Social networks from the perspective of Physics J

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Social networks from the perspective of
Physics János Kertész1,2 Jukka-Pekka Onnela2,
Jari Saramäki2, Jörkki Hyvönen2, Kimmo Kaski2,
Jussi Kumpula2 David Lazer3 Gábor Szabó3,4,
Albert-László Barabási3,41Budapest
University of Technology and Economics, Hungary
2Helsinki University of Technology,
Finland3Harvard University4University of Notre
Dame, USA
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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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Introduction
Complex systems Many interacting units such that
the resulting behavior is more than a mere sum
(brain, internet, society) Much is known about
the interactions but complex behavior often still
puzzling N 3 can be many! See Three-body
problem of mechanics Statistical physics N
1023 Social sciences N 3 109
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Introduction
Complex systems More input needed than mere
interactions ? Forget about interactions Network
s Scaffold of complexity Useful to concentrate
on the carrying NW structure (nodes and links)
Holistic approach with very general
statements Spectacular recent development Abunda
nce of data due to IT new concepts
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Introduction
Phenomenon Nodes Links
Cell metabolism Molecules Chemical reactions
Scientific collaboration Scientists Joint papers
WWW Pages URL links
Air traffic Airports Airline connections
Economy Firms Trading
Language Words Synonymous meaning
Society People Acquaintances

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Introduction

• Characterization of many empirical NW-s
• BROAD DEGREE DISTRIBUTION in many natural and
  human made NW-s
• - SMALL WORLD property Average distance between
  two nodes usually very small ( log( N ) ) 6
  degrees of separation
• - HIGH CLUSTERING The number of triangles is
  significantly high
• Studied in many networks WWW, Internet, actor,
  citation, metabolic etc

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World not only small and scale free but
clustered! Friends of friends are often friends.
Clustering coeff. C ratio of connected neighbors
ER graph too small clustering!
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Introduction
WEIGHTED NW-S Step 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)
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Introduction
SOCIAL NW-S Much has been taken from Sociology
betweennes, clustering, assortativity Main
method Questionnaires (10 - 10 000) Weighted
social nw-s Strength of social relationships
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
How to measure?
Scale? Subjectivity?
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Introduction
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 often M.Gonzales et al. Physica A 379,
307-316. (2007) Alternative approach Use
communication databases (email, phone etc)
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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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Constructing the Network

• Use a network constructed from mobile phone calls
  as a proxy for a social network
• In the network
• Nodes ? individuals
• Links ? voice calls
• Link weights
• Number of calls
• Total call duration (time money)

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Constructing the Network

• Over 7 million private mobile phone subscriptions
• Focus voice calls within the home operator
• Data aggregated from a period of 18 weeks
• Require reciprocity (X?Y AND Y?X) for a link
• Customers are anonymous (hash codes)
• Data from an European mobile operator

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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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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!
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Basic Statistics Distributions
Vertex degree distribution
Link weight distribution
Fat tail
Dunbar number (monkeysphere) max 150 connections
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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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Granovetters 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 individuals
  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.

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Granovetters 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

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Overlap

• Definition relative neighborhood overlap
  (topological)
• where the number of triangles around edge (vi,
  vj) is nij
• Illustration of the concept

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Empirical Verification

• Let ltOgtw denote Oij averaged over a bin of
  w-values
• Use cumulative link weight distribution
• (the fraction of links with weights less than
  w)

• Relative neighbourhood overlap increases as a
  function of link weight
• ?Verifies Granovetters hypothesis (95)
• (Exception Top 5 of weights)
• Blue curve empirical network
• Red curve weight randomised network

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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
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A Piece of the Network
weak links
strong links
community
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Overlap
Global optimization to transport would put high
weights to links with high betweenness
centrality ( passing shortest paths)
In contrast, ltO gt decreases with b
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High Weight Links?

• (a) Average Oij as a function of weight w
• w ? 104 stronger tie ? larger overlap
• w ? 104 stronger tie ? smaller overlap
• Contradicts the weak ties hypothesis !
• Links in the decreasing part correspond to over
  3h of communication over the period
• (b) Putting it into perspective
• - For only 5 of links w ? 104
• - Corresponds to 325 000 links, cannot be
  insufficient statistics

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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 ltOgtw

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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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Thresholding Analysis Introduction

• Childrens 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
  (f0) and the set of isolated nodes (f1)
• 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?

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Thresholding

• Initial connected network (f0)
• ? All links are intact, i.e. the network is in
  its initial stage

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Thresholding

• Increasing weight thresholded network (f0.8)
• ? 80 of the weakest links removed, strongest
  20 remain

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Thresholding

• Initial connected network (f0)
• ? All links are intact, i.e. the network is in
  its initial stage

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Thresholding

• Decreasing weight thresholded network (f0.8)
• ? 80 of the strongest links removed, weakest
  20 remain

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Thresholding

• We will study, as a function of the control
  parameter f, the following
• Order parameter (size of the largest component)
• Susceptibility (average size of other
  components)
• Average path lengths (in LCC)
• Average clustering coefficient in the LCC

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Thresholding Size of Largest Component

• 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
(c)
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Thresholding Size of Other Components

• 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?8 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

remove low first remove high first
(c)
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Thresholding Path Lengths in LCC

• Granovetter refers to interpersonal flow
  (information, rumour) from one person to another
• In order for a flow to exist, the two people
  (nodes) need to be connected at least through one
  path
• The size of the LCC says nothing about how
  tightly connected the component is, only that it
  is connected
• Granovetters corollary
• Weak ties create a large number of short paths
  between nodes in different communities, and
  thus removing them should increase average path
  lengths and make it more difficult for the flow
  to happen

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Thresholding Path Lengths in LCC

• Connectedness necessary but not sufficient
  condition for flow
• But how is the LCC connected?
• Use a.p.l. ltlgt to study the role of different
  links for global paths
• Removing weak links leads to longer paths
  f0.75 ltlgt45 vs. ltlgt30
• Supports the weak ties conjecture on path lengths
  
• (communities are locally connected by weak ties)

remove low first remove high first
(c)
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Thresholding Clustering in LCC

• Effect of different links on the structure of
  communities?
• Quantify this with ltCgt, average clustering
  coefficient
• Strong links are mostly within communities
  (triangles abundant), and thus removing them
  lowers clustering
• Weak links are mostly between communities (rarely
  participate in triangles), and thus removing them
  has little effect
• Removing high Oij links shatters communities
  quickly
• Removing low Oij links brings out communities

remove low first remove high first
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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Diffusion of infromation
• Modeling
• Conclusions

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Diffusion of information

• Knowledge of information diffusion based on
  unweighted networks
• Use 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 ? ltwgt
• Spreading significantly faster on the reference
  (average weight) network
• Information gets trapped in communities in the
  real network

Reference
Empirical
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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

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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

Best search results Reach out of your own
community
Empirical
Reference
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Spreading

• In simplified terms, we can think of each link as
  transmitting information locally between the two
  individuals it connects
• Strong links involve larger time commitments, so
  natural to assume that information flow through a
  link is proportional to its weight wij
• Flow through weak (high wij) links
• (i) Low per se (by definition)
• (ii) Low overlap Oij ? Few alternative paths of
  length 2, so information can easily get trapped
• Flow through strong (high wij) links
• (i) High per se (by definition)
• (ii) High overlap Oij ? Many alternative paths
  enhance flow further, so particularly well suited
  to efficient local transfer

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Searching

• Fix a set of search strategies
• Study which strategies are successful in finding
  information
• Best search results Reach out of your own
  community!

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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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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

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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
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Microscopic mechanisms in sociology

• Network sociology
• Cyclic closure
• Exponential decay for growing geodesic distance
• Focal closure
• Distance independent
• Sample window

• Network model
• Local attachment (LA)
• Special case of cyclic closure Triadic closure
• Global attachment (GA)
• Node deletion (ND)

M. Kossinets et al., Empirical Analysis of an
Evolving Social Network, Science 311, 88 (2006)
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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.
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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 lttgt2
  lttwgt(pd)-1
• Model parameters
• d Free weight reinforcement parameter
• pr 10-3 Sets the time scale of the model lt tN
  gt 1/pd
• (average node lifetime of 1000 time steps)
• pr 510-4 Global connections results not
  sensitive for it
• (one random link per node during 1000 time
  steps)
• p? Adjusted in relation to d to keep ltkgt
  constant
• (structure changes due to only link
  re-organisations)

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Modeling
pd 0.001 pr 0.0005 N 30 000
Changing ? by keeping ltkgt fixed by adjusting p?.
Communities emerge, with strong internal
links. Communities are interconnected by weak
links
? 0.001 0.1 0.5
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Social network model
Samples of N105 network for variable
weight-increase d
Tie strength weak ? intermediate ? strong tie
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Communities by inspection

• Average number of links
• constant ltLgt N ltkgt/2
• (ltkgt 10 )
• gt All changes in structure due to
  re-organisation of links
• Increasing d traps search
• in communities, further enhancing
  trapping effect
• gt Clear communities form
• Triangles accumulate weight and act as
  nuclei for communities to emerge

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Communities by k-clique method

• k-clique algorithm as definition for communities
• Focus on 4-cliques (smallest non-trivial cliques)
• Relative largest community size Rk4 ? 0,1
• Average community size ltnsgt (excl. largest)
• Observe clique percolation through the system for
  small d
• Increasing d leads to condensation of communities

G. Palla et al., Uncovering the overlapping
community structure..., Nature 435, 814 (2005)
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Global consequences
Remove weak strong links first
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Global consequences
Model network
Phone network
Ascending link removal
Descending link removal
Ascending Descending
Fraction of links, f
0
1
f
f
Phase transition for ascending tie removal
(weaker first)
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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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Outline

• 0. Introduction
• Constructing the social network
• Basic statistics
• Granovetters hypothesis
• Thresholding (percolation)
• Spreading
• Modeling
• Conclusions

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Discussion and Conclusion

• Weak ties maintain networks 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/
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Marc Granovetter, Connections, 1990
gtgtIn the history of public speaking, there have
been many famous denials. One sunny day in 1880,
Karl Marx declared "I am not a Marxist". On a
less auspicious occasion in 1973, Richard Nixon
insisted "I am not a crook". Neither Marx nor
Nixons audience gave much credence to their
denials, and you too may respond with disbelief
when I tell you that "I am not a networker".ltlt
gtgtInstead, the slogan of the day will be "We are
all networkers now".ltlt