Structural Equivalence and Assortative Mixing
Leonid E. Zhukov
School of Data Analysis and Artificial IntelligenceDepartment of Computer Science
National Research University Higher School of Economics
Structural Analysis and Visualization of Networks
Leonid E. Zhukov (HSE) Lecture 7 24.02.2015 1 / 22
Lecture outline
1 Node equivalenceStructural equivalenceRegular equivalence
2 Node similarityJaccard similarityCosine similarityPearson correlation
3 Assortative mixingMixing by valueDegree correlation
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Patterns of relations
Global, statistical properties of the networks:- average node degree (degree distribution)- average clustering- average path length
Local, per vertex properties:- node centrality- page rank
Pairwise properties:- node equivalence- node similarity- correlation between pairs of vertices (node values)
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Structural equivalence
Definition
Structural equivalence: two vertices are structurally equivalent if theirrespective sets of in-neighbors and out-neighbors are the same
u1 u2 v1 v2 w
u1 0 0 1 1 0
u2 0 0 1 1 0
v1 0 0 0 1 1
v2 0 0 1 0 1
w 0 0 0 0 0
rows and columns of adjacency matrix of structurally equivalent nodes areidentical, ”connect to the same neighbors”
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Structural equivalence
In order for adjacent vertices to be structurally equivalent, then mighthave self loops.
Sometimes called ”strong structural equivalence”
Sometimes relax requirements for self loops for adjacent nodes
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Similarity measures
Jaccard similarity
J(vi , vj) =|N (vi ) ∩N (vj)||N (vi ) ∪N (vj)|
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Similarity measures
Undirected graph
Cosine similarity (vectors in n-dim space)
σ(vi , vj) = cos(θij) =vTi vj|vi ||vj |
=
∑k AikAkj√∑A2ik
√∑A2jk
Pearson correlation coefficient:
rij =
∑k(Aik − 〈Ai 〉)(Ajk − 〈Aj〉)√∑
k(Aik − 〈Ai 〉)2√∑
k(Ajk − 〈Aj〉)2
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Similarity measures
Unweighted undirected graph Aik = Aki , binary matrix, only 0 and 1
ki =∑
k Aik =∑
k A2ik - node degree
nij =∑
k AikAkj = (A2)ij - number of shared neighbors
〈Ai 〉 = 1n
∑k Aik
Cosine similarity (vectors in n-dim space)
σ(vi , vj) = cos(θij) =nij√kikj
Pearson correlation coefficient:
rij =nij −
kikjn√
ki −k2in
√kj −
k2j
n
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Similarity matrix
Graph Node similarity matrix
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Regular equivalence
Definition
Regular equivalence: two vertices are regularly equivalent if they areequally related to equivalent others.
Equivalent definition in terms of role assignment (coloring):vertices that are colored the same, have the same colors of theirneighborhoods
White and Reitz,1983; Everette and Borgatti, 1991
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Equivalence example
structural equivalence
regular equivalence
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Vertex similairty
Recursive definition: two vertices are regularly equivalent if they areequally related to equivalent others. Quantitative measure of regularequivalence - σij , similarity score
σij = α∑k,l
AikAjlσkl
σσσ = αAσA
should have high σii - self similarity
σij = α∑k,l
AikAjlσkl + δij
σσσ = αAσA + I
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Vertex similarity
A vertex j is similar to vertex i (dashed line) if i has a networkneighbor v (solid line) that is itself similar to j
σij = α∑v
Aivσvj + δij
σσσ = αAσ + I
Closed form solution:σσσ = (I− αA)−1
Leicht, Holme, and Newman, 2006
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SimRank
s(a, b) - similarity between a and b
I () - set of in-neighbours
s(a, b) =C
|I (a)||I (b)|
I (a)∑i=1
I (b)∑j=1
s(Ii (a), Ij(b)), a 6= b
s(a, a) = 1
Matrix notation:
Sij =C
kikj
∑k,m
AkiAmjSkm
Jeh and Widom, 2002
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Mixing patterns
Network mixing patterns
Assortative mixing, ”like links with like”, attributed of connectednodes tend to be more similar than if there were no such edge
Disassortative mixing, ”like links with dislike”, attributed ofconnected nodes tend to be less similar than if there were no suchedge
Vertices can mix on any vertex attributes (age, sex, geography in socialnetworks), unobserved attributes, vertex degrees
Examples:assortative mixing - in social networks political beliefs, obesity, racedisassortative mixing - dating network, food web (predator/prey),economic networks (producers/consumers)
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Assortative mixing
Political polarization on Twitter: political retweet network ,red color -”right-learning” users, blue color - ”left learning” users
Assortative mixing = homophily
Conover et al., 2011
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Assortative mixing
The Spread of Obesity in a Large Social Network over 32 Years
Node colors - person’s obesity status: yellow denotes an obese person(body-mass index > 30) and green denotes a nonobese person.Edge colors - relationship between them: purple denotes a friendship ormarital tie and orange denotes a familial tie.Christakis and Fowler, 2007
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Mixing by categorical attributes
Vertex categorical attribute (ci -label): color, gender, ethnicity
How much more often do attributes match across edges thanexpected at random?
Modularity :
Q =mc − 〈mc〉
m=
1
2m
∑ij
(Aij −
kikj2m
)δ(ci , cj)
mc - number of edges between vertices with same attributes〈mc〉 - expected number of edges within the same class in randomnetwork
Assortativity coefficient:
Q
Qmax=
∑ij
(Aij −
kikj2m
)δ(ci , cj)
2m −∑
ijkikj2m δ(ci , cj)
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Mixing by scalar values
Vertex scalar value (attribute) - xiHow much more similar are attributes across edges than expected atrandom?Average and covariance over edges
〈x〉 =
∑i kixi∑i ki
=1
2m
∑i
kixi =1
2m
∑ij
Aijxi
var =1
2m
∑ij
Aij(xi − 〈x〉)2 =1
2m
∑i
ki (xi − 〈x〉)2
cov =1
2m
∑ij
Aij(xi − 〈x〉)(xj − 〈x〉)
Assortativity coefficient
r =cov
var=
∑ij
(Aij −
kikj2m
)xixj∑
ij
(kiδij −
kikj2m
)xixj
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Mixing by node degree
Assortative mixing by node degree, xi ← ki
r =
∑ij
(Aij −
kikj2m
)kikj∑
ij
(kiδij −
kikj2m
)kikj
Computations:S1 =
∑i ki = 2m
S2 =∑
i k2i
S3 =∑
i k3i
Se =∑
ij Aijkikj
Assortatitivity coefficient
r =SeS1 − S2
2
S3S1 − S22
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Mixing by node degree
Assortative network: interconnected high degree nodes - core, lowdegree nodes - peripheryDisassortative network: high degree nodes connected to low degreenodes, star-like structure
Assortative network Disassortative network
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References
White, D., Reitz, K.P. Measuring role distance: structural, regularand relational equivalence. Technical report, University of California,Irvine, 1985S. Borgatti, M. Everett. The class of all regular equivalences:algebraic structure and computations. Social Networks, v 11, p65-68,1989E. A. Leicht, P.Holme, and M. E. J. Newman. Vertex similarity innetworks. Phys. Rev. E 73, 026120, 2006G. Jeh and J. Widom. SimRank: A Measure of Structural-ContextSimilarity. Proceedings of the eighth ACM SIGKDD , p 538-543.ACM Press, 2002M. E. J. Newman. Assortative mixing in networks. Phys. Rev. Lett.89, 208701, 2002.M. Newman. Mixing patterns in networks. Phys. Rev. E, Vol. 67, p026126, 2003M. McPherson, L. Smith-Lovin, and J. Cook. Birds of a Feather:Homophily in Social Networks, Annu. Rev. Sociol, 27:415-44, 2001.M. D. Conover, J. Ratkiewicz, et al, Political Polarization on Twitter.Fifth International AAAI Conference on Weblogs and Social Media,2011N. Christakis and J. Fowler. The spread of obesity in a large socialnetwork over 32 years. Engl J Med v 357:370-379, 2007
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