (power law waiting times and number of events in BP but wrong exponents

We have a to do list, which contain the tasks in a hierarchical order. Always the highest priority task is executed. Tasks arrive at random and get a hierarchy paramater at random.

Summary

- Mobile phone call network used as a proxy for the human network at the societal level- Structure of the society follows Granovetter’s picture (up to 95%)- Micro and macro structures are related- Several different types of correlations- Spreading slowed down mainly by weight-topology correlations and burstiness of human activities- Bursty are highly correlated events (not explainable by circadic patterns)- Strong short time correlations, no model – new explanation needed

J.-P. Onnela, et al. PNAS 104, 7332-7336 (2007)J.-P. Onnela, et al. New J. Phys. 9, 179 (2007)J. Kumpula et al. PRL 99, 228701 (2007) M. Karsai et al. arXiv:1006.2125