Social networks from the perspective of Physics János Kertész 1,2 Jukka-Pekka Onnela 2, Jari...

Social networks from the Social networks from the perspective of Physicsperspective of Physics

János KertészJános Kertész1,21,2 Jukka-Pekka OnnelaJukka-Pekka Onnela22, Jari Saramäki, Jari Saramäki22,, Jörkki Hyvönen Jörkki Hyvönen22, Kimmo Kaski, Kimmo Kaski22,, Jussi Kumpula Jussi Kumpula22 David David

LazerLazer33 Gábor SzabóGábor Szabó33,4,4, Albert-László Barabási, Albert-László Barabási33,4,4

11Budapest University of Technology and Economics, HungaryBudapest University of Technology and Economics, Hungary 22Helsinki University of Technology, FinlandHelsinki University of Technology, Finland

33Harvard Harvard UniversityUniversity44University of Notre Dame, USA University of Notre Dame, USA

OutlineOutline

0. Introduction0. Introduction1.1. Constructing the social network Constructing the social network 2.2. Basic statisticsBasic statistics3.3. Granovetter’s hypothesisGranovetter’s hypothesis4.4. Thresholding (percolation)Thresholding (percolation)5.5. SpreadingSpreading6.6. ModelingModeling7.7. ConclusionsConclusions

IntroductionIntroduction

Complex systems: More input needed than mere interactions Forget about interactions

Networks: Scaffold of complexity

Useful to concentrate on the carrying NW structure (nodes and links): Holistic approach with very general statements

Spectacular recent development:Abundance of data due to IT + new concepts


WEIGHTED NW-SStep toward reductionism: Interactions have different strength weights on links Weights: Fluxes (traffic or chemical reactions), correlation based networks, etc.

(Often no negative weights, wij 0.)

How to characterize weighted NW-s? E.g. STRENGTH of node i: si = j wij

Intensity, coherence of subgraphs; clustering, motifs etc. (see: Onnela et al. PRE 71, 065103(R) (2005)


SOCIAL NW-S: Much has been taken from Sociology: betweennes, clustering, assortativity…Main method: Questionnaires (10 - 10 000)

Weighted social nw-s: Strength of socialrelationships varies over wide range„I know him/her”

„We are on first name basis”„We are friends”„We are good friends”„We are very good friends”…

Scale?Subjectivity?

Howtomeasure?


Advantage of questionnaires: Ask whatever you are interested in. It enables complex studies, multi-factor analyses.Disadvantage: Difficulty in quantification and subjectivity

E.g., AddHealth: Quantification of tie strength by number of joint activities

Mutuality test fails very oftenM.Gonzales et al. Physica A 379, 307-316.

(2007)

Alternative approach: Use communication

databases (email, phone etc)

OutlineOutline

0. 0. IntroductionIntroduction1.1. Constructing the social networkConstructing the social network 2.2. Basic statisticsBasic statistics3.3. Granovetter’s hypothesisGranovetter’s hypothesis4.4. Thresholding (percolation)Thresholding (percolation)5.5. SpreadingSpreading6.6. ModelingModeling7.7. ConclusionsConclusions

Constructing the NetworkConstructing the Network

• Use a network constructed from mobile phone Use a network constructed from mobile phone calls as a proxy for a social networkcalls as a proxy for a social network

• In the network:In the network:

Nodes Nodes individuals individuals

Links Links voice calls voice calls

• Link weights: Link weights:

• Number of calls Number of calls

• Total call duration (time & money)(time & money)

• 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 European mobile operator

Constructing the NetworkConstructing the Network

Y

X 15 min

5 min

20 minX

Y

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Basic Statistics: Visualisation

Largest connected component dominates

3.9M / 4.6M nodes

6.5M / 7.0M links

Use it for analysis!Use it for analysis!

Basic Statistics: Distributions

Fat tailFat tail

Vertex degree distributionVertex degree distribution Link weight distributionLink weight distribution

Dunbar number (monkeysphere):max ~150 connections

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Granovetter’s Weak Ties Hypothesis

• Granovetter* suggests analysis of social networks as a tool for linking micro and macro levels of sociological theory

• Considers the macro level implications of tie (micro level) strengths:

“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.”

• Formulates a hypothesis:The relative overlap of two individual’s friendship networks varies directly with the strength of their tie to one another

• Explores the impact of the hypothesis on, e.g. diffusion of information, stressing the cohesive power of weak ties

* M. Granovetter, The Strength of Weak Ties, The American Journal of Sociology 78, 1360-1380, 1973.

Granovetter’s Weak Ties Hypothesis

• Hypothesis based on theoretical work and some direct evidence

• Present network is suitable for testing the hypothesis: (i) Call durations time commitment tie strength(ii) Call durations monetary commitment tie strength (iii) Largest weighted social network so far

(Problem: Other factors, such as emotional intensity or reciprocal services?)

• What is the coupling between network topology and link weights?

• Consider two connected nodes. We would like to characterize their relative neighborhood overlap, i.e. proportion of common friends

• This leads naturally to link neighborhood overlap

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

Local Implications

• Implication for strong links?

Neighbourhood overlap is high

People form strongly connected communities

• Implication for weak links?

Neighbourhood overlap is low

Communities are connected by weak links

A Piece of the NetworkA Piece of the Network

communitycommunity

weak linksweak links

strong strong linkslinks

Overlap

ijji

ijij nkk

nO

)1()1(Global optimization to transport would put high weights to links with high betweenness centrality(# passing shortest paths)

In contrast, <O > decreases with b

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(

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• Children’s approach: Break to learn!

• We do this systematically using thresholding analysis:• Order the links by weight • Delete the links, one by one, based on their order

• Control parameter f is the fraction of removed links

• We can continuously interpolate, in either direction, between the initial connected network (f=0) and the set of isolated nodes (f=1)

• We use two different thresholding schemes(i) Increasing thresholding (remove low wij/Oij links

first)(ii) Descending thresholding (remove high wij/Oij links

first)

• Question: How does the network respond to link removal?• How similar is the response to wij and Oij driven thresholding?

Thresholding Analysis: Introduction

Thresholding

Initial connected network (f=0) All links are intact, i.e. the network is in its initial stage

Thresholding

Increasing weight thresholded network (f=0.8) 80% of the weakest links removed, strongest 20% remain

Thresholding

Initial connected network (f=0) All links are intact, i.e. the network is in its initial stage

Thresholding

Decreasing weight thresholded network (f=0.8) 80% of the strongest links removed, weakest 20% remain

We will study, as a function of the control parameter f, the following:

1. Order parameter (size of the largest component)2. “Susceptibility” (average size of other components)3. Average path lengths (in LCC)4. Average clustering coefficient in the LCC

Thresholding

Thresholding: Size of Largest Component

(c)

• RLCC is the fraction of nodes in the largest connected component

• LCC is able to sustain its integrity for moderate values of f • Least affected by removal of high Oij links (in tight

communities)• Most affected by removal of low Oij links (between

communities)• Difference between removal of low and high wij links is small,

but LCC breaks earlier if weak links are removed (Granovetter) • Very few links are required for global connectivity

remove low first remove high first

Thresholding: Size of Other Components

(c)

• Collapse for different values of f, but what is its nature?• “Susceptibility” (average cluster size excl. LCC) ns is the number of clusters with s nodes• Percolation theory: S→∞ as f→fc

Finite signature of divergence: fc ≈ 0.60 (incr. o.) fc ≈ 0.82 (incr. w.) • Demarcation between weak and strong links given by fc ≈ 0.82 • Qualitatively different role for weak and strong links

s

snsS 2

remove low first remove high first

0. Introduction0. Introduction

1.1. Constructing the social network Constructing the social network

2.2. Basic statisticsBasic statistics

3.3. Granovetter’s hypothesisGranovetter’s hypothesis

4.4. Thresholding (percolation)Thresholding (percolation)

5.5. Diffusion of infromationDiffusion of infromation

6.6. ModelingModeling

7.7. ConclusionsConclusions

OutlineOutline

Diffusion 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) Empirical: pij wij

(2) Reference: pij <w>

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

ijij xwp

Reference

Empirical

Diffusion of information

• Where do individuals get their information? Majority of infections through(1) Empirical: ties of intermediate strength(2) Reference: (would be) weak ties

• Both weak and strong ties have a diminishing role as information sources: The weakness of weak and strong ties

Reference

Empirical

Best Best search search results: results: Reach out Reach out of your of your own own communitcommunityyEmpirical

Diffusion of information

- Start spreading 100 times (large red node)- Information flows differently due to the local organizational principle

(1) Empirical: information flows along a strong tie backbone(2) Reference: information mainly flows along the shortest paths

Reference

0. Introduction0. Introduction

1.1. Constructing the social network Constructing the social network

2.2. Basic statisticsBasic statistics

3.3. Granovetter’s hypothesisGranovetter’s hypothesis

4.4. Thresholding (percolation)Thresholding (percolation)

5.5. SpreadingSpreading

6.6. ModelingModeling

7.7. ConclusionsConclusions

OutlineOutline

Modeling

What is all this good for?• Understanding structure and mechanisms of the society• Improving spreading of news and opinions(Developing marketing strategies and other tools of mass manipulation)

MODELING needed

Modeling

Needed: Weighted network model, which reflects the observations with possibly limited input

Links created by random encounters on acquaintance basis

Weights generated by one-to-one activities (phone calls)

Take into account the different time scales:

Encounter (call) frequency

Lifetime of relationships

Lifetime of nodes treated together

Modeling

i meets j with prob. wij , who meets k with prob. wjk. If k is a common friend wij, wjk wki are increased by (a). If k is not connected to i, wik = w0 ( = 1) is created with probability p (b).

With prob. pr new links with w0 weight are created (c).With prob. pd a node with all links is deleted and a

new one is born with no links.

Microscopic rules in the model

Summary of the model• Weighted local search for new acquaintances• Reinforcement of existing (popular) links• Unweighted global search for new acquaintances• Node removal, exp.link & weight lifetimes: <τ>=2

<τw>=(pd)-1

Model parametersδ Free weight reinforcement parameter

pr = 10-3 Sets the time scale of the model < τN > =1/pd

(average node lifetime of 1000 time steps)

pr = 5×10-4 Global connections; results not sensitive for it(one random link per node during 1000 time steps)

pΔ Adjusted in relation to δ to keep <k> constant(structure changes due to only link re-organisations)

Social network model

Tie strength:Tie strength: weak weak →→ intermediate intermediate →→ strong strong tietie

Samples of N=105 network for variable weight-increase δ

No communities

0Communities

start nucleating

1.0 Communities

forming

5.0Communities with dense & strong internal and sparse & weak external connections (cf. phone network)

1

Communities by inspection

• Average number of links constant: <L> = N <k>/2

(<k> ≈ 10 ) => All changes in structure

due to re-organisation of links

• Increasing δ traps search in communities, further

enhancing trapping effect

=> Clear communities form

• Triangles accumulate weight and act as nuclei for communities to emerge

δ = 0.1δ = 0

δ = 0.5 δ = 1

Communities by k-clique method

• k-clique algorithm as definition for communities*• Focus on 4-cliques (smallest non-trivial cliques)

• Relative largest community size Rk=4 [0,1]

• Average community size <ns> (excl. largest)

• Observe clique percolation through the system for small δ

• Increasing δ leads to condensation of communities

* G. Palla et al., “Uncovering the overlapping community structure...”, Nature 435, 814 (2005)

Global consequences

Ascending link removal

Model networkDescending link

removal

Phone networkAscending & Descending

Phase transition for ascending tie removal (weaker first)

Fraction of links, ff f 0 1

Modeling

The model fulfills essential criteria of social nw-s:

• Broad (but not scale free degree) distribution• Assortative mixing (popular people attract each

other)• High clustering: many triangles (by construction)• Community structure with strong links inside and weak ones between them

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Discussion and Conclusion• Weak ties maintain network’s structural integrity; Strong ties maintain local

communities; Intermediate ties mostly responsible for first-time infections• How can one efficiently search for information in a social network? ”Go out

of your community!”• Social networks seem better suited to local processing than global

transmission of information• Are there simple rules or mechanisms that lead to observed properties?• Efficient modeling possible

Publications: J.-P. Onnela, et al. PNAS 104, 7332-7336 (2007) J.-P. Onnela, et al. New J. Phys. 9, 179 (2007)

J.M. Kumpula, et al. PRL (to be published) www.phy.bme.hu/~kertesz/

Social networks from the perspective of Physics János Kertész 1,2 Jukka-Pekka Onnela 2, Jari...

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