centrality, influence, consensus, polarization in network...
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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
Lecture 1: Centrality, influence, consensus 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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
1
Marriages among prominent Florentine families
in the 15th century (from Padgett and Ansell)
’Lorenzo de’ Medici’
G. Vasari, 1534
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Who is the most central node?
1
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Degree centrality
di = number of links in node i
1
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
More general networks
In many (social) networks, links may be unilateral.
i j
1
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Who is the most central node?
1
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Degree centrality revisited
d−i = number of incoming links to node i
1
Example: Number of citation received is a measure of theimportance of an author.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
πj .
Example: Citation received by author j should count in proportion to the
centrality of j .
Does it exist such a vector π?
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
πj .
Example: Citation received by author j should count in proportion to the
centrality of j .
Does it exist such a vector π?
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
1
I Strongly connected:∀i , j ∈ V ∃ path from i to j ;
I Aperiodic: lengths of cyclesthrough a node are coprime;
I Adjacency matrix:
Aij =
{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 ;
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
1
I Strongly connected:∀i , j ∈ V ∃ path from i to j ;
I Aperiodic: lengths of cyclesthrough a node are coprime;
I Adjacency matrix:
Aij =
{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 ;
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Back to centrality
G = (V, E)
πi ∝∑j→i
πj =∑j
πjAji
m
λπ = A′π
λ must be chosen to be a (non negative) eigenvalue of A.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Back to centrality
G = (V, E)
πi ∝∑j→i
πj =∑j
πjAji
m
λπ = A′π
λ must be chosen to be a (non negative) eigenvalue of A.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
1
djπj .
Example: Citation received by authors parsimonious in citing,should count more in evaluating the importance of an author.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
1
djπj .
Example: Citation received by authors parsimonious in citing,should count more in evaluating the importance of an author.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Beyond eigenvalue centrality
G = (V, E), A adjacency matrix. Put Pij = 1diAij .
πi ∝∑j→i
1
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Beyond eigenvalue centrality
G = (V, E), A adjacency matrix. Put Pij = 1diAij .
πi ∝∑j→i
1
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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| (
∑i did
−1i Aij =
∑i Aij = dj)
For undirected strongly connected graphs:
Bonacich centrality = degree centrality.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
A comparison of the various centralities
1718
1920
21
1112
1314
15
16
67
8 9 10
1
2
3 45
1
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
A comparison of the various centralities
Eigenvalue centrality
1718
1920
21
1112
1314
15
16
67
8 9 11
1
2
3 45
1
Bonacich centrality
1718
1920
21
1112
1314
15
16
67
8 9 11
1
2
3 45
1
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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 .
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Centrality measures in 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
Pij = 1wiWij : random walk on G.
I degree centrality πdeg = w−
I eigenvector centrality λWπeig = W ′πeig
I Bonacich centrality π = P ′π
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Centrality measures
Centrality measures in 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
Pij = 1wiWij : random walk on G.
I degree centrality πdeg = w−
I eigenvector centrality λWπeig = W ′πeig
I Bonacich centrality π = P ′π
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
De Groot learning model
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
De Groot learning model
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
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+∑
πiNi
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)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
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+∑
πiNi
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)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
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+∑
πiNi
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)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
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+∑
πiNi
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)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
De Groot learning model, averaging dynamics
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
p = π ⇒ p(t) = π for every t.
Theorem
GP str. connected, aperiodic. Then, π is unique and p(t)→ π forevery p(0) = p.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
Markov chains
Markov chains
Theorem (Ergodic theorem)
GP str. connected. For every f : V → R
limt→+∞
1
t + 1
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
A local perturbation of the network
G = (V, E)
1
πi = di/|E|
G = (V, E)
1
πi =?
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
A local perturbation of the network
G = (V, E)
1
πi = di/|E|
G = (V, E)
1
πi =?
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
A local perturbation of the network
G = (V, E)
1718
1920
21
1112
1314
15
16
67
8 9 11
1
2
3 45
1
π7 = 0.069π8 = 0.069π14 = 0.083
G = (V, E)
1718
1920
21
1112
1314
15
16
67
8 9 11
1
2
3 45
1
π7 = 0.080π8 = 0.077π14 = 0, 076
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
τ+i
Mean hitting time from i to j : τij = Ei [min{t > 0 |V (t) = j}]A useful formula: τ+
i =∑jPij(1 + τji )
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
τ+i
Mean hitting time from i to j : τij = Ei [min{t > 0 |V (t) = j}]A useful formula: τ+
i =∑jPij(1 + τji )
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
τ+i
Mean hitting time from i to j : τij = Ei [min{t > 0 |V (t) = j}]
A useful formula: τ+i =
∑jPij(1 + τji )
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
τ+i
Mean hitting time from i to j : τij = Ei [min{t > 0 |V (t) = j}]A useful formula: τ+
i =∑jPij(1 + τji )
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
An intrinsic fragility
G = (V, E ,W ) st.connected
1
Pij = w−1i Wij , π = P ′π
G = (V, E , W ), W = W + qevv
v
q
1
Pij = w−1i Wij , π = P ′π
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
An intrinsic fragility
G = (V, E ,W ) st.connected
1
Pij = w−1i Wij , π = P ′π
G = (V, E , W ), W = W + qevv
v
q
1
Pij = w−1i Wij , π = P ′π
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
An intrinsic fragility
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
An intrinsic fragility
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
An intrinsic fragility
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Proof of Theorem
Pij =
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).
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Proof of Theorem
Pij =
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).
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Proof of Theorem
Pij =
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).
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
The effect of adding an edge
G = (V, E ,W ) st.connected
1
Pij = w−1i Wij , π = P ′π
G = (V, E , W ), W = W + qevw
w
v
1
Pij = w−1i Wij , π = P ′π
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
w
v
1
Pij = w−1i Wij , π = P ′π
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
The effect of adding an edgeAs expected: πw > πw : G = (V, E , W ), W = W + qevw
w
v
1
Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)
α =∑
j Wvj∑j Wvj+q
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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+
w .
G = (V, E , W ), W = W + qevw
w
v
1
Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)
α =∑
j Wvj∑j Wvj+q
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Network engineeringCan v increase its centrality byselecting a new outgoing edge?
What would be the best choice?
G = (V, E , W ), W = W + qevw
w
v
1
Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)
α =∑
j Wvj∑j Wvj+q
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Network engineeringIs there a choice for v to increaseits centrality by selecting anoutgoing edge?
What would be the best choice?
Theorem
argmaxw∈V
πv = argminw :(w ,v)∈E
τwv
G = (V, E , W ), W = W + qevw
w
v
1
Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)
α =∑
j Wvj∑j Wvj+q
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Network engineeringIs there a choice for v to increaseits centrality by selecting anoutgoing edge?
What would be the best choice?
Theorem
argmaxw∈V
πv = argminw :(w ,v)∈E
τwv
The best choice is linking to anin-neighbor w for which thereturn time τwv is minimal.
G = (V, E , W ), W = W + qevw
v
1
Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)
α =∑
j Wvj∑j Wvj+q
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Network engineering
Theorem
argmaxw∈V
πv = argminw :(w ,v)∈E
τwv
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
v
1
Pvi = αPvi if i 6= w ,Pvw = αPvw + (1− α)
α =∑
j Wvj∑j Wvj+q
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
The computation of the entrance times
How do we practically solve argminw :(w ,v)∈E
τwv?
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
↓
τwv
(same complexity than computing π)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
The computation of the entrance times
How do we practically solve argminw :(w ,v)∈E
τwv?
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
↓
τwv
(same complexity than computing π)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
The computation of the entrance times
How do we practically solve argminw :(w ,v)∈E
τwv?
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
↓
τwv
(same complexity than computing π)
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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
Zwv
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
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!
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Example
w3
w1 vw2
1
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Example
w3
w1 vw2
1
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
A deeper analysis on the centrality vector π
Example
w3
w1 vw2
1
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models
Lecture 1: Centrality, influence, consensus in network models
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.
Fabio Fagnani, DISMA Department of Mathematical Science Politecnico di Torino UCSB May 2017
Centrality, influence, consensus, polarization in network models