These are the slides for a tutorial talk about "multilayer networks" that I gave at NetSci 2014.
I walk people through a review article that I wrote with my PLEXMATH collaborators: http://comnet.oxfordjournals.org/content/2/3/203
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
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”)
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).
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
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.
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
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.
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.)
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!
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!