Centralities (2) Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California rgera@nps.edu Excellence Through Knowledge

A periodic table of centralities An interactive periodic table of centralities: http://schochastics.net/sna/periodic.html

Different types of centralities: Betweenness Centrality Closeness Centrality Eigenvector Centrality Degree Centrality Source: Discovering Sets of Key Players in Social Networks – Daniel Ortiz-Arroyo – Springer 2010/

Recall What makes a vertex central in a network? (one or more ideas) How do you describe it mathematically? When is it appropriate to use it? How can we capture it? Lots of one-hop connections from 𝑣 The number of vertices that 𝑣 influences directly Local influence matters Small diameter Degree centrality (or simply the deg⁡(𝑣)) Lots of one-hop connections from 𝑣 relative to the size of the graph The proportion of the vertices that 𝑣 influences directly Degree centrality (Normalized) deg⁡(𝑣) |V(G)| In the “middle” of the graph HOW?

Recall What makes a vertex central in a network? (one or more ideas) How do you describe it mathematically? When is it appropriate to use it? How can we capture it? Lots of one-hop connections from 𝑣 The number of vertices that 𝑣 influences directly Local influence matters Small diameter Degree centrality (or simply the deg⁡(𝑣)) Lots of one-hop connections from 𝑣 relative to the size of the graph The proportion of the vertices that 𝑣 influences directly Degree centrality (Normalized) deg⁡(𝑣) |V(G)| In the “middle” of the graph - closeness centrality Close to everyone at the same time The efficiency of a vertex of reaching everyone quickly (spreading news or a virus for example) 𝐶 𝑖 = 1 𝑗=1 𝑛 𝑑(𝑖,𝑗)

Closeness Centrality

What if it’s not so important to have many direct friends? Closeness What if it’s not so important to have many direct friends? But one still wants to be in the “middle” of the network by being close to many friends. What metric could identify these central nodes? Graph theory: Cen(G) = {v : e(v) is the smallest of all vertices in G} Complex networks: Closeness centrality

Closeness centrality: definition Closeness is based on the average distance between a vertex i and all vertices in the graph (consider vertices in the same component only): 𝐶 𝑖 = 1 𝑗=1 𝑛 𝑑(𝑖,𝑗) depends on inverse distance to other vertices Closeness centrality can be viewed as the efficiency of a vertex in spreading information to all other vertices. Drawback: computed per component b/c of the denominator References: Everett and Borgatti 1999

Closeness Centrality 𝐶 𝑖 = 1 𝑗=1 𝑛 𝑑(𝑖,𝑗) 𝐶 𝑖 = 1 𝑗=1 𝑛 𝑑(𝑖,𝑗) 𝐶 𝐴 = 1 𝑑 𝐴𝐵 +𝑑 𝐴𝐶 +𝑑 𝐴𝐷 +𝑑 𝐴𝐸 +𝑑 𝐴𝐹 +𝑑 𝐴𝐺 𝐶 𝐴 = 1 1+1+1+2+1+2 𝐶 𝐴 = 1 8 =0.125 References: Everett and Borgatti 1999

In class exercise: closeness centrality What is the centrality of a vertex in 𝐾 4 ? What is the centrality of a vertex in 𝐾 14 ? What is the centrality of a vertex in 𝐾 𝑛 ? Should they be the same regardless of the n? Sometimes, we care for a relative centrality, so it should be the same for all n values, since it identifies a certain structure. How would you fix the “problem” so that it scales with n?

Normalized closeness centrality 𝐶 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑖 = 𝑛−1 𝑗=1 𝑛 𝑑 𝑖,𝑗 where n is number of vertices in the graph. In class exercise: What is the normalized closeness centrality of a vertex in 𝐾 𝑛 ?

What happens if the network is disconnected? Closeness centrality In a typical network the closeness centrality might span a factor of five or less It is difficult to distinguish between central and less central vertices a small change in network might considerably affect the centrality ranking b/c of the small values What happens if the network is disconnected? Is closeness centrality well defined? 𝐶 𝑖 = 1 𝑗=1 𝑛 𝑑(𝑖,𝑗)

Harmonic (or closeness) centrality Alternative computations to exist but they have their own problems: The harmonic mean: C i = 𝑗 1 𝑑 𝑖,𝑗 Defined for disconnected graphs since 1 𝑑 𝑖,𝑗  0 if i and j are in different components. But still small range of values for most networks, Also known as Harary Index (Zhoug et al., 2008) Normalized C i = 1 𝑛 𝑗 1 𝑑 𝑖,𝑗 (Crucitti et al., 2004) 𝐶 𝑖 = 1 𝑗=1 𝑛 𝑑(𝑖,𝑗) References: Boldi and Vigna 2014, Rochat 2009

Small values of δ: more weight on close nodes, Decay Centrality A node is rewarded for how close it is to other nodes, yet distant nodes weigh less than closer ones 𝐶 𝑖 = 𝑗≠𝑖 𝛿 𝑑(𝑖,j) where 0 < δ < 1 is a decay parameter, and 𝑑(𝑖,j) is the distance between nodes 𝑖 and j. Small values of δ: more weight on close nodes, High values of δ: emphasize all the nodes of the component that j belongs to. Jackson, M.O. Social and Economic Networks. Princeton, NJ: Princeton University Press (2008) Tsakas, Nikolas. "On decay centrality." The BE Journal of Theoretical Economics (2016)

How would you generalize any of these centrality? Generalization How would you generalize any of these centrality? This is a good time to share your thoughts Closeness centrality of a set rather than a vertex? Everett and Borgatti 1999: 𝐶 𝑆 = 𝑣∈𝑉 𝐺 −𝑆 𝑑(𝑣, 𝑆)

Extensions to sets: Closeness (Everett and Borgatti 1999): 𝐶 𝑆 = 𝑣∈𝑉 𝐺 −𝑆 𝑑(𝑣, 𝑆) Harmonic (Rochat 2009): Decay (Jackson 2010): 𝐶 𝑆 = 𝑢∉𝑆 𝛿 𝑑(𝑢,𝑆) where 0 < δ < 1 is a decay parameter and 𝑑(𝑢,𝑣) is the distance between nodes u and v. k-step reach (Borgatti et al. 2013): 𝐶 𝑆 = 𝑣∈𝑉 𝐺 −𝑆 1 {𝑑 𝑣, 𝑆 ≤𝑘 } - where 1 {} is 1 if inequality is satisfied, otherwise it is 0. Eccentricity-like (Hage and Harary 1995): 𝐶 𝑆 = max 𝑣∈𝑉−𝑆 𝑑(𝑣, 𝑆)

EigenVector Centrality

Recall: What makes a vertex central in a network? (one or more ideas) How do you describe it mathematically? When is it appropriate to use it? How can we capture it? Lots of one-hop connections from 𝑣 The number of vertices that 𝑣 influences directly Local influence matters Small diameter Degree centrality (or simply the C i = deg⁡(𝑖)) Lots of one-hop connections from 𝑣 relative to the size of the graph The proportion of the vertices that 𝑣 influences directly Normalized degree centrality C i = deg⁡(𝑖) |V(G)| Lots of one-hop connections to high centrality vertices A weighted degree centrality based on the weight of the neighbors (instead of a weight of 1 as in degree centrality) For example when the people you are connected to matter.

Recall: HOW? What makes a vertex central in a network? (one or more ideas) How do you describe it mathematically? When is it appropriate to use it? How can we capture it? Lots of one-hop connections from 𝑣 The number of vertices that 𝑣 influences directly Local influence matters Small diameter Degree centrality (or simply the C i = deg⁡(𝑖)) Lots of one-hop connections from 𝑣 relative to the size of the graph The proportion of the vertices that 𝑣 influences directly Normalized degree centrality C i = deg⁡(𝑖) |V(G)| Lots of one-hop connections to high centrality vertices A weighted degree centrality based on the weight of the neighbors (instead of a weight of 1 as in degree centrality) For example when the people you are connected to matter. HOW?

Recall: What makes a vertex central in a network? (one or more ideas) How do you describe it mathematically? When is it appropriate to use it? How can we capture it? Lots of one-hop connections from 𝑣 The number of vertices that 𝑣 influences directly Local influence matters Small diameter Degree centrality (or simply the C i = deg⁡(𝑖)) Lots of one-hop connections from 𝑣 relative to the size of the graph The proportion of the vertices that 𝑣 influences directly Normalized degree centrality C i = deg⁡(𝑖) |V(G)| Lots of one-hop connections to high centrality vertices A weighted degree centrality based on the weight of the neighbors (instead of a weight of 1 as in degree centrality) For example when the people you are connected to matter. Eigenvector centrality (recursive formula): 𝐶 𝑖 = 𝑖𝑗 є 𝐸(𝐺) 𝐶 𝑗

Eigenvector Centrality A generalization of the degree centrality: a weighted degree vector that depends on the centrality of its neighbors (rather than every neighbor having a centrality of 1) How do we find it? By finding the largest eigenvalue and its eigenvectors (i.e. leading eignvec) of the adjacency matrix (i.e. 𝐴𝑥=λ 𝑥) We will get to the “why eigenvectors work” part. Bonacich, Phillip. "Factoring and weighting approaches to status scores and clique identification." Journal of mathematical sociology 2.1 (1972): 113-120.

Example 1 (Eigenvector centrality) Node 𝑖: Eigenvect. centrality 𝐶 𝑖 0: 0, 1: 0.5298987782873977, 2: 0.3577513877490464, 3: 0.5298987782873977, 4: 0.3577513877490464, 5: 0.4271328349194304

Example 1 (Eigenvector centrality) Node 𝑖: Eigenvect. centrality 𝐶 𝑖 0: 0, 1: 0.5298987782873977, 2: 0.3577513877490464, 3: 0.5298987782873977, 4: 0.3577513877490464, 5: 0.4271328349194304 Notice that deg(5) < deg (4). Why 𝐶 5 > 𝐶 4 ?

Example 2 (Eigenvector centrality) Node 𝑖: Eigenvect. centrality 𝐶 𝑖 0: 0.5842167062067959, 1: 0.4569862980311799, 2: 0.18307279191182163, 3: 0.4569862980311799, 4: 0.41711700125458717, 5: 0.18307279191182163 Slightly lower for vertex 4 here, but in a large graph it may make a bigger difference. deg(4) > deg(3)

Example 3 (Adjacency matrix) 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 1. 0. 0. 1. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 1. 1. 0. 0. 0. 0. 0. 0. 0. 1. 0. 1. 1. 0. 0. 0. 0. 1. 0. 0. 1. 0. 0. 0. 0. 0. 1. 0. 1. 0. 1. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 1. 1. 0. 0. 1. 0. 0. 0. 0. 0. 1. 1. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 1. 0. 0. 0. 1. 0. 0. 1. 0. 0. 1. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 1. 0. 0. 0. 0. 0. 0. 0. 1. 1. 1. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 1. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 1. 0. 1. 0. 0. 0. 0. 0. 0.

Example 3 (Eigenvector centrality) 0: 0.08448651593556764, 1: 0.1928608426462633, 2: 0.3011603786470362, 3: 0.17530527234882679, 4: 0.40835121533077895, 5: 0.2865100597893966, 6: 0.2791290343288953, 7: 0.1931920790704947, 8: 0.24881035953707603, 9: 0.13868390351302598, 10: 0.336067959653752, 11: 0.16407815738375933, 12: 0.33838887484747293, 13: 0.2871391639624871, 14: 0.22848023925633135

Example 3 (Eigenvector centrality) 2: 0.3011603786470362, 4: 0.40835121533077895, 5: 0.2865100597893966, 6: 0.2791290343288953, 8: 0.24881035953707603, 10: 0.336067959653752, 12: 0.33838887484747293, 13: 0.2871391639624871, 14: 0.22848023925633135 Also, deg(4) = deg(2) = Δ(G), yet…

Example 3 (Eigenvector centrality) 2: 0.3011603786470362, 4: 0.40835121533077895, 5: 0.2865100597893966, 6: 0.2791290343288953, 8: 0.24881035953707603, 10: 0.336067959653752, 12: 0.33838887484747293, 13: 0.2871391639624871, 14: 0.22848023925633135 Also, deg(4) = deg(2) = Δ(G), yet 𝐶 4 > 𝐶 2 .

Example 3 (Eigenvector centrality) Adjacent to vertices of small degree Adjacent to vertices of large degree

Example 3 (Eigenvector centrality) {0: 0.08448651593556764, 1: 0.1928608426462633, 2: 0.3011603786470362, 3: 0.17530527234882679, 4: 0.40835121533077895, 5: 0.2865100597893966, 6: 0.2791290343288953, 7: 0.1931920790704947, 8: 0.24881035953707603, 9: 0.13868390351302598, 10: 0.336067959653752, 11: 0.16407815738375933, 12: 0.33838887484747293, 13: 0.2871391639624871, 14: 0.22848023925633135} deg(13) < deg(8)

Example 3 (Eigenvector centrality) {0: 0.08448651593556764, 1: 0.1928608426462633, 2: 0.3011603786470362, 3: 0.17530527234882679, 4: 0.40835121533077895, 5: 0.2865100597893966, 6: 0.2791290343288953, 7: 0.1931920790704947, 8: 0.24881035953707603, 9: 0.13868390351302598, 10: 0.336067959653752, 11: 0.16407815738375933, 12: 0.33838887484747293, 13: 0.2871391639624871, 14: 0.22848023925633135}

Eigenvector Centrality Define the centrality 𝑥′ 𝑖 of 𝑖 recursively in terms of the centrality of its neighbors 𝑥 𝑖 ′ = 𝑘∈𝑁(𝑖) 𝑥 𝑘 𝑜𝑟 𝑥 𝑖 ′ = 𝑗 𝐴 𝑖𝑗 𝑥 𝑗 With initial vertex centrality 𝑥 𝑗 =1, ∀𝑗 (𝑖𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 𝑖)—we’ll see why on next slide That is equivalent to: 𝑥′ 𝑖 (𝑡)= 𝑗 𝐴 𝑖𝑗 𝑥 𝑗 (𝑡−1) with the centrality at time 𝑡=0 being 𝑥 𝑗 0 =1, ∀𝑗 The centrality of vertices 𝑖 and 𝑗 at time t and t-1, respectively Bonacich, Phillip. "Factoring and weighting approaches to status scores and clique identification." Journal of mathematical sociology 2.1 (1972): 113-120.

In class: Eigenvector Centrality Adjacency matrix A for the graph to the right: A= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 Then the vector x(t) = 𝑥 1 𝑥 2 : 𝑥 𝑛 gives a random surfer’s behavior. Answer the following questions based on the information above

In class: Eigenvector Centrality Q1: Find x(1). What does it represent? Answ: 𝑥(1)= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 = ?

In class: Eigenvector Centrality Q1: Find x(1). What does it represent? Answ: 𝑥(1)= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1 = 3 3 3 0 3 2 The Degree Centrality vector

In class: Eigenvector Centrality Q2: Find x(2). What does it represent? Answ: 𝑥(2)= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 3 3 3 0 3 2 = ?

In class: Eigenvector Centrality Q2: Find x(2). What does it represent? Answ: 𝑥(2)= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 3 3 3 0 3 2 = 9 9 8 0 8 6 A weighted Degree Centrality vector (distance 2 or less)

In class: Eigenvector Centrality Q3: Find x(3). What does it represent? Answ: 𝑥(3)= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 9 9 8 0 8 6 = ?

In class: Eigenvector Centrality Q3: Find x(3). What does it represent? Answ: 𝑥(3)= 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 9 9 8 0 8 6 = 25 25 24 0 24 16 Another weighted Degree Centrality vector (distance 3 or less)

In class: Eigenvector Centrality 0: 0.49122209552166, 1: 0.49122209552166, 2: 0.4557991200411896, 3: 0, 4: 0.4557991200411896, 5: 0.31921157573304415

Convergence of the entries Note that 𝑥 𝑖 2 𝑥 𝑖 1 ≠ 𝑥 𝑖 3 𝑥 𝑖 2 , where 𝑥 𝑖 is the 𝑖 𝑡ℎ entry. For example: 1 1 1 1 1 1 , 3 3 3 0 3 2 , 9 9 8 0 8 6 , 25 25 24 0 24 16 (in finding the eigenvectors, these vectors get normalized as they are computed using the power method from Lin Alg - see page 248 in Newman’s book) However, the ratios will converge to λ 1 : 3 1 , 9 3 , 25 9 , …→ λ 1

Discussion: What did you notice? What is x(3)? Answ: 𝑥 3 =𝐴(𝐴(𝐴 1 1 1 1 1 1 ))) = 𝐴 3 𝑥 0 𝑥 3 depends on the centrality of its distance 3 or less neighbors

Discussion: Can you generalize it? What is x(t)? Answ: 𝑥 t =𝐴(𝐴…(𝐴 1 1 1 1 1 1 ))) = 𝐴 𝑡 𝑥 0 , 𝑡>0 𝑥 t depends on the centrality of its distance t or less neighbors

Eigenvector Centrality Derivation We can consolidate the eigenvector centralities of all the nodes in a recursive formula with vectors: x 𝑡 =𝐴 ∙ 𝐱(𝑡−1) with the centrality at time t=0 being x 0 =𝟏 (as a vector) Then, we solve: x 𝑡 = 𝐴 𝑡 ∙ 𝐱(0), with x 0 =𝟏 Let 𝒗 𝑘 be the eigenvectors of the adjacency matrix 𝐴 Let 𝜆 1 be the largest eigenvalue. Let x 0 = 𝑘 𝑐 𝑘 𝒗 𝑘 , be a linear combination of 𝒗 𝑘 (eigenvectors are orthogonal since 𝐴 is real and symmetric)

Eigenvector Centrality Derivation Facts from previous page: x 𝑡 = 𝐴 𝑡 ∙ 𝐱(0), with x 0 =1 𝒗 𝒌 are the eigenvectors of the adjacency matrix A x 0 = 𝑘 𝑐 𝑘 𝒗 𝑘 is a linear combination of 𝒗 𝒌 𝜆 1 be the largest eigenvalue. Then x 𝑡 = 𝐴 𝑡 ∙ 𝐱 0 = 𝐴 𝑡 𝑘 𝑐 𝑘 𝑣 𝑘 = 𝑘 𝑐 𝑘 𝜆 𝑘 𝑡 𝑣 𝑘 = = 𝜆 1 𝑡 𝑘 𝑐 𝑘 𝜆 𝑘 𝑡 𝜆 1 𝑡 𝑣 𝑘 = 𝜆 1 𝑡 ( 𝑐 1 𝜆 1 𝑡 𝜆 1 𝑡 𝑣 1 + 𝑐 2 𝜆 2 𝑡 𝜆 1 𝑡 𝑣 2 + 𝑐 3 𝜆 3 𝑡 𝜆 1 𝑡 𝑣 3 …) x 𝑡  𝜆 1 𝑡 𝑐 1 𝑣 1 since 𝜆 𝑘 𝑡 𝜆 1 𝑡 0 as t ∞ (as you repeat the process)

Eigenvector Centrality Thus the eigenvector centrality is 𝒙 𝑡 = 𝜆 1 𝑡 𝑐 1 𝒗 1 where 𝒗 𝟏 is the eigenvector corresponding to the largest eigenvalue 𝜆 1 𝑡 So the eigenvector centrality (as a vector), 𝐱 𝑡 , is a multiple of the eigenvector 𝑣 1 , i.e. 𝐱 𝑡 is an eigenvector of 𝜆 1 𝑡 . A x 𝑡 = 𝜆 1 𝑡 x 𝑡 Meaning that the eigenvector centrality of each node is given by the entries of the leading eigenvector (the one corresponding to the largest eigenvalue λ= 𝜆 1 𝑡 )

Is it well defined? That is: Is the vector guaranteed to exist? Is it unique? Is the eigenvalue unique? Can we have negative entries in the eigenvector? We say that a matrix/vector is positive if all of its entries are positive Perron-Frobenius theorem: A real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components Perron-Frobenius theorem applies to positive matrices (but it gives similar information for nonnegative ones)

Perron-Frobenius theorem for nonnegative symmetric (0,1)-matrices Let A ∈ 𝑅 𝑛 𝑋 𝑛 be symmetric (0,1)-nonnegative, then there is a unique maximal eigenvalue λ 1 of the matrix A (for any other eigenvalue λ, we have λ < λ 1 , with the possibility of |λ| = λ 1 for nonnegative matrices) λ 1 is real, simple (i.e., has multiplicity one), and positive (trace is zero so some are positive and some negative), the associated eigenvector is nonnegative (and there are no other nonnegative ones since all eigenvectors are orthogonal) If you have not seen this and its proof in linear algebra, see a proof on pages 346-347 of Newman’s textbook

In class problem: Let G be an r-regular graph. Find the eigenvector centrality, Find the leading eigenvalue. Soln: use a matrix A and the vector 1 to begin with, and follow a similar argument as the earlier in class problem Take away: even this centrality may not differentiate vertices that it should, such as:

Eigenvector Centrality it is a generalized degree centrality (takes into consideration the global network using the degree of the neighbors) It is extremely useful, one of the most common ones used for non-oriented networks 𝐶 𝑖 = 𝑗 𝐴 𝑖𝑗 𝐶 𝑗 or 𝐶 𝑖 = 𝑖𝑗 є 𝐸(𝐺) 𝐶 𝑗 How would you extend it to directed graphs? Next: Katz centrality for directed graphs

References Jackson, M.O. 2008. Social and Economic Networks. Princeton, NJ: Princeton University Press Wasserman, Stanley, and Katherine Faust. Social network analysis: Methods and applications. Vol. 8. Cambridge university press, 1994. Bonacich, Phillip. "Factoring and weighting approaches to status scores and clique identification." Journal of mathematical sociology 2.1 (1972): 113-120. Bonacich, Phillip, and Paulette Lloyd. "Eigenvector-like measures of centrality for asymmetric relations." Social networks 23.3 (2001): 191-201. Bonacich, Phillip. "Some unique properties of eigenvector centrality." Social networks 29.4 (2007): 555-564.