Introduction
Voter model
Voter model on networks: role of dimensionality,
disorder and degree distribution
Luca Caniparoli
SISSA, Trieste
March 11th, 2011
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Statistical physics of social system: what, why and how
A paradigmatic example: the Voter model
Voter model on lattices
Complex networks and voter model
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Statistical physics of social dynamics
Application of the statistical physics to a “human ensemble”
Many different phenomena have been studyed:
opinion (voter model!) and cultural dynamics
language dynamics: evolution, interaction, extinction of languages
crowd behavior: flocking, pedestrian behavior, applause dynamics
formation of hierarchies
human dynamics: queuing models, social web
social spreading phenomena: rumors and news
Luca Caniparoli
Voter model on networks
Introduction
Voter model
The idea of a physical modelling of social phenomena is very old,
roots in the discovery of quantitative laws in the properties of a
population (e.g. birth/death ratios, crime statistics. . . )
Widespread studies are recent: new large databases are now
available and brand new phenomena emerged (Internet!)
Luca Caniparoli
Voter model on networks
Introduction
Voter model
The idea of a physical modelling of social phenomena is very old,
roots in the discovery of quantitative laws in the properties of a
population (e.g. birth/death ratios, crime statistics. . . )
Widespread studies are recent: new large databases are now
available and brand new phenomena emerged (Internet!)
What is the main question?
How do the interactions between social agents create order out of
an initial disordered situation?
Luca Caniparoli
Voter model on networks
Introduction
Voter model
The idea of a physical modelling of social phenomena is very old,
roots in the discovery of quantitative laws in the properties of a
population (e.g. birth/death ratios, crime statistics. . . )
Widespread studies are recent: new large databases are now
available and brand new phenomena emerged (Internet!)
What is the main question?
How do the interactions between social agents create order out of
an initial disordered situation?
Remember that
order ⇔ consensus, agreement, uniformity
disorder ⇔ fragmentation, disagreement
Luca Caniparoli
Voter model on networks
Introduction
Voter model
What are the “atoms”?
In statistical mechanics, the “actors” are simple objects
(atoms and molecules)
Macroscopic phenomena are NOT due to a complex behavior
of the single entities
Luca Caniparoli
Voter model on networks
Introduction
Voter model
What are the “atoms”?
In statistical mechanics, the “actors” are simple objects
(atoms and molecules)
Macroscopic phenomena are NOT due to a complex behavior
of the single entities
Humans are exactly the opposite!
No one knows precisely the dynamics of the single
We have to chop off much detail. . .
Luca Caniparoli
Voter model on networks
Introduction
Voter model
What are the “atoms”?
In statistical mechanics, the “actors” are simple objects
(atoms and molecules)
Macroscopic phenomena are NOT due to a complex behavior
of the single entities
Humans are exactly the opposite!
No one knows precisely the dynamics of the single
We have to chop off much detail. . .
Then, tractable models are too simplified to describe real systems?
Qualitative large scale properties (usually)
do not depend on the microscopic detail:
we hope that universality is at work. . .
Luca Caniparoli
Voter model on networks
Introduction
Voter model
What are the relevant interactions?
The dynamics tend to reduce the variability, leading to an ordered
or a fragmented state
Social influence
The agents feel the tendency to become more alike
Topology of the network
Traditional statistical physics models are on regular lattices:
we can solve analytically, but it is quite far from reality. . .
Complex networks are more realistic and their effects are
usually nontrivial!
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Opinion dynamics
The aim is to understand what are the elementary processes that
make a population of agents reach consensus
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Opinion dynamics
The aim is to understand what are the elementary processes that
make a population of agents reach consensus
But opinion are complex!
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Opinion dynamics
The aim is to understand what are the elementary processes that
make a population of agents reach consensus
But opinion are complex!
Everyday life, people are confronted with a limited number of
positions on a specific issue
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model
The idea
We assign a spin varable to every agent. The social influence is
modelled copying the opinion of a neighbor: we are conformists!
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model
The idea
We assign a spin varable to every agent. The social influence is
modelled copying the opinion of a neighbor: we are conformists!
Definition
Each agent is a binary spin variable s = ±1
An agent i is chosen along with one of its neighbors j
We set si = sj , i.e. the agent takes the opinion of the selected
neighbor
It is rather crude, but can be solved on regular lattices
Agents feel the pressure of the majority in an average sense
Noise is absent: the “full consensus states” are absorbing
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Luca Caniparoli
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model on networks
Introduction
Voter model
Luca Caniparoli
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model doesn’t exibit surface tension
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model on regular lattices
[Frachebourg, Krapivsky, 1996]
On a d-dimensional hypercubic lattice, the transition rate for a
spin k to flip is


X
d
1
Wk (S) ≡ W (sk → −sk ) = 1 −
sk
sj  .
4
2d
j
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model on regular lattices
[Frachebourg, Krapivsky, 1996]
On a d-dimensional hypercubic lattice, the transition rate for a
spin k to flip is


X
d
1
Wk (S) ≡ W (sk → −sk ) = 1 −
sk
sj  .
4
2d
j
We can write a master equation for the probability of a state:
dP(S, t) X =
Wk (S k )P(S k , t) − Wk (S)P(S, t)
dt
k
where S k differs from S only for the flip of the spin k.
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
We can derive a set ofP
differential equations for the spin correlation
functions hsk . . . sl i = S P(S, t)sk . . . sl .
Two-spins correlation function
dhsk sl i
= (∆k + ∆l )hsk sl i
dt
P
where ∆k hsk sl i = (−2dhsk sl i + j∼k hsj sl i)/2 is the laplacian
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
We can derive a set ofP
differential equations for the spin correlation
functions hsk . . . sl i = S P(S, t)sk . . . sl .
Two-spins correlation function
dhsk sl i
= (∆k + ∆l )hsk sl i
dt
P
where ∆k hsk sl i = (−2dhsk sl i + j∼k hsj sl i)/2 is the laplacian
This equation can be solved analytically using Laplace transform
If we define na (t) = (1 − hsk sk+1 i)/2 as the density of active
interfaces, we find the asymptotic behavior

−(2−d)/2

d <2
t
na (t) ∼ 1/ log t
d =2


a − bt −d/2 d > 2
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
In infinite systems consensus is reached only if d ≤ 2
Voter model can be mapped on a model of random walkers
that coalesce upon encounter: RW recurrence/transience is
linked to consensus/fragmentation!!
Consensus can be reached in finite systems in d > 2 as a large
fluctuation
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
In infinite systems consensus is reached only if d ≤ 2
Voter model can be mapped on a model of random walkers
that coalesce upon encounter: RW recurrence/transience is
linked to consensus/fragmentation!!
Consensus can be reached in finite systems in d > 2 as a large
fluctuation
This model can be modified in many directions:
presence of quenched disorder: the zealot never changes his
mind
“spin-1” model: a centrist state is added
introduction of memory. . .
Real social networks are far from regular lattices!
What is the effect of the network on the behavior of the model?
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Complex networks
Real networks show some peculiar properties (e.g. power law
degree distribution, clustering, small path lengths. . . ) which could
heavily influence the behavior of the models
We will use three network models: Small-Word, Barabasi-Albert
and Structured Scale Free
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Complex networks
Real networks show some peculiar properties (e.g. power law
degree distribution, clustering, small path lengths. . . ) which could
heavily influence the behavior of the models
We will use three network models: Small-Word, Barabasi-Albert
and Structured Scale Free
Small World Networks (Watts-Strogatz)
Starting from a 1-D ring lattice with 2k NN links, every node is
visited and each of its link is rewired with probability p
Small average path length (small world!), high clustering
Non-scale free degree distribution
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Scale Free network: Barabási-Albert model
Start with a connected network of mo nodes. AddP
nodes one at a
time, linking it to m existing nodes w.p. pi = ki /( kj ). Nodes
with high connectivity tend to accumulate even more links
Power law degree distribution: P(k) ∼ k −3
Path length grows with size: ℓ ∼ log N
Clustering diminishes with size
Preferential attachement leads to the formation of hubs
This model fits quite well technological networks
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
[Klemm, Eguiluz, 2002]
Structured Scale Free network
Start from a fully connected graph with m active vertices. Add a
new node and attach it to all the m active nodes and deactivate
one of the active nodes w.p. p(k) ∝ k −1
Power law degree distribution P(k) ∼ k −3+1/m
1-D network [Eguı́luz et al., 2003]
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
[Klemm, Eguiluz, 2002]
Structured Scale Free network
Start from a fully connected graph with m active vertices. Add a
new node and attach it to all the m active nodes and deactivate
one of the active nodes w.p. p(k) ∝ k −1
Power law degree distribution P(k) ∼ k −3+1/m
1-D network [Eguı́luz et al., 2003]
Let’s see how the voter model behaves on these networks. . .
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Voter model on complex networks
We will analyze:
Dimensionality and ordering
Role of disorder
Role of degree distribution
Order parameter
The order parameter is the average interface density na :
na =
1 − hsi sj ij∼i
2
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Dimensionality and ordering
Does the network dimensionality influence the ordering dynamics?
Qualitatively, the voter model on BA networks does not order:
The height of the plateau ξ is a measure of the disorder!
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
. . . and neither on Small World networks:
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
But these are infinite dimensional networks, and we know that the
VM does not order on regular lattices if d > 2. . .
Maybe it is just a matter of dimensionality!
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
But these are infinite dimensional networks, and we know that the
VM does not order on regular lattices if d > 2. . .
Maybe it is just a matter of dimensionality!
What happens on the SSF “1D” networks?
The dimensionality seems to be
the important ingredient ruling
the ordering dynamics.
The degree distribution is
qualitatively not crucial
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Role of disorder
What is the effect of the disorder on the ordering dynamics?
To address the question, it’s useful to introduce a network
interpolating between the SSF and the BA:
Small World Scale Free
Take a SSF network and rewire a random link with probability p,
permuting the link (i, j) with (k, l)
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Role of disorder
What is the effect of the disorder on the ordering dynamics?
To address the question, it’s useful to introduce a network
interpolating between the SSF and the BA:
Small World Scale Free
Take a SSF network and rewire a random link with probability p,
permuting the link (i, j) with (k, l)
Survival time
An interesting quantity is the time τ the system takes to reach an
absorbing state
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Role of network disorder
Plateau heigth increases as p grows: network disorder influence the
disorder of the metastable state
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Varying p one smoothly
interpolates between the SSF
and the random SF networks
Increasing randomness the
system approaches the BA
behavior
Higher disorder means higher
plateau but shorter survival
time
Similar results hold for SW
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Role of heterogeneity of degree distribution
Does a scale free degree distribution influence the dynamics?
Small World Scale Free vs Small World:
Similar plateau value, decay
faster for scale free network
Finite size fluctuations
seem to be more efficient
when hubs are present,
causing faster ordering
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Plotting the survival time and the plateau height, we see that the
Scale-Free network sistematically orders faster:
It happens that τSW > τSWSF and ξSW ≃ ξSWSF
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Finally, for the BA network:
EN: like BA without
preferential attachement
RN: p = 1 limit of SW
Plateau height remains
constant among the
different distributions
Again, power law in the
degree distribution shortens
the survival time
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Ordering
The effective dimensionality of the network seems to be the
relevant parameter that determines whether the model orders or
not.
Network disorder
Disorder decreases the lifetime of metastable non-consensus states
Network degree distribution
Heterogeneity of the degree distribution also reduces the lifetime of
the metastable state: finite size fluctuations ordering the system
are more efficient when there are “hubs”
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
The arena of social dynamics models
There is plenty of models around: thousands modification to the voter,
majority rule, social impact, Sznajd, bounded confidence, Axelrod. . .
Despite the intense research in the field, the discussion mostly
stayed on the theoretical level and had been a little self-referring
The field seems mature and in recent years large databases and adequate
means to analyse them became available (Facebook phenomenology)
Maybe it’s time to start a serious review of the result and a
quantitative comparison with the reality
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
The arena of social dynamics models
There is plenty of models around: thousands modification to the voter,
majority rule, social impact, Sznajd, bounded confidence, Axelrod. . .
Despite the intense research in the field, the discussion mostly
stayed on the theoretical level and had been a little self-referring
The field seems mature and in recent years large databases and adequate
means to analyse them became available (Facebook phenomenology)
Maybe it’s time to start a serious review of the result and a
quantitative comparison with the reality
. . . but however hard we study and however deeply we understand,
there will be someone able to exploit the “bugs” of the system. . .
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
Thank you!
Luca Caniparoli
Voter model on networks
Introduction
Voter model
Voter model on regular lattices
Complex Networks
Voter model on complex networks
References
Sucheki, Eguı́luz, San Miguel, Phys. Rev. E 72, 036132 (2005)
Castellano, Vilone, Vespignani, Europys. Lett. 63 (1), 153 (2003)
Castellano, Fortunato, Loreto, Rev. Mod. Phys. 81, (2009)
Klemm, Eguı́luz, Phys. Rev. E 65, 036123 (2002)
Eguı́luz, Hernádez-Garcia, Piro, Klemm, Phys. Rev. E 68,
055102(R), (2003)
Frachebourg, Krapivsky, Phys. Rev. E 53 (4), 3009, (1996)
Vilone, Castellano, Phys. Rev. E 69, 016109, (2004)
Luca Caniparoli
Voter model on networks