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Mason 
A. 
Porter 
Mathematical 
Institute, 
University 
of 
Oxford 
(@masonporter, 
masonporter.blogspot.co.uk) 
Mostly, 
we’ll 
be 
“following” 
(i.e. 
skimming 
through) 
our 
new 
review 
article: 
M 
Kivelä, 
A. 
Arenas, 
M. 
Barthelemy, 
J. 
P. 
Gleeson, 
Y. 
Moreno, 
& 
MAP, 
“Multilayer 
Networks”, 
Journal 
of 
Complex 
Networks, 
Vol. 
2, 
No. 
3: 
203–271 
[2014].
5VGRUVQ*CRRKPGUU 
• 1. 
Go 
to 
http://people.maths.ox.ac.uk/porterm/ 
temp/netsci2014/ 
and 
download 
the 
.pdf 
file 
of 
this 
presentation. 
• 2. 
Download 
the 
review 
article 
from 
http://comnet.oxfordjournals.org/content/ 
2/3/203 
and 
(just 
in 
case) 
download 
our 
earlier 
article 
on 
the 
tensorial 
formalism 
– http://people.maths.ox.ac.uk/porterm/papers/ 
PhysRevX.3.041022.pdf 
• 3. 
Use 
these 
materials 
and 
be 
happy.
1WVNKPG 
• Browsing 
through 
the 
mega-­‐review 
article 
– 1. 
Introduction 
– 2. 
Conceptual 
and 
Mathematical 
Framework 
– 3. 
Data 
– 4. 
Models, 
Methods, 
Diagnostics, 
and 
Dynamics 
– 5. 
Conclusions 
and 
Outlook 
• Some 
Advertisements 
– Journals, 
workshops/conferences
ň$WYQTF$KPIQʼn
DWYQTFDKPIQ
“How 
Candide 
was 
multilayer 
networks 
were 
brought 
up 
in 
a 
magnificent 
castle 
and 
how 
he 
was 
they 
were 
driven 
thence” 
+06417%6+10
'ZCORNG/WNVKRNGZ0GVYQTM 
• The concept of “multiplex network” 
has been around for many decades.!
'ZCORNG'FIG%QNQTGF/WNVKITCRJ 
• Monster 
movement 
in 
the 
game 
“Munchkin 
Quest”
'ZCORNG0GVYQTMQH0GVYQTMU 
• The notion (and terminology) “network of 
networks” is also several decades old.! 
(Craven 
and 
Wellman, 
1973)
#0GVYQTMQH0GVYQTMUa 
7-+PHTCUVTWEVWTG 
(Courtesy 
of 
Sco; 
Thacker, 
ITRC, 
University 
of 
Oxford)
'ZCORNG%QIPKVKXG5QEKCN5VTWEVWTG 
(David 
Krackhardt, 
1987)
/WNVKNC[GT0GVYQTM
ň
CEJCT[-CTCVG%NWD%NWDʼn 
0GVYQTM
2KEVWTGUEQWTVGU[QH#CTQP%NCWUGV
3WGUVKQPU!
“What 
befell 
Candide 
multilayer 
networks 
among 
the 
mathematicians” 
%10%'267#.#0 
/#6*'/#6+%#.(4#/'914-5
2.1. 
General 
Form
5QHVYCTGHQT8KUWCNKCVKQPCPF#PCN[UKU 
• See 
h;p://www.plexmath.eu/?page_id=327 
• M. 
De 
Domenico, 
M. 
A. 
Porter, 
 
A. 
Arenas, 
arXiv:1405.0843
2.2. 
Tensorial 
Representaon 
• Adjacency 
tensor 
for 
unweighted 
case: 
• Elements 
of 
adjacency 
tensor: 
– Auvαβ 
= 
Auvα1β1 
… 
αdβd 
= 
1 
iff 
((u,α), 
(v,β)) 
is 
an 
element 
of 
EM 
(else 
Auvαβ 
= 
0) 
• Important 
note: 
‘padding’ 
layers 
with 
empty 
nodes 
– One 
needs 
to 
disnguish 
between 
a 
node 
not 
present 
in 
a 
layer 
and 
nodes 
exisng 
but 
edges 
not 
present 
(use 
a 
supplementary 
tensor 
with 
labels 
for 
edges 
that 
could 
exist), 
as 
this 
is 
important 
for 
normalizaon 
in 
many 
quanes.
#FLCEGPE[6GPUQTYKVJF#URGEV 
• One 
can 
write 
a 
general 
(rank-­‐4) 
mullayer 
adjacency 
tensor 
M 
in 
terms 
of 
a 
tensor 
product 
between 
single-­‐layer 
adjacency 
tensors 
[C(l) 
in 
upper 
right] 
and 
canonical 
basis 
tensors 
[see 
lower 
right] 
• w: 
weights 
• E: 
canonical 
basis 
tensors 
• Weighted 
edge 
from 
node 
ni 
in 
layer 
h 
to 
node 
nj 
in 
layer 
k 
• Note: 
Einstein 
summaon 
convenon 
• Page 
3 
of 
De 
Domenico 
et 
al., 
PRX, 
2013
/WNVKNKPGCT#NIGDTCCPF.CRNCEKCP6GPUQTU 
• Explored 
in 
several 
papers. 
Examples: 
– Supra-­‐Laplacian 
matrices: 
S. 
Gómez, 
A. 
Díaz-­‐Guilera, 
J. 
Gómez-­‐Gardeñes, 
C. 
J. 
Pérez-­‐Vicent, 
Y. 
Moreno, 
 
A. 
Arenas, 
Physical 
Review 
Le8ers, 
Vol. 
110, 
028701 
(2013) 
– Mullayer 
Laplacian 
tensors: 
De 
Domenico 
et 
al, 
Physical 
Review 
X, 
2013 
– Spectral 
properes 
of 
mullayer 
Laplacians: 
A. 
Solé-­‐ 
Ribalta, 
M. 
De 
Domenico, 
N. 
E. 
Kouvaris, 
A. 
Díaz-­‐ 
Guilera, 
S. 
Gómez, 
 
A. 
Arenas, 
Physical 
Review 
E, 
Vol. 
88, 
032807 
(2013) 
– Also 
see 
summary 
in 
the 
review 
arcle.
/WNVKNC[GT%QODKPCVQTKCN.CRNCEKCP 
• Mullayer 
combinatorial 
Laplacian: 
– First 
term: 
strength 
(i.e. 
weighted 
degree) 
tensor 
– A 
bit 
more 
on 
degree 
tensor 
later 
– Second 
term: 
mullayer 
adjacency 
tensor 
(recall) 
– U: 
tensor 
with 
all 
entries 
equal 
to 
1 
– E: 
canonical 
basis 
for 
tensors 
(recall) 
– δ: 
Kronecker 
delta
'ZCORNGň/WNVKUNKEGʼn0GVYQTMU 
• P. 
J. 
Mucha, 
T. 
Richardson, 
K. 
Macon, 
MAP, 
 
J.-­‐P. 
Onnela, 
“Community 
Structure 
in 
Time-­‐Dependent, 
Mulscale, 
and 
Mulplex 
Networks”, 
Science, 
Vol. 
328, 
No. 
5980, 
876–878 
(2010) 
• Simple 
idea: 
Glue 
common 
nodes 
across 
“slices” 
(i.e. 
“layers”)
ň(NCVVGPGFʼn/WNVKUNKEG0GVYQTMU 

UWRTCCFLCEGPE[TGRTGUGPVCVKQP 
• Schematic from M. Bazzi, MAP, 
S. Williams, M. McDonald, D. J. 
Fenn,  S. D. Howison, in 
preparation!
%NCUUKH[KPI/WNVKNC[GT0GVYQTMU 
• Special 
cases 
of 
mullayer 
networks 
include: 
mulplex 
networks, 
interdependent 
networks, 
networks 
of 
networks, 
node-­‐colored 
networks, 
edge-­‐colored 
mulgraphs, 
… 
• To 
obtain 
one 
of 
these 
special 
cases, 
we 
impose 
constraints 
on 
the 
general 
structure 
defined 
earlier. 
• See 
the 
review 
arcle 
for 
details.
%QPUVTCKPVU
HTQOVJG6CDNG 
• 1. 
Node-­‐aligned 
(or 
fully 
interconnected): 
All 
layers 
contain 
all 
nodes. 
• 2. 
Layer 
disjoint: 
Each 
node 
exists 
in 
at 
most 
one 
layer. 
• 3. 
Equal 
size: 
Each 
layer 
has 
the 
same 
number 
of 
nodes 
(but 
they 
need 
not 
be 
the 
same 
ones). 
• 4. 
Diagonal 
coupling: 
Inter-­‐layer 
edges 
only 
can 
exist 
between 
nodes 
and 
their 
counterparts. 
• 5. 
Layer 
coupling: 
coupling 
between 
layers 
is 
independent 
of 
node 
identy 
– Note: 
special 
case 
of 
“diagonal 
coupling” 
• 6. 
Categorical 
coupling: 
diagonal 
couplings 
in 
which 
inter-­‐layer 
edges 
can 
be 
present 
between 
any 
pair 
of 
layers 
– Contrast: 
“ordinal” 
coupling 
for 
tensorial 
representaon 
of 
temporal 
networks 
• Example 
1: 
Most 
–– 
but 
not 
all! 
–– 
“mul@plex 
networks” 
studied 
in 
the 
literature 
sasfy 
(1,3,4,5,6) 
and 
include 
d 
= 
1 
aspects. 
– Note: 
Many 
important 
situaons 
need 
(1,3) 
to 
be 
relaxed. 
(E.g. 
Some 
people 
have 
Facebook 
accounts 
but 
not 
Twi;er 
accounts.) 
• Example 
2: 
The 
“networks 
of 
networks” 
that 
have 
been 
invesgated 
thus 
far 
sasfy 
(3) 
and 
have 
addional 
constraints 
(which 
can 
be 
relaxed).
The literature is messy. #makeitstop
#0QFG%QNQTGF0GVYQTM 
• Node-­‐colored 
network: 
also 
known 
as 
interconnected 
network, 
network 
of 
networks, 
etc. 
• (three 
alternative 
representations)
/WNVKNC[GT0GVYQTM
ň
CEJCT[-CTCVG%NWD%NWDʼn 
0GVYQTM
0QFG%QNQTGF0GVYQTMU
/WNVKRNGZ0GVYQTMU 
• Networks 
with 
multiple 
types 
of 
edges 
– Also 
known 
as 
multirelational 
networks, 
edge-­‐colored 
multigraphs, 
etc. 
• Many 
studies 
in 
practice 
use 
the 
same 
sets 
of 
nodes 
in 
each 
layer, 
but 
this 
isn’t 
required. 
– Challenge 
for 
tensorial 
representation: 
need 
to 
keep 
track 
of 
lack 
of 
presence 
of 
a 
tie 
versus 
a 
node 
not 
being 
present 
in 
a 
layer 
(relevant 
e.g. 
for 
normalization 
of 
multiplex 
clustering 
coefficients) 
• Question: 
When 
should 
you 
include 
inter-­‐ 
layer 
edges 
and 
when 
should 
you 
ignore 
them?
*[RGTITCRJU 
• Hyperedges 
generalize 
edges. 
A 
hyperedge 
can 
include 
any 
(nonzero) 
number 
of 
nodes. 
• Example: 
A 
k-­‐uniform 
hypergraph 
has 
cardinality 
k 
for 
each 
hyperedge 
(e.g. 
a 
folksonomy 
like 
Flickr). 
– One 
can 
represent 
a 
k-­‐uniform 
hypergraph 
using 
adjacency 
tensors, 
and 
there 
have 
been 
some 
studies 
of 
multiplex 
networks 
by 
mapping 
them 
into 
k-­‐uniform 
hypergraphs. 
– A 
nice 
paper: 
Michoel 
 
Nachtergaele, 
PRE, 
2012 
• Note 
that 
multilayer 
networks 
are 
still 
formulated 
for 
pairwise 
connections 
(but 
a 
more 
general 
type 
of 
pairwise 
connections 
than 
usual).
1TFKPCN%QWRNKPIUCPF 
6GORQTCN0GVYQTMU 
• Ordinal 
coupling: 
diagonal 
inter-­‐layer 
edges 
among 
consecutive 
layers 
(e.g. 
multilayer 
representation 
of 
a 
temporal 
network) 
• Categorical 
coupling: 
diagonal 
inter-­‐layer 
edges 
between 
all 
pairs 
of 
edges 
• Both 
can 
be 
present 
in 
a 
multilayer 
network, 
and 
both 
can 
be 
generalized
1VJGT6[RGUQH 
/WNVKNC[GT0GVYQTMU
GZCORNGU 
• k-­‐partite 
graphs 
– Bipartite 
networks 
are 
most 
commonly 
studied 
• Coupled-­‐cell 
networks 
– Associate 
a 
dynamical 
system 
with 
each 
node 
of 
a 
multigraph. 
Network 
structure 
through 
coupling 
terms. 
• Multilevel 
networks 
– Very 
popular 
in 
social 
statistics 
literature 
(upcoming 
special 
issue 
of 
Social 
Networks) 
– Each 
level 
is 
a 
layer 
– Think 
‘hierarchical’ 
situations. 
Example: 
‘micro-­‐ 
level’ 
social 
network 
of 
researchers 
and 
a 
‘macro-­‐ 
level’ 
for 
a 
research-­‐exchange 
network 
between 
laboratories 
to 
which 
the 
researchers 
belong
3WGUVKQPU!
“What 
they 
saw 
in 
the 
Country 
of 
El 
Dorado 
real 
world” 
#6#
5QOGCVC5GVU
5QOG/QTGCVC5GVU
2TCEVKECNKVKGUCPF/GUUCIGU 
• Lots 
of 
reliable 
data 
on 
intra-­‐layer 
relations 
(i.e. 
the 
usual 
kind 
of 
edges) 
• It’s 
much 
more 
challenging 
to 
collect 
reliable 
data 
for 
inter-­‐layer 
edges. 
We 
need 
more 
data. 
– E.g. 
Transportation 
data 
should 
be 
a 
very 
good 
resource. 
Think 
about 
the 
amount 
of 
time 
to 
change 
gates 
during 
a 
layover 
in 
an 
airport. 
– E.g. 
Transition 
probabilities 
of 
a 
person 
using 
different 
social 
media 
(each 
medium 
is 
a 
layer). 
• Most 
empirical 
multilayer-­‐network 
studies 
thus 
far 
have 
tended 
to 
be 
multiplex 
networks. 
• Determining 
inter-­‐layer 
edges 
as 
a 
problem 
in 
trying 
to 
reconcile 
node 
identities 
across 
networks. 
(Can 
you 
figure 
out 
that 
a 
Twitter 
account 
and 
Facebook 
account 
belong 
to 
the 
same 
person?) 
– Major 
implications 
for 
privacy 
issues 
• Take-­‐home 
message: 
Be 
creative 
about 
how 
you 
construct 
multilayer 
networks 
and 
define 
layers!
Multilayer tutorial-netsci2014-slightlyupdated
3WGUVKQPU!
“Candide’s 
Our 
voyage 
to 
Constantinople 
Istanbul 
measuring 
and 
modeling” 
/1'.5/'6*15 
+#)0156+%5#0;0#/+%5
#IITGICVKQPQH 
/WNVKNC[GT0GVYQTMU 
• Construct 
single-­‐layer 
(i.e. 
“monoplex”) 
networks 
and 
apply 
the 
usual 
tools. 
– Obtain 
edge 
weights 
as 
weighted 
average 
of 
connections 
in 
different 
layers. 
You 
get 
a 
different 
weighted 
network 
with 
a 
different 
weighting 
vector. 
• E.g. 
Zachary 
Karate 
Club 
– Information 
loss 
• Is 
there 
a 
way 
to 
do 
this 
to 
minimize 
information 
loss? 
• Important: 
Loss 
of 
“Markovianity” 
(a 
la 
temporal 
networks) 
– Processes 
that 
are 
Markovian 
on 
a 
multilayer 
network 
may 
yield 
non-­‐Markovian 
processes 
after 
aggregating 
the 
network
KCIPQUVKEU 
• Generalizations 
of 
the 
usual 
suspects 
– Degree/strength 
– Neighborhood 
• Which 
layers 
should 
you 
consider? 
– Centralities 
– Walks, 
paths, 
and 
distances 
– Transitivity 
and 
local 
clustering 
• Important 
note: 
Sometimes 
you 
want 
to 
define 
different 
values 
for 
different 
node-­‐layers 
(e.g. 
a 
vector 
of 
centralities 
for 
each 
entity) 
and 
sometimes 
you 
want 
a 
scalar. 
• Need 
to 
be 
able 
to 
consider 
different 
subsets 
of 
the 
layers 
• Need 
more 
genuinely 
multilayer 
diagnostics 
– It 
is 
important 
to 
go 
beyond 
“bigger 
and 
better” 
versions 
of 
the 
usual 
concepts.
GITGGUCPF0GKIJDQTJQQFU 
• Simplest 
way: 
Use 
aggregation 
and 
then 
measure 
degree, 
strength, 
and 
neighborhoods 
on 
a 
monoplex 
network 
obtained 
from 
aggregation. 
– Possibly 
only 
consider 
a 
subset 
of 
the 
layers 
• More 
sophisticated: 
Define 
a 
multi-­‐edge 
as 
a 
vector 
to 
track 
the 
information 
in 
each 
layer. 
With 
weighted 
multilayer 
networks, 
you 
can 
keep 
track 
of 
different 
weights 
in 
intra-­‐layer 
versus 
inter-­‐layer 
edges. 
• Towards 
multilayer 
measures: 
overlap 
multiplicity 
for 
a 
multiplex 
network 
can 
track 
how 
often 
an 
edge 
between 
entities 
i 
and 
j 
occurs 
in 
multiple 
layers
#FLCEGPE[6GPUQTYKVJF#URGEV
TGECNNVJKUUNKFG 
• One 
can 
write 
a 
general 
(rank-­‐4) 
mullayer 
adjacency 
tensor 
M 
in 
terms 
of 
a 
tensor 
product 
between 
single-­‐layer 
adjacency 
tensors 
[C(l) 
in 
upper 
right] 
and 
canonical 
basis 
tensors 
[see 
lower 
right] 
• w: 
weights 
• E: 
canonical 
basis 
tensors 
• Weighted 
edge 
from 
node 
ni 
in 
layer 
h 
to 
node 
nj 
in 
layer 
k 
• Note: 
Einstein 
summaon 
convenon 
• Page 
3 
of 
De 
Domenico 
et 
al., 
PRX, 
2013
'ZCORNG6GPUQTKCN0QVCVKQPHQTGITGG 

GQOGPKEQGVCN24:R
'ZCORNG/WNVKFGITGGEGPVTCNKV[a 

GQOGPKEQGVCN24:R
9CNMU2CVJUCPFKUVCPEGU 
• To 
define 
a 
walk 
(or 
a 
path) 
on 
a 
multilayer 
network, 
we 
need 
to 
consider 
the 
following: 
– Is 
changing 
layers 
considered 
to 
be 
a 
step? 
Is 
there 
a 
“cost” 
to 
changing 
layers? 
How 
do 
you 
measure 
this 
cost? 
• E.g. 
transportation 
networks 
vs 
social 
networks 
– Are 
intra-­‐layer 
steps 
different 
in 
different 
layers? 
• Example: 
labeled 
walks 
(i.e. 
compound 
relations) 
are 
walks 
in 
a 
multiplex 
network 
that 
are 
associated 
with 
a 
sequence 
of 
layer 
labels 
• Generalizing 
walks 
and 
paths 
is 
necessary 
to 
develop 
generalizations 
for 
ideas 
like 
clustering 
coefficients, 
transitivity, 
communicability, 
random 
walks, 
graph 
distance, 
connected 
components, 
betweenness 
centralities, 
motifs, 
etc. 
• Towards 
multilayer 
measures: 
Interdependence 
is 
the 
ratio 
of 
the 
number 
of 
shortest 
paths 
that 
traverse 
more 
than 
one 
layer 
to 
the 
number 
of 
shortest 
paths
%NWUVGTKPI%QGHHKEKGPVU 
CPF6TCPUKVKXKV[ 
• Our 
approach: 
Cozzo 
et 
al., 
2013 
– Use 
the 
idea 
of 
multilayer 
walks. 
Keep 
track 
of 
returning 
to 
entity 
i 
(possibly 
in 
a 
different 
layer 
from 
where 
we 
started) 
separately 
for 
1 
total 
layer, 
2 
total 
layers, 
3 
total 
layers 
(and 
in 
principle 
more). 
• Insight: 
Need 
different 
types 
of 
transitivity 
for 
different 
types 
of 
multiplex 
networks. 
– Example 
(again): 
transportation 
vs 
social 
networks 
– There 
are 
several 
different 
clustering 
coefficients 
for 
monoplex 
weighted 
networks, 
and 
this 
situation 
is 
even 
more 
extreme 
for 
multilayer 
networks.
'ZCORNG%NWUVGTKPI%QGHHKEKGPV 

%QQGVCNCT:KX 
• Our perspective: 
multilayer walks, 
which can return 
to node i on 
different layers 
and traverse 
different numbers 
of layers!
%GPVTCNKV[/GCUWTGU 
• In 
studies 
of 
networks, 
people 
compute 
a 
crapload 
of 
centralities. 
• The 
common 
ones 
have 
been 
generalized 
in 
various 
ways 
for 
multilayer 
networks. 
– Again, 
one 
needs 
to 
ask 
whether 
you 
want 
a 
centrality 
for 
a 
node-­‐layer 
or 
for 
a 
given 
entity 
(across 
all 
layers 
or 
a 
subset 
of 
layers). 
• Eigenvector 
centralities 
and 
related 
ideas 
can 
be 
derived 
from 
random 
walks 
on 
multilayer 
networks. 
– Consider 
different 
spreading 
weights 
for 
different 
types 
of 
edges 
(e.g. 
intra-­‐layer 
vs 
inter-­‐layer 
edges; 
or 
different 
in 
different 
layers) 
• Betweenness 
centralities 
can 
be 
calculated 
for 
different 
generalizations 
of 
short 
paths. 
• A 
point 
of 
caution: 
“What 
the 
world 
needs 
now 
is 
another 
centrality 
measure.” 
– I.e. 
although 
they 
can 
be 
very 
useful, 
please 
don’t 
go 
too 
crazy 
with 
them.
+PVGTNC[GTKCIPQUVKEU 
• The 
community 
needs 
to 
construct 
genuinely 
multilayer 
diagnostics 
and 
go 
beyond 
‘bigger 
and 
better’ 
versions 
of 
the 
concepts 
we 
know 
and 
(presumably) 
love. 
– Not 
very 
many 
yet 
• Correlations 
of 
network 
structures 
between 
layers 
– E.g. 
interlayer 
degree-­‐degree 
correlations 
(or 
any 
other 
diagnostic) 
• ! 
Interpreting 
communities 
as 
layers, 
quantities 
like 
assortativity 
can 
be 
construed 
as 
inter-­‐layer 
diagnostics 
• Interdependence 
is 
the 
ratio 
of 
the 
number 
of 
shortest 
paths 
that 
traverse 
more 
than 
one 
layer 
to 
the 
number 
of 
shortest 
paths
/QFGNUQH/WNVKRNGZ0GVYQTMU 
• Straightforward: 
Use 
your 
favorite 
monoplex 
model 
for 
intra-­‐layer 
connections 
and 
then 
construct 
inter-­‐layer 
edges 
in 
some 
way. 
– E.g. 
random-­‐graph 
models 
like 
Erdös-­‐Rényi, 
network 
growth 
models 
like 
preferential 
attachment 
• Correlated 
layers: 
Include 
correlations 
between 
properties 
in 
different 
intra-­‐layer 
networks 
in 
the 
construction 
of 
random-­‐graph 
ensembles. 
– E.g. 
Include 
intra-­‐layer 
degree-­‐degree 
correlations 
ρ 
in 
[-­‐1,1] 
• Exponential 
Random 
Graph 
Models 
(ERGMs) 
for 
multiplex 
networks 
– Used 
a 
lot 
for 
multilevel 
networks
/QFGNUQH/WNVKRNGZ0GVYQTMU 
• Statistical-­‐mechanical 
ensembles 
of 
multiplex 
networks 
• Generalize 
growth 
mechanisms 
like 
preferential 
attachment 
– Again, 
one 
can 
include 
inter-­‐layer 
correlations 
in 
designing 
a 
model 
• It 
would 
be 
good 
to 
go 
beyond 
“bigger 
and 
better” 
versions 
of 
the 
usual 
ideas. 
– Including 
simple 
inter-­‐layer 
correlations 
(especially 
between 
intra-­‐ 
layer 
degrees) 
has 
been 
the 
main 
approach 
so 
far.
/QFGNUQH 
+PVGTEQPPGEVGF0GVYQTMU 
• Straightforward: 
Construct 
different 
layers 
separately 
using 
your 
favorite 
model 
(or 
even 
one 
that 
you 
hate) 
and 
then 
add 
inter-­‐layer 
edges 
uniformly 
at 
random. 
• More 
sophisticated: 
Be 
more 
strategic 
in 
adding 
inter-­‐layer 
edges. 
• Some 
random-­‐graph 
modules 
with 
community 
structure 
can 
be 
useful, 
where 
we 
think 
of 
each 
community 
as 
a 
separate 
layer 
(i.e. 
as 
a 
separate 
network 
in 
a 
network 
of 
networks) 
– E.g. 
Melnik 
et 
al’s 
paper 
(Chaos, 
2014) 
on 
random 
graphs 
with 
heterogeneous 
degree 
assortativity 
• The 
homophily 
is 
different 
in 
different 
layers 
and 
there 
is 
a 
mixing 
matrix 
for 
inter-­‐layer 
connections
%QOOWPKVKGUCPF1VJGT 
/GUQUECNG5VTWEVWTGU 
• Communities 
are 
dense 
sets 
of 
nodes 
in 
a 
network 
(typically 
relative 
to 
some 
null 
model). 
– One 
can 
use 
these 
ideas 
for 
multilayer 
networks 
(e.g. 
multislice 
modularity). 
• Interpreting 
communities 
as 
roadblocks 
to 
some 
dynamical 
process 
(e.g. 
starting 
from 
some 
initial 
condition), 
one 
can 
have 
such 
a 
process 
on 
a 
multilayer 
network—with 
different 
spreading 
rates 
in 
different 
types 
of 
edges—to 
algorithmically 
find 
communities 
in 
multilayer 
networks. 
• Most 
work 
thus 
far 
on 
multilayer 
representation 
of 
temporal 
networks. 
– One 
exception 
is 
recent 
work 
on 
“Kantian 
fractionalization” 
in 
international 
relations. 
• Challenge: 
Develop 
multilayer 
null 
models 
for 
community 
detection 
(different 
for 
ordinal 
vs. 
categorical 
coupling) 
• Blockmodels 
• Spectral 
clustering 
(e.g. 
Michoel 
 
Nachtergaele) 
• Note: 
Because 
I 
have 
done 
a 
lot 
of 
work 
in 
this 
area, 
I 
will 
go 
through 
a 
bit 
in 
some 
detail 
to 
help 
illustrate 
some 
general 
points 
that 
are 
also 
relevant 
in 
other 
studies 
of 
multilayer 
networks.
! Communities = Cohesive 
groups/modules/ 
mesoscopic structures 
› In stat phys, you try to 
derive macroscopic and 
mesoscopic insights from 
microscopic information 
! Community structure 
consists of complicated 
interactions between 
modular (horizontal) 
and hierarchical 
(vertical) structures 
! communities have denser 
set of Internal edges 
relative to some null 
model for what edges 
are present at random 
› “Modularity”
'ZCORNGň/WNVKUNKEGʼn0GVYQTMU 
• P. 
J. 
Mucha, 
T. 
Richardson, 
K. 
Macon, 
MAP, 
 
J.-­‐P. 
Onnela, 
“Community 
Structure 
in 
Time-­‐Dependent, 
Mulscale, 
and 
Mulplex 
Networks”, 
Science, 
Vol. 
328, 
No. 
5980, 
876–878 
(2010) 
• Simple 
idea: 
Glue 
common 
nodes 
across 
“slices” 
(i.e. 
“layers”) 
• “Diagonal” 
coupling
'ZCORNGKCIPQUVKE/WNVKUNKEG/QFWNCTKV[ 
• Find 
communies 
algorithmically 
by 
opmizing 
“mulslice 
modularity” 
– We 
derived 
this 
funcon 
in 
Mucha 
et 
al, 
2010 
• Laplacian 
dynamics: 
find 
communies 
based 
on 
how 
long 
random 
walkers 
are 
trapped 
there. 
Exponenate 
and 
then 
linearize 
to 
derive 
modularity. 
• Generalizes 
derivaon 
of 
monoplex 
modularity 
from 
R. 
Lambio;e, 
J.-­‐C. 
Delvenne, 
. 
M 
Barahona, 
arXiv:0812.1770 
• Different 
spreading 
weights 
on 
different 
types 
of 
edges 
– Node 
x 
in 
layer 
r 
is 
a 
different 
node-­‐layer 
from 
node 
x 
in 
layer 
s
Example: 
Zachary 
Karate 
Club
4QNN%CNN8QVKPI0GVYQTMUa 

GZCORNGVQKNNWUVTCVGGHHGEVQHRCTCOGVGTʩ 
• A. 
S. 
Waugh, 
L. 
Pei, 
J. 
H. 
Fowler, 
P. 
J. 
Mucha, 
 
M. 
A. 
Porter 
[2012], 
arXiv:0907.3509 
(without 
multilayer 
formulation) 
• Modularity 
Q 
as 
a 
measure 
of 
polarization 
• Can 
calculate 
how 
closely 
each 
legislator 
is 
tied 
to 
their 
community 
(e.g. 
by 
looking 
at 
magnitude 
of 
corresponding 
component 
of 
leading 
eigenvector 
of 
modularity 
matrix 
if 
using 
a 
spectral 
optimization 
method) 
• Medium 
levels 
of 
optimized 
modularity 
as 
a 
predictor 
of 
majority 
turnover 
– By 
contrast, 
leading 
political 
science 
measure 
doesn’t 
give 
statistically 
significant 
indication 
• One 
network 
slice 
for 
each 
two-­‐year 
Congress
P. 
J. 
Mucha 
 
M. 
A. 
Porter, 
Chaos, 
Vol. 
20, 
No. 
4, 
041108 
(2010)
Multilayer tutorial-netsci2014-slightlyupdated
Multilayer tutorial-netsci2014-slightlyupdated
Multilayer tutorial-netsci2014-slightlyupdated
Braiiiiiiiiiiiiins
Multilayer tutorial-netsci2014-slightlyupdated
Construcng 
Time-­‐Dependent 
Networks
[PCOKE4GEQPHKIWTCVKQPQH*WOCP 
$TCKP0GVYQTMUWTKPI.GCTPKPIa 

$CUUGVVGVCN20#5 
• fMRI 
data: 
network 
from 
correlated 
time 
series 
• Examine 
role 
of 
modularity 
in 
human 
learning 
by 
identifying 
dynamic 
changes 
in 
modular 
organization 
over 
multiple 
time 
scales 
• Main 
result: 
flexibility, 
as 
measured 
by 
allegiance 
of 
nodes 
to 
communities, 
in 
one 
session 
predicts 
amount 
of 
learning 
in 
subsequent 
session
Staonarity 
and 
Flexibility 
• Community 
staonarity 
ζ 
(autocorrelaon 
over 
me 
of 
community 
membership): 
• Node 
flexibility: 
– fi 
= 
number 
of 
mes 
node 
i 
changed 
communies 
divided 
by 
total 
number 
of 
possible 
changes 
– Flexibility 
f 
= 
fi
[PCOKE%QOOWPKV[5VTWEVWTG 
• Investigating 
community 
structure 
in 
a 
multilayer 
framework 
requires 
consideration 
of 
new 
null 
models 
• Many 
more 
details! 
– E.g., 
Robustness 
of 
results 
to 
choice 
of 
size 
of 
time 
window, 
size 
of 
inter-­‐slice 
coupling, 
particular 
definition 
of 
flexibility, 
complicated 
modularity 
landscape, 
definition 
of 
‘similarity’ 
of 
time 
series, 
etc.
Dynamic 
Reconfiguraon 
of 
Human 
Brain 
Networks 
During 
Learning 
(Basse; 
et 
al, 
PNAS, 
2011) 
• fMRI 
data: 
network 
from 
correlated 
me 
series 
• Examine 
role 
of 
modularity 
in 
human 
learning 
by 
idenfying 
dynamic 
changes 
in 
modular 
organizaon 
over 
mulple 
me 
scales 
• Main 
result: 
flexibility, 
as 
measured 
by 
allegiance 
of 
nodes 
to 
communies, 
in 
one 
session 
predicts 
amount 
of 
learning 
in 
subsequent 
session
Development 
of 
Null 
Models 
for 
Mullayer 
Networks 
• D. 
S. 
Basse;, 
M. 
A. 
Porter, 
N. 
F. 
Wymbs, 
S. 
T. 
Graƒon, 
J. 
M. 
Carlson, 
 
P. 
J. 
Mucha, 
Chaos, 
23(1): 
013142 
(2013) 
• Addional 
structure 
in 
adjacency 
tensors 
gives 
more 
freedom 
(and 
responsibility) 
for 
choosing 
null 
models. 
• Null 
models 
that 
incorporate 
informaon 
about 
a 
system 
• E.g. 
chain 
null 
model 
fixes 
network 
topology 
but 
randomizes 
network 
“geometry” 
(edge 
weights) 
• Also: 
Examine 
null 
models 
from 
shuffling 
me 
series 
directly 
(before 
turning 
into 
a 
network) 
• Structural 
(γ) 
versus 
temporal 
resoluon 
parameter 
(ω) 
• More 
generally, 
how 
to 
choose 
inter-­‐layer 
(off-­‐ 
diagonal) 
terms 
Cjrs 
• Time 
series 
from 
experiments 
as 
well 
as 
output 
of 
a 
dynamical 
system 
(e.g. 
Kuramoto 
model). 
Analogous 
to 
structural 
vs 
funconal 
brain 
networks.
/GVJQFU$CUGFQP 
6GPUQTGEQORQUKVKQP 
• Many 
different 
generalizations 
of 
singular 
value 
decomposition 
(SVD) 
to 
tensors 
– Every 
matrix 
has 
a 
unique 
SVD, 
but 
we 
have 
to 
relax 
this 
for 
tensors. 
– See 
Kolda 
and 
Bader, 
SIAM 
Review, 
2009 
– Tensor 
rank 
vs 
matrix 
rank: 
hard 
to 
determine 
that 
rank 
of 
tensors 
of 
order 
3+ 
• Note: 
“rank” 
is 
also 
used 
as 
a 
synonym 
for 
“order” 
(see 
earlier). 
Here, 
“rank” 
is 
the 
generalization 
of 
matrix 
rank: 
the 
minimum 
number 
of 
column 
vectors 
needed 
to 
span 
the 
range 
of 
a 
matrix. 
The 
tensor 
rank 
is 
the 
minimum 
number 
of 
rank-­‐1 
tensors 
with 
which 
one 
can 
express 
a 
tensor 
as 
a 
sum. 
The 
purpose 
of 
an 
SVD 
(and 
generalizations) 
is 
to 
find 
a 
low-­‐rank 
approximation. 
• Non-­‐negative 
tensor 
factorization
[PCOKECN5[UVGOUQP 
/WNVKNC[GT0GVYQTMU 
• Basic 
question: 
How 
do 
multilayer 
structures 
affect 
dynamical 
systems 
on 
networks? 
– Effects 
of 
multiplexity? 
(edge 
colorings) 
– Effects 
of 
interconnectedness? 
(node 
colorings) 
• Important 
goal: 
Find 
new 
phenomena 
that 
cannot 
occur 
without 
multilayer 
structures. 
– Example: 
Speeding 
up 
vs 
slowing 
down 
spreading? 
– Example: 
Multiplexity-­‐induced 
correlations 
in 
dynamics? 
– Example: 
Effect 
of 
different 
costs 
for 
changing 
layers?
%QPPGEVGF%QORQPGPVU 
CPF2GTEQNCVKQP 
• Connected 
component 
defined 
as 
in 
monoplex 
networks, 
except 
that 
multiple 
types 
of 
edges 
can 
occur 
in 
a 
path. 
• In 
multilayer 
networks, 
one 
again 
uses 
branching-­‐process 
approximations 
that 
allow 
the 
use 
of 
generating 
function 
technology. 
– Same 
fundamental 
idea 
(and 
limitations) 
as 
in 
monoplex 
networks, 
but 
the 
calculations 
are 
more 
intricate 
• More 
flavors 
of 
giant 
connected 
components 
(GCCs) 
that 
can 
be 
defined
2GTEQNCVKQP%CUECFGU 
• Example 
(from 
Buldyrev 
et 
al, 
Nature, 
2010)
2GTEQNCVKQP%CUECFGU
2GTEQNCVKQP%CUECFGU 
• Numerous 
papers 
for 
both 
multiplex 
networks 
and 
interconnected 
networks 
• A 
few 
interesting 
ideas 
– Localized 
attack 
• More 
generally, 
multilayer 
networks 
allow 
more 
creativity 
in 
targeted 
attacks. 
Why 
in 
Hell 
is 
it 
almost 
always 
by 
degree 
(even 
for 
monoplex 
networks)? 
Be 
creative! 
– Viable 
cluster: 
mutually 
connected 
giant 
component
%QORCTVOGPVCN5RTGCFKPI 
/QFGNUCPFKHHWUKQP 
• Random 
walks 
and 
Laplacians 
– Different 
spreading 
rates 
on 
different 
types 
of 
edges 
• See 
earlier 
discussions 
of 
multislice 
community 
structure 
• Strong 
vs 
weak 
inter-­‐layer 
coupling 
– Examine 
generic 
properties 
of 
phase 
transitions 
(e.g. 
as 
a 
function 
of 
weights 
of 
inter-­‐layer 
edges) 
• Competing 
(toy 
models 
of) 
biological 
contagions 
– Your 
favorite 
toy 
models 
(SI, 
SIS, 
SIR, 
SIRS, 
etc.) 
• Layers 
with 
biological 
contagions 
interacting 
with 
layers 
of 
information 
diffusion 
(e.g. 
of 
awareness)
%QORCTVOGPVCN5RTGCFKPI 
/QFGNUCPFKHHWUKQP
%QORCTVOGPVCN5RTGCFKPI 
/QFGNUCPFKHHWUKQP
%QORCTVOGPVCN5RTGCFKPI 
/QFGNUCPFKHHWUKQP 
• Metapopulation 
models 
as 
biological 
epidemics 
on 
networks 
of 
networks 
– E.g. 
Melnik 
et 
al. 
random-­‐graph 
model 
(different 
degree 
assortativities 
in 
different 
layers), 
similar 
model 
by 
Joel 
Miller 
and 
collaborators 
(explicitly 
in 
a 
metapopulation 
context)
%QWRNGFEGNN0GVYQTMU 
• Each 
node 
is 
associated 
with 
a 
dynamical 
system, 
and 
two 
nodes 
have 
the 
same 
color 
if 
they 
have 
the 
same 
state 
space 
and 
an 
identical 
dynamical 
system. 
• The 
couplings 
between 
dynamical 
systems 
are 
the 
edges 
(or 
hyperedges). 
Two 
edges 
have 
the 
same 
color 
if 
the 
couplings 
are 
equivalent 
• There 
exist 
many 
nice 
results 
for 
generic 
bifurcations 
in 
small 
coupled-­‐celled 
networks. 
– Spiritually 
similar 
results 
for 
generic 
phase 
transitions 
in 
random 
walks 
and 
Laplacians, 
but 
for 
very 
low-­‐dimensional 
systems 
instead 
of 
high-­‐ 
dimensional 
ones 
• Surgeon 
General’s 
warning: 
The 
papers 
on 
coupled-­‐ 
cell 
networks 
(many 
by 
Marty 
Golubitsky 
and 
company) 
are 
very 
mathematical.
1VJGT[PCOKECN5[UVGOU 
• The 
usual 
suspects. 
Pick 
your 
favorite. 
:) 
• Kuramoto 
model 
• Threshold 
models 
of 
social 
influence 
– Percolation-­‐like 
– E.g. 
Watts 
model 
• Games 
on 
networks 
• Sandpiles 
• Others
%QPVTQNCPF[PCOKEU 
• It’s 
important 
to 
consider 
feedback 
loops. 
• Maybe 
one 
is 
only 
allowed 
to 
apply 
controls 
to 
a 
subset 
of 
the 
layers? 
• Layer 
decompositions: 
Start 
with 
a 
network 
and 
try 
to 
infer 
layers 
– Reminiscent 
of 
community 
detection, 
but 
with 
layers 
instead 
of 
dense 
modules 
– E.g. 
research 
by 
Prescott 
and 
Papachristodoulou 
on 
biochemical 
networks 
– Similar 
problem 
in 
social 
networks 
• “Control 
network” 
used 
to 
influence 
an 
“open-­‐loop 
network” 
(which 
doesn’t 
include 
feedback) 
• “Pinning 
control”, 
in 
which 
one 
controls 
a 
small 
fraction 
of 
nodes 
to 
try 
to 
influence 
the 
dynamics 
of 
other 
nodes, 
in 
the 
context 
of 
interconnected 
networks.
3WGUVKQPU!
4GOKPFGT)NQUUCT[
“What 
befell 
Candide 
us 
at 
the 
end 
of 
his 
our 
journey” 
%10%.75+105#0 
176.11-
%QPENWUKQPU 
• Multilayer networks are interesting and important objects to study.! 
• We have developed a unified framework that allows a 
classification of different types of multilayer networks.! 
• Many real networks have multilayer structures.! 
• Multilayer networks make it possible to throw away less data. 
Additionally, they have interesting structural features and 
have interesting effects in dynamical processes.! 
• Adjacency tensors: their time has come! 
– We need to use tools from multilinear algebra. Tensors generalize 
matrices, but there are important differences to consider.! 
• Challenge: Need to collect good data, especially w.r.t. realiable 
quantitative values for inter-layer edges! 
• Challenge: Need more genuinely interlayer diagnostics! 
• Not just “bigger and better” version of monoplex objects! 
• Challenge: Need additional general results on dynamical processes 
(bifurcations, phase transitions). There are some, but we need 
more.! 
• Challenge: Need to move farther beyond the usual percolation-like 
models! 
• Not just “bigger and better” versions of monoplex processes! 
• Review article of multilayer networks: Journal of Complex Networks, 
in press (arXiv:1309.7233)! 
• Code for visualization and analysis of multilayer networks: 
http://www.plexmath.eu/?page_id=327! 
• Thanks: James S. McDonnell Foundation, EPSRC, FET-Proactive project 
“PLEXMATH”!
1WVNQQM 
• All 
is 
for 
the 
best 
in 
this 
best 
of 
all 
possible 
worlds. 
• (Also: 
The 
future’s 
so 
bright, 
we 
gotta 
to 
wear 
shades.)
3WGUVKQPU!
“What 
befell 
Candide 
us 
after 
the 
story 
ended” 
#8'46+5'/'065
#FXGTVKUGOGPVa 
,QWTPCNUGFKECVGFVQ0GVYQTM5EKGPEG 
• Ones with me on the editorial 
board:! 
– Journal of Complex Networks (OUP)! 
– IEEE Transactions on Network 
Science and Engineering (IEEE)! 
• Without me:! 
– Network Science (CUP)!
#FXGTVKUGOGPV.CMG%QOQ5EJQQNQP 
%QORNGZ0GVYQTMU
/C[ł 
• Lake Como School of Advanced 
Studies: ! 
– http://lakecomoschool.org/! 
• School on Complex Networks! 
– The Boss: Carlo Piccardi! 
– Scientific Board: Stefano 
Battiston, Vittoria Colizza, Peter 
Holme, Yamir Moreno, Mason Porter!
#FXGTVKUGOGPV9QTMUJQRQPVJG/CVJGOCVKEUCPF2J[UKEU 
QH/WNVKNC[GT%QORNGZ0GVYQTMU
/#2%1/ 
• Organizers: Alex Arenas, Mason 
Porter! 
• July 6–8, 2015, Max Planck 
Institute for the Physics of 
Complex Systems, Dresden, Germany! 
• Watch this space:! 
– http://www.mpipks-dresden.mpg.de/ 
pages/veranstaltungen/ 
frames_veranst_en.html!
#FXGTVKUGOGPV/$+5GOGUVGT2TQITCO 
QP0GVYQTMU
5RTKPI 
• Mathematical Biosciences 
Institute, The Ohio State 
University, USA! 
• Semester program on “Dynamics of 
Biologically Inspired Networks”! 
– http://mbi.osu.edu/programs/ 
emphasis-programs/future-programs/ 
spring-2016-dynamics-biologically-inspired- 
networks/! 
• Focuses on theoretical questions 
on networks that arise from 
biology!
#FXGTVKUGOGPV/$+92a 
ň)GPGTCNKGF0GVYQTM5VTWEVWTGUCPF[PCOKEUʼn 
• March 21–25, 2016! 
• http://mbi.osu.edu/event/?id=898!

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Multilayer tutorial-netsci2014-slightlyupdated

  • 1. Mason A. Porter Mathematical Institute, University of Oxford (@masonporter, masonporter.blogspot.co.uk) Mostly, we’ll be “following” (i.e. skimming through) our new review article: M Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, & MAP, “Multilayer Networks”, Journal of Complex Networks, Vol. 2, No. 3: 203–271 [2014].
  • 2. 5VGRUVQ*CRRKPGUU • 1. Go to http://people.maths.ox.ac.uk/porterm/ temp/netsci2014/ and download the .pdf file of this presentation. • 2. Download the review article from http://comnet.oxfordjournals.org/content/ 2/3/203 and (just in case) download our earlier article on the tensorial formalism – http://people.maths.ox.ac.uk/porterm/papers/ PhysRevX.3.041022.pdf • 3. Use these materials and be happy.
  • 3. 1WVNKPG • Browsing through the mega-­‐review article – 1. Introduction – 2. Conceptual and Mathematical Framework – 3. Data – 4. Models, Methods, Diagnostics, and Dynamics – 5. Conclusions and Outlook • Some Advertisements – Journals, workshops/conferences
  • 5. “How Candide was multilayer networks were brought up in a magnificent castle and how he was they were driven thence” +06417%6+10
  • 6. 'ZCORNG/WNVKRNGZ0GVYQTM • The concept of “multiplex network” has been around for many decades.!
  • 7. 'ZCORNG'FIG%QNQTGF/WNVKITCRJ • Monster movement in the game “Munchkin Quest”
  • 8. 'ZCORNG0GVYQTMQH0GVYQTMU • The notion (and terminology) “network of networks” is also several decades old.! (Craven and Wellman, 1973)
  • 9. #0GVYQTMQH0GVYQTMUa 7-+PHTCUVTWEVWTG (Courtesy of Sco; Thacker, ITRC, University of Oxford)
  • 15. “What befell Candide multilayer networks among the mathematicians” %10%'267#.#0 /#6*'/#6+%#.(4#/'914-5
  • 17. 5QHVYCTGHQT8KUWCNKCVKQPCPF#PCN[UKU • See h;p://www.plexmath.eu/?page_id=327 • M. De Domenico, M. A. Porter, A. Arenas, arXiv:1405.0843
  • 18. 2.2. Tensorial Representaon • Adjacency tensor for unweighted case: • Elements of adjacency tensor: – Auvαβ = Auvα1β1 … αdβd = 1 iff ((u,α), (v,β)) is an element of EM (else Auvαβ = 0) • Important note: ‘padding’ layers with empty nodes – One needs to disnguish between a node not present in a layer and nodes exisng but edges not present (use a supplementary tensor with labels for edges that could exist), as this is important for normalizaon in many quanes.
  • 19. #FLCEGPE[6GPUQTYKVJF#URGEV • One can write a general (rank-­‐4) mullayer adjacency tensor M in terms of a tensor product between single-­‐layer adjacency tensors [C(l) in upper right] and canonical basis tensors [see lower right] • w: weights • E: canonical basis tensors • Weighted edge from node ni in layer h to node nj in layer k • Note: Einstein summaon convenon • Page 3 of De Domenico et al., PRX, 2013
  • 20. /WNVKNKPGCT#NIGDTCCPF.CRNCEKCP6GPUQTU • Explored in several papers. Examples: – Supra-­‐Laplacian matrices: S. Gómez, A. Díaz-­‐Guilera, J. Gómez-­‐Gardeñes, C. J. Pérez-­‐Vicent, Y. Moreno, A. Arenas, Physical Review Le8ers, Vol. 110, 028701 (2013) – Mullayer Laplacian tensors: De Domenico et al, Physical Review X, 2013 – Spectral properes of mullayer Laplacians: A. Solé-­‐ Ribalta, M. De Domenico, N. E. Kouvaris, A. Díaz-­‐ Guilera, S. Gómez, A. Arenas, Physical Review E, Vol. 88, 032807 (2013) – Also see summary in the review arcle.
  • 21. /WNVKNC[GT%QODKPCVQTKCN.CRNCEKCP • Mullayer combinatorial Laplacian: – First term: strength (i.e. weighted degree) tensor – A bit more on degree tensor later – Second term: mullayer adjacency tensor (recall) – U: tensor with all entries equal to 1 – E: canonical basis for tensors (recall) – δ: Kronecker delta
  • 22. 'ZCORNGň/WNVKUNKEGʼn0GVYQTMU • P. J. Mucha, T. Richardson, K. Macon, MAP, J.-­‐P. Onnela, “Community Structure in Time-­‐Dependent, Mulscale, and Mulplex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010) • Simple idea: Glue common nodes across “slices” (i.e. “layers”)
  • 23. ň(NCVVGPGFʼn/WNVKUNKEG0GVYQTMU UWRTCCFLCEGPE[TGRTGUGPVCVKQP • Schematic from M. Bazzi, MAP, S. Williams, M. McDonald, D. J. Fenn, S. D. Howison, in preparation!
  • 24. %NCUUKH[KPI/WNVKNC[GT0GVYQTMU • Special cases of mullayer networks include: mulplex networks, interdependent networks, networks of networks, node-­‐colored networks, edge-­‐colored mulgraphs, … • To obtain one of these special cases, we impose constraints on the general structure defined earlier. • See the review arcle for details.
  • 25. %QPUVTCKPVU HTQOVJG6CDNG • 1. Node-­‐aligned (or fully interconnected): All layers contain all nodes. • 2. Layer disjoint: Each node exists in at most one layer. • 3. Equal size: Each layer has the same number of nodes (but they need not be the same ones). • 4. Diagonal coupling: Inter-­‐layer edges only can exist between nodes and their counterparts. • 5. Layer coupling: coupling between layers is independent of node identy – Note: special case of “diagonal coupling” • 6. Categorical coupling: diagonal couplings in which inter-­‐layer edges can be present between any pair of layers – Contrast: “ordinal” coupling for tensorial representaon of temporal networks • Example 1: Most –– but not all! –– “mul@plex networks” studied in the literature sasfy (1,3,4,5,6) and include d = 1 aspects. – Note: Many important situaons need (1,3) to be relaxed. (E.g. Some people have Facebook accounts but not Twi;er accounts.) • Example 2: The “networks of networks” that have been invesgated thus far sasfy (3) and have addional constraints (which can be relaxed).
  • 26. The literature is messy. #makeitstop
  • 27. #0QFG%QNQTGF0GVYQTM • Node-­‐colored network: also known as interconnected network, network of networks, etc. • (three alternative representations)
  • 31. /WNVKRNGZ0GVYQTMU • Networks with multiple types of edges – Also known as multirelational networks, edge-­‐colored multigraphs, etc. • Many studies in practice use the same sets of nodes in each layer, but this isn’t required. – Challenge for tensorial representation: need to keep track of lack of presence of a tie versus a node not being present in a layer (relevant e.g. for normalization of multiplex clustering coefficients) • Question: When should you include inter-­‐ layer edges and when should you ignore them?
  • 32. *[RGTITCRJU • Hyperedges generalize edges. A hyperedge can include any (nonzero) number of nodes. • Example: A k-­‐uniform hypergraph has cardinality k for each hyperedge (e.g. a folksonomy like Flickr). – One can represent a k-­‐uniform hypergraph using adjacency tensors, and there have been some studies of multiplex networks by mapping them into k-­‐uniform hypergraphs. – A nice paper: Michoel Nachtergaele, PRE, 2012 • Note that multilayer networks are still formulated for pairwise connections (but a more general type of pairwise connections than usual).
  • 33. 1TFKPCN%QWRNKPIUCPF 6GORQTCN0GVYQTMU • Ordinal coupling: diagonal inter-­‐layer edges among consecutive layers (e.g. multilayer representation of a temporal network) • Categorical coupling: diagonal inter-­‐layer edges between all pairs of edges • Both can be present in a multilayer network, and both can be generalized
  • 34. 1VJGT6[RGUQH /WNVKNC[GT0GVYQTMU GZCORNGU • k-­‐partite graphs – Bipartite networks are most commonly studied • Coupled-­‐cell networks – Associate a dynamical system with each node of a multigraph. Network structure through coupling terms. • Multilevel networks – Very popular in social statistics literature (upcoming special issue of Social Networks) – Each level is a layer – Think ‘hierarchical’ situations. Example: ‘micro-­‐ level’ social network of researchers and a ‘macro-­‐ level’ for a research-­‐exchange network between laboratories to which the researchers belong
  • 36. “What they saw in the Country of El Dorado real world” #6#
  • 39. 2TCEVKECNKVKGUCPF/GUUCIGU • Lots of reliable data on intra-­‐layer relations (i.e. the usual kind of edges) • It’s much more challenging to collect reliable data for inter-­‐layer edges. We need more data. – E.g. Transportation data should be a very good resource. Think about the amount of time to change gates during a layover in an airport. – E.g. Transition probabilities of a person using different social media (each medium is a layer). • Most empirical multilayer-­‐network studies thus far have tended to be multiplex networks. • Determining inter-­‐layer edges as a problem in trying to reconcile node identities across networks. (Can you figure out that a Twitter account and Facebook account belong to the same person?) – Major implications for privacy issues • Take-­‐home message: Be creative about how you construct multilayer networks and define layers!
  • 42. “Candide’s Our voyage to Constantinople Istanbul measuring and modeling” /1'.5/'6*15 +#)0156+%5#0;0#/+%5
  • 43. #IITGICVKQPQH /WNVKNC[GT0GVYQTMU • Construct single-­‐layer (i.e. “monoplex”) networks and apply the usual tools. – Obtain edge weights as weighted average of connections in different layers. You get a different weighted network with a different weighting vector. • E.g. Zachary Karate Club – Information loss • Is there a way to do this to minimize information loss? • Important: Loss of “Markovianity” (a la temporal networks) – Processes that are Markovian on a multilayer network may yield non-­‐Markovian processes after aggregating the network
  • 44. KCIPQUVKEU • Generalizations of the usual suspects – Degree/strength – Neighborhood • Which layers should you consider? – Centralities – Walks, paths, and distances – Transitivity and local clustering • Important note: Sometimes you want to define different values for different node-­‐layers (e.g. a vector of centralities for each entity) and sometimes you want a scalar. • Need to be able to consider different subsets of the layers • Need more genuinely multilayer diagnostics – It is important to go beyond “bigger and better” versions of the usual concepts.
  • 45. GITGGUCPF0GKIJDQTJQQFU • Simplest way: Use aggregation and then measure degree, strength, and neighborhoods on a monoplex network obtained from aggregation. – Possibly only consider a subset of the layers • More sophisticated: Define a multi-­‐edge as a vector to track the information in each layer. With weighted multilayer networks, you can keep track of different weights in intra-­‐layer versus inter-­‐layer edges. • Towards multilayer measures: overlap multiplicity for a multiplex network can track how often an edge between entities i and j occurs in multiple layers
  • 46. #FLCEGPE[6GPUQTYKVJF#URGEV TGECNNVJKUUNKFG • One can write a general (rank-­‐4) mullayer adjacency tensor M in terms of a tensor product between single-­‐layer adjacency tensors [C(l) in upper right] and canonical basis tensors [see lower right] • w: weights • E: canonical basis tensors • Weighted edge from node ni in layer h to node nj in layer k • Note: Einstein summaon convenon • Page 3 of De Domenico et al., PRX, 2013
  • 49. 9CNMU2CVJUCPFKUVCPEGU • To define a walk (or a path) on a multilayer network, we need to consider the following: – Is changing layers considered to be a step? Is there a “cost” to changing layers? How do you measure this cost? • E.g. transportation networks vs social networks – Are intra-­‐layer steps different in different layers? • Example: labeled walks (i.e. compound relations) are walks in a multiplex network that are associated with a sequence of layer labels • Generalizing walks and paths is necessary to develop generalizations for ideas like clustering coefficients, transitivity, communicability, random walks, graph distance, connected components, betweenness centralities, motifs, etc. • Towards multilayer measures: Interdependence is the ratio of the number of shortest paths that traverse more than one layer to the number of shortest paths
  • 50. %NWUVGTKPI%QGHHKEKGPVU CPF6TCPUKVKXKV[ • Our approach: Cozzo et al., 2013 – Use the idea of multilayer walks. Keep track of returning to entity i (possibly in a different layer from where we started) separately for 1 total layer, 2 total layers, 3 total layers (and in principle more). • Insight: Need different types of transitivity for different types of multiplex networks. – Example (again): transportation vs social networks – There are several different clustering coefficients for monoplex weighted networks, and this situation is even more extreme for multilayer networks.
  • 51. 'ZCORNG%NWUVGTKPI%QGHHKEKGPV %QQGVCNCT:KX • Our perspective: multilayer walks, which can return to node i on different layers and traverse different numbers of layers!
  • 52. %GPVTCNKV[/GCUWTGU • In studies of networks, people compute a crapload of centralities. • The common ones have been generalized in various ways for multilayer networks. – Again, one needs to ask whether you want a centrality for a node-­‐layer or for a given entity (across all layers or a subset of layers). • Eigenvector centralities and related ideas can be derived from random walks on multilayer networks. – Consider different spreading weights for different types of edges (e.g. intra-­‐layer vs inter-­‐layer edges; or different in different layers) • Betweenness centralities can be calculated for different generalizations of short paths. • A point of caution: “What the world needs now is another centrality measure.” – I.e. although they can be very useful, please don’t go too crazy with them.
  • 53. +PVGTNC[GTKCIPQUVKEU • The community needs to construct genuinely multilayer diagnostics and go beyond ‘bigger and better’ versions of the concepts we know and (presumably) love. – Not very many yet • Correlations of network structures between layers – E.g. interlayer degree-­‐degree correlations (or any other diagnostic) • ! Interpreting communities as layers, quantities like assortativity can be construed as inter-­‐layer diagnostics • Interdependence is the ratio of the number of shortest paths that traverse more than one layer to the number of shortest paths
  • 54. /QFGNUQH/WNVKRNGZ0GVYQTMU • Straightforward: Use your favorite monoplex model for intra-­‐layer connections and then construct inter-­‐layer edges in some way. – E.g. random-­‐graph models like Erdös-­‐Rényi, network growth models like preferential attachment • Correlated layers: Include correlations between properties in different intra-­‐layer networks in the construction of random-­‐graph ensembles. – E.g. Include intra-­‐layer degree-­‐degree correlations ρ in [-­‐1,1] • Exponential Random Graph Models (ERGMs) for multiplex networks – Used a lot for multilevel networks
  • 55. /QFGNUQH/WNVKRNGZ0GVYQTMU • Statistical-­‐mechanical ensembles of multiplex networks • Generalize growth mechanisms like preferential attachment – Again, one can include inter-­‐layer correlations in designing a model • It would be good to go beyond “bigger and better” versions of the usual ideas. – Including simple inter-­‐layer correlations (especially between intra-­‐ layer degrees) has been the main approach so far.
  • 56. /QFGNUQH +PVGTEQPPGEVGF0GVYQTMU • Straightforward: Construct different layers separately using your favorite model (or even one that you hate) and then add inter-­‐layer edges uniformly at random. • More sophisticated: Be more strategic in adding inter-­‐layer edges. • Some random-­‐graph modules with community structure can be useful, where we think of each community as a separate layer (i.e. as a separate network in a network of networks) – E.g. Melnik et al’s paper (Chaos, 2014) on random graphs with heterogeneous degree assortativity • The homophily is different in different layers and there is a mixing matrix for inter-­‐layer connections
  • 57. %QOOWPKVKGUCPF1VJGT /GUQUECNG5VTWEVWTGU • Communities are dense sets of nodes in a network (typically relative to some null model). – One can use these ideas for multilayer networks (e.g. multislice modularity). • Interpreting communities as roadblocks to some dynamical process (e.g. starting from some initial condition), one can have such a process on a multilayer network—with different spreading rates in different types of edges—to algorithmically find communities in multilayer networks. • Most work thus far on multilayer representation of temporal networks. – One exception is recent work on “Kantian fractionalization” in international relations. • Challenge: Develop multilayer null models for community detection (different for ordinal vs. categorical coupling) • Blockmodels • Spectral clustering (e.g. Michoel Nachtergaele) • Note: Because I have done a lot of work in this area, I will go through a bit in some detail to help illustrate some general points that are also relevant in other studies of multilayer networks.
  • 58. ! Communities = Cohesive groups/modules/ mesoscopic structures › In stat phys, you try to derive macroscopic and mesoscopic insights from microscopic information ! Community structure consists of complicated interactions between modular (horizontal) and hierarchical (vertical) structures ! communities have denser set of Internal edges relative to some null model for what edges are present at random › “Modularity”
  • 59. 'ZCORNGň/WNVKUNKEGʼn0GVYQTMU • P. J. Mucha, T. Richardson, K. Macon, MAP, J.-­‐P. Onnela, “Community Structure in Time-­‐Dependent, Mulscale, and Mulplex Networks”, Science, Vol. 328, No. 5980, 876–878 (2010) • Simple idea: Glue common nodes across “slices” (i.e. “layers”) • “Diagonal” coupling
  • 60. 'ZCORNGKCIPQUVKE/WNVKUNKEG/QFWNCTKV[ • Find communies algorithmically by opmizing “mulslice modularity” – We derived this funcon in Mucha et al, 2010 • Laplacian dynamics: find communies based on how long random walkers are trapped there. Exponenate and then linearize to derive modularity. • Generalizes derivaon of monoplex modularity from R. Lambio;e, J.-­‐C. Delvenne, . M Barahona, arXiv:0812.1770 • Different spreading weights on different types of edges – Node x in layer r is a different node-­‐layer from node x in layer s
  • 62. 4QNN%CNN8QVKPI0GVYQTMUa GZCORNGVQKNNWUVTCVGGHHGEVQHRCTCOGVGTʩ • A. S. Waugh, L. Pei, J. H. Fowler, P. J. Mucha, M. A. Porter [2012], arXiv:0907.3509 (without multilayer formulation) • Modularity Q as a measure of polarization • Can calculate how closely each legislator is tied to their community (e.g. by looking at magnitude of corresponding component of leading eigenvector of modularity matrix if using a spectral optimization method) • Medium levels of optimized modularity as a predictor of majority turnover – By contrast, leading political science measure doesn’t give statistically significant indication • One network slice for each two-­‐year Congress
  • 63. P. J. Mucha M. A. Porter, Chaos, Vol. 20, No. 4, 041108 (2010)
  • 70. [PCOKE4GEQPHKIWTCVKQPQH*WOCP $TCKP0GVYQTMUWTKPI.GCTPKPIa $CUUGVVGVCN20#5 • fMRI data: network from correlated time series • Examine role of modularity in human learning by identifying dynamic changes in modular organization over multiple time scales • Main result: flexibility, as measured by allegiance of nodes to communities, in one session predicts amount of learning in subsequent session
  • 71. Staonarity and Flexibility • Community staonarity ζ (autocorrelaon over me of community membership): • Node flexibility: – fi = number of mes node i changed communies divided by total number of possible changes – Flexibility f = fi
  • 72. [PCOKE%QOOWPKV[5VTWEVWTG • Investigating community structure in a multilayer framework requires consideration of new null models • Many more details! – E.g., Robustness of results to choice of size of time window, size of inter-­‐slice coupling, particular definition of flexibility, complicated modularity landscape, definition of ‘similarity’ of time series, etc.
  • 73. Dynamic Reconfiguraon of Human Brain Networks During Learning (Basse; et al, PNAS, 2011) • fMRI data: network from correlated me series • Examine role of modularity in human learning by idenfying dynamic changes in modular organizaon over mulple me scales • Main result: flexibility, as measured by allegiance of nodes to communies, in one session predicts amount of learning in subsequent session
  • 74. Development of Null Models for Mullayer Networks • D. S. Basse;, M. A. Porter, N. F. Wymbs, S. T. Graƒon, J. M. Carlson, P. J. Mucha, Chaos, 23(1): 013142 (2013) • Addional structure in adjacency tensors gives more freedom (and responsibility) for choosing null models. • Null models that incorporate informaon about a system • E.g. chain null model fixes network topology but randomizes network “geometry” (edge weights) • Also: Examine null models from shuffling me series directly (before turning into a network) • Structural (γ) versus temporal resoluon parameter (ω) • More generally, how to choose inter-­‐layer (off-­‐ diagonal) terms Cjrs • Time series from experiments as well as output of a dynamical system (e.g. Kuramoto model). Analogous to structural vs funconal brain networks.
  • 75. /GVJQFU$CUGFQP 6GPUQTGEQORQUKVKQP • Many different generalizations of singular value decomposition (SVD) to tensors – Every matrix has a unique SVD, but we have to relax this for tensors. – See Kolda and Bader, SIAM Review, 2009 – Tensor rank vs matrix rank: hard to determine that rank of tensors of order 3+ • Note: “rank” is also used as a synonym for “order” (see earlier). Here, “rank” is the generalization of matrix rank: the minimum number of column vectors needed to span the range of a matrix. The tensor rank is the minimum number of rank-­‐1 tensors with which one can express a tensor as a sum. The purpose of an SVD (and generalizations) is to find a low-­‐rank approximation. • Non-­‐negative tensor factorization
  • 76. [PCOKECN5[UVGOUQP /WNVKNC[GT0GVYQTMU • Basic question: How do multilayer structures affect dynamical systems on networks? – Effects of multiplexity? (edge colorings) – Effects of interconnectedness? (node colorings) • Important goal: Find new phenomena that cannot occur without multilayer structures. – Example: Speeding up vs slowing down spreading? – Example: Multiplexity-­‐induced correlations in dynamics? – Example: Effect of different costs for changing layers?
  • 77. %QPPGEVGF%QORQPGPVU CPF2GTEQNCVKQP • Connected component defined as in monoplex networks, except that multiple types of edges can occur in a path. • In multilayer networks, one again uses branching-­‐process approximations that allow the use of generating function technology. – Same fundamental idea (and limitations) as in monoplex networks, but the calculations are more intricate • More flavors of giant connected components (GCCs) that can be defined
  • 78. 2GTEQNCVKQP%CUECFGU • Example (from Buldyrev et al, Nature, 2010)
  • 80. 2GTEQNCVKQP%CUECFGU • Numerous papers for both multiplex networks and interconnected networks • A few interesting ideas – Localized attack • More generally, multilayer networks allow more creativity in targeted attacks. Why in Hell is it almost always by degree (even for monoplex networks)? Be creative! – Viable cluster: mutually connected giant component
  • 81. %QORCTVOGPVCN5RTGCFKPI /QFGNUCPFKHHWUKQP • Random walks and Laplacians – Different spreading rates on different types of edges • See earlier discussions of multislice community structure • Strong vs weak inter-­‐layer coupling – Examine generic properties of phase transitions (e.g. as a function of weights of inter-­‐layer edges) • Competing (toy models of) biological contagions – Your favorite toy models (SI, SIS, SIR, SIRS, etc.) • Layers with biological contagions interacting with layers of information diffusion (e.g. of awareness)
  • 84. %QORCTVOGPVCN5RTGCFKPI /QFGNUCPFKHHWUKQP • Metapopulation models as biological epidemics on networks of networks – E.g. Melnik et al. random-­‐graph model (different degree assortativities in different layers), similar model by Joel Miller and collaborators (explicitly in a metapopulation context)
  • 85. %QWRNGFEGNN0GVYQTMU • Each node is associated with a dynamical system, and two nodes have the same color if they have the same state space and an identical dynamical system. • The couplings between dynamical systems are the edges (or hyperedges). Two edges have the same color if the couplings are equivalent • There exist many nice results for generic bifurcations in small coupled-­‐celled networks. – Spiritually similar results for generic phase transitions in random walks and Laplacians, but for very low-­‐dimensional systems instead of high-­‐ dimensional ones • Surgeon General’s warning: The papers on coupled-­‐ cell networks (many by Marty Golubitsky and company) are very mathematical.
  • 86. 1VJGT[PCOKECN5[UVGOU • The usual suspects. Pick your favorite. :) • Kuramoto model • Threshold models of social influence – Percolation-­‐like – E.g. Watts model • Games on networks • Sandpiles • Others
  • 87. %QPVTQNCPF[PCOKEU • It’s important to consider feedback loops. • Maybe one is only allowed to apply controls to a subset of the layers? • Layer decompositions: Start with a network and try to infer layers – Reminiscent of community detection, but with layers instead of dense modules – E.g. research by Prescott and Papachristodoulou on biochemical networks – Similar problem in social networks • “Control network” used to influence an “open-­‐loop network” (which doesn’t include feedback) • “Pinning control”, in which one controls a small fraction of nodes to try to influence the dynamics of other nodes, in the context of interconnected networks.
  • 90. “What befell Candide us at the end of his our journey” %10%.75+105#0 176.11-
  • 91. %QPENWUKQPU • Multilayer networks are interesting and important objects to study.! • We have developed a unified framework that allows a classification of different types of multilayer networks.! • Many real networks have multilayer structures.! • Multilayer networks make it possible to throw away less data. Additionally, they have interesting structural features and have interesting effects in dynamical processes.! • Adjacency tensors: their time has come! – We need to use tools from multilinear algebra. Tensors generalize matrices, but there are important differences to consider.! • Challenge: Need to collect good data, especially w.r.t. realiable quantitative values for inter-layer edges! • Challenge: Need more genuinely interlayer diagnostics! • Not just “bigger and better” version of monoplex objects! • Challenge: Need additional general results on dynamical processes (bifurcations, phase transitions). There are some, but we need more.! • Challenge: Need to move farther beyond the usual percolation-like models! • Not just “bigger and better” versions of monoplex processes! • Review article of multilayer networks: Journal of Complex Networks, in press (arXiv:1309.7233)! • Code for visualization and analysis of multilayer networks: http://www.plexmath.eu/?page_id=327! • Thanks: James S. McDonnell Foundation, EPSRC, FET-Proactive project “PLEXMATH”!
  • 92. 1WVNQQM • All is for the best in this best of all possible worlds. • (Also: The future’s so bright, we gotta to wear shades.)
  • 94. “What befell Candide us after the story ended” #8'46+5'/'065
  • 95. #FXGTVKUGOGPVa ,QWTPCNUGFKECVGFVQ0GVYQTM5EKGPEG • Ones with me on the editorial board:! – Journal of Complex Networks (OUP)! – IEEE Transactions on Network Science and Engineering (IEEE)! • Without me:! – Network Science (CUP)!
  • 96. #FXGTVKUGOGPV.CMG%QOQ5EJQQNQP %QORNGZ0GVYQTMU /C[ł • Lake Como School of Advanced Studies: ! – http://lakecomoschool.org/! • School on Complex Networks! – The Boss: Carlo Piccardi! – Scientific Board: Stefano Battiston, Vittoria Colizza, Peter Holme, Yamir Moreno, Mason Porter!
  • 97. #FXGTVKUGOGPV9QTMUJQRQPVJG/CVJGOCVKEUCPF2J[UKEU QH/WNVKNC[GT%QORNGZ0GVYQTMU /#2%1/ • Organizers: Alex Arenas, Mason Porter! • July 6–8, 2015, Max Planck Institute for the Physics of Complex Systems, Dresden, Germany! • Watch this space:! – http://www.mpipks-dresden.mpg.de/ pages/veranstaltungen/ frames_veranst_en.html!
  • 98. #FXGTVKUGOGPV/$+5GOGUVGT2TQITCO QP0GVYQTMU 5RTKPI • Mathematical Biosciences Institute, The Ohio State University, USA! • Semester program on “Dynamics of Biologically Inspired Networks”! – http://mbi.osu.edu/programs/ emphasis-programs/future-programs/ spring-2016-dynamics-biologically-inspired- networks/! • Focuses on theoretical questions on networks that arise from biology!
  • 99. #FXGTVKUGOGPV/$+92a ň)GPGTCNKGF0GVYQTM5VTWEVWTGUCPF[PCOKEUʼn • March 21–25, 2016! • http://mbi.osu.edu/event/?id=898!