Consensus, Flocking and Opinion Dynamics Antoine Girard Laboratoire - - PowerPoint PPT Presentation

Consensus, Flocking and Opinion Dynamics

Antoine Girard

Laboratoire Jean Kuntzmann, Universit´ e de Grenoble antoine.girard@imag.fr International Summer School of Automatic Control GIPSA Lab, Grenoble, France, September 2010

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 1 / 88

Scientific context

Networks everywhere: Biological networks (genetic regulation, ecosystems...) Technological networks (internet, sensor networks...) Economical networks (production and distribution networks, financial networks...) Social networks (scientific collaboration networks, Facebook...)

• A. Girard (LJK - U. Grenoble)

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Emerging behaviors in networks

Distributed decision making in a network. Each agent collaborates/negotiates locally with its neighbors in a network. The process succeeds if all agents eventually agree globally on some quantity of interest. Examples: bird flocks, fish schools, market prices...

• A. Girard (LJK - U. Grenoble)

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Example: flocking in mobile networks

Consider a set of agents willing to move in a common direction: Agent i is characterized by its position xi and velocity vi.

• A. Girard (LJK - U. Grenoble)

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Example: flocking in mobile networks

Consider a set of agents willing to move in a common direction: Agent i has limited communication or sensing capabilities.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 4 / 88

Example: flocking in mobile networks

Consider a set of agents willing to move in a common direction: Agent i tries to align its velocity on its neighbors: ˙ vi =

j∈Ni(vj − vi).

• A. Girard (LJK - U. Grenoble)

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Example: flocking in mobile networks

Consider a set of agents willing to move in a common direction: The communication network is described by a (dynamic) graph.

• A. Girard (LJK - U. Grenoble)

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Example: flocking in mobile networks

Consider a set of agents willing to move in a common direction: Global linear dynamics with structure given by the graph: ˙ v = −Lv.

• A. Girard (LJK - U. Grenoble)

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Example: flocking in mobile networks

Consider a set of agents willing to move in a common direction: Do the agents eventually agree on a common velocity?

• A. Girard (LJK - U. Grenoble)

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What we will see in this lecture

This lecture is not meant to give an exhaustive description of the area... Instead, we will provide a deeper insight on a small number of representative results. Main references used while preparing the lecture:

• C. Godsil & G. Royle, Algebraic Graph Theory, Springer 2001.
• R. Olfati-Saber, J.A. Fax & R.M. Murray, Consensus and cooperation in networked

multi-agent systems, Proc. IEEE, 2007. V.D. Blondel, J.M. Hendrickx, A. Olshevsky & J.N. Tsitsiklis, Convergence in multiagent coordination, consensus, and flocking, Proc. CDC, 2005.

• L. Moreau, Stability of continuous-time distributed consensus algorithms, Proc. CDC,

2004.

• S. Martin & A. Girard, Sufficient conditions for flocking via graph robustness analysis,
• Proc. CDC, 2010.
• C. Morarescu & A. Girard, Opinion dynamics with decaying confidence: application to

community detection in graphs, ArXiv, 2009.

• A. Girard (LJK - U. Grenoble)

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Lecture outline

1 Algebraic graph theory

Basic graph notions Laplacian matrix Normalized Laplacian matrix

2 Consensus Algorithms:

Discrete time and continuous time Agreement in networks with fixed topology Agreement in networks with dynamic topology

3 Applications:

Flocking in mobile networks Opinion dynamics and community detection in social networks

• A. Girard (LJK - U. Grenoble)

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Graphs

Definition

A graph is couple G = (V , E) consisting of: A finite set of vertices V = {1, . . . , n}; A set of edges, E ⊆ V × V . We assume G has no self-loops (∀i ∈ V , (i, i) / ∈ E) and is undirected (∀i, j ∈ V , (i, j) ∈ E ⇐ ⇒ (j, i) ∈ E).

Definition

In an undirected graph G = (V , E): The neighborhood of a vertex i ∈ V is the set Ni = {j ∈ V | (i, j) ∈ E}. The degree of a vertex i ∈ V is di = |Ni|.

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Example

A simple graph: V = {1, 2, 3, 4, 5} E = {(1, 2), (1, 3), (2, 4), (2, 5), (3, 5), (4, 5)... (2, 1), (3, 1), (4, 2), (5, 2), (5, 3), (5, 4)}

1 2 3 4 5

N1 = {2, 3}, d1 = 2 N2 = {1, 4, 5}, d2 = 3 N3 = {1, 5}, d3 = 2 N4 = {2, 5}, d4 = 2 N5 = {2, 3, 4}, d5 = 3

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Subgraphs

Definition

A graph G ′ = (V ′, E ′) is a subgraph of G = (V , E) if V ′ ⊆ V and E ′ ⊆ E. In addition, if V ′ = V then G ′ is a spanning subgraph of G. The subgraph of G induced by a set of vertices V ′ ⊆ V is the graph G ′ = (V ′, E ′) where E ′ = E ∩ V ′ × V ′.

initial graph subgraph spanning subgraph induced subgraph

1 2 3 4 5 2 4 5 1 2 3 4 5 2 4 5

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Connectivity notions

Definition

A path in a graph G = (V , E) is a finite sequence of edges (i1, i2), (i2, i3), . . . , (ip, ip+1) such that (ik, ik+1) ∈ E for all k ∈ {1, . . . , p}.

Definition

In a graph G = (V , E), two vertices i, j ∈ V are connected if there exists a path joining i and j (i.e. i1 = i, ip+1 = j). G is connected if for all i, j ∈ V , i and j are connected. A subset of vertices V ′ ⊆ V is a connected component of G if:

1 For all i, j ∈ V ′, i and j are connected; 2 For all i ∈ V ′, for all j ∈ V \ V ′, i and j are not connected.

• A. Girard (LJK - U. Grenoble)

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Example

1 2 3 4 5

Path joining 1 and 5

1 2 3 4 5 1 2 3 4 5

Graph is connected Graph is not connected

1 2 3 4 5

Connected components

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Adjacency and degree matrices

Definition

The adjacency matrix of a graph G = (V , E) is the n × n symmetric matrix A = (aij) given for all i, j ∈ V by: aij = 1 if (i, j) ∈ E,

• therwise.

Definition

The degree matrix of G is the n × n diagonal matrix D = (dij) given for all i, j ∈ V by: dij = di if i = j,

• therwise.
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Example

1 2 3 4 5

A =       1 1 1 1 1 1 1 1 1 1 1 1       D =       2 3 2 2 3      

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Laplacian matrix

Definition

The Laplacian matrix of a graph G = (V , E) is the n × n symmetric matrix L = (lij) given for all i, j ∈ V by: lij =    di if i = j, −1 if (i, j) ∈ E,

• therwise.

We have L = D − A. 1 2 3 4 5 L =       2 −1 −1 −1 3 −1 −1 −1 2 −1 −1 2 −1 −1 −1 −1 3      

• A. Girard (LJK - U. Grenoble)

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Normalized Laplacian matrix

Definition

The normalized Laplacian matrix of a graph G = (V , E) is the n × n symmetric matrix L = (ℓij) given for all i, j ∈ V by: ℓij =    1 if i = j and di = 0, −1/

• didj

if (i, j) ∈ E,

• therwise.

If di > 0 for all i ∈ V , then L = I − D−1/2AD−1/2 = D−1/2LD−1/2. 1 2 3 4 5 L =         1 − 1

√ 6

−1

2

− 1

√ 6

1 − 1

√ 6

−1

3

−1

2

1 − 1

√ 6

− 1

√ 6

1 − 1

√ 6

−1

3

− 1

√ 6

− 1

√ 6

1        

• A. Girard (LJK - U. Grenoble)

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Fundamental property of the Laplacian matrix

Theorem (Sum of squares property)

Let L be the Laplacian matrix of a graph G = (V , E) then, for all x ∈ Rn: x⊤Lx = 1 2

• (i,j)∈E

(xi − xj)2. Proof: For all x ∈ Rn, x⊤Lx =

• i∈V

xi

• j∈V

lijxj =

• i∈V

xi(dixi −

• (i,j)∈E

xj) =

• i∈V

xi

• (i,j)∈E

(xi − xj) =

• i∈V
• (i,j)∈E

(x2

i − xixj)

=

• (i,j)∈E

(x2

i − xixj)

• A. Girard (LJK - U. Grenoble)

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Fundamental property of the Laplacian matrix

Theorem (Sum of squares property)

Let L be the Laplacian matrix of a graph G = (V , E) then, for all x ∈ Rn: x⊤Lx = 1 2

• (i,j)∈E

(xi − xj)2. Proof: Since whenever (i, j) ∈ E, (j, i) ∈ E we have

• (i,j)∈E

(x2

i − xixj) =

• (i,j)∈E

(x2

j − xixj).

It follows that x⊤Lx = 1 2

• (i,j)∈E

(x2

i − 2xixj + x2 j ) = 1

2

• (i,j)∈E

(xi − xj)2.

• A. Girard (LJK - U. Grenoble)

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Laplacian matrix of a subgraph

Corollary

Let G = (V , E) be a graph and L its Laplacian matrix, let G ′ = (V , E ′) be a spanning subgraph of G and L′ its Laplacian matrix. Then, for all x ∈ Rn, x⊤Lx ≥ x⊤L′x Proof: By the SOS property, x⊤Lx = 1 2

• (i,j)∈E

(xi − xj)2 = 1 2

• (i,j)∈E ′

(xi − xj)2 + 1 2

• (i,j)∈E\E ′

(xi − xj)2 ≥ 1 2

• (i,j)∈E ′

(xi − xj)2 = x⊤L′x.

• A. Girard (LJK - U. Grenoble)

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Eigenvalues of the Laplacian matrix

Proposition

The Laplacian matrix L of an undirected graph G = (V , E) is symmetric

• positive. 0 is an eigenvalue of L with associated eigenvector 1n.

Proof: Symmetry is consequence of G being undirected. By the SOS property, ∀x ∈ Rn, xTLx = 1 2

• (i,j)∈E

(xi − xj)2 ≥ 0 which gives L positive. Let y = L1n, then yi =

• j∈V

lij = di +

• (i,j)∈E

−1 = di − di = 0.

• A. Girard (LJK - U. Grenoble)

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Eigenvalues of the Laplacian matrix

Proposition

0 is a simple eigenvalue of L if and only if the graph G is connected. Proof: ( = ⇒ ) Assume the graph is not connected, then there exists a connected component of G, V ′ V . Let x ∈ Rn, such that xi = 1 if i ∈ V ′ and xi = 0 otherwise, let y = Lx. Then, ∀i ∈ V ′, yi =

• j∈V

lijxj =

• j∈V ′

lij =

• j∈V

lij −

• j∈V \V ′

lij = 0 ∀i ∈ V \ V ′, yi =

• j∈V

lijxj =

• j∈V ′

lij = 0. Then x = α1n is an eigenvector of L for eigenvalue 0. 0 is not simple.

• A. Girard (LJK - U. Grenoble)

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Eigenvalues of the Laplacian matrix

Proposition

0 is a simple eigenvalue of L if and only if the graph G is connected. Proof: (⇐ =) Assume the graph is connected. Let x such that Lx = 0. Then the SOS property gives:

• (i,j)∈E

(xi − xj)2 = x⊤Lx = 0. Thus, for all (i, j) ∈ E, xi = xj. Let i = 1, since G is connected there exists a path (i1, i2), . . . , (ip, ip+1) joining 1 and i. It follows that x1 = xi1 = xi2 = · · · = xip+1 = xi. The eigenvector x is of the form α1n. 0 is simple.

• A. Girard (LJK - U. Grenoble)

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Eigenvalues of the Laplacian matrix

Proposition

We denote the eigenvalues of L by 0 = λ1 ≤ λ2 ≤ · · · ≤ λn. Let ∆ = maxi∈V di, then, for all k, λk ≤ 2∆. Proof: Let λ be an eigenvalue of L and x an associated eigenvector. Let i ∈ V , such that for all j ∈ V , |xi| ≥ |xj|. Then, λxi =

• j∈V

lijxj =

• j∈V \{i}

lijxj + dixi. It follows that |λ − di| ≤

• j∈V \{i}

|lij||xj| |xi| ≤

• j∈V \{i}

|lij| = di. Then λ ≤ 2di ≤ 2∆.

• A. Girard (LJK - U. Grenoble)

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Algebraic connectivity

Definition

The second smallest eigenvalue λ2 of L is referred to as the algebraic connectivity of the graph G.

Theorem

λ2 > 0 if and only if G is connected. It satisfies: λ2 = min

x⊥1n

x⊤Lx x⊤x . Proof: Let v1, . . . , vn be an orthonormal basis of eigenvectors. Then ∀x⊥1n, x⊤Lx =

n

• k=2

λk(v⊤

k x)2 ≥ λ2 n

• k=2

(v⊤

k x)2 = λ2x⊤x.

Moreover, for x = v2, v⊤

2 Lv2 = λ2.

• A. Girard (LJK - U. Grenoble)

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Algebraic connectivity of a subgraph

Theorem

Let G = (V , E) be a graph and λ2 its algebraic connectivity, let G ′ = (V , E ′) be a spanning subgraph of G and λ′

2 its algebraic

• connectivity. Then, λ2 ≥ λ′

2.

Proof: Since for all x ∈ Rn, x⊤Lx ≥ x⊤L′x, we have min

x⊥1n

x⊤Lx x⊤x ≥ min

x⊥1n

x⊤L′x x⊤x from which follows λ2 ≥ λ′

2.

Remark

Removing (adding) edges to graph can only make the algebraic connectivity decrease (increase).

• A. Girard (LJK - U. Grenoble)

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Normalized Laplacian matrix

Using the fact the L = D−1/2LD−1/2 we can obtain similar results.

Theorem (Sum of squares property)

Let L be the normalized Laplacian matrix of a graph G = (V , E) then, for all x ∈ Rn: x⊤Lx = 1 2

• (i,j)∈E
• xi

√di − xj

• dj

2 .

Proposition

The normalized Laplacian matrix L of an undirected graph G = (V , E) is symmetric positive. 0 is an eigenvalue of L with associated eigenvector D1/21n.

• A. Girard (LJK - U. Grenoble)

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Normalized Laplacian matrix

Proposition

0 is a simple eigenvalue of L if and only if the graph G is connected.

Proposition

We denote the eigenvalues of L by 0 = ˜ λ1 ≤ ˜ λ2 ≤ · · · ≤ ˜ λn. Then, for all k, ˜ λk ≤ 2.

Theorem

˜ λ2 > 0 if and only if G is connected. It satisfies: ˜ λ2 = min

x⊥D1/21n

x⊤Lx x⊤x = min

x⊥D1n

x⊤Lx x⊤Dx .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 24 / 88

Example

Second smallest eigenvalues of L and L for a graph and 2 subgraphs:

1 2 3 4 5 1 2 3 4 5

λ2 = 1.38 ˜ λ2 = 0.67

1 2 3 4 5

λ2 = 1.38 ˜ λ2 = 0.69 λ2 = 0.83 ˜ λ2 = 0.59

• A. Girard (LJK - U. Grenoble)

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Summary

We have defined fundamental notions such as graphs, subgraphs, pathes, connectivity... We have introduced Laplacian and normalized Laplacian matrices:

We have shown the SOS property for both types of matrices. We have shown that the connectivity of a graph can be determined by second smallest eigenvalue of these matrices (λ2 and ˜ λ2).

Some results that are valid for the Laplacian matrix cannot be extended to the normalized Laplacian matrix e.g.: If G ′ is a subgraph of G then λ′

2 ≤ λ2.

Though, in practice, the eigenvalue ˜ λ2 is often a good measure of the connectivity of the graph (less dependent to the size of the graph).

• A. Girard (LJK - U. Grenoble)

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Lecture outline

1 Algebraic graph theory

Basic graph notions Laplacian matrix Normalized Laplacian matrix

2 Consensus Algorithms:

Discrete time and continuous time Agreement in networks with fixed topology Agreement in networks with dynamic topology

3 Applications:

Flocking in mobile networks Opinion dynamics and community detection in social networks

• A. Girard (LJK - U. Grenoble)

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Consensus algorithms

Consider a set of agents V organized in a network with (possibly dynamic) topology described by an undirected graph G(t) = (V , E(t)). Each agent i ∈ V has a state xi(t) ∈ R which is updated according to a simple local rule, e.g. ˙ xi(t) =

• j∈Ni(t)

(xj(t) − xi(t)),

• r in matrix form

˙ x(t) = −L(t)x(t). We say that the agents achieve a consensus if lim

t→+∞ x1(t) = · · · =

lim

t→+∞ xn(t).

• A. Girard (LJK - U. Grenoble)

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Various types of consensus algorithms

We will consider the following consensus algorithms:

continuous time discrete time ˙ xi(t) = X

j∈Ni (t)

(xj(t) − xi(t)) xi(t + 1) = xi(t) + ε X

j∈Ni (t)

(xj(t) − xi(t)) ˙ xi(t) = 1 di(t) X

j∈Ni (t)

(xj(t) − xi(t)) xi(t + 1) = xi(t) + ε di(t) X

j∈Ni (t)

(xj(t) − xi(t))

Equivalent algorithms in matrix form:

continuous time discrete time ˙ x(t) = −L(t)x(t) x(t + 1) = (I − εL(t))x(t) ˙ x(t) = −D−1(t)L(t)x(t) x(t + 1) = (I − εD−1(t)L(t))x(t)

• A. Girard (LJK - U. Grenoble)

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Example

Consider the following network with fixed topology:

1 2 3 4 5

We run continuous time consensus algorithms over this network.

Standard consensus algorithm: ˙ x(t) = −Lx(t)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5

Normalized consensus algorithm: ˙ x(t) = −D−1Lx(t)

1 2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 3.5 4 4.5 5

• A. Girard (LJK - U. Grenoble)

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Consensus in networks with fixed topology

We assume that for all t ∈ R, G(t) = (V , E) and consider the following consensus algorithm: ˙ xi(t) =

• j∈Ni

(xj(t) − xi(t))

• r in matrix form: ˙

x(t) = −Lx(t).

Lemma

The quantity 1⊤

n x(t) = i∈V xi(t) is invariant.

Proof: Let us compute the derivative d dt

• 1⊤

n x(t)

• =

1⊤

n ˙

x(t) = −1⊤

n (Lx(t)) = −(1⊤ n L)x(t) = 0.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 31 / 88

Consensus value

Proposition

Let x∗ = 1

n1⊤ n x(0), if a consensus is achieved, then

lim

t→+∞ x(t) = x∗1n.

Proof: If a consensus is achieved, then for all i ∈ V , lim

t→+∞ xi(t) =

lim

t→+∞

1 n1⊤

n x(t) = 1

n1⊤

n x(0) = x∗.

Remark

The consensus value is the average of the initial values. It is independent

• f the topology of the graph G = (V , E).
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Convergence

Theorem

If the graph G is connected, then the consensus is achieved. Moreover, ∀t ≥ 0, x(t) − x∗1n ≤ e−λ2t x(0) − x∗1n . Proof: Let V (t) = x(t) − x∗1n2 . Then, d dt V (t) = 2 (x(t) − x∗1n)⊤ ˙ x(t) = −2 (x(t) − x∗1n)⊤ Lx(t) = −2 (x(t) − x∗1n)⊤ L (x(t) − x∗1n) . Let us remark that 1⊤

n (x(t) − x∗1n) = 1⊤ n x(t) − x∗1⊤ n 1n = 1⊤ n x(0) − x∗n = 0

which gives (x(t) − x∗1n) ⊥1n.

• A. Girard (LJK - U. Grenoble)

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Convergence

Theorem

If the graph G is connected, then the consensus is achieved. Moreover, ∀t ≥ 0, x(t) − x∗1n ≤ e−λ2t x(0) − x∗1n . Proof: Therefore, (x(t) − x∗1n)⊤ L (x(t) − x∗1n) ≥ λ2 x(t) − x∗1n2 which gives d dt V (t) ≤ −2λ2 x(t) − x∗1n2 = −2λ2V (t). Thus, V (t) ≤ V (0)e−2λ2t. If the graph G is connected then λ2 > 0 and the consensus is achieved.

• A. Girard (LJK - U. Grenoble)

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Consensus in discrete time

We assume that for all t ∈ N, G(t) = (V , E), let ε ∈ (0,

1 2∆), we consider

the consensus algorithm: ∀i ∈ V , xi(t + 1) = xi(t) + ε

• j∈Ni

(xj(t) − xi(t))

• r in matrix form: x(t + 1) = Px(t) where P = I − εL.

Proposition

The quantity 1⊤

n x(t) = i∈V xi(t) is invariant. Let x∗ = 1 n1⊤ n x(0), if a

consensus is achieved, then lim

t→+∞ x(t) = x∗1n.

Proof: Let us remark that 1⊤

n x(t + 1) = 1⊤ n (I − εL)x(t) = 1⊤ n x(t) − ε1⊤ n Lx(t) = 1⊤ n x(t).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 34 / 88

Convergence

Lemma

We have P1n = 1n and for all x⊥1n, Px ≤ (1 − ελ2)x. Proof: First, P1n = (I − εL)1n = 1n − εL1n = 1n. Let x⊥1n and v1, . . . , vn be an orthonormal basis of eigenvectors of L. Then, x =

n

• k=2

(v⊤

k x)vk and Px = n

• k=2

(v⊤

k x)(1 − ελk)vk.

Thus Px2 =

n

• k=2

(v⊤

k x)2(1 − ελk)2 ≤ n

max

k=2 (1 − ελk)2x2.

Moreover, from 0 ≤ λ2 ≤ · · · ≤ λn ≤ 2∆ and ε ∈ (0,

1 2∆), we obtain n

max

k=2 (1 − ελk)2 = (1 − ελ2)2 < 1.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 35 / 88

Convergence

Theorem

If the graph G is connected, then the consensus is achieved. Moreover, ∀t ≥ 0, x(t) − x∗1n ≤ (1 − ελ2)t x(0) − x∗1n . Proof: Let V (t) = x(t) − x∗1n2 . Then, V (t + 1) = Px(t) − x∗1n2 = P(x(t) − x∗1n)2 Let us remark that 1⊤

n (x(t) − x∗1n) = 1⊤ n x(t) − x∗1⊤ n 1n = 1⊤ n x(0) − x∗n = 0

which gives (x(t) − x∗1n) ⊥1n. Then, V (t + 1) ≤ (1 − ελ2)2 x(t) − x∗1n2 = (1 − ελ2)2V (t).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 36 / 88

Other consensus algorithms

Using the fact the D−1L = D−1/2LD1/2 we obtain similar results for the consensus algorithm: ˙ xi(t) = 1 di

• j∈Ni

(xj(t) − xi(t))

• r in matrix form: ˙

x(t) = −D−1Lx(t).

Theorem

If the graph G is connected, then the consensus is achieved. The consensus value is x∗ =

• i∈V dixi(0)
• i∈V di

. Moreover, ∀t ≥ 0, x(t) − x∗1nD ≤ e−˜

λ2t x(0) − x∗1nD .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 37 / 88

Other consensus algorithms

For the discrete time case, let ε ∈ (0, 1

2) and consider the consensus

algorithm: xi(t + 1) = xi(t) + ε di

• j∈Ni

(xj(t) − xi(t))

• r in matrix form: x(t + 1) = (I − εD−1L)x(t).

Theorem

If the graph G is connected, then the consensus is achieved. The consensus value is x∗ =

• i∈V dixi(0)
• i∈V di

. Moreover, ∀t ≥ 0, x(t) − x∗1nD ≤ (1 − ε˜ λ2)t x(0) − x∗1nD .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 38 / 88

Summary

We have considered several consensus algorithms in continuous or discrete time for networks with fixed topology. For the standard consensus algorithm, the consensus value is independent of the network topology (average of initial values) whereas for the normalized consensus algorithm, it depends on the network topology (weighted average where the weights are the degrees of the vertices of the network). Consensus is achieved if the network is connected. The consensus is approached at exponential speed. Moreover, the convergence speed is determined by the second smallest eigenvalue of the Laplacian or normalized Laplacian matrix: the more connected the network the faster the consensus.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 39 / 88

Consensus in networks with dynamic topology

We now assume that the graph is time-varying G(t) = (V , E(t)) and consider the following consensus algorithm: ˙ xi(t) =

• j∈Ni(t)

(xj(t) − xi(t))

• r in matrix form: ˙

x(t) = −L(t)x(t). A lot of results are actually similar to the fixed topology case:

Proposition

The quantity 1⊤

n x(t) = i∈V xi(t) is invariant. Let x∗ = 1 n1⊤ n x(0), if a

consensus is achieved, then lim

t→+∞ x(t) = x∗1n.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 40 / 88

Convergence

Theorem

If the graph G(t) is connected for all t ∈ R, then the consensus is

• achieved. Moreover, for λ2 ≤ mint∈R+ λ2(t),

∀t ≥ 0, x(t) − x∗1n ≤ e−λ2t x(0) − x∗1n . Proof: Let V (t) = x(t) − x∗1n2 . Then, d dt V (t) = 2 (x(t) − x∗1n)⊤ ˙ x(t) = −2 (x(t) − x∗1n)⊤ Lx(t) = −2 (x(t) − x∗1n)⊤ L(t) (x(t) − x∗1n) ≤ −2λ2(t) x(t) − x∗1n2 ≤ −2λ2V (t) Thus, V (t) ≤ V (0)e−2λ2t. If the graph G is connected, we can choose λ2 > 0 and the consensus is achieved.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 41 / 88

Discrete time algorithm

For the discrete time case, let ε ∈ (0,

1 2(n−1)) and consider the consensus

algorithm: xi(t + 1) = xi(t) + ε

• j∈Ni

(xj(t) − xi(t))

• r in matrix form: x(t + 1) = (I − εL)x(t).

Theorem

If the graph G(t) is connected for all t ∈ R, then the consensus is

• achieved. The consensus value is x∗ = 1

n1⊤ n x(0).

Moreover, for λ2 ≤ mint∈R+ λ2(t), ∀t ≥ 0, x(t) − x∗1n ≤ (1 − ελ2)t x(0) − x∗1n .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 42 / 88

Some remarks

For the standard Laplacian consensus algorithm, it is straightforward to extend the results from fixed topology to dynamic topology under the assumption that the graph G(t) remains connected for all t. For the normalized Laplacian consensus algorithm, the results cannot be extended in a straightforward manner. Indeed, even the consensus value is dependent on the graph sequence... Another approach is needed! The assumption that the graph remains connected for all time is actually quite strong. Is it possible to relax this assumption ?

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 43 / 88

A general consensus algorithm

Let us assume that the graph is time-varying G(t) = (V , E(t)) and consider the following discrete time consensus algorithm: xi(t + 1) =

• j∈V

pij(t)xj(t) under the following assumptions (for some α > 0):

Assumption

pii(t) ≥ α, ∀i ∈ V , t ∈ N. pij(t) = 0 if and only if (i, j) ∈ E(t), ∀i, j ∈ V , i = j, t ∈ N. pij(t) ∈ {0} ∪ [α, 1], ∀i, j ∈ V , i = j, t ∈ N.

• j∈V pij(t) = 1, ∀i ∈ V , t ∈ N.

Let us remark that the previous discrete time consensus algorithms satisfy these assumptions.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 44 / 88

Convergence analysis

For a subset of agents V ′ ⊆ V , let mV ′(t) = mini∈V ′ xi(t) and MV ′(t) = maxi∈V ′ xi(t).

Proposition

Let V ′ ⊆ V such that for all i ∈ V , for all j ∈ V \ V ′, (i, j) / ∈ E(t). Then, mV ′(t + 1) ≥ mV ′(t) and MV ′(t + 1) ≤ MV ′(t). Proof: Let i ∈ V ′, then xi(t + 1) =

• j∈V

pij(t)xj(t) =

• j∈V ′

pij(t)xj(t) ≥ m′

V (t)

• j∈V ′

pij(t) = mV ′(t). Then, mV ′(t + 1) ≥ mV ′(t). Similarly, MV ′(t + 1) ≤ MV ′(t).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 45 / 88

Convergence analysis

Remark

The sequences MV (t) and mV (t) are monotonic and bounded, therefore convergent. The consensus is achieved if and only if lim

t→+∞ MV (t) =

lim

t→+∞ mV (t)

• r equivalently

lim

t→+∞ MV (t) − mV (t) = 0.

We will prove this under an assumption of asymptotic connectivity:

Assumption

For all t ∈ N, the graph (V , ∪s≥tE(s)) is connected.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 46 / 88

Convergence analysis

Lemma

For all t ∈ N there exists T ≥ t such that MV (T) − mV (T) ≤ (1 − αn)(MV (t) − mV (t)). Proof: Let us remark that for all s ≥ t, i ∈ V , mV (t) ≤ xi(s) ≤ MV (t). Consider the following property for k ∈ {1, . . . , n}: Pk : ∃tk ≥ t, Vk ⊆ V , such that |Vk| = k and mVk(t) ≥ mV (t) + αk(MV (t) − mV (t)). If Pn is true then necessarily Vn = V and since MV (tn) ≤ MV (t): MV (tn) − mV (tn) ≤ MV (t) − mV (t) − αn(MV (t) − mV (t)) ≤ (1 − αn)(MV (t) − mV (t)).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 47 / 88

Convergence analysis

Proof: Let i1 ∈ V such that xi1(t) = MV (t). Let t1 = t and V1 = {i1}, then mV1(t1) = xi1(t) = MV (t) ≥ mV (t) + α(MV (t) − mV (t)). Thus, P1 is true. Assume Pk is true for some k ∈ {1, . . . , n − 1}. Let Tk ≥ tk be the first time such that ∃(ik+1, jk+1) ∈ E(Tk), for some ik+1 ∈ V \ Vk, jk+1 ∈ Vk. For all tk ≤ s ≤ Tk − 1, i ∈ Vk, j ∈ V \ Vk, (i, j) / ∈ E(s). Then, ∀i ∈ Vk, xi(Tk) ≥ mVk(Tk) ≥ mVk(tk) ≥ mV (t) + αk(MV (t) − mV (t)).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 48 / 88

Convergence analysis

Proof: Then, for all i ∈ Vk, xi(Tk + 1) − mV (t) =  

j∈V

pij(Tk)xj(Tk)   − mV (t) =

• j∈V

pij(Tk) (xj(Tk) − mV (t)) ≥ pii(Tk) (xi(Tk) − mV (t)) ≥ α (xi(Tk) − mV (t)) ≥ α

• mV (t) + αk(MV (t) − mV (t)) − mV (t)
• ≥

αk+1(MV (t) − mV (t)).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 49 / 88

Convergence analysis

Proof: Moreover, xik+1(Tk + 1) − mV (t) =  

j∈V

pik+1j(Tk)xj(Tk)   − mV (t) =

• j∈V

pik+1j(Tk) (xj(Tk) − mV (t)) ≥ pik+1jk+1(Tk)

• xjk+1(Tk) − mV (t)
• ≥

α

• xjk+1(Tk) − mV (t)
• ≥

α

• mV (t) + αk(MV (t) − mV (t)) − mV (t)
• ≥

αk+1(MV (t) − mV (t)). Hence, Pk+1 holds for tk+1 = Tk + 1 and Vk+1 = Vk ∪ {ik+1}. Then, Pn holds.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 50 / 88

Convergence analysis

Theorem

If, for all t ∈ N, the graph (V , ∪s≥tE(s)) is connected, then the consensus is achieved. Proof: From the previous lemma, there exists an increasing sequence Tk ∈ N such that T0 = 0 and ∀k ∈ N, 0 ≤ MV (Tk) − mV (Tk) ≤ (1 − αn)k(MV (0) − mV (0)). Therefore, lim

k→+∞ MV (Tk) − mV (Tk) = 0.

Since in addition MV (t) and mV (t) are convergent it necessarily follows that lim

t→+∞ MV (t) − mV (t) = 0.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 51 / 88

Continuous time consensus

The previous result is not valid for continuous time algorithms. Consider e.g.: ˙ xi(t) =

• j∈Ni(t)

(xj(t) − xi(t)), in a network of two agents with dynamic topology G(t) = (V , E(t)) where V = {1, 2}, E(t) = {(1, 2), (2, 1)}, if t ∈ [k, k + 1/2k) ∅, if t ∈ [k + 1/2k, k + 1) , k ∈ N. Let us remark that for all t ∈ R, the graph (V , ∪s≥tE(s)) is connected.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 52 / 88

Continuous time consensus

It follows that, ˙ x2(t) − ˙ x1(t) = −2(x2(t) − x1(t)), if t ∈ [k, k + 1/2k) 0, if t ∈ [k + 1/2k, k + 1) Then, for all k ∈ N: x2(k + 1) − x1(k + 1) = x2(k + 1/2k) − x1(k + 1/2k) = (x2(k) − x1(k))e−2/2k = (x2(0) − x1(0))e−2e−2/2e−2/22 . . . e−2/2k = (x2(0) − x1(0))e−2(1+1/2+1/22+...+1/2k) = (x2(0) − x1(0))e−4(1−1/2k+1).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 53 / 88

Continuous time consensus

It follows that lim

k→+∞ x2(k + 1) − x1(k + 1) = (x2(0) − x1(0))e−4 = 0.

The consensus is not achieved despite the fact that for all t ∈ R, the graph (V , ∪s≥tE(s)) is connected. The key observation is that even though the edge (1, 2) appears infinitely often, it is present only for a finite time... If one impose a dwell time (when an edge appears it remains present for a duration at least τ > 0), one can avoid this kind of phenomenom.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 54 / 88

A general consensus algorithm

Let us assume that the graph is time-varying G(t) = (V , E(t)) and consider the following continuous time consensus algorithm: ˙ xi(t) =

• j∈V \{i}

pij(t)(xj(t) − xi(t)) under the following assumptions (for some α > 0, β > 0):

Assumption

pij(t) = 0 if and only if (i, j) ∈ E(t), ∀i, j ∈ V , i = j, t ∈ R. pij(t) ∈ {0} ∪ [α, β], ∀i, j ∈ V , i = j, t ∈ R.

• j∈V \{i} pij(t) ≤ β, ∀i ∈ V , t ∈ R.

Let us remark that the previous continuous time consensus algorithms satisfy these assumptions.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 55 / 88

A general consensus algorithm

We add the following dwell time assumption:

Assumption

For all i, j ∈ V , if the edge (i, j) appears at time t, i.e.: (i, j) ∈ E(t) and ∃ε > 0, ∀s ∈ [t − ε, t), (i, j) / ∈ E(s) then (i, j) remains at for a duration τ, i.e.: ∀s ∈ [t, t + τ], (i, j) ∈ E(s). Then, we have the convergence result:

Theorem

If, for all t ∈ R, the graph (V , ∪s≥tE(s)) is connected, then the consensus is achieved.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 56 / 88

Summary

Discrete time consensus algorithms:

Algorithm Topology Consensus if Convergence rate x(t + 1) = (I − εL)x(t) fixed G is connected (1 − ελ2)t x(t + 1) = (I − εD−1L)x(t) fixed G is connected (1 − ε˜ λ2)t x(t + 1) = (I − εL)x(t) dynamic G(t) is connected (1 − ελ2)t with for all t ∈ R λ2 ≤ mint∈N λ2(t) x(t + 1) = (I − εL)x(t) dynamic (V , ∪s≥tE(s)) is

• connected for all t ∈ R

x(t + 1) = (I − εD−1L)x(t) dynamic (V , ∪s≥tE(s)) is

• connected for all t ∈ R

Connectivity properties are crucial for convergence of consensus algorithms. Some convergence rates are determined by the second smallest eigenvalue of the Laplacian or normalized Laplacian matrix: the more connected the network the faster the consensus.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 57 / 88

Summary

Continuous time consensus algorithms:

Algorithm Topology Consensus if Convergence rate ˙ x(t) = −Lx(t) fixed G is connected e−λ2t ˙ x(t) = −D−1Lx(t) fixed G is connected e−˜

λ2t

˙ x(t) = −Lx(t) dynamic G(t) is connected e−λ2t with for all t ∈ R λ2 ≤ mint∈R+ λ2(t) ˙ x(t) = −Lx(t) dynamic Dwell time and

• (V , ∪s≥tE(s)) is

connected for all t ∈ R ˙ x(t) = −D−1Lx(t) dynamic Dwell time and

• (V , ∪s≥tE(s)) is

connected for all t ∈ R

Results are very similar to those for discrete time algorithms. For dynamic topologies, a supplementary dwell time assumption is needed in order to prove that the consensus is achieved.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 58 / 88

Lecture outline

1 Algebraic graph theory

Basic graph notions Laplacian matrix Normalized Laplacian matrix

2 Consensus Algorithms:

Discrete time and continuous time Agreement in networks with fixed topology Agreement in networks with dynamic topology

3 Applications:

Flocking in mobile networks Opinion dynamics and community detection in social networks

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 59 / 88

Flocking in mobile agents network

Flocking is the behavior exhibited when a group of birds are in flight: More precisely, a flocking behavior is characterized by three properties:

1 Alignement: the birds have the same velocity. 2 Cohesion: the birds remain together. 3 Separation: there is a minimum distance between birds.

In the following, we focus on the first and second property.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 60 / 88

A flocking model

We consider a set of mobile agents V = {1, . . . , n}: for each i ∈ V , xi(t) ∈ Rd and vi(t) ∈ Rd denote its position and its velocity. An agent can communicate only with agents that are sufficiently close: the interaction graph G(t) = (V , E(t)) is given by E(t) = {(i, j) ∈ V × V | i = j and xi(t) − xj(t) ≤ R}. The agents try to agree on their velocity using the continuous time consensus algorithm: ˙ vi(t) =

• j∈Ni(t)

(vj(t) − vi(t)).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 61 / 88

A flocking model

The considered model is then: ˙ xi(t) = vi(t) ˙ vi(t) =

• j∈Ni(t)(vj(t) − vi(t))

We want to determine a set of initial conditions ensuring that a flocking behavior is achieved. We have already shown the following result:

Theorem

If the graph G(t) is connected for all t ∈ R, then the consensus is

• achieved. Moreover, for λ2 ≤ mint∈R+ λ2(t),

∀t ≥ 0, v(t) − 1n ⊗ v∗ ≤ e−λ2t v(0) − 1n ⊗ v∗ with v∗ =

i∈V vi(0).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 62 / 88

Graph robustness analysis

We define a notion of robustness for interaction graphs. For i ∈ V , let xi ∈ Rd be the position of agent i, let x = (x1, . . . , xn), we define the associated graph Gx = (V , Ex) with Ex = {(i, j) ∈ V × V | i = j and xi − xj ≤ R}. Let x0 ∈ Rnd be a reference configuration. Assuming that Gx0 is a connected graph, we are interested in characterizing a neighborhood

• f x0 such that for any perturbed configuration y in this

neighborhood the graph Gy is connected. We introduce a measure of robustness for Gx0 which allows us to identify such a neighborhood.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 63 / 88

Graph robustness analysis

Let (i1, i2), . . . , (ip, ip+1) be a path of Gx0, we define the slackening of the path: s((i1, i2), . . . , (ip, ip+1)) =

p

min

k=1(R − x0 ik − x0 ik+1).

If the distances between agents do not change more than s((i1, i2), . . . , (ip, ip+1)) then the path is preserved. The path-robustness ρij between two agents i, j ∈ V with i = j, is the maximal slackening of all paths between i and j: ρij = max

(i1,i2),...,(ip,ip+1)∈Paths(i,j) s((i1, i2), . . . , (ip, ip+1)).

If the distances between agents do not change more than ρij there remains a path between i and j. We set ρii = R.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 64 / 88

Graph robustness analysis

The robustness of the graph ρGx0 is the minimal path-robustness between all pair of nodes: ρGx0 = min

(i,j)∈V 2 ρij

If the distances between agents do not change more than ρGx0 then the graph remains connected. The core robust subgraph of Gx0 is the graph M(Gx0) = (V , M(Ex0)) where M(Ex0) =

• (i, j) ∈ V × V | i = j and xi − xj ≤ R − ρGx0
• .

Since ρGx0 ≥ 0, M(Gx0) is clearly a subgraph of Gx0.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 65 / 88

Graph robustness analysis

Example of a core robust subgraph:

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 66 / 88

Graph robustness analysis

Lemma

Let x0 ∈ Rnd be a reference configuration such that the graph Gx0 is

• connected. Then, the core robust subgraph M(Gx0) is connected.

Proof: Let i, j ∈ V , then ρij ≥ ρGx0. Let (i1, i2), . . . , (ip, ip+1) be a path of Gx0 between i and j with maximal slackening, then s((i1, i2), . . . , (ip, ip+1)) = ρij. Then, for all k ∈ {1, . . . , p}, R − x0

ik − x0 ik+1 ≥ s((i1, i2), . . . , (ip, ip+1)) ≥ ρGx0.

Therefore, for all k ∈ {1, . . . , p}, (ik, ik+1) ∈ M(Ex0). Then, (i1, i2), . . . , (ip, ip+1) is a path of M(Gx0) between i and j. Thus, M(Gx0) is connected.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 67 / 88

Graph robustness analysis

Proposition

Let x0 ∈ Rn×d be a reference configuration such that the associated graph Gx0 is connected. Let y ∈ Rn×d be a perturbed configuration such that y − x0 ≤ ρGx0 √ 2 . Then, M(Gx0) is a subgraph of Gy and Gy is connected. Proof: Let z = (z1, . . . , zn) such that z = y − x0. For all i, j ∈ V , we have −2z⊤

i zj ≤ zi2 + zj2. Then,

zi − zj2 = zi2 + zj2 − 2z⊤

i zj ≤ 2(zi2 + zj2)

≤ 2z2 = 2y − x02 ≤ ρ2

Gx0.

Then, zi − zj ≤ ρGx0.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 68 / 88

Graph robustness analysis

Proposition

Let x0 ∈ Rn×d be a reference configuration such that the associated graph Gx0 is connected. Let y ∈ Rn×d be a perturbed configuration such that y − x0 ≤ ρGx0 √ 2 . Then, M(Gx0) is a subgraph of Gy and Gy is connected. Proof: Let (i, j) ∈ M(Ex0), then x0

i − x0 j ≤ R − ρGx0. Thus,

yi − yj = x0

i − x0 j + zi − zj ≤ x0 i − x0 j + zi − zj

≤ R − ρGx0 + ρGx0 = R. Then, (i, j) ∈ Ey. Therefore M(Gx0) is a subgraph of Gy. Since M(Gx0) is connected, so is Gy.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 68 / 88

Graph robustness computation

Algorithm (Computation of the robustness ρGx0)

// Initialization: ∀(i, j) ∈ V 2, ρ0

ij ← R − x0 i − x0 j ;

// Main loop: for k ∈ V do for i ∈ V do for j ∈ V do ρk

ij ← max

• ρk−1

ij

, min

• ρk−1

ik

, ρk−1

kj

• ;

end for end for end for // Computation of robustness: ρGx0 = min

(i,j)∈V 2 ρn ij;

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 69 / 88

Sufficient conditions for flocking

Theorem

Let x(0) ∈ Rnd be a vector of initial positions of the agents such that the associated graph Gx(0) is connected and its robustness ρGx(0) > 0. Let v(0) ∈ Rnd be a vector of initial velocities such that v(0) − 1n ⊗ v∗ ≤ λ∗

2ρGx(0)

√ 2 where λ∗

2 is the algebraic connectivity of M(Gx(0)).

Then, for all t ∈ R+, M(Gx(0)) is a subgraph of G(t). Moreover, v(t) − 1n ⊗ v∗ ≤ e−λ∗

2t v(0) − 1n ⊗ v∗ .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 70 / 88

Sufficient conditions for flocking

Proof: Let Π be the set of graphs with n nodes which have M(Gx(0)) as a

• subgraph. Since Gx(0) is connected, we have that M(Gx(0)) is connected.

Therefore, all graphs in Π are connected and since M(Gx(0)) ∈ Π, min

G∈Π λ2(G) = λ2(M(Gx(0))) = λ∗ 2 > 0.

Let us assume that there exists t > 0 such that M(Gx(0)) is not a subgraph of G(t) (i.e. G(t) / ∈ Π). Let t∗ = inf{t ∈ R+; G(t) / ∈ Π}. For i ∈ V , let yi(t) = xi(t) − v∗t. Let us remark that for all i, j ∈ V , yi(t) − yj(t) = xi(t) − xj(t). Therefore, for all t ∈ R+, G(t) = Gy(t).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 71 / 88

Sufficient conditions for flocking

Proof: If t∗ > 0, it follows that for all t ∈ [0, t∗) v(t) − 1n ⊗ v∗ ≤ e−λ∗

2t v(0) − 1n ⊗ v∗ ≤ e−λ∗ 2t λ∗

2ρGx(0)

√ 2 . By remarking that y(t) = x0 + t (v(t) − 1n ⊗ v∗) ds we have for all t ∈ [0, t∗) y(t) − x0 ≤ λ∗

2ρGx0

√ 2 t e−λ∗

2sds <

ρGx0 √ 2 . Then, by continuity of y, there exists ε > 0 such that for all t ∈ [0, t∗ + ε], y(t) − x0 ≤ ρGx0 √ 2 .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 72 / 88

Sufficient conditions for flocking

Proof: If t∗ = 0, since y(0) = x0 and by continuity of y, the same kind of property holds. Then, we have for all t ∈ [0, t∗ + ε], G(t) = Gy(t) ∈ Π. This contradicts the definition of t∗. Therefore, for all t ∈ R+, G(t) ∈ Π. Thus for all t ∈ R+, M(Gx(0)) as a subgraph of G(t). This proves the first part of the theorem. Moreover, it follows that for all t ∈ R+, λ2(t) ≥ λ∗

2, and then

v(t) − 1n ⊗ v∗ ≤ e−λ∗

2t v(0) − 1n ⊗ v∗ .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 73 / 88

Example

Example in a network of 6 agents with communication radius R = 3.2:

‚ ‚v(0) − 1n ⊗ v∗‚ ‚ =

λ∗ 2 ρGx(0) √ 2

‚ ‚v(0) − 1n ⊗ v∗‚ ‚ =

5λ∗ 2 ρGx(0) √ 2

‚ ‚v(0) − 1n ⊗ v∗‚ ‚ =

4λ∗ 2 ρGx(0) √ 2

1) 2)

−6 −4 −2 2 4 6

3) 1) 2)

−6 −4 −2 2 4 6

3) 1) 2)

−6 −4 −2 2 4 6

3)

The bound that we compute is often conservative as shown on the example above. However, we can find examples where the bound is tight.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 74 / 88

Summary

We have considered a model of flocking behavior based on communication graphs given by a proximity rule. We have established a set of initial conditions for which the flocking behavior is achieved. The conditions are only sufficient but can be checked algorithmically. The main concepts are those of graph robustness and the associated core robust subgraph that remains for all time ensuring that the communication graph remains connected for all time.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 75 / 88

Lecture outline

1 Algebraic graph theory

Basic graph notions Laplacian matrix Normalized Laplacian matrix

2 Consensus Algorithms:

Discrete time and continuous time Agreement in networks with fixed topology Agreement in networks with dynamic topology

3 Applications:

Flocking in mobile networks Opinion dynamics and community detection in social networks

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 76 / 88

Opinion dynamics in social networks

Opinion dynamics studies the emergence of consensus in social networks: In real social networks, it is often the case that a global consensus cannot be reached. Instead, several consensus are reached locally by subsets of agents forming communities. Can we propose a model of opinion dynamics reproducing this phenomenom ? Can we learn something on real social networks using this model ?

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 77 / 88

A model of opinion dynamics

We consider a set of agents i ∈ V in a network G = (V , E). Each agent i ∈ V has an opinion xi(t) ∈ R. We propose a discrete time model of opinion dynamics as follows: At each time step, agent i receives the opinion of its neighbors in G. Agent i gives confidence only to his neighbor that have an opinion close from its own. Agent i updates its opinion accordingly to its confidence neighborhood. Due to loss of patience, the confidence of each agent decreases at each time step: agent i requires that, at each negotiation round, the

• pinion of agent j moves significantly towards its opinion in order to

keep negotiating with j.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 78 / 88

A model of opinion dynamics

Formally, the opinion dynamics model is described as follows: xi(t + 1) = xi(t) + ε di(t)

• j∈Ni(t)

(xj(t) − xi(t)) where the interaction graph at time t is G(t) = (V , E(t)) with E(t) =

• (i, j) ∈ E|
• |xi(t) − xj(t)| ≤ Rρt

where ε ∈ (0, 1/2), R ≥ 0 and ρ ∈ (0, 1) are model parameters. The parameter ρ characterizes the confidence decay of the agents.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 79 / 88

Convergence analysis

Proposition

For all i ∈ V , the sequence (xi(t))t∈N is convergent. We denote x∗

i its

• limit. Furthermore, we have for all t ∈ N,

|xi(t) − x∗

i | ≤

εR 1 − ρρt. Proof: Let i ∈ V , t ∈ N, we have |xi(t + 1) − xi(t)| =

• ε

di(t)

• j∈Ni(t)

(xj(t) − xi(t))

• ≤

ε di(t)

• j∈Ni(t)

|xj(t) − xi(t)| ≤ ε di(t)

• j∈Ni(t)

Rρt = εRρt.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 80 / 88

Convergence analysis

Proposition

For all i ∈ V , the sequence (xi(t))t∈N is convergent. We denote x∗

i its

• limit. Furthermore, we have for all t ∈ N,

|xi(t) − x∗

i | ≤

εR 1 − ρρt. Proof: Let t ∈ N, τ ∈ N, then |xi(t + τ) − xi(t)| ≤

τ−1

• k=0

|xi(t + k + 1) − xi(t + k)| ≤

τ−1

• k=0

εRρt+k ≤ εR 1 − ρρt which shows, since ρ ∈ (0, 1), that the sequence (xi(t))t∈N is a Cauchy sequence in R. Therefore, it is convergent. The inequality of the proposition is obtained by letting τ go to +∞.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 80 / 88

Communities

Definition

A community is subset of agent C ⊆ V such that for all i, j ∈ C, x∗

i = x∗ j .

The graph of a community C is GC = (C, EC) where EC = {(i, j) ∈ E| i ∈ C, j ∈ C}. The set of communities is C , it is a partition of V . The graph of communities is GC = (V , EC ) where EC = {(i, j) ∈ E| x∗

i = x∗ j }.

Let us remark that EC =

• C∈C

EC.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 81 / 88

Characterization of communities

Before giving an algebraic characterization of communities, we need to make a supplementary assumption. We have seen that the opinions converge at a speed O(ρt). In practice, we observe that the convergence is often slightly faster and this motivates the following assumption:

Assumption (Fast convergence)

There exists ρ < ρ and M ≥ 0 such that for all i ∈ V , for all t ∈ N, |xi(t) − x∗

i | ≤ Mρt.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 82 / 88

Characterization of communities

Proposition

There exists T ∈ N, such that for all t ≥ T, E(t) = EC . Proof: (E(t) ⊆ EC ) E can be splitted into two subets: E f consists of edges that disappear in finite time, E ∞ consists of edges that appear infinitely often. Then, there exists T1 such that for all t ≥ T1, E(t) ⊆ E ∞. Let (i, j) ∈ E ∞ then there exists an unbounded sequence of times τk such that (i, j) ∈ E(τk). This gives |xi(τk) − xj(τk)| ≤ Rρt and x∗

i = x∗ j . Therefore (i, j) ∈ EC .

Hence, for all t ≥ T1, E(t) ⊆ E ∞ ⊆ EC .

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 83 / 88

Characterization of communities

Proposition

There exists T ∈ N, such that for all t ≥ T, E(t) = EC . Proof: (EC ⊆ E(t)) Let (i, j) ∈ EC , then for all t ∈ N |xi(t) − xj(t)| ≤ |xi(t) − x∗

i | + |x∗ i − x∗ j | + |x∗ j − xj(t)|

≤ |xi(t) − x∗

i | + |x∗ j − xj(t)| ≤ 2Mρt.

Since ρ < ρ, there exists T2 such that for all t ≥ T2, 2Mρt ≤ Rρt. This implies that for all t ≥ T2, (i, j) ∈ E(t). Hence, for all t ≥ T2, EC ⊆ E(t).

Remark

We have shown that after a finite time, the graph G(t) is fixed.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 83 / 88

Characterization of communities

Theorem

For almost all vectors of initial opinions, for all communities C ∈ C , such that |C| ≥ 2, ˜ λ2(GC) > (1 − ρ)/ε. Main ideas of the proof: Let C ∈ C and let us assume that ˜ λ2(GC) ≤ (1 − ρ)/ε. Let xC(t) denote the vector of opinions of agents in C. For all t ≥ T, since G(t) = GC , we have xC(t + 1) = (I − εD(GC)−1L(GC))xC(t). We have seen that the rate of convergence of xC(t) is 1 − ε˜ λ2(GC) ≥ ρ except if xC(T) and thus x(T) belongs to a specific subspace of zero measure HC.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 84 / 88

Characterization of communities

Theorem

For almost all vectors of initial opinions, for all communities C ∈ C , such that |C| ≥ 2, ˜ λ2(GC) > (1 − ρ)/ε. Main ideas of the proof: By assumption the rate of convergence of xC(t) is smaller than ρ < ρ. Thus x(T) necessarily belongs to HC. Then, by remarking that all matrices (I − εD(t)−1L(t)) are invertible, we can move backward in time and show that the initial conditions leading to HC at time T are included in a set of zero measure (consisting of a countable union of subspaces) that is independent of C and T.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 84 / 88

Example: karate club network

A social network of 34 agents:

(1 − ρ)/ε = 0.1 (1 − ρ)/ε = 0.2 (1 − ρ)/ε = 0.3 (1 − ρ)/ε = 0.4 min

C∈C

˜ λ2(GC ) = 0.12 min

C∈C

˜ λ2(GC ) = 0.25 min

C∈C

˜ λ2(GC ) = 0.33 min

C∈C

˜ λ2(GC ) = 0.57

The second partition is almost that observed by Zachary in its original study (1973).

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 85 / 88

Example: book network

A network of 105 books on American politics:

Original (1 − ρ)/ε = 0.1 (1 − ρ)/ε = 0.15 (1 − ρ)/ε = 0.2 network min

C∈C

˜ λ2(GC ) = 0.13 min

C∈C

˜ λ2(GC ) = 0.18 min

C∈C

˜ λ2(GC ) = 0.27

One recovers the information on the political alignment (democrat, republican, centrist) of the books.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 86 / 88

Summary

We have introduced a model of opinion dynamics with decaying confidence. In this model, a global consensus may not be achieved and only local agreements can be reached. The agents organize themselves in communities. The analysis of the model provided an algebraic characterization of the communities under a fast convergence assumption. The experimental results tend to confirm the validity of the algebraic

• characterization. Moreover, the communities that are obtained on

real examples are meaningful and allows to uncover information hidden in the network structure.

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 87 / 88

Conclusions

We have presented a collection of representative results on consensus algorithms in continuous and discrete time, and we have shown the relation to graph theory. We have shown two different applications of consensus algorithms:

Flocking in mobile networks Opinion dynamics in social networks

Consensus algorithms are currently pretty well understood... but there are still a lot of things to do on consensus applications. Thank you for your attention!

• A. Girard (LJK - U. Grenoble)

Consensus, Flocking and Opinion Dynamics 88 / 88