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Co-evolution of networks and opinions phase transitions in social - - PowerPoint PPT Presentation

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Co-evolution of networks and opinions PETTER HOLME Co-evolution of networks and opinions phase transitions in social systems? Petter Holme coevolution of networks and opinions KTH, CSC, Computational Biology validation November 4,

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SLIDE 1

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

Co-evolution of networks and opinions

Petter Holme

KTH, CSC, Computational Biology

November 4, 2008, DIMACS

http://www.csc.kth.se/∼pholme/

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SLIDE 2

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

dynamics of the network dynamics on the network

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SLIDE 3

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

dynamics of the network

  • pinions, information

disease, religion, norms dynamics on the network friendships, trust business contacts

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SLIDE 4

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

phase transitions in social systems?

  • ur models

verify empirically / experimentally what can we learn?

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SLIDE 5

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

phase transitions in social systems?

  • ur models

verify empirically / experimentally what can we learn?

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SLIDE 6

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

phase transitions in social systems?

  • ur models

verify empirically / experimentally what can we learn?

slide-7
SLIDE 7

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

phase transitions in social systems?

  • ur models

verify empirically / experimentally what can we learn?

slide-8
SLIDE 8

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

  • utline

phase transitions in social systems?

  • ur models

verify empirically / experimentally what can we learn?

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SLIDE 9

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

phase transitions

system’s environment quantity describing system

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SLIDE 10

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

. . . in social systems?

quantities describing the system — census statistics, election results, . . . parameters describing the environment (should be “the same” for all the agents) — gas price, . . . does social systems fit this framework? phase transitions can be categorized by their “critical exponents”, which depends only on symmetries in the system (not boundary conditions, dynamic properties, etc.)

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SLIDE 11

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

. . . in social systems?

quantities describing the system — census statistics, election results, . . . parameters describing the environment (should be “the same” for all the agents) — gas price, . . . does social systems fit this framework? phase transitions can be categorized by their “critical exponents”, which depends only on symmetries in the system (not boundary conditions, dynamic properties, etc.)

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SLIDE 12

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

. . . in social systems?

quantities describing the system — census statistics, election results, . . . parameters describing the environment (should be “the same” for all the agents) — gas price, . . . does social systems fit this framework? phase transitions can be categorized by their “critical exponents”, which depends only on symmetries in the system (not boundary conditions, dynamic properties, etc.)

slide-13
SLIDE 13

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

. . . in social systems?

quantities describing the system — census statistics, election results, . . . parameters describing the environment (should be “the same” for all the agents) — gas price, . . . does social systems fit this framework? phase transitions can be categorized by their “critical exponents”, which depends only on symmetries in the system (not boundary conditions, dynamic properties, etc.)

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SLIDE 14

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

. . . in social systems?

quantities describing the system — census statistics, election results, . . . parameters describing the environment (should be “the same” for all the agents) — gas price, . . . does social systems fit this framework? phase transitions can be categorized by their “critical exponents”, which depends only on symmetries in the system (not boundary conditions, dynamic properties, etc.)

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SLIDE 15

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the idea

P . Holme & M. E. J. Newman, Phys. Rev. E 74, 056108 (2006). Opinions spread over social networks. People with the same opinion are likely to become acquainted. We try to combine these points into a simple model of simultaneous opinion spreading and network evolution.

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SLIDE 16

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the idea

P . Holme & M. E. J. Newman, Phys. Rev. E 74, 056108 (2006). Opinions spread over social networks. People with the same opinion are likely to become acquainted. We try to combine these points into a simple model of simultaneous opinion spreading and network evolution.

slide-17
SLIDE 17

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the idea

P . Holme & M. E. J. Newman, Phys. Rev. E 74, 056108 (2006). Opinions spread over social networks. People with the same opinion are likely to become acquainted. We try to combine these points into a simple model of simultaneous opinion spreading and network evolution.

slide-18
SLIDE 18

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the idea

P . Holme & M. E. J. Newman, Phys. Rev. E 74, 056108 (2006). Opinions spread over social networks. People with the same opinion are likely to become acquainted. We try to combine these points into a simple model of simultaneous opinion spreading and network evolution.

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SLIDE 19

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model Clifford & Sudbury, Biometrika 60, 581 (1973). Holley & Liggett, Ann. Probab. 3, 643 (1975).

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SLIDE 20

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model choose one vertex randomly

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SLIDE 21

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model copy the opinion of a random neighbor

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SLIDE 22

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model and so on . . .

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SLIDE 23

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model and so on . . .

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SLIDE 24

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model and so on . . .

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SLIDE 25

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

the voter model and so on . . .

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SLIDE 26

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics: precepts

People of similar interests are likely to get acquainted. The number of edges is constant.

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SLIDE 27

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics: precepts

People of similar interests are likely to get acquainted. The number of edges is constant.

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SLIDE 28

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics: precepts

People of similar interests are likely to get acquainted. The number of edges is constant.

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SLIDE 29

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics

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SLIDE 30

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics choose one vertex randomly

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SLIDE 31

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics rewire an edge to a vertex w same opinion

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SLIDE 32

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics and so on . . .

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SLIDE 33

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics and so on . . .

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SLIDE 34

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics and so on . . .

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SLIDE 35

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

acquaintance dynamics and so on . . .

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SLIDE 36

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

model definition

1

Start with a random network of N vertices M = ¯ kN/2 edges and G = N/γ randomly assigned opinions.

2

Pick a vertex i at random.

3

With a probability φ make an acquaintance formation step from i.

4

. . . otherwise make a voter model step from i.

5

If there are edges leading between vertices of different

  • pinions—iterate from step 2.
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SLIDE 37

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

model definition

1

Start with a random network of N vertices M = ¯ kN/2 edges and G = N/γ randomly assigned opinions.

2

Pick a vertex i at random.

3

With a probability φ make an acquaintance formation step from i.

4

. . . otherwise make a voter model step from i.

5

If there are edges leading between vertices of different

  • pinions—iterate from step 2.
slide-38
SLIDE 38

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

model definition

1

Start with a random network of N vertices M = ¯ kN/2 edges and G = N/γ randomly assigned opinions.

2

Pick a vertex i at random.

3

With a probability φ make an acquaintance formation step from i.

4

. . . otherwise make a voter model step from i.

5

If there are edges leading between vertices of different

  • pinions—iterate from step 2.
slide-39
SLIDE 39

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

model definition

1

Start with a random network of N vertices M = ¯ kN/2 edges and G = N/γ randomly assigned opinions.

2

Pick a vertex i at random.

3

With a probability φ make an acquaintance formation step from i.

4

. . . otherwise make a voter model step from i.

5

If there are edges leading between vertices of different

  • pinions—iterate from step 2.
slide-40
SLIDE 40

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

model definition

1

Start with a random network of N vertices M = ¯ kN/2 edges and G = N/γ randomly assigned opinions.

2

Pick a vertex i at random.

3

With a probability φ make an acquaintance formation step from i.

4

. . . otherwise make a voter model step from i.

5

If there are edges leading between vertices of different

  • pinions—iterate from step 2.
slide-41
SLIDE 41

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

model definition

1

Start with a random network of N vertices M = ¯ kN/2 edges and G = N/γ randomly assigned opinions.

2

Pick a vertex i at random.

3

With a probability φ make an acquaintance formation step from i.

4

. . . otherwise make a voter model step from i.

5

If there are edges leading between vertices of different

  • pinions—iterate from step 2.
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SLIDE 42

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

phases low φ—one dominant cluster

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SLIDE 43

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

phases high φ—clusters of similar sizes

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SLIDE 44

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

quantities we measure

The relative largest size S of a cluster (of vertices with the same opinion). The average time τ to reach consensus.

slide-45
SLIDE 45

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

quantities we measure

The relative largest size S of a cluster (of vertices with the same opinion). The average time τ to reach consensus.

slide-46
SLIDE 46

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

quantities we measure

The relative largest size S of a cluster (of vertices with the same opinion). The average time τ to reach consensus.

slide-47
SLIDE 47

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

cluster size distribution

φ = 0.458 φ = 0.04 φ = 0.96

P(s) P(s) P(s) 10−4 10−6 10−8 0.01 s 10−4 0.01 10−6 0.01 10 1 100 1000 10−4

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SLIDE 48

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

finding the phase transition

Assume a critical scaling form: scaling form S = N−a F

  • Nb(φ − φc)
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SLIDE 49

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

finding the phase transition

40 10 20 30 5 6 7 0.45 0.46 0.47 1 0.2 0.4 0.6 0.8 φ S 1N−a φ S 1N−a

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SLIDE 50

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

finding the phase transition

(φ − φc)Nb 5.0 5.5 6.0 6.5 N = 200 N = 400 N = 800 N = 1600 N = 3200 −0.2 0.2 0.4 S 1N−a −0.4

a = 0.61 ± 0.05, φc = 0.458 ± 0.008, b = 0.7 ± 0.1 random graph percolation: a = b = 1/3

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SLIDE 51

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

finding the phase transition

(φ − φc)Nb 5.0 5.5 6.0 6.5 N = 200 N = 400 N = 800 N = 1600 N = 3200 −0.2 0.2 0.4 S 1N−a −0.4

a = 0.61 ± 0.05, φc = 0.458 ± 0.008, b = 0.7 ± 0.1 random graph percolation: a = b = 1/3

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SLIDE 52

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

finding the phase transition

(φ − φc)Nb 5.0 5.5 6.0 6.5 N = 200 N = 400 N = 800 N = 1600 N = 3200 −0.2 0.2 0.4 S 1N−a −0.4

a = 0.61 ± 0.05, φc = 0.458 ± 0.008, b = 0.7 ± 0.1 random graph percolation: a = b = 1/3

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SLIDE 53

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

dynamic critical behavior

10 15 0.4 0.5 20 5 φ τN−z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.75 1 0.25 0.5 φ Vτ

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SLIDE 54

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

conclusions

We have proposed a simple, non-equilibrium model for the coevolution of networks and opinions. The model undergoes a second order phase transition between: One state of clusters of similar sizes. One state with one dominant cluster. The universality class is not the same as random graph percolation. In society, a tiny change in the social dynamics may cause a large change in the diversity of opinions.

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SLIDE 55

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

conclusions

We have proposed a simple, non-equilibrium model for the coevolution of networks and opinions. The model undergoes a second order phase transition between: One state of clusters of similar sizes. One state with one dominant cluster. The universality class is not the same as random graph percolation. In society, a tiny change in the social dynamics may cause a large change in the diversity of opinions.

slide-56
SLIDE 56

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

conclusions

We have proposed a simple, non-equilibrium model for the coevolution of networks and opinions. The model undergoes a second order phase transition between: One state of clusters of similar sizes. One state with one dominant cluster. The universality class is not the same as random graph percolation. In society, a tiny change in the social dynamics may cause a large change in the diversity of opinions.

slide-57
SLIDE 57

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

conclusions

We have proposed a simple, non-equilibrium model for the coevolution of networks and opinions. The model undergoes a second order phase transition between: One state of clusters of similar sizes. One state with one dominant cluster. The universality class is not the same as random graph percolation. In society, a tiny change in the social dynamics may cause a large change in the diversity of opinions.

slide-58
SLIDE 58

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

conclusions

We have proposed a simple, non-equilibrium model for the coevolution of networks and opinions. The model undergoes a second order phase transition between: One state of clusters of similar sizes. One state with one dominant cluster. The universality class is not the same as random graph percolation. In society, a tiny change in the social dynamics may cause a large change in the diversity of opinions.

slide-59
SLIDE 59

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

an equilibrium model

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SLIDE 60

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

an equilibrium model

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SLIDE 61

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

an equilibrium model

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SLIDE 62

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

an equilibrium model

slide-63
SLIDE 63

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

methodology of mechanistic models behavior of the individual model capturing macroscopic properties

  • bservations

consistent with

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SLIDE 64

Co-evolution

  • f networks

and opinions PETTER HOLME phase transitions in social systems? coevolution of networks and

  • pinions

validation

thank you!

Mark Newman Zhi-Xi Wu Gourab Ghoshal Andreas Gr¨

  • nlund

Lu´ ıs Enrique Correa da Rocha Fredrik Liljeros