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Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
Co-evolution of networks and opinions
Petter Holme
KTH, CSC, Computational Biology
November 4, 2008, DIMACS
http://www.csc.kth.se/∼pholme/
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
dynamics of the network
dynamics on the network
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
dynamics of the network
opinions, information
disease, religion, norms
dynamics on the network
friendships, trust
business contacts
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
phase transitions in social systems?
our models
verify empirically / experimentally
what can we learn?
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
phase transitions in social systems?
our models
verify empirically / experimentally
what can we learn?
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
phase transitions in social systems?
our models
verify empirically / experimentally
what can we learn?
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
phase transitions in social systems?
our models
verify empirically / experimentally
what can we learn?
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
outline
phase transitions in social systems?
our models
verify empirically / experimentally
what can we learn?
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
phase transitions
system’s environmentqu
an
tity
de
scrib
ing
syste
m
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
. . . in social systems?
quantities describing the system — census statistics,election results, . . .
parameters describing the environment (should be “thesame” for all the agents) — gas price, . . .
does social systems fit this framework?
phase transitions can be categorized by their “criticalexponents”, which depends only on symmetries in thesystem (not boundary conditions, dynamic properties, etc.)
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
. . . in social systems?
quantities describing the system — census statistics,election results, . . .
parameters describing the environment (should be “thesame” for all the agents) — gas price, . . .
does social systems fit this framework?
phase transitions can be categorized by their “criticalexponents”, which depends only on symmetries in thesystem (not boundary conditions, dynamic properties, etc.)
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
. . . in social systems?
quantities describing the system — census statistics,election results, . . .
parameters describing the environment (should be “thesame” for all the agents) — gas price, . . .
does social systems fit this framework?
phase transitions can be categorized by their “criticalexponents”, which depends only on symmetries in thesystem (not boundary conditions, dynamic properties, etc.)
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
. . . in social systems?
quantities describing the system — census statistics,election results, . . .
parameters describing the environment (should be “thesame” for all the agents) — gas price, . . .
does social systems fit this framework?
phase transitions can be categorized by their “criticalexponents”, which depends only on symmetries in thesystem (not boundary conditions, dynamic properties, etc.)
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
. . . in social systems?
quantities describing the system — census statistics,election results, . . .
parameters describing the environment (should be “thesame” for all the agents) — gas price, . . .
does social systems fit this framework?
phase transitions can be categorized by their “criticalexponents”, which depends only on symmetries in thesystem (not boundary conditions, dynamic properties, etc.)
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
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 becomeacquainted.
We try to combine these points into a simple model ofsimultaneous opinion spreading and network evolution.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
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 becomeacquainted.
We try to combine these points into a simple model ofsimultaneous opinion spreading and network evolution.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
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 becomeacquainted.
We try to combine these points into a simple model ofsimultaneous opinion spreading and network evolution.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
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 becomeacquainted.
We try to combine these points into a simple model ofsimultaneous opinion spreading and network evolution.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
Clifford & Sudbury, Biometrika 60, 581 (1973).Holley & Liggett, Ann. Probab. 3, 643 (1975).
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
choose one vertex randomly
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
copy the opinion of a random neighbor
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
the voter model
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics: precepts
People of similar interests are likely to get acquainted.
The number of edges is constant.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics: precepts
People of similar interests are likely to get acquainted.
The number of edges is constant.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics: precepts
People of similar interests are likely to get acquainted.
The number of edges is constant.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
choose one vertex randomly
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
rewire an edge to a vertex w same opinion
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
acquaintance dynamics
and so on . . .
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
model definition
1 Start with a random network of N vertices M = kN/2edges 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
opinions—iterate from step 2.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
model definition
1 Start with a random network of N vertices M = kN/2edges 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
opinions—iterate from step 2.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
model definition
1 Start with a random network of N vertices M = kN/2edges 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
opinions—iterate from step 2.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
model definition
1 Start with a random network of N vertices M = kN/2edges 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
opinions—iterate from step 2.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
model definition
1 Start with a random network of N vertices M = kN/2edges 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
opinions—iterate from step 2.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
model definition
1 Start with a random network of N vertices M = kN/2edges 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
opinions—iterate from step 2.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
phases
low φ—one dominant cluster
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
phases
high φ—clusters of similar sizes
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
quantities we measure
The relative largest size S of a cluster (of vertices with thesame opinion).
The average time τ to reach consensus.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
quantities we measure
The relative largest size S of a cluster (of vertices with thesame opinion).
The average time τ to reach consensus.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
quantities we measure
The relative largest size S of a cluster (of vertices with thesame opinion).
The average time τ to reach consensus.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
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
101 100 1000
10−4
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
finding the phase transition
Assume a critical scaling form:
scaling form
S = N−a F(Nb(φ − φc)
)
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
finding the phase transition
40
0
10
20
30
5
6
7
0.45 0.46 0.47
10 0.2 0.4 0.6 0.8φ
S1N−a
φ
S1N−a
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
finding the phase transition
(φ − φc)Nb
5.0
5.5
6.0
6.5N = 200N = 400
N = 800N = 1600N = 3200
−0.2 0 0.2 0.4
S1N−a
−0.4
a = 0.61 ± 0.05, φc = 0.458 ± 0.008, b = 0.7 ± 0.1random graph percolation: a = b = 1/3
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
finding the phase transition
(φ − φc)Nb
5.0
5.5
6.0
6.5N = 200N = 400
N = 800N = 1600N = 3200
−0.2 0 0.2 0.4
S1N−a
−0.4
a = 0.61 ± 0.05, φc = 0.458 ± 0.008, b = 0.7 ± 0.1random graph percolation: a = b = 1/3
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
finding the phase transition
(φ − φc)Nb
5.0
5.5
6.0
6.5N = 200N = 400
N = 800N = 1600N = 3200
−0.2 0 0.2 0.4
S1N−a
−0.4
a = 0.61 ± 0.05, φc = 0.458 ± 0.008, b = 0.7 ± 0.1random graph percolation: a = b = 1/3
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
dynamic critical behavior
10
15
0.4 0.5
20
0
5
φτN−z
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.75 10.25 0.5φ
Vτ
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
conclusions
We have proposed a simple, non-equilibrium model for thecoevolution of networks and opinions.
The model undergoes a second order phase transitionbetween: One state of clusters of similar sizes. One statewith one dominant cluster.
The universality class is not the same as random graphpercolation.
In society, a tiny change in the social dynamics may causea large change in the diversity of opinions.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
conclusions
We have proposed a simple, non-equilibrium model for thecoevolution of networks and opinions.
The model undergoes a second order phase transitionbetween: One state of clusters of similar sizes. One statewith one dominant cluster.
The universality class is not the same as random graphpercolation.
In society, a tiny change in the social dynamics may causea large change in the diversity of opinions.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
conclusions
We have proposed a simple, non-equilibrium model for thecoevolution of networks and opinions.
The model undergoes a second order phase transitionbetween: One state of clusters of similar sizes. One statewith one dominant cluster.
The universality class is not the same as random graphpercolation.
In society, a tiny change in the social dynamics may causea large change in the diversity of opinions.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
conclusions
We have proposed a simple, non-equilibrium model for thecoevolution of networks and opinions.
The model undergoes a second order phase transitionbetween: One state of clusters of similar sizes. One statewith one dominant cluster.
The universality class is not the same as random graphpercolation.
In society, a tiny change in the social dynamics may causea large change in the diversity of opinions.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
conclusions
We have proposed a simple, non-equilibrium model for thecoevolution of networks and opinions.
The model undergoes a second order phase transitionbetween: One state of clusters of similar sizes. One statewith one dominant cluster.
The universality class is not the same as random graphpercolation.
In society, a tiny change in the social dynamics may causea large change in the diversity of opinions.
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
an equilibrium model
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
an equilibrium model
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
an equilibrium model
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
an equilibrium model
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
methodology of mechanistic models
behavior of the individual
model capturingmacroscopic properties
observations
consistent with
Co-evolutionof networks
and opinions
PETTERHOLME
phasetransitions insocialsystems?
coevolution ofnetworks andopinions
validation
thank you!
Mark NewmanZhi-Xi Wu
Gourab GhoshalAndreas GronlundLuıs Enrique Correa da RochaFredrik Liljeros
![arXiv:1112.6414v2 [physics.soc-ph] 23 Apr 2012 · arXiv:1112.6414v2 [physics.soc-ph] 23 Apr 2012 1 Evolution of opinions on social networks in the presence of competing committed](https://img.pdfslide.net/doc/110x75/5f8b2a6a9291be2b8c314725/arxiv11126414v2-23-apr-2012-arxiv11126414v2-23-apr-2012-1-evolution.jpg)
