My talk to the LC2S2 satellite at NetSci2015

1. Structure of triadic relations in multiplex (social) networks
Emanuele Cozzo
BiFi Institute, Universidad de Zaragoza
June 2, 2015

2. Collaborators
• Mikko Kivela
• Manlio De Domenico
• Albert Sol´e-Ribalta
• Alex Arenas
• Sergio G´omez
• Mason A. Porter
• Yamir Moreno
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3. Why Multiplex Social Networks?
• People live in multiple social worlds
• Individual have partial membership in multiple networks
• Different networks operate in different ways
• Relationships depend on context
◦ Individuals present different faces in different circumstances
◦ They still have a core being that emphasizes different identities in
each milieu1
1
Networked, The New Social Operating System. Lee Rainie and Barry Wellman
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4. Multiplex Networks Representation
The structure of each layer is represented by an adjacency matrix Aα
A: intra-layer adjacency matrix. A = α Aα
Cαβ stores the connections between layers α and β
Supra-node VS node-layer pair
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5. How to, properly, generalize structural metrics to
multiplex networks?
A structural metric for multiplex networks should
• reduce to the ordinary monoplex metric when layers reduce to one
• be defined for node-layer pairs
• have the same value of the monoplex one for a multiplex of
identical layers (if intensive)
• be defined for non-node-aligned multiplex networks
Start from first principles
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6. In a monoplex network:
define
the local clustering coefficient Cu as the number of 3-cycles tu that
start and end at the focal node u divided by the number of 3-cycles
du such that the second step of the cycle occurs in a complete graph
tu = (A3
)uu, du = (AFA)uu (1)
local clustering coefficient
Cu =
tu
du
(2)
global clustering coefficient
C = u tu
u du
(3)
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7. Multiplex networks contain cycles that can traverse different additional
layers but still have 3 intra-layer steps.
Define
a supra-walk as a walk on a multiplex network in which, either before
or after each intra-layer step, a walk can either continue on the same
layer or change to an adjacent layer
C = αI + βC (4)
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8. A supra-step consists either of only a single intra-layer step or of a
step that includes both an intra-layer step changing from one layer to
another (either before or after having an intra-layer step)
tM,i = [(AC)3
+ (CA)3
]ii = 2[(AC)3
]ii (5)
dM,i = 2[ACFCAC]ii (6)
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9. Elementary Cycles
tM,i =
E∈E
wE(E)ii (7)
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10. Local and Global clustering coefficient for Multiplex
Networks
We can calculate a natural multiplex analog to the usual monoplex local
clustering coefficient for any node i of the supra-graph.
A node u allows an intermediate description for clustering between local and
the global clustering coefficients
c∗,i =
t∗,i
d∗,i
, (8)
C∗,u =
i∈l(u) t∗,i
i∈l(u) d∗,i
, (9)
C∗ = i t∗,i
i d∗,i
, (10)
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11. Layer-decomposed clustering coefficients
We can decompose the expression in Eq. 10 in terms of the
contributions from cycles that traverse exactly one, two, and three
layers (i.e., for m = 1, 2, 3) to give
t∗,ı = t∗,1,i α3
+ t∗,2,i αβ2
+ t∗,3,i β3
, (11)
d∗,i = d∗,1,i α3
+ d∗,2,i αβ2
+ d∗,3,i β3
, (12)
C
(m)
∗ = i t∗,m,i
i d∗,m,i
. (13)
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12. Clustering Coefficients in Erd˝os-R´enyi (ER) Multiplex
Networks
0.2
0.4
0.6
0.8
C∗
AC(1)
M
C(2)
M
C(3)
M
p
B C
0.2 0.4 0.6 0.8
x
0.2
0.4
0.6
0.8
c∗
DcAAA
cAACAC
cACAAC
cACACA
cACACAC
p
0.2 0.4 0.6 0.8
x
E
0.2 0.4 0.6 0.8
x
F
(A, B, C) Global and (D, E, F) local multiplex clustering coefficients in multiplex networks that consist of ER layers.
The markers give the results of simulations of 100-node ER node-aligned multiplex networks that we average over 10
realizations. The solid curves are theoretical approximations. Panels (A, C, D, F) show the results for three-layer
networks, and panels (B, E) show the results for six-layer networks. The ER edge probabilities of the layers are (A, D)
{0.1, 0.1, x}, (B, E) {0.1, 0.1, 0.1, 0.1, x, x}, and (C, F) {0.1, x, 1 − x}
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13. Clustering Coefficient in Social Network is Context
Dependent
For each social network we analysed
CM < C
(1)
M and C
(1)
M > C
(2)
M > C
(3)
M
The primary contribution to the triadic structure in multiplex social
networks arises from 3-cycles that stay within a given layer.
Tailor Shop Management Families Bank Tube Airline
CM
orig. 0.319** 0.206** 0.223’ 0.293** 0.056 0.101**
ER 0.186 ± 0.003 0.124 ± 0.001 0.138 ± 0.035 0.195 ± 0.009 0.053 ± 0.011 0.038 ± 0.000
C
(1)
M
orig. 0.406** 0.436** 0.289’ 0.537** 0.013” 0.100**
ER 0.244 ± 0.010 0.196 ± 0.015 0.135 ± 0.066 0.227 ± 0.038 0.053 ± 0.013 0.064 ± 0.001
C
(2)
M
orig. 0.327** 0.273** 0.198 0.349** 0.043* 0.150**
ER 0.191 ± 0.004 0.147 ± 0.002 0.138 ± 0.040 0.203 ± 0.011 0.053 ± 0.020 0.041 ± 0.000
C
(3)
M
orig. 0.288** 0.192** - 0.227** 0.314** 0.086**
ER 0.165 ± 0.004 0.120 ± 0.001 - 0.186 ± 0.010 0.051 ± 0.043 0.037 ± 0.000
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14. Context Matter
Triadic-closure mechanisms in social networks cannot be considered
purely at the aggregated network level.
These mechanisms appear to be more effective inside of layers than
between layers.
0 0.4 0.8 1
cx
0.0
0.2
0.4
0.6
0.8
1.0
cy
c(1)
M,i / c(2)
M,i
c(2)
M,i / c(3)
M,i
c(1)
M,i / c(3)
M,i
0.5 0.0 0.5
cx − cx
0.5
0.0
0.5
cy−cy
A B
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15. Thanks for the attention!
Reference: Structure of Triadic Relations in Multiplex Networks
E. Cozzo, M. Kivela, M. De Domenico, A. Sol´e, A. Arenas, S. G´omez, M. A. Porter, Y. Moreno
http://arxiv.org/abs/1307.6780 (submitted)
Contacts:
@ecozzo
emcozzo@gmail.com
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