Centrality, influence, consensus, polarization in network...

Lecture 1: Centrality, influence, consensus in network models

Centrality, influence, consensus, polarization innetwork models

Fabio Fagnani,DISMA Department of Mathematical Science

Politecnico di Torino

UCSB May 2017

Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017

Centrality, influence, consensus, polarization in network models

The network group at DISMA POLITO

I Giacomo Como

I F. F.

I Rosario Maggistro, Post-doc

I Barbara Franci, PhD

I Lorenzo Zino, PhD

Main collaborators for the results in these lectures

I Daron Acemoglu, MIT

I Jean-Charles Delvenne, UCLouvain

I Paolo Frasca, CNRS Grenoble

I Asuman Ozdaglar, MIT

I Sandro Zampieri, University of Padova

Outline of the minicourse

Lecture 1: Centrality, influence, consensus in network models: theglobal effect of local specifications.

Lecture 2: From opinion dynamics to randomized networkalgorithms.

Lecture 3: Fragility and resilience in centrality and consensusmodels.

Lecture 4: When consensus breaks down: influential nodes,dissensus, polarization.

Lecture 1: Centrality, influence, consensus in networkmodels: the global effect of local specifications.

I Centrality measures;

I De-Groot averaging model. Consensus.

I Large scale networks. Wise societies.

I The probabilistic view point.

I Sensitivity to local perturbations.

I Network engineering.

Centrality measures

Who is the most central node?

1: Pazzi

2: Salviati3: Acciaiuoli

4: Medici5: Barbadori

6: Ginori

7: Albizzi

8: Tornabuoni9: Ridolfi

10: Castellani

11: Guadagni12: Strozzi

13: Lamberteschi14: Bischeri

15: Peruzzi

Marriages among prominent Florentine families

in the 15th century (from Padgett and Ansell)

’Lorenzo de’ Medici’

G. Vasari, 1534

Centrality measures

Degree centrality

di = number of links in node i

Centrality measures

More general networks

In many (social) networks, links may be unilateral.

Example

I Twitter: i → j if user i follows user j ;

I Citation: i → j if author i cites author j ;

I Web: i → j if in page i there is a hyperlink to page j .

Centrality measures

Degree centrality revisited

d−i = number of incoming links to node i

Example: Number of citation received is a measure of theimportance of an author.

Centrality measures

Beyond degree centrality

Drawback of degree-centrality: all incoming links are consideredthe same.

We would like the centrality πi of a node i to depend on thecentrality of the nodes linking to i :

πi ∝∑j→i

Example: Citation received by author j should count in proportion to the

centrality of j .

Does it exist such a vector π?

Centrality measures

Beyond degree centrality

Drawback of degree-centrality: all incoming links are consideredthe same.

We would like the centrality πi of a node i to depend on thecentrality of the nodes linking to i :

πi ∝∑j→i

Example: Citation received by author j should count in proportion to the

centrality of j .

Does it exist such a vector π?

Centrality measures

Some basic notions of graphs

Directed graph: G = (V, E)

I V set of nodes (units),

I E ⊆ V × V set of (directed)links.

I Strongly connected:∀i , j ∈ V ∃ path from i to j ;

I Aperiodic: lengths of cyclesthrough a node are coprime;

I Adjacency matrix:

{1 if (i , j) ∈ E0 if (i , j) 6∈ E ;

I Degrees: 1 vector of all 1’s,A1 = d A′1 = d−

di out-degree of node i ,d−i in-degree of node i ;

Centrality measures

Some basic notions of graphs

Directed graph: G = (V, E)

I V set of nodes (units),

I E ⊆ V × V set of (directed)links.

I Strongly connected:∀i , j ∈ V ∃ path from i to j ;

I Aperiodic: lengths of cyclesthrough a node are coprime;

I Adjacency matrix:

{1 if (i , j) ∈ E0 if (i , j) 6∈ E ;

I Degrees: 1 vector of all 1’s,A1 = d A′1 = d−

di out-degree of node i ,d−i in-degree of node i ;

Centrality measures

Back to centrality

G = (V, E)

πi ∝∑j→i

πj =∑j

πjAji

λπ = A′π

λ must be chosen to be a (non negative) eigenvalue of A.

Centrality measures

Back to centrality

G = (V, E)

πi ∝∑j→i

πj =∑j

πjAji

λπ = A′π

λ must be chosen to be a (non negative) eigenvalue of A.

Centrality measures

Perron-Frobenius theoryW non-negative matrix (Wij ≥ 0 for all i , j ∈ V)

GW = (V, E) with E = {(i , j) |Wij > 0} graph associated with W .

Theorem (Perron-Frobenius)

There exists λW ≥ 0 and non-negative vectors x 6= 0, y 6= 0 s.t.

I Wx = λW x , W ′y = λW y ;

I every eigenvalue µ of W is such that |µ| ≤ λW ;

I If GW is strongly connected, then 1 is simple;

I If GW is strongly connected and aperiodic, then everyeigenvalue µ 6= λW is s.t. |µ| < λW .

λW dominant eigenvalue of W .

Centrality measures

λW dominant eigenvalue of W .Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017

Centrality measures

Eigenvector centrality

G strongly connected.

πeig eigenvector centrality:

λAπeig = A′πeig ,

πeig unique up to normalization (∑

i πeigi = 1)

If all nodes have the same in-degree: d−i = δ for all i

A′1 = δ1

λA = δ, πeig = n−11

Centrality measures

Eigenvector centrality

G strongly connected.

πeig eigenvector centrality:

λAπeig = A′πeig ,

πeig unique up to normalization (∑

i πeigi = 1)

If all nodes have the same in-degree: d−i = δ for all i

A′1 = δ1

λA = δ, πeig = n−11

Centrality measures

Beyond eigenvector centrality

Drawback of eigenvector centrality: nodes contribute to thecentrality of all their out-neighbors irrespective of their out-degree

We would like the centrality πi of a node i to depend on thecentrality of the nodes that link to i scaled by their out-degrees:

πi ∝∑j→i

djπj .

Example: Citation received by authors parsimonious in citing,should count more in evaluating the importance of an author.

Centrality measures

Beyond eigenvector centrality

Drawback of eigenvector centrality: nodes contribute to thecentrality of all their out-neighbors irrespective of their out-degree

We would like the centrality πi of a node i to depend on thecentrality of the nodes that link to i scaled by their out-degrees:

πi ∝∑j→i

djπj .

Example: Citation received by authors parsimonious in citing,should count more in evaluating the importance of an author.

Centrality measures

Beyond eigenvalue centrality

G = (V, E), A adjacency matrix. Put Pij = 1diAij .

πi ∝∑j→i

djπj =

∑j→i

Pjiπj ⇔ λπ = P ′π

I P is a stochastic matrix (non-negative and P1 = 1);

I λP = 1;

I ∃π non negative s.t. P ′π = π;

I If G is strongly connected, π is unique up to normalization.

P simple random walk (SRW) on G.

Centrality measures

Beyond eigenvalue centrality

G = (V, E), A adjacency matrix. Put Pij = 1diAij .

πi ∝∑j→i

djπj =

∑j→i

Pjiπj ⇔ λπ = P ′π

I P is a stochastic matrix (non-negative and P1 = 1);

I λP = 1;

I ∃π non negative s.t. P ′π = π;

I If G is strongly connected, π is unique up to normalization.

P simple random walk (SRW) on G.

Centrality measures

Bonacich centrality

G strongly connected. P SRW on Gπ Bonacich centrality:

π = P ′π,∑i

πi = 1

G = (V, E) undirected ((i , j) ∈ E ⇔ (j , i) ∈ E):

P ′d = d ⇒ πi = di|E|

For undirected strongly connected graphs:

Bonacich centrality = degree centrality.

Centrality measures

Bonacich centrality

π = P ′π,∑i

πi = 1

P ′d = d ⇒ πi = di|E|

Centrality measures

Bonacich centrality

π = P ′π,∑i

πi = 1

P ′d = d ⇒ πi = di|E| (

∑i did

−1i Aij =

∑i Aij = dj)

Centrality measures

Bonacich centrality

π = P ′π,∑i

πi = 1

P ′d = d ⇒ πi = di|E|

Centrality measures

A comparison of the various centralities

8 9 10

Deg Eig Bon

1 0.0345 0.0348 0.03132 0.0517 0.0581 0.04513 0.0517 0.0664 0.06134 0.0517 0.0689 0.06805 0.0517 0.0680 0.08696 0.0517 0.0430 0.0490

7 0.0690 0.0678 0.05148 0.0517 0.0661 0.04449 0.0517 0.0659 0.0491

10 0.0517 0.0627 0.076111 0.0345 0.0226 0.032412 0.0345 0.0215 0.024013 0.0517 0.0399 0.031714 0.0690 0.0640 0.054815 0.0517 0.0613 0.046416 0.0517 0.0484 0.081717 0.0517 0.0225 0.048118 0.0345 0.0215 0.024019 0.0345 0.0307 0.030020 0.0517 0.0492 0.044121 0.0172 0.0166 0.0204

Centrality measures

A comparison of the various centralities

Eigenvalue centrality

8 9 11

Bonacich centrality

8 9 11

Centrality measures

A variation of Bonacich centralityG strongly connected. P SRW on GPage-rank centrality: πpr = (1− α)P ′πpr + αµ

µ > 0 intrinsic centrality,∑

i µi = 1, α ∈ [0, 1]

It is the centrality used by web engines like Google (α = 0.15)

Vantages:

I It does not need the graph to be connected.

πpr = [(1− α)P ′ + αµ1′]πpr , 1′πpr = 1

Q = (1− α)P + α1µ′ stochastic. GQ is complete.Hence, πpr exists and is unique.

I It is more robust: πpri ≥ αµi for all i .

Centrality measures

i µi = 1, α ∈ [0, 1]

Vantages:

πpr = [(1− α)P ′ + αµ1′]πpr , 1′πpr = 1

Centrality measures

i µi = 1, α ∈ [0, 1]

Vantages:

πpr = [(1− α)P ′ + αµ1′]πpr , 1′πpr = 1

Centrality measures

π = P ′π: a long story....

Phillip Bonacich

1987: Power and Centrality: A Family of Measures, AmericanJournal of Sociology.

Morris Herman DeGroot

1974: Reaching a consensus, Journal of the American StatisticalAssociation.

Centrality measures

π = P ′π: a long story....

Phillip Bonacich

1987: Power and Centrality: A Family of Measures, AmericanJournal of Sociology.

Morris Herman DeGroot

1974: Reaching a consensus, Journal of the American StatisticalAssociation.

Centrality measures

Weighted graphs

G = (V, E ,W ), W ∈ RV×V+ s. t. (i , j) ∈ E ⇔ Wij > 0

Weighted degrees: wi =∑

j Wij , w−i =

∑j Wji

Interpretations for Wij :

I Strength of the connection between i and j ;

I How much i trusts j ;

I The number of times author i has cited author j .

Centrality measures

Centrality measures in weighted graphs

j Wij , w−i =

∑j Wji

Pij = 1wiWij : random walk on G.

I degree centrality πdeg = w−

I eigenvector centrality λWπeig = W ′πeig

I Bonacich centrality π = P ′π

Centrality measures

Centrality measures in weighted graphs

j Wij , w−i =

∑j Wji

Pij = 1wiWij : random walk on G.

I degree centrality πdeg = w−

I eigenvector centrality λWπeig = W ′πeig

I Bonacich centrality π = P ′π

De Groot learning model, averaging dynamics

De Groot learning model

G = (V, E ,W ) social network. Wij influence strength of j on i .

Each agent i ∈ V has an opinion on some fact or event.

xi (t) ∈ R opinion of agent i at time t.

Updating rule: xi (t + 1) =∑

j Pijxj(t) Averaging dynamics

Compact notation:x(t + 1) = Px(t)

x(t) = Ptx(0)

G = (V, E ,W ) social network. Wij influence strength of j on i .

Each agent i ∈ V has an opinion on some fact or event.

xi (t) ∈ R opinion of agent i at time t.

Updating rule: xi (t + 1) =∑

j Pijxj(t) Averaging dynamics

Compact notation:x(t + 1) = Px(t)

x(t) = Ptx(0)

Theorem

G = (V, E ,W ) strongly connected, aperiodic.P random walk on G, π = P ′π

limt→+∞

Pt = 1π′

limt→+∞

x(t) = limt→+∞

Ptx(0) = 1π′x(0)

All units have their opinion converging to the common valueπ′x(0): CONSENSUS!

Theorem

G = (V, E ,W ) strongly connected, aperiodic.P random walk on G, π = P ′π

limt→+∞

Pt = 1π′

limt→+∞

x(t) = limt→+∞

Ptx(0) = 1π′x(0)

All units have their opinion converging to the common valueπ′x(0): CONSENSUS!

Wisdom of crowds and wise societiesx(0)i = µ+ Ni : µ true state, Ni indep. E[Ni ] = 0, Var(Ni ) = σ2.

π′x(0) = µ+∑

Var(∑πiNi ) = σ2

∑π2i

G strongly connected, πi > 0 for all i , then,∑π2i <

∑πi = 1.

Var(∑πiNi ) < σ2 Crowd is wiser than a single!

Society: sequence of graphs with increasing size n:

Wise society: limn→+∞

π′x(0) = µ ⇔ limn→+∞

maxi πi = 0

(Golub and Jackson, 2010)

π′x(0) = µ+∑

∑π2i

∑πi = 1.

π′x(0) = µ ⇔ limn→+∞

maxi πi = 0

π′x(0) = µ+∑

∑π2i

∑πi = 1.

π′x(0) = µ

⇔ limn→+∞

maxi πi = 0

π′x(0) = µ+∑

∑π2i

∑πi = 1.

π′x(0) = µ ⇔ limn→+∞

maxi πi = 0

Other applications of averaging dynamics

I Load balancing in computer networks

I Inferential cooperative algorithms in sensor networks

I Clock syncronization

I Relative localization

I Coordination dynamics of robot networks.

Markov chains

π = P ′π: a long story

Andrei Andreevich Markov

1906: Extension of the law of largenumbers to dependent quantities, IzvestiiaFiz.-Matem. Obsch. Kazan Univ

Beginning 20th century: a debate in Russia regarding the interpretationof certain regularity observed in social behaviors.

Quetelet: laws governing social phenomena exactly as in physics.

Nekrasov: theological arguments (free will) against social physics. Law oflarge numbers only holds for independent variables.

Markov invented the chains just to disprove this affirmation!

Markov chains

π = P ′π: a long story

Andrei Andreevich Markov

1906: Extension of the law of largenumbers to dependent quantities, IzvestiiaFiz.-Matem. Obsch. Kazan Univ

Beginning 20th century: a debate in Russia regarding the interpretationof certain regularity observed in social behaviors.

Quetelet: laws governing social phenomena exactly as in physics.

Nekrasov: theological arguments (free will) against social physics. Law oflarge numbers only holds for independent variables.

Markov invented the chains just to disprove this affirmation!

Markov chains

Markov chainsP stochastic matrix, p stochastic vector on V.

(P, p) −→ V (0),V (1), . . . ,V (t), . . . Markov chain (MC) on V

P(V (t + 1) = it+1, |V (1) = i1, . . .V (t) = it) = Pit it+1 ;P(V (0) = i) = pi

p(t)i = P(V (t) = i) = (P ′tp)i marginal distribution of V (t).

π = P ′π equilibrium distribution

Theorem

GP str. connected, aperiodic. Then, π is unique and p(t)→ π forevery p(0) = p.

Markov chains

P(V (t + 1) = it+1, |V (1) = i1, . . .V (t) = it) = Pit it+1 ;

P(V (0) = i) = pi

Theorem

Markov chains

Theorem

Markov chains

Theorem

Markov chains

Theorem

Markov chains

p = π ⇒ p(t) = π for every t.

Theorem

Markov chains

Theorem

Markov chains

Theorem (Ergodic theorem)

GP str. connected. For every f : V → R

limt→+∞

t∑s=0

f (V (s)) =∑i∈V

πi f (i)

A generalization of the law of large numbers:

limt→+∞

number of visits in i0 before time t

t + 1= πi0

Markov chains

Markov chains: terminology

G = (V, E ,W ) graph. Pij = w−1i Wij

V (t) Markov chain associated with (p,P) (for some p).

We call random walk on G both V (t) and P.

Simple random walk if W = A adjacency matrix.

A deeper analysis on the centrality vector π

The many roles of π = P ′π

I it measures centrality in networks;

I it describes the fraction of time spent in the various nodes bya random walk on the graph;

I it determines the consensus point in averaging dynamics;

I it describes the asymptotics of other network dynamics(gossip dynamics, voter model) (details in Lecture 2)

π has many names:

Bonacich centrality, invariant probability, equilibrium probability

π has many names:

Some fundamental problems on π

I π can be analytically computed in very special cases(undirected graphs, G = (V, E ,W ), W = W ′)

I topology of the graph ↔ properties of π (e.g. wise society)?;

I Sensitivity or resilience to local perturbations.

I Centrality optimization by local rewiring (networkengineering).

Further discussion in Lecture 3

Some fundamental problems on π

I π can be analytically computed in very special cases(undirected graphs, G = (V, E ,W ), W = W ′)

I topology of the graph ↔ properties of π (e.g. wise society)?;

I Sensitivity or resilience to local perturbations.

I Centrality optimization by local rewiring (networkengineering).

Further discussion in Lecture 3

A local perturbation of the network

G = (V, E)

πi = di/|E|

G = (V, E)

πi =?

G = (V, E)

πi = di/|E|

G = (V, E)

πi =?

G = (V, E)

8 9 11

π7 = 0.069π8 = 0.069π14 = 0.083

G = (V, E)

8 9 11

π7 = 0.080π8 = 0.077π14 = 0, 076

A useful tool for the analysis of πG = (V, E ,W ) st. connected. Pij = w−1

i Wij , π = P ′π

V (t) random walk on G starting from i .

Mean return time to i : τ+i = Ei [min{t > 0 |V (t) = i}]

Theorem

πi =1

Mean hitting time from i to j : τij = Ei [min{t > 0 |V (t) = j}]A useful formula: τ+

i =∑jPij(1 + τji )

Theorem

πi =1

Theorem

πi =1

Mean hitting time from i to j : τij = Ei [min{t > 0 |V (t) = j}]

A useful formula: τ+i =

∑jPij(1 + τji )

Theorem

πi =1

An intrinsic fragility

G = (V, E ,W ) st.connected

Pij = w−1i Wij , π = P ′π

G = (V, E , W ), W = W + qevv

Theorem

πv = πvπv+α(1−πv )

πi = απiπv+α(1−πv ) for i 6= v

where α =

∑j Wvj∑

j Wvj + q

I πv > πv , πi < πi for i 6= v ;

I πi/πj = πi/πj for all i , j 6= v ;

I q → +∞ ⇒ α→ 0 ⇒ πv → 1

Theorem

where α =

∑j Wvj∑

j Wvj + q

I q → +∞ ⇒ α→ 0 ⇒ πv → 1

Theorem

where α =

∑j Wvj∑

j Wvj + q

I q → +∞ ⇒ α→ 0 ⇒ πv → 1

Proof of Theorem

Pij i 6= vαPij i = v , j 6= vαPij + (1− α) i = v , j = v

α =∑

j Wvj∑j Wvj+q

τ+v =

∑j Pvj(1 + τjv ) =

∑j 6=v αPvj(1 + τjv ) + (αPvv + 1− α)

= ατ+v + (1− α)

⇒ πv = πvπv+α(1−πv )

(απv , πi i 6= v) invariant for P ′

⇒ πi = απiπv+α(1−πv ) , i 6= v

Is probability really necessary for the proof? Linear algebraic proofavailable (but more boring and less insightful).

Proof of Theorem

α =∑

j Wvj∑j Wvj+q

τ+v =

∑j Pvj(1 + τjv ) =

∑j 6=v αPvj(1 + τjv ) + (αPvv + 1− α)

= ατ+v + (1− α)

Proof of Theorem

α =∑

j Wvj∑j Wvj+q

τ+v =

∑j Pvj(1 + τjv ) =

∑j 6=v αPvj(1 + τjv ) + (αPvv + 1− α)

= ατ+v + (1− α)

The effect of adding an edge

G = (V, E , W ), W = W + qevw

The effect of adding an edge

I As expected: πw > πw

I Nothing in general can besaid on the centrality of theother nodes.

I Can v increase its centralityby selecting a new outgoingedge?What would be the bestchoice?

G = (V, E , W ), W = W + qevw

The effect of adding an edgeAs expected: πw > πw : G = (V, E , W ), W = W + qevw

Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)

α =∑

j Wvj∑j Wvj+q

The effect of adding an edgeAs expected: πw > πw :Proof by coupling: V (t), V (t) MC’s

I V (0) = V (0) = w

I Move jointly according to P aslong they do not touch v ;

I In v , move jointly with prob.α, while with prob. 1− α,V (t) moves to w and V (t)moves ind. according to P

I V (t), V (t) are MC w.r to Pand P.

I τ+w > τ+

G = (V, E , W ), W = W + qevw

α =∑

j Wvj∑j Wvj+q

Network engineeringCan v increase its centrality byselecting a new outgoing edge?

What would be the best choice?

G = (V, E , W ), W = W + qevw

α =∑

j Wvj∑j Wvj+q

Network engineeringIs there a choice for v to increaseits centrality by selecting anoutgoing edge?

Theorem

argmaxw∈V

πv = argminw :(w ,v)∈E

G = (V, E , W ), W = W + qevw

α =∑

j Wvj∑j Wvj+q

Network engineeringIs there a choice for v to increaseits centrality by selecting anoutgoing edge?

Theorem

argmaxw∈V

The best choice is linking to anin-neighbor w for which thereturn time τwv is minimal.

G = (V, E , W ), W = W + qevw

α =∑

j Wvj∑j Wvj+q

Network engineering

Theorem

argmaxw∈V

Proof: πv = (τ+v )−1

τ+v =

∑i 6=w αPvi (τiv + 1) +

(αPvw + (1− α))(τwv + 1) =ατ+

v + (1− α)(τwv + 1)

argminw∈V τ+v = argminw∈V τwv

G = (V, E , W ), W = W + qevw

α =∑

j Wvj∑j Wvj+q

The computation of the entrance times

How do we practically solve argminw :(w ,v)∈E

Need to compute the entrance times τwv .

Consider a vector τ·v whose components are the τwv ’s.

Recursive relation:

(I − P)τ·v = 1, τvv = 0

(same complexity than computing π)

Recursive relation:

(I − P)τ·v = 1, τvv = 0

Recursive relation:

(I − P)τ·v = 1, τvv = 0

A very important object: the Green functionG = (V, E ,W ) st. connected, P random walk on G, π = P ′π.

Zij :=+∞∑t=0

[Ptij − πj ]

Theorem

πjτij = Zjj − Zij

Hence,

argminw :(w ,v)∈E

τwv = argmaxw :(w ,v)∈E

The computation of the Green function and suboptimalproblemsG = (V, E ,W ) st. connected, P random walk on G, π = P ′π.

Zij :=+∞∑t=0

[Ptij − πj ], argmax

w :(w ,v)∈EZwv

I Z is not analytically computable in general;I we can approximate Z truncating the series;I suboptimal choices:

argmaxw :(w ,v)∈E

Pwv , argmaxw :(w ,v)∈E

[Pwv + P2wv ], . . . ;

I a crucial point: the speed of convergence to consensusPt → 1π′. In Lecture 2!

Zij :=+∞∑t=0

w :(w ,v)∈EZwv

I Z is not analytically computable in general;

I we can approximate Z truncating the series;I suboptimal choices:

argmaxw :(w ,v)∈E

[Pwv + P2wv ], . . . ;

Zij :=+∞∑t=0

w :(w ,v)∈EZwv

I Z is not analytically computable in general;I we can approximate Z truncating the series;

I suboptimal choices:argmaxw :(w ,v)∈E

[Pwv + P2wv ], . . . ;

Zij :=+∞∑t=0

w :(w ,v)∈EZwv

argmaxw :(w ,v)∈E

[Pwv + P2wv ], . . . ;

Zij :=+∞∑t=0

w :(w ,v)∈EZwv

argmaxw :(w ,v)∈E

[Pwv + P2wv ], . . . ;

Example

w1 vw2

Q(t) = P + P2 + · · ·+ Pt

Q(1)wv Q(2)wv Q(3)wv Q(4)wv

w1 0.333 0.333 0.518 0.586w2 0.333 0.333 0.407 0.455w3 0.333 0.333 0.472 0.510

argmaxw∈V

πv = w1

Example

w1 vw2

Q(t) = P + P2 + · · ·+ Pt

w1 0.333 0.333 0.518 0.586w2 0.333 0.333 0.407 0.455w3 0.333 0.333 0.472 0.510

argmaxw∈V

πv = w1

Example

w1 vw2

Q(t) = P + P2 + · · ·+ Pt

w1 0.333 0.333 0.518 0.586w2 0.333 0.333 0.407 0.455w3 0.333 0.333 0.472 0.510

argmaxw∈V

πv = w1

Summary and next lectures

Summary

I Centralities: degree, eigenvalue, Bonacich.

I De-Groot model

I How Bonacich centrality is affected by network perturbations

I Network engineering: how to shape centrality by localmodifications

I Probabilistic tools

Summary and next lectures

What is next?

Lecture 1: Centrality, influence, consensus in network models: theglobal effect of local specifications.

Lecture 2: From opinion dynamics to randomized networkalgorithms.

Lecture 3: Fragility and resilience in centrality and consensusmodels.

Lecture 4: When consensus breaks down: influential nodes,dissensus, polarization.

Centrality, influence, consensus, polarization in network...

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