Opinion dynamics on social networks · Opinion dynamics on social networks Giacomo Albi The opinion...

Opiniondynamics on

social networks

Giacomo Albi

The opiniondynamicsEmergence ofconsensus

Symmetricinteractions

Non-symmetricinteractions

Opinion dynamics on social networks

Giacomo Albiwww.giacomoalbi.com

Department of Computer Science,University of Verona, Italy

ECMI Modelling courseVerona, October, 2017

October 2017Giacomo Albi (University of Verona) Opinion dynamics on social networks October, 2017 1 / 20

Opiniondynamics on

social networks

Giacomo Albi




Outline

Problem posingWe want to model, simulate and control an interacting systemdescribing the evolution of individuals’ opinions over a social network

The objects of study areIndividual’s opinion, which evolves according to the exchange ofinformations with other individuals.The interaction network, the social system ruling the interactionsamong individuals (Facebook, Twitter, personal network,. . . )Influence by external factors: how advertisement, politicalpolicies affect the global opinion?

Giacomo Albi (University of Verona) Opinion dynamics on social networks October, 2017 2 / 20

Opiniondynamics on

social networks

Giacomo Albi




Modelling opinion dynamics

We propose the following model for opinion formation

ddt wi =

N∑j=1

Pij(t)(wj − wi), i = 1, . . . , N

where,wi ∈ [−1, +1] represents the opinion of the i-agent, where −1 and+1 represent two opposite opinions.Pij ≥ 0 is the communication function, quantifying the influencebetween agent i and j.

The underling process of such model represents a generic way to describealignment, where opinion of agent i aligns toward the opinion of agent j,

wi → wj


Opiniondynamics on

social networks

Giacomo Albi




The communication function

In a general setting we assume the communication function Pij(t) to be anonlinear model of the agents’ positions, and the associated graph, G,

Pij(t) = Pij(w(t),G(t)), w(t) = {wk(t)}k

Note that Pij(t) can be seen as weights that naturally induce a directgraph structure on the set of agents.Hence we can define the graph induced by Pij , for any ε ≥ 0 andt ≥ 0 the graph Pε(t) as

Pε = {(i, j) ∈ {1, . . . , N}2|Pij(t) > ε}.

Thus P0(t) is the the set of edges (i, j) for which the communicationchannel from i to j is active at time t.


Opiniondynamics on

social networks

Giacomo Albi




The communication function

In what follows, we will assume that interactions Pij(t) = Pij(w(t),G(t)) areruled by two main mechanisms

Network based interactions, where Pij is the adjacency matrix of anassociated graph, G, describing the set of direct connections among agents,

Pij(t) = Pij(G(t))

Metric based interactions, where the communication among agents Pij is afunction of the relative distance, dij = |wi − wj |.

Pij(|wi − wj |)


Opiniondynamics on

social networks

Giacomo Albi




Network based interactions

We considered N = 30 agents with different adjacency matrix Pij


Opiniondynamics on

social networks

Giacomo Albi




Metric based interactions

As an example we can consider Bounded Confidence type of interactions,

Pij =1Nχ(|wi − wj | ≤ C),

where agents interact only within a confidence level C .

Left: C = 0.25, Center: C = 0.45 Right: C = 0.65


Opiniondynamics on

social networks

Giacomo Albi




Emergence of consensus

Definition: Global consensus

Let w : R+ → [−1, +1]N be a solution of the opinion model with initialdatum w(0) = w0. We say that w(·) converges to consensus if there existsw∞ ∈ [+1,−1] such that, for every i = 1, . . . , N it holds

limt→+∞

|wi(t)− w∞| = 0,

where | · | is the Euclidean norm. The value w∞ is called the consensusstate.

wi(t)→ w∞.

We will see that the w∞ is an emergent property of the global dynamics ofthe system.


Opiniondynamics on

social networks

Giacomo Albi




Emergence of consensus

The average opinionIf the communication function is symmetric, i.e. Pij = Pji , theaverage opinion is an invariant of the system.

Prove that

ddt w̄(t) = 0, with w̄(t) = 1

N

N∑i=1

wi(t)

As a consequence, if the solution of the opinion model converges toconsensus, we have

w∞ = 1N

N∑i=1

w0i .


Opiniondynamics on

social networks

Giacomo Albi




The Laplacian formulation

We consider the following reformulation of the opinion model

ddt

wi =N∑

j=1

Pij(t)wj − wi

N∑j=1

Pij(t) = − (LP(t)w)i , i = 1, . . . ,N

where LP(t) is defined as the Laplacian matrix associated to the communicationfunction P(t) as follows

LP =

∑

j 6=1 P1j −P12 −P13 . . . −P1N−P21

∑j 6=2 P2j −P23 . . . −P2N

−P31 −P32∑

j 6=3 P3j . . . −P3N...

......

. . ....

−PN1 −PN2 −PN3 . . .∑

j 6=N PNj

For symmetric interactions, Pij = Pji , the Laplacian matrix is symmetric.The laplacian formulation is connected to the heat equation

∂tu = ∂xxu

in particular if we consider a finite difference discretization of the lastequation.


Opiniondynamics on

social networks

Giacomo Albi




The Laplacian matrix of a graph

The Laplacian matrix L associated to the (simple) graph G is defined as follows

L = D −A

where:A is the adjacency matrix of G,and D is the (diagonal) degree matrix of G, defined as Dii =

∑j 6=iAij , for

i = 1, . . . ,N .

L =

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

Write the Laplacian matrix associated to the left graph.


Opiniondynamics on

social networks

Giacomo Albi




The Laplacian matrix of an undirect graph

Symmetric interactionsFor undirect graphs, G, the Laplacian matrix is a symmetric semi-positve definitematrix.

The Laplacian matrix L has eigenvalues

0 = λ1 ≤ λ2 ≤ . . . ≤ λN

Indeed, every row sum and column sum of L is zero,

L1 = 0,

1 = (1, 1, . . . , 1)T , 0 = (0, 0, . . . , 0)T

thus we have that L is singular with eigenvalue 0.The number of connected components in the graph G is the geometricalmultiplicity of the 0 eigenvalue.The second smallest eigenvalue of L (could be zero), λ2(L) is the Fiedlernumber of G and it is associated to the algebraic connectivity.


Opiniondynamics on

social networks

Giacomo Albi




On the algebraic connectivity

Algebraic connectivityThe algebraic connectivity, or Fiedler eigenvalue, of a graph G is thesecond-smallest eigenvalue of the Laplacian matrix of G.

This eigenvalue is greater than 0 if and only if G is a connected graph.Indeed the number of times 0 appears as an eigenvalue in the Laplacian isthe number of connected components of G.The magnitude of this value reflects how well connected the overall graph is.

Write the Laplacian matrix of both graphs and compute its eigenvalues.


Opiniondynamics on

social networks

Giacomo Albi




Unconditional consensus under symmetry

Let us introduce the following new variable

vi(t) := wi(t)− w̄

where w̄ is the average of the system which is conserved over time thanks to symmetry.

We can write,

ddt

vi =N∑

j=1

Pij(vj − vi), vi(0) = w0i − w̄.

Thus we can compute the following chain of equalities

12

ddt

∑i

|vi(t)|2 =12

∑i

2〈ddt

vi , vi〉

=∑

i

∑j

Pij〈vj − vi , vi〉

=12

∑i

∑j

Pij〈vj − vi , vi〉 −12

∑i

∑j

Pij〈vj − vi , vj〉

= −12

∑i

∑j

Pij |vj − vi |2


Opiniondynamics on

social networks

Giacomo Albi





Courant-Fischer-Weyl min-max principle

We can characterize the Fiedler number of the Laplacian matrix, λ2(LP) in termsof the vector v ∈ RN orthogonal to the first eigenvector 1 = (1, 1, . . . , 1)T , asfollows

λ2(LP) = min1T v=0

〈LPv, v〉〈v, v〉

≤(1/2)

∑i

∑j Pij |vi − vj |2∑

i |vi |2

Thanks to this result we can infer from the previous equalities the followingestimate

12

ddt

∑i

|vi(t)|2 = −12

∑i

∑j

Pij |vj − vi |2 ≤ −λ2(LP)∑

i

|vi(t)|2.

Thus we have an estimate for the following quantity

1N

∑i

|vi(t)|2 =1N

N∑j=1

|w(t)− w̄(0)|2 =: Vw(t).


Opiniondynamics on

social networks

Giacomo Albi





In terms of Vw(t) we haveddt

Vw(t) ≤ −2λ2(LP)Vw(t)

then the following theorem holds

Theorem (S. Motsch- E. Tadmor ’14)

Given the opinion model

ddt

w = −LP(t)w, w(0) = w0

with symmetric communication P. Then the following concentration estimate holds

Vw(t) ≤ exp

(−2

∫ t

0

λ2(LP(s)) ds

)Vw(0).

In particular, if the interactions remain “strong enough” so that

limt→∞

∫ t

0

λ2(LP(s)) ds = +∞,

then there is convergence toward consensus, i.e. w(t)→ w∞ = w̄(0).Giacomo Albi (University of Verona) Opinion dynamics on social networks October, 2017 16 / 20

Opiniondynamics on

social networks

Giacomo Albi




Connectivity

A direct path in G from Vi to Vj is a sequence of edges

Γij =(

eii1 , ei1i2 , . . . , eik j)∈ G.

A graph G is connected when there is a path between every pair of vertices.Conversely a graph G is said to be disconnected if there exist two nodes inG such that no path in G has those nodes as endpoints.


Opiniondynamics on

social networks

Giacomo Albi




Strong connectivity

The graph G is said to be strongly connected if for any pair of vertexes Vi ,Vjthere exists a direct path from Vi to Vj


Opiniondynamics on

social networks

Giacomo Albi




Unconditional consensus under strong connectivity

Let Pε(t) be the graph induced by the communication functionPij(t), we recall that

Pε = {(i, j) ∈ {1, . . . , N}2|Pij(t) > ε}.

Theorem (Haskovec, ’15)

Given the opinion model

ddt

w = −LP(t)w, w(0) = w0

with associated graph Pε(t). If there exists an ε > 0 and a strongly connectedgraph G on the set of agents on which the system spends an infinite amount oftime, i.e.

T = {t ≥ 0|Pε(t) ≡ G}, `1(T ) = +∞.Then there is convergence toward consensus, i.e. w(t)→ w∞.

where `1 is the Lebesgue measure.


Opiniondynamics on

social networks

Giacomo Albi




Bibliography

Bongini, M. (2016), Sparse Optimal Control of Multiagent Systems,PhD thesis, TUM.Watts, D. J., Dodds, P. S. (2007). Influentials, networks, and publicopinion formation. Journal of consumer research, 34(4), 441-458.Hegselmann, R., Krause, U. (2002). Opinion dynamics and boundedconfidence models, analysis, and simulation. Journal of artificialsocieties and social simulation, 5(3).Cucker, F., Smale, S. (2007). Emergent behavior in flocks. IEEETransactions on automatic control, 52(5), 852-862.Motsch, S., Tadmor, E. (2014). Heterophilious dynamics enhancesconsensus. SIAM review, 56(4), 577-621.Moreau, L. (2005). Stability of multiagent systems withtime-dependent communication links. IEEE Trans. Automat. Control, 50(2):169-182.Haskovec, J. (2015). A note on the consensus finding problem incommunication networks with switching topologies. Appl. Anal.,94(5):991-998.


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