arXiv:1507.00695v1 [cs.SI] 2 Jul 2015
Network models that reflect multiplex dynamics
Daryl DeFord
Scott Pauls
Department of Mathematics
Dartmouth College
Hanover, NH 03755
Email: [email protected]
Department of Mathematics
Dartmouth College
Hanover, NH 03755
Email: [email protected]
only express a single type of relationship, representing a
multiplex as a single network can lead to distortion of the
very topological properties we are trying to understand. A
meaningful global dynamic model should respect the individual topological behaviors of the layers without introducing
confounding structural effects.
In this paper, we revisit some recent multiplex structural
representations, examining them through the lens of dynamical consistency to show that in many cases the structural
definitions introduce factors that confound the analysis of the
multiplex dynamics. To remedy this, we develop a new method
for modeling multiplex systems which respects dynamics and
derive spectral results similar to those known for monoplex
networks.
Abstract—Many of the systems that are traditionally analyzed
as complex networks have natural interpretations as multiplex
structures. While these formulations retain more information
than standard network models, there is not yet a fully developed
theory for computing network metrics and statistics on these
objects. As many of the structural representations associated to
these models can distort the underlying characteristics of dynamical process, we introduce an algebraic method for modeling
arbitrary dynamics on multiplex networks. Since several network
metrics are based on generalized notions of dynamical transfer,
we can use this framework to extend many of the standard
network metrics to multiplex structures in a consistent fashion.
Index Terms—Multiplex Networks, Network Dynamics
I. I NTRODUCTION
Mathematical analyses of simple network models of complex systems provide a surprising amount of information determining important components of the system [5], finding
hidden communities [27], demonstrating the robustness of the
system to failures [2], [15], among many others. But the
simplest network models, where nodes represent components
of the system and edges represent interaction or association
between two nodes, are limited as they conflate different types
of interaction between nodes.
A nascent theory of multiplex networks (see [23] for a
thorough review of the current research) deals with this
limitation by allowing for an all-encompassing structure with
multiple layers, one for each type of interaction, where each
layer is a network on the same set of nodes. These structures
arise naturally in many settings: trade networks in economics
[3], [4], social networks [22], [25], and transportation networks
[14], [17], among others.
While researchers have recently focused on describing structural representations of multiplex networks and analyzing their
properties [9], [11], only a relatively small portion of the literature engages questions about dynamics on multiplex networks.
As structural representations of simple networks that reflect
dynamics are essential to some aspects of network analysis
- including diffusion processes, random walk probabilities,
clustering, and percolation - transporting these ideas to the
multiplex setting requires structural representations consistent
with multiplex dynamics.
Studying multiplex structures using tools from complex
networks exposes a tension between structural representations
of the entire multiplex and the dynamical interpretations of
the individual layers. Since the edges in traditional networks
Related Work
Kivelä et al. [23] provide a descriptive overview of many
of the methods and techniques that are being developed for
studying multiplex structures. Currently, these methods favor
tensorial representations [11], as they contain all available
data, in contrast to earlier attempts which aggregated multiplex
data into a single network [35].
Many standard monoplex metrics and statistics have been
extended to multiplex structures. For example Cozzo et al.
describe a generalized notion of clustering coefficients for multiplex structures in [9], while De Domenico et al. generalized
a wider variety of single network techniques to the multilayer
setting [11]. A study of the redundant information captured
by these representations is presented in [10].
A first approach, [26], studied dynamics on disaggregated
multiplex structures via percolation theory. Buldyrev et al. [7]
considered cascading failures on related networks. Building
on this work, Gao et al. characterized robustness of interconnected networks [18] and developed models for studying
the connectivity of interconnected networks [19]. Recently, a
k−core approach to these problems in the multiplex setting
was discussed in [1].
Several studies have extended monoplex centralities, such
as eigenvector centrality, Katz centrality, and various geodesic
centralities, to multiplex networks [11]–[13]. Our dynamical
method presented in this paper is a generalization of the
Khatri-Rao product approach to centrality problems introduced
by Sol et al. in [32].
1
To study distributed consensus problems on multiplex structures Trpevski et al. [34] proposed a supratransition matrix as
an analogue of the standard stochastic random walk matrix
for networks. Their stochastic model relates to De Domenico
et al.’s model of random walks on multiplex structures [14].
Both of these constructions are special cases of the method
we develop, see Section IV. Additionally, our work provides a
general framework for interpreting similar results on arbitrary
multiplex structures not arising from these specific applications.
In [21], the authors describe a process for modeling diffusion in multiplex networks by constructing a generalized
Laplacian. They used pertubation theory to characterize the
eigenvalues of the supra–Laplacian in order to discuss rates
of diffusion across multiplex networks [33]. Further spectral
considerations of multiplex structures were studied in [31].
Additionally, the structural behavior of multiplex networks
has been connected to the eigenvalues of this generalized
Laplacian [29], [30], although these results may be artifacts
of the composition of the operator under consideration [20].
We analyze the global network statistics of the Krackhardt
High–Tech Managers data [25] and a collection of random
multiplex networks, comparing the values derived from the
original layers with the values associated to the single network
representations. For the random networks, we follow [34]
and use a mixture of Erdös–Renyi, Barabasi–Albert, and
Watts–Strogatz networks on 100 nodes per layer. The results,
summarized in Table 1 and Figure 2 respectively, show that
these multiplex models obscure the heterogeneity of the layer
statistics.
The Krackhardt data gives rise to a multiplex with directed
layers, which poses a particular problem for the matched
sum model as the structural conflation can be particularly
misleading. Each of the three layers in the Krackhardt data
represents a particular type of association: advice relations,
friendship relations, and direct report relations in the company.
However, the matched sum does not distinguish between these
types of connections. Additionally, the edges between copies
of the same individual do not have a natural interpretation.
TABLE I
C OMPARISON OF M ULTIPLEX G LOBAL S TATISTICS
II. T OPOLOGICAL C ONSEQUENCES OF S TRUCTURAL
R EPRESENTATIONS
Statistic
Density
Transitivity
Reciprocity
Mean Degree
There are two main constructions or structural models in
the literature for generating such a network, aggregation and
matched summation. Aggregation combines all edges between
pairs of nodes into a single edge, while matched summation
(called diagonal, categorical multilayer networks in [23]) creates a single network from the layers by introducing new edges
connecting the nodes to copies of themselves in other layers.
Figure 1 shows aggregate and matched sum networks for an
example two-layer multiplex. Both of these structural models
distort network statistics relative to those of the original layers.
The main drawback to aggregation is that it loses information by conflating different types of interactions. The World
Trade Web network provides an important example where
aggregation obscures much of the topological heterogeneity
of the individual layers, particularly with respect to clustering
and reciprocity [3].
Unlike aggregate networks, no information is lost when
forming the matched sum network of a multiplex. Matched
summation corresponds to unfolding/flattening the tensorial
representation of a multiplex structure [11], which underlies
the centrality measures presented in [12], as well as many of
the network metrics discussed in [11], [23]. Although the tensorial construction preserves all of the structural information
about the multiplex, unfolding distorts the original data.
There are two main problems with matched summation.
First, the added interlayer edges have a different role than the
intralayer edges, but the two are conflated. Second, connecting
all the copies of a given node to one another creates dense
cliques throughout the matched sum. Many of the dynamics
based on flattening/unfolding tensorial representations of multiplex structures implicitly incorporate these distortions in a
fundamental way.
Advice
0.452
0.465
0.236
18.1
Friend
0.243
0.276
0.225
9.71
Report
0.0476
0.000
0.000
1.90
Aggregate
0.552
0.561
0.314
22.1
Matched Sum
0.112
0.311
0.299
13.9
The distortions we witness in these examples are not isolated: given an arbitrary multiplex with n nodes and k layers,
we compute bounds on many of the global statistics of the
aggregate and matched sum networks in terms of the layer
statistics.
Density: The density of the aggregate monoplex is greater
than or equal to the density of the densest layer. Matched sum
constructions tend to be less dense than the original layers
since the only inter–layer connections occur between copies
of the same node. Thus, the maximum number of neighbors of
a node is (n − 1) + (k − 1) out of the nk − 1 total nodes in
the matched sum.
Transitivity: The transitivity of the aggregate monoplex is
greater than or equal to the transitivity of the most transitive
layer. Matched sum networks tend to be less transitive than
the original layers since the only inter layer triangles occur
between the copies of each node. None of the connected triples
formed by one inter–layer edge and one intra-layer edge can
be closed, diluting the proportion of transitive triples.
Reciprocity: The reciprocity of the aggregate monoplex is
greater than or equal to the reciprocity of the most reciprocal
layer. The reciprocity of the matched sum is greater than or
equal to the reciprocity of the least reciprocal layer since all
of the connecting arcs are bidirectional.
Mean Degree: The mean degree of the aggregate monoplex
is greater than or equal to the maximum mean degree of the
layers. The mean degree of the aggregate monoplex is greater
than or equal to the minimum mean degree of the layers
because each node receives k−1 new incident edges. Although
2
(a) Layer 1
(b) Layer 2
(c) Aggregate Network
(d) Matched Sum
Fig. 1. A toy multiplex example with two layers and ten nodes. Figures (a) and (b) show the individual layers, while figure (c) shows the associated aggregate
monoplex and figure (d) shows the associated matched sum.
(a) Density
(b) Transitivity
(c) Mean Degree
(d) Path Length
Fig. 2. Global statistic comparison for single network representations. We constructed a nine layer random multiplex on 100 nodes and computed global
network statistics: (a) density, (b) transitivity, (c) mean degree, and (d) path length, for the individual layers as well as for the corresponding aggregate and
matched sum networks. The layer values are reported as the blue bars, while the red line represents the matched sum statistic and the green line represents
the aggregate statistic. Note that in addition to over or underestimating the layer values, the metrics associated to the single network representations do not
capture the heterogeneity of the layer statistics.
III. DYNAMICAL C ONSEQUENCES OF S TRUCTURAL
R EPRESENTATIONS
the mean degree and density capture the same information
about the original layers, the matched sum perturbs each
statistic differently.
In addition to distorting the basic topological properties,
forming aggregate or matched sum networks from multiplex
data also distorts multiplex dynamics. We identify two issues
– instances where interpretations of the dynamics are difficult,
and where these models provide incorrect generalizations of
standard dynamics. The structural distortions we observed in
the previous section interfere with such optimizations. As the
problems with aggregate models have been previously noted
by Barigozzi et al. in the context of the World Trade Web
network [3], [4], and more generally by Kivelä et al. in their
survey [23], we focus here on the effects of matched summation models and, consequently, of tensorial representations.
To illustrate the problem of interpretability of tensorial representations, we demonstrate the consequences of conflating
edge types by considering the simplest associated dynamical
model. Here, the tensor acts on supra–vector of quantities,
representing flow along the “edges” of the structure. The
resulting quantity at node j layer i is the sum of the neighbors
of j along layer i combined with the sum of the values at
β for all other layers. Although the intra–layer component
corresponds directly to familiar monoplex dynamics, the inter–
layer component reflects the effect of adding edges between
the node copies in the matched sum. Additionally, all of the
copies of node j represent the same object: it is unclear what
meaning should be attached to the edges connecting the layers.
Together these problems of interpretation make it difficult to
describe meaningful objective functions by directly viewing
Average Path Length: The average path length of the
aggregate monoplex is less than or equal to the shortest
average path length of the layers because the shortest paths
from each layer are still viable in the aggregate network. In
matched sum networks, the average path length also tends to
decrease since it requires at most two interlayer steps to make
use of the shortest intra–layer path between two nodes.
Clique Numbers: The clique numbers of the aggregate
and matched sum networks are both be greater than or equal
to the maximum clique number among the layers.
Except in the case of degenerate examples, such as duplicate
or empty layers, the bounds above become strict inequalities.
We observe that increasing the number of layers magnifies the
affects on these statistics. For example, as the number of layers
increases with a fixed number of nodes and average density,
the aggregate density increases while the matched sum density
decreases. Similar results hold for the other statistics.
Taken together, these results emphasize the toplogical damage inherent in using aggregate or matched sum representations of multiplex networks. Although for some applications,
such as the transportation networks studied in [14], these
structural additions accurately model the system, researchers
should consider possible confounding effects before applying
these techniques.
3
Notationally, v(i−1)n+j is the quantity at node j in layer i.
T
To extract layer quantities, we write v = v 1 , v 2 , · · · v k
where each v i is a 1 × n vector associated to layer i. That
is, vji ≡ v i (j) = v(i−1)n+j is the amount at node j in layer i.
the tensorial representation as an operator.
This problem also arises when considering heterogeneous
multiplex data, for example when some of the layers of the
multiplex are directed while others are undirected. While it is
true that an undirected edge may be considered as a pair of
opposing directed edges with no loss of structural information,
the dynamical interpretation of such relations can be quite
different and is entirely obscured by a naı̈ve model. Indeed,
the more heterogeneous the layers, the more important it is not
to add confounding structural factors that obscure the effects
of the topology on dynamical models.
Questions of interpretation are not the only issue inherent in
these tensorial dynamics. The structural distortions described
in the previous section have clear dynamical consequences. In
a matched sum, each node belongs to a clique representing all
its copies across the layers, vastly distoring local topologies
in networks with sparse layers. This in turn distorts metrics
and dynamics computed with walks and geodesics, since
the number of possible paths between two nodes is greatly
magnified. For example, measures like Katz centrality and
Pagerank penalize longer paths by an exponential factor, so
most of the centrality value accrues directly from the other
copies of the same node, overwhelming the actual connections
of interest.
The combinatorial supra–Laplacian, as defined in [11] and
[23] suffers from these deficiencies. Interpreted as a linear
differential operator on the tensor space, it represents the
Laplacian of the matched sum of the multiplex and hence
incorporates these structural distortions at a fundamental level.
Although possibly a reasonable model in some applications, it
is not a clear dynamical generalization of the standard graph
Laplacian. Consequently, we cannot derive the relations between the eigenvalues of this structural representation and the
multiplex properties analogously to the monoplex derivation.
Finally, as discussed above as the number of layers increases, the effects of these distortions increase. This explains
the behaviors noted in several papers based on dynamical
models associated to tensorial or supra– constructions that for
large portions of the parameter space, the dynamical process
is controlled by the layer mixing [12]–[14], [21], [29]–[31],
[33], [34]. It is unclear at this point what information about
multiplex models is actually related to the spectral structures of
the associated tensor representations since the corresponding
dynamic operators do not always have a natural interpretation.
Motivation
We want to interleave the intra–layer dynamics with an
operation that gathers the quantity stored at each copy of a
given node and redistributes among the other copies of that
node. We model this as an iterative two step process: we first
apply D to v and second enforce the interlayer dynamics by
setting (v 0 )ij to a proportion of a convex combination of the
{(Dv)`j }k`=1 ,
(v 0 )ij = αji
k
X
`
ci,`
j (Dv)j .
(1)
`=1
The constants ci,`
j provide the rate at which quantities pass
from layer ` to layer i through node j, while the αji allow
node j in layer i to send or receive quantities external to the
network. The restriction to a convex combination of the (Dv)`j
Pk
i,`
enforces that ci,`
j ≥ 0 and
`=1 cj = 1 for all 1 ≤ j ≤ n
and 1 ≤ i, ` ≤ k. Scaling by αji allows any linear combination
of the (Dv)ij with non–negative coefficients. As we will see,
we can perform this second operation algebraically using a
collection of scaled orthogonal projection operators, one for
each node in the multiplex.
This dynamic process preserves the intra–layer dynamics,
while also allowing effects to pass between layers through
the copies of each node. Unless the intra–layer dynamics
have a component that incorporates the value at the node, the
copies of a node do not interact directly. Instead the dynamical
effects acting on any particular copy are passed through to
the other copies, with proportions dictated by the coupling
constraints, the αji and ci,`
j . Consequently, we avoid the pitfalls
of aggregate and matched sum networks described above.
Derivation of the Operator
We accomplish step one of our method with the block
diagonal matrix D, so it suffices to construct a matrix that
redistributes the values between the copies of the node as described in equation (1). Let Pji = P(i−1)n+j be the projection
onto the (i − 1)n + j coordinate, Pji is a 1 × nk vector with
a one in the (i − 1)n + j entry and zeros elsewhere. Then,
Pji Dv = (Dv)ij . Using the αji and ci,`
j , we proportionally
distribute this value among all copies of node j using the
nk × 1 column vector
IV. M ULTIPLEX DYNAMICS
We present a model for multiplex dynamics that avoids the
structural conflation of single network models by allowing effects to pass between layers without artificially adding edges to
the network. For a multiplex structure on n nodes and k layers
with a matrix dynamical operator Di associated to each layer
1 ≤ i ≤ k we model the intra–layer dynamics by defining the
block diagonal matrix D = diag(D1 , D2 , . . . , Dk ). The matrix
D acts on an nk × 1 vector v representing the “quantities”
contained at k copies of each if the n nodes.
h
iT
k,i
1,i
k,i
.
αji c1,i
j δ1 (j), . . . , cj δn (j), . . . , cj δ1 (j), . . . , cj δn (j)
The projections are pairwise orthogonal, so we perform
the projection and redistribution steps concurrently using the
4
matrix
defined by the network structure or application. This is equivi,m
alent to the condition ci,`
for all 1 ≤ i, `, m ≤ k and
j = cj
1 ≤ j ≤ n.
For our second simplification, the hierarchical layer model,
we assume there is a hierarchy of layers where the effect of
layer i on layer ` is fixed for all the nodes. In this case, each of
the matrices C i,` is just a linear homothety or scalar multiple
i,`
`
`
of the identity, i.e. ci,`
a = cb and αa = αb for all layers a and
b. We see two natural approaches to defining these coefficients
in the absence of an application specific determination. The
first is to use the density of the layers as the C i,` . Second, we
set C i,` to be the ratio of the number of edges in layer i to
the number of edges in layer ` or more globally define C i,` to
be equal to the proportion of all of the edges in the multiplex
that occur in layer `.
This model is the assymmetric influence matrix W introduced in [32] for measuring eigenvector centrality in multiplex
networks (we recover the asymmetric version by assuming
ci,`
= ci,`
= c`,i
= c`,i
for all 1 ≤ a, b ≤ n). When
a
a
b
b
the Di are the adjacency matrices for the layers, we recover
their computation of the global heterogeneous eigenvector
centrality. Thus, our model is a direct generalization which
allows for arbitrary dynamics and more complex layer ranking
behavior individualized to each node.
In the next section, we use a still simpler version of this
i
model where we further assume that ci,`
j ≡ c for 1 ≤ ` ≤
k, 1 ≤ j ≤ n.
Our third and simplest version, the equidistribution model,
occurs when D is closed and the quantities at each copy of
a node are distributed equally among other copies. We set
1
ci,`
j = k for all 1 ≤ j ≤ n and 1 ≤ i, ` ≤ k and name the
resulting operator as E. This formulation is a natural model for
applications where the dynamic flow is equally likely to move
between layers, or when the quantities at each node represent
the outcomes of a binary process. In the next section we will
show that E has a particularly simple spectral structure related
to that of the aggregate network.

c1,i
j δ1 (j)


..


.


c1,i δ (j)
n


j
k
n
XX 
 i
..
i 
 Pj .
M=
αj 
.


 k,i
i=1 j=1
 cj δ1 (j) 


..




.
k,i
cj δn (j)

We realize M as block matrix,
 1,1
C
C 1,2
2,1
C
C 2,2

M = .
..
 ..
.
C k,1
C k,2
···
···
..
.
···

C 1,k
C 2,k 

..  ,
. 
C k,k
` i,`
` i,`
where C i,` = diag(α1` ci,`
1 , α2 c2 , . . . , αn cn ).
The final multiplex dynamic operator is a product of the
layer dynamics matrix D and the redistribution matrix M
 1,1

C D1 C 1,2 D2 · · · C 1,k Dk
C 2,1 D1 C 2,2 D2 · · · C 2,k Dk 


D = MD =  .
.
..
.
.
.
.
.
 .

.
.
.
C k,1 D1
C k,2 D2
···
C k,k Dk
Before examining the spectral properties of D we discuss three special cases of D reflecting natural modeling
constraints. We begin with a useful definition, describing a
hypothesis that enforces a conservation law for the quantities
under consideration.
Definition 1. The operator D is called closed if αji = 1 for
all 1 ≤ i ≤ k and 1 ≤ j ≤ n
When D is closed, the matrix M is stochastic, which we
will use in the next section when analyzing multiplex random
walk dynamics. When D is closed and the Di are random
walk matrices we recover the models used in [14] and [34].
We motivate our first simplification, which we call the
unified node model, with an example using economic exchange
networks. Consider trade relationships between countries,
where bilateral trade occurs within commodity based layers,
but parties inside the countries redistribute money internally
redistributed before international trade begins anew. This allows countries to use surpluses in one area to fund deficits
in another. Further, the α coefficients represent unmodeled
sources or sinks of income external to the trade network.
To adjust our model, we enforce a homogeneity condition
at each node: instead of computing
Pk M from nk projections
we use the n projections Qj = `=1 αj` Pj` , j ∈ {1, . . . , n}.
The value of the product Qj Dv is the weighted sum of the
quantities at node j on each layer. When D is closed, Qj
simply becomes the uniform projection onto the subspace
spanned by all of the copies of node j.
For each node j, we redistribute Qj Dv proportionally
among its copies according to their relative importance as
V. S PECTRAL R ESULTS
We present basic spectral results for our multiplex formulation and show that our operators arise as natural multiplex
interpretations of standard network dynamic models, such as
random walks and heat diffusion, derived from first principles.
One particularly valuable aspect of D is that under reasonable assumptions, it preserves matrix properties of the original
layer dynamics, such as primitivity, stochasticness, and postive
semi–definiteness. These properties provide the link between
the spectrum of the operator and the multiplex structure. We
begin with a simple result, relating the eigenvalues of the
equidistribution operator E to the spectrum of the aggregate
network.
Proposition 1. Except for P
zero, the eigenvalues of E are
k
exactly the eigenvalues of k1 `=1 D` .
Similar results exist for the other formulations, but they are
more complex. In the simplest version of the hierarchical layer
5
i
model, where ci,`
j ≡ c , we have that the non–zero eigenvalues
Pk
i
of D are exactly the eigenvalues of
i=1 c Di . Although
determining the spectral structure of sums of matrices in
terms of the spectra of the summands is difficult, when the
matrices are symmetric, we can use standard results due to
Weyl described in [16], [24] to obtain upper and lower bounds
on the individual eigenvalues of the derived operators (as in
Proposition 4).
its neighbors:
k
X
X
dvji
`
(vj` − vm
=K
).
ci,`
j
dt
`
`
`=1
nj ∼nm
Here K is the diffusion constant and the ci,`
j represent the
proportion of the effect on layer ` that passes through to nij .
Linear algebraically, this is:
" k
#
X i,`
dvji
=K
cj L` v `
(2)
dt
Stochastic Dynamics
`=1
A left stochastic matrix is a non–negative matrix where the
entries in each column sum to one. Such a matrix defines a
Markov process on a network, with the entries Di,j corresponding to the probability of moving from from node j to
node i at each time step. Stochastic processes on networks
are commonly used for determining centrality and community
detection.
If a stochastic operator is primitive or irreducible, the
Perron–Fröbenius Theorem guarantees that the largest eigenvalue of that matrix is 1 and that the entries in one of
the corresponding eigenvectors are non–negative. This vector
represents the limiting state of the dynamical system for
arbitrary input.
j
which agrees with (1) when the layer dynamics are given by
the respective layer Laplacians, Li .
This model implicitly assumes that each copy of each node
can be assigned a separate temperature. This is a reasonable
assumption for applications such as some economic exchange
networks or the transportation models considered in [14].
However, in examples such as our social multiplex where the
node copies are only representing different interaction types
associated to the same individual, we enforce the additional
condition that vji = vj` for all 1 ≤ j ≤ n and 1 ≤ i, ` ≤ k.
Intuitively, at each time step, each individual has some fixed
amount of information regardless of what types of interactions
they are performing.
The equidistribution operator E, encodes this additional
condition. As we’ve reduced to n independent variables, we
restate Equation (2) in terms of a n × q vector
hPw recording
i
dw
k
`
.
the temperature at each node as dtj = K
`=1 L w
k
Proposition 2. If each Di is stochastic and D is closed then
D is stochastic. Additionally, if each Di is irreducible then D
is irreducible and if each Di is primitive then D is primitive.
The first statement follows from the fact that the hypotheses
guarantee that D is the product of two stochastic operators,
and the second two conclusions can be easily shown from
the graph theoretic interpretation of the conditions. We view
a stochasitic D as describing a random walk on the multiplex
with the probability of transitioning from the copy of node j
on layer ` to a neighbor of node j on any arbitrary layer i
1
i
given by ci,`
j deg(nij ) , where nj is the copy of node i in layer
j.
This Markov formulation is equivalent to the problem
considered in [34] for modeling distributed concensus which
provides additional confirmation of our methods. Under similar hypotheses to those in [14], [34], our operator reduces to
a version of their supratransition matrix.
j
The ability to models these types of dynamics distinguishes
our analysis from previous studies of diffusion on multiplex
networks [21].
Using tools from the theory of Hermitian matrices we can
prove the following bounds for the operator E when the intra–
layer dynamics are Laplacians.
Proposition 3. If each Di is the graph Laplacian then E is
positive semi–definite and the eigenvectors of E corresponding
to distinct non–zero eigenvalues are orthogonal.
Consequently, the solutions to the differential equation ddtϕ̂ +
Eϕ̂ = 0 have the same algebraic and analytic structure as the
standard Laplacian: the solution consists of constant elements
and terms that decay exponentially. We now find eigenvalue
bounds for E in this case. Our first result shows that diffusion
across the multiplex network should happen at least as fast
as diffusion across the best connected layer. We assume that
the layer networks are connected - although similar bounds
exist in the case where the layer networks are disconnected,
the formulas become more complex.
We also introduce some additional notation. Let λij be the
j th eigenvalue of Di written in descending order with λif
representing the Fiedler value corresponding to Di . Let the
eigenvalues of E be λ1 ≥ λ2 · · · ≥ λkn . As rank(E) = n − 1,
for ` > n − 1 we have λ` = 0. Finally, let m be the index
i
such that λm
1 = maxi (λ1 ).
Diffusion Dynamics
We often model diffusion on networks as a discretization
of the continuous heat flow problem, which yields the graph
Laplacian, L = D − A, whose spectral structure has strong
connections to important graph properties such as connectivity,
communities, and random walks [6], [8], [28].
To extend this model to the multiplex setting, we consider
an initial vector v representing the temperature at each node
copy and define the change in the value of vji with respect
to time to be proportional to the sum of the differences in
temperature between each copy of node j (including nij ) and
6
Proposition 4. If each Di is the graph Laplacian then we
have the following eigenvalue bounds for the operator E
P
j
i
m
• Fiedler Value: maxi (λf ) ≤ kλf ≤ λf +
j6=` λ1 ,
P
i
i
• Leading Value: maxi (λ1 ) ≤ kλ1 ≤
i λ1 ,
[5] Stephen P. Borgatti. Centrality and network flow. Social Networks,
27(1):55–71, January 2005.
[6] Richard A. Brualdi. The Mutually Beneficial Relationship of Graphs
and Matrices. American Mathematical Society, Providence, R.I, July
2011.
[7] Sergey V. Buldyrev, Roni Parshani, Gerald Paul, H. Eugene Stanley,
and Shlomo Havlin. Catastrophic cascade of failures in interdependent
networks. Nature, 464(7291):1025–1028, April 2010.
[8] Fan R. K. Chung. Spectral Graph Theory. American Mathematical
Society, Providence, R.I, December 1996.
[9] Emanuele Cozzo, Mikko Kivela, Manlio De Domenico, Albert Sole,
Alex Arenas, Sergio Gomez, Mason A. Porter, and Yamir Moreno.
Structure of Triadic Relations in Multiplex Networks. arXiv:1307.6780
[cond-mat, physics:physics], July 2013. arXiv: 1307.6780.
[10] Manlio De Domenico, Vincenzo Nicosia, Alexandre Arenas, and Vito
Latora. Structural reducibility of multilayer networks. Nature Communications, 6, April 2015.
[11] Manlio De Domenico, Albert Sole-Ribalta, Emanuele Cozzo, Mikko
Kivela, Yamir Moreno, Mason A. Porter, Sergio Gomez, and Alex
Arenas. Mathematical Formulation of Multilayer Networks. Physical
Review X, 3(4):041022, December 2013.
[12] Manlio De Domenico, Albert Sole-Ribalta, Elisa Omodei, Sergio
Gomez, and Alex Arenas. Centrality in Interconnected Multilayer Networks. Nature Communications, 6:6868, April 2015. arXiv: 1311.2906.
[13] Manlio De Domenico, Albert Sole-Ribalta, Elisa Omodei, Sergio
Gomez, and Alex Arenas. Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications, 6, April 2015.
[14] Manlio De Domenico, Albert Sole-Ribalta, Sergio Gomez, and Alex
Arenas. Navigability of interconnected networks under random failures.
Proceedings of the National Academy of Sciences, 111(23):8351–8356,
June 2014.
[15] Nicholas J. Foti, Scott Pauls, and Daniel N. Rockmore. Stability of
the World Trade Web over time An extinction analysis. Journal of
Economic Dynamics and Control, 37(9):1889–1910, September 2013.
[16] William Fulton. Eigenvalues, invariant factors, highest weights, and
Schubert calculus. Bulletin of the American Mathematical Society,
37(3):209–249, 2000.
[17] Riccardo Gallotti and Marc Barthelemy. The multilayer temporal
network of public transport in Great Britain. Scientific Data, 2:140056,
January 2015.
[18] Jianxi Gao, Sergey V. Buldyrev, Shlomo Havlin, and H. Eugene Stanley.
Robustness of a Network of Networks. Physical Review Letters,
107(19):195701, November 2011.
[19] Jianxi Gao, Sergey V. Buldyrev, H. Eugene Stanley, and Shlomo Havlin.
Networks formed from interdependent networks. Nature Physics,
8(1):40–48, January 2012.
[20] Juan P. Garrahan and Igor Lesanovsky. Comment on ”Abrupt transition
in the structural formation of interconnected networks”, F. Radicchi and
A. Arenas, Nature Phys. 9, 717 (2013). arXiv:1406.4706 [cond-mat],
June 2014. arXiv: 1406.4706.
[21] S. Gomez, A. Diaz-Guilera, J. Gomez-Gardenes, C. J. Perez-Vicente,
Y. Moreno, and A. Arenas. Diffusion Dynamics on Multiplex Networks.
Physical Review Letters, 110(2):028701, January 2013.
[22] B. Kapferer. Strategy and transaction in an African factory. Manchester
University Press, Manchester, 1972.
[23] Mikko Kivela, Alex Arenas, Marc Barthelemy, James P. Gleeson, Yamir
Moreno, and Mason A. Porter. Multilayer networks. Journal of Complex
Networks, page cnu016, July 2014.
[24] Allen Knutson and Terence Tao. The honeycomb model of (c) tensor
products I: Proof of the saturation conjecture. Journal of the American
Mathematical Society, 12(4):1055–1090, 1999.
[25] David Krackhardt. Cognitive social structures. Social Networks,
9(2):109–134, June 1987.
[26] E. A. Leicht and Raissa M. D’Souza. Percolation on interacting
networks. arXiv:0907.0894 [cond-mat], July 2009. arXiv: 0907.0894.
[27] M. E. J. Newman. Detecting community structure in networks. The
European Physical Journal B - Condensed Matter and Complex Systems,
38(2):321–330, March 2004.
[28] Mark Newman. Networks: An Introduction. Oxford University Press,
Oxford ; New York, 1 edition edition, May 2010.
[29] Filippo Radicchi. Driving Interconnected Networks to Supercriticality.
Physical Review X, 4(2):021014, April 2014.
These bounds are special cases of the following more
general but less computationally feasible bounds:
!!
k
X
σ(i)
i
maxi (λn−` ) ≤ kλn−` ≤
min
min
λ ji
,
J`n+k−(`+1)
σ∈Sn
i=1
Pk
where J = (j1 , j2 , . . . , jk ) such that i=1 ji = n + k − (` +
1). As remarked after Proposition 1, we can give a similar
characterization of the eigenvalues of the simplest case of the
hierarchical layer model, where we obtain results equivalent to
Propositions 3 and 4, replacing λij with (ci λi )j (after possible
reordering) in each occurrence.
VI. D ISCUSSION AND C ONCLUSIONS
The field of multiplex networks is rapidly becoming an
important segment of the complex systems community. Although many natural structural representations for multiplex
structures have been presented, there is still a great deal left
to learn about multiplex structures and their associated dynamics. To faithfully represent dynamical processes on multiplex
networks, we have shown it is sometimes necessary to avoid
the confounding structural features introduced by tensorial
and supra– constructions. To obtain meaningful results in the
multiplex setting we must verify that the correct analogy with
the monoplex network case holds.
While there may be particular applications for which these
structurally motivated dynamics are well suited, in general we
recommend that researchers choose dynamical models that do
not introduce structural features in order to more accurately
capture the dynamics of interest. It is particularly important
when relying on spectral computations and invariants that the
associated operator have a natural interpretation that allows
the appropriate Rayleigh quotient to be derived as a solution
to the given objective function.
Our model avoids the structural defects inherent in single
network multiplex representations and allows the inherent
layer dynamics to interact without explicitly confounding
structural effects, generalizing several successful models to a
wide class of network dynamics.
R EFERENCES
[1] N. Azimi-Tafreshi, J. Gomez-Gardenes, and S. N. Dorogovtsev. k-core
percolation on multiplex networks. Physical Review E, 90(3):032816,
September 2014.
[2] Albert-Laszlo Barabasi and Eric Bonabeau. Scale-free networks. Scientific American, 288(5):60–69, May 2003.
[3] Matteo Barigozzi, Giorgio Fagiolo, and Diego Garlaschelli. Multinetwork of international trade: A commodity-specific analysis. Physical
Review E, 81(4):046104, April 2010.
[4] Matteo Barigozzi, Giorgio Fagiolo, and Giuseppe Mangioni. Community
Structure in the Multi-network of International Trade. In Luciano
da F. Costa, Alexandre Evsukoff, Giuseppe Mangioni, and Ronaldo
Menezes, editors, Complex Networks, number 116 in Communications
in Computer and Information Science, pages 163–175. Springer Berlin
Heidelberg, 2011.
7
[30] Filippo Radicchi and Alex Arenas. Abrupt transition in the structural
formation of interconnected networks. Nature Physics, 9(11):717–720,
September 2013. arXiv: 1307.4544.
[31] Ruben J. Sanchez-Garcia, Emanuele Cozzo, and Yamir Moreno. Dimensionality reduction and spectral properties of multilayer networks.
Physical Review E, 89(5):052815, May 2014.
[32] Luis Sola, Miguel Romance, Regino Criado, Julio Flores, Alejandro
Garcia del Amo, and Stefano Boccaletti. Eigenvector centrality of nodes
in multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear
Science, 23(3):033131, September 2013.
[33] A. Sole-Ribalta, M. De Domenico, N. E. Kouvaris, A. Diaz-Guilera,
S. Gomez, and A. Arenas. Spectral properties of the Laplacian of
multiplex networks. Physical Review E, 88(3):032807, September 2013.
[34] I. Trpevski, A. Stanoev, A. Koseska, and L. Kocarev. Discrete-time
distributed consensus on multiplex networks. New Journal of Physics,
16(11):113063, November 2014.
[35] W. Zachary. An information flow model for conflict and fission in small
groups. Journal of Anthropological Research, 33:452–473, 1977.
8