Observability transition in multiplex networks

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by  Saeed Osat

Observability transition in multiplex networks
Saeed Osat1 and Filippo Radicchi2
1Molecular Simulation Laboratory, Department of Physics, Faculty of Basic Sciences,
Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran
2Center for Complex Networks and Systems Research, School of Informatics and Computing,
Indiana University, Bloomington, Indiana 47408, USA∗
Weextendtheobservabilitymodeltomultiplexnetworks. Wepresentmathematicalframeworks,
validunderthetreelikeansatz,abletodescribetheemergenceofthemacroscopicclusterofmutually
observable nodes in both synthetic and real-world multiplex networks. We show that the observ-
abilitytransitioninsyntheticmultiplexnetworksisdiscontinuous. Inreal-worldmultiplexnetworks
instead,edgeoverlapamonglayersisresponsibleforthedisappearanceofanysignofabruptnessin
the emergence of the the macroscopic cluster of mutually observable nodes.
I. INTRODUCTION model finds its motivation in the study of some dynami-
cal processes where the state of the system can be deter-
7
mined by monitoring or dominating the states of a lim-
1 Complex systems where elementary units have differ-
0 ited number of nodes in the network [35]. Examples in-
enttypesofinteractionscanbeconvenientlymodelledas
2 clude, among others, real-time monitoring of power-grid
multiplexnetworks[1–3]. Thisisaverygenericrepresen-
networks [33] and mobile ad-hoc networks [36]. The ob-
n tation, where elements of a system are organized in mul-
servability model has been considered in synthetic mod-
a tiple network layers, each standing for a specific color or
J els [33] and real-world topologies [34]. In the simplest
flavour of interaction. Systems that can be represented
5 versionofthemodel, everynodeinthenetworkcanhost
in this way are abundant in the real world. Examples
1 an observer with probability φ. Placing an observer on
include, among others, social networks sharing the same
one node can make the node itself and all its nearest
actors[4,5],andmultimodaltransportationgraphsshar-
] neighbors observable. Nodes in the network can there-
h ing common geographical locations [6, 7].
p fore assume three different states: (i) directly observ-
- Several analyses of multiplex networks have been per- able, if hosting an observer; (ii) indirectly observable, if
c formed recently [1–3]. A common result, shared by the being the first neighbor of an observer; (iii) or not ob-
o
vastmajorityofthesestudies, isthataprocessesdefined servable, otherwise. Observable, either directly or indi-
s
. on a multiplex network is characterized by features rad- rectly,nearest-neighbornodesformclustersofconnected
s
c ically different from those observed for the same type of observablenodes. Asinthecaseofpercolation,theques-
i process when this is applied to an isolated network. Ex- tionofinterestinnetworkobservabilityisunderstanding
s
y amples regard dynamical processes taking place on mul- the macroscopic formation of observable clusters in the
h tiplexnetworks,suchasdiffusion[8,9],epidemicspread- network on the basis of microscopic changes in the state
p ing[10–13],synchronization[14],andcontrollability[15]. of its individual nodes. In synthetic infinite graphs, the
[ Examples include also structural processes as those typ- macroscopic cluster of observable nodes obeys a contin-

1 ically framed in terms of percolation models. For in- uous phase transition as a function of the probability φ,
v stance in their seminal paper, Buldyrev et al. consid- and the critical threshold is generally very small [33]. In
2 eredasite-percolationmodelaimedatunderstandingthe most real-world networks also, the largest cluster grows
4 role of interdependencies among two network layers [16]. smoothly, andthetransitionpointisgenerallyveryclose
0 Themacroscopicclusterofmutuallyconnectednodesina to zero [34].
4 multiplexnetworkemergesdiscontinuously, atoddswith
0
whatinsteadobservedforthesameprocessonanisolated In our extension of the observability model to multi-
.
1 network where the percolation transition is always con- plex networks, we focus our attention on the emergence
0 tinuous. Alargenumberofsubsequentstudieshavethen of clusters of mutually observable nodes. The definition
7
analyzed in detail the features of the percolation transi- of these clusters is a straightforward combination of the
1
tioninmultiplexnetworks[17–28]. Severalvariantsofthe notions of clusters of observable nodes in isolated net-
:
v percolation model have been also considered, including works [33] and clusters of mutually connected nodes in
Xi redundant site percolation [29], k-core percolation [30], multiplex networks [16]. The extension of the observ-
weak percolation [31], and bond percolation [32]. ability model to multiplex networks finds its rationale in
r
a In this paper, we focus our attention on an additional any situation where the goal is monitoring or controlling
variant of the percolation model usually named as the thedynamicsofasystemstructuredinmultiplelayersof
observability model [33, 34]. In isolated networks, the interactions. A genuine example could be tracking the
spread of information on two coupled social media, such
as Facebook and Twitter. Actors are shared by the two
social networks. Observing an individual means getting
∗Electronicaddress: fi[email protected] fullaccesstoher/hisaccountsthusbeingabletoreadthe
ii
content of the messages that the individual, and her/his changes as a function of the microscopic probability φ of
friends, are posting on the two platforms. Clusters of nodes to be directly observable. To generate the mathe-
mutually observable nodes in this sense correspond to matical framework necessary to study such a model, we
connected portions of the systems where the content of makeuseofasimpleself-consistentcondition: anobserv-
information that is exchanged by people within the clus- able node belongs to the LMOC if it has, in both layers,
ter can be completely monitored. at least one neighbor that is part of the LMOC. Such a
The paper is organized as follows. In section II, we condition can be expressed in terms of elementary con-
define the observability model in multiplex networks. In ditionalprobabilitiesdefinedfortheedgesandthenodes
sectionIII,westudythemodelinensemblesofmultiplex of the multiplex network. In the following, we discuss
networkswhoselayersaregeneratedaccordingtothecon- the details of how the mathematical framework can be
figuration model. In section IV, we introduce a message- solved exactly under two conditions: (i) absence of link
passing method to deal with the observability model in overlapamonglayers,and(ii)validityofthelocallytree-
multiplex networks with specified topology, such as real- like ansatz. In the Supplemental Material, we report the
world multiplex networks. The framework described in full mathematical framework valid when condition (i) is
this section is based on the assumption that the number removed.
of edges shared by the different layers of the multiplex is
negligible. The message-passing method valid for multi-
plex networks in presence of edge overlap is much more III. OBSERVABILITY TRANSITION IN
cumbersome, and, for this reason, presented only in the RANDOM MULTIPLEX NETWORKS
Supplemental Material. Finally, in section V, we sum-
marize the main findings of the paper. Let us consider the case of a multiplex composed of
twonetworklayersgeneratedindependentlyaccordingto
theconfigurationmodel[37]. Ifthelayersaresufficiently
sparse, the fact that the layers are generated indepen-
II. OBSERVABILITY MODEL IN MULTIPLEX
NETWORKS dentlyallowustoconsidertheoverlapamonglayers(i.e.,
the simultaneous existence of the same edge in both lay-
ers happens with vanishing probability) negligible. The
WeconsideramultiplexnetworkcomposedofN nodes
only inputs required to study such a system are the de-
structured in two layers, namely α and β. Node labels
gree distributions of the two layers, namely P[α](k) and
take integer values from 1 to N in both layers. Nodes
P[β](k).
with identical labels correspond to copies (or replicas) of
The analytic treatment for this special types of net-
the same individual or unit in the two layers. The ob-
works is similar to one presented in Ref. [33] for isolated
servability model we consider here is a natural extension
networks. To generate a self-consistent set of equations
to multiplex networks of the same model already consid-
abletodescribehowtheLMOCchangesasafunctionofφ
ered on isolated networks [33, 34]. Observers or sensors
in this special type of multiplex networks, we define the
are placed at random with probability φ on every node
following conditional probabilities valid for a randomly
in the system. If an observer is placed on a node i, the
selected edge in layer α:
node i is directly observable in both layers. A node i,
thatisnotdirectlyobservable,butisattachedtoatleast
1. u[α], that is the probability that one of the nodes
onedirectlyobservablenodej inlayerαandatleastone
at the end of the edge is in the LMOC given that
directly observable node k in layer β, with j not neces-
the other node at the end of the edge is directly
sarily equal to k, is indirectly observable. We focus our
observable.
attention on mutually observable clusters of nodes. The
way these clusters are defined is identical to the one in 2. v[α], that is the probability that one of the nodes
whichclustersofmutuallyconnectednodesaredefinedin
at the end of the edge is in the LMOC given that
site percolation [16]. The only difference comes from the
thenodeattheotherendoftheedgeisnotdirectly
fact that a node in order to be “occupied” can be either
observable.
in the directly or indirectly observable state. Note that
mutuallyconnectedandmutuallyobservableclustersco- 3. z[α], that is the probability that one of the nodes
incide for φ = 1. In particular, a cluster of mutually at the end of the edge is in the LMOC given that
observable nodes is defined in a recursive manner and both nodes at the end of the edge are not directly
is composed by all the nodes that are connected by at observable.
least by one path (internal to the cluster itself) in both
layers. Our model can be seen as a depth-one percola- The same exact definitions are valid for the conditional
tion model on a multiplex network. As we are interested probabilities u[β],v[β], and z[β] for layer β.
in understanding the extent of the system that can be As mentioned above, a generic node is part of the
monitored by placing random points of observation, the LMOC if observable, either directly or indirectly, and
focus of our analysis is centered on quantifying how the attached to at least one other node that is part of the
sizeoftheLargestMutuallyObservableCluster(LMOC) LMOC. The way this happens depends however on the
iii
state of the node. We make therefore a distinction be- our indirectly observable node is part of the LMOC. If
tween(i)adirectlyobservablenodeand(ii)anotdirectly we consider all nodes, we have that
observable node.
Let us consider first case (i). If one of the nodes at r[α] =(cid:80) P[α](k)r[α] =
k k
theendofagenericedgeisdirectlyobservable,theprob- 1 G[α](cid:0)1 φu[α] (1 φ)z[α](cid:1)+ .
ability that the node is not connected to the LMOC is +−G[α]0(cid:0)(1 −φ)(1 −z[α])(cid:1)− G[α](1 φ)
φ(1 u[α]) if the other node is directly observable, or 0 − − − 0 −
(1 −φ)(1 v[α]) if the other node is not directly observ- Forlayerβ,theargumentsareidentical. Theprobability
− −
able. If the node we are considering has degree k, then thatagenericnodethatisnotdirectlyobservableispart
thenprobabilitythatthenodeisconnectedtotheLMOC of the LMOC is given by the product that the node is
in layer α is attached to the LMOC simultaneously in both layers,
that is
k (cid:18) (cid:19)
q[α] =1 (cid:88) k [φ(1 u[α])]m[(1 φ)(1 v[α])]k−m . r =(1 φ)r[α]r[β] , (2)
k − m − − − −
m=0
where the extra factor 1 φ comes from the fact that
The sum on the r.h.s. of the equation above quantifies we are considering the cas−e of a node that is not directly
the probability for the none of the neighbors of a node observable. Combiningcases(i)and(ii)together,wecan
with degree k is part of the LMOC. This probability is finally write
then discounted from 1 to estimate the probability that
at least one neighbor of the node is part of the LMOC. P =q+r , (3)
∞
If we consider all nodes, we have that
for the average size of the LMOC. To compute Eq. (3),
q[α] =(cid:88)P[α](k)q[α] =1 G[α](cid:16)1 φu[α] (1 φ)v[α](cid:17) , westillrequireawaytoestimateproperlytheconditional
k − 0 − − − probabilities u[α],u[β],v[α],v[β],u[α], and z[β]. Following
k
a similar approach as the one described above, we have
whereG[α](x)=(cid:80) P[α](k)xk isthegeneratingfunction that
0 k
of the degree distribution for layer α. We can repeat u[α] = (cid:2)1 G[α](1 φu[α] (1 φ)v[α])(cid:3)
the same exact arguments for layer β. In particular, the (cid:2)1−G[1β](1−φu[β] −(1−φ)v[β])(cid:3)× , (4)
probability that a generic node is part of the LMOC is − 0 − − −
given by the product that the node is attached to the
LMOC simultaneously in both layers, that is v[α] = (cid:2)1 G[α](1 φu[α] (1 φ)z[α])(cid:3)
q =φq[α]q[β] , (1) (cid:2)1G−−[βG](1[01β](1φ−−)+φuG[β[β]]−−(((11−−φφ))(z1[β])z+[×β]))(cid:3) , (5)
− 0 − 0 − −
wheretheextrafactorφcomesfromthefactthatweare
and
consideringthecaseofanodethatisdirectlyobservable.
Forcase(ii),wecanproceedinasimilarwayasabove. z[α] = (cid:2)1 G[α](1 φu[α] (1 φ)z[α])+
Ifournodeisnotdirectlyobservable,andweselectoneof G−[α](11 φ−)+G[α]−((1 −φ)(1 z[α]))(cid:3)
iitssneodtgceosnantecratenddotmo t,hteheLnMtOheCpirsoφba(1bilituy[αt]h)aitftthhiesontohdeer −(cid:2)1 1G[β]−(1 φu[β1] (1− φ)z[β−])+ × . (6)
nodeattheendoftheedgeisdirectly−observable,or(1 G−[β](01 φ−)+G[β]−((1 −φ)(1 z[β]))(cid:3)
φ)(1 z[α])iftheothernodeisnotdirectlyobservable. −If − 0 − 0 − −
then−odeweareconsideringhasdegreek,theprobability The intuition behind the previous equations is straight-
that this node is connected to the LMOC in layer α is is forward. When looking at an edge on layer α, the
probability that this edge will bring to the LMOC
r[α] = 1 (1 φ)k+ will depend on the generating function of the ex-
k −(cid:80)k − (cid:0)k(cid:1)[φ(1 u[α])]m[(1 φ)(1 z[α])]k−m . cess degree distribution of the layer, i.e., G[1α](x) =
− m=1 m − − − [d/dxG[α](x)]/[d/dxG[α](1)]. On the other hand, the
0 0
The term (1 φ)k comes from the fact that if none of probability that a node at the end of this edge is also
the nodes at−the end of the k edges departing from our attached to the LMOC on layer β will depend only on
node are directly observable, then necessarily our node thedegreeofthenodeitselfonlayerβ,accountedbythe
will be not indirectly observable and thus surely out of generating function of the degree distribution, namely
any cluster. The remaining part of the r.h.s. instead G[β](x). Note that equations for u[β],v[β], and z[β] can
0
quantifies the probability that none of the neighbors of beobtainedbysimplyswappingαandβ inEqs.(4), (5),
ournodeispartoftheLMOC,assumingthatatleastone and (6).
of them is directly observable. Both these probabilities Atthispoint,wehaveeverythingnecessarytoestimate
are discounted from 1 to compute the probability that Eq.(3). WehavefirsttosolveEqs.(4), (5),and (6),and
iv
1.0 20
a
2 1
c −
˜φ
2 2
−
2 3
−
21 22 23 24 25
∞
0.5
P
hki=3.0 20
k =5.0
h i ) b
k =7.0 + 2 2
hki=10.0 φc −
hki=15.0 (∞2 4
h i −
k =20.0 P
0.0 h i 2 6
−
0.0 0.5 1.0 21 22 23 24 25
φ k
h i
Figure 1: Observability transition in random Poisson multi- Figure 2: Observability transition in random Poisson multi-
plex networks. We consider multiplex networks with layers plex networks. (a) We plot the rescaled value of the critical
generatedindependentlyandgivenbyrealizationsofthecon- threshold φ˜ = φ +(1−φ )[1−(1−φ )(cid:104)k(cid:105)] (black circles)
c c c c
figurationmodelwithN =10,000nodesanddegreesequence as a function of the average degree (cid:104)k(cid:105) for Poisson multiplex
generatedaccordingtoaPoissondistributionwithaveragede- networksofinfinitesize. Wefindthatφ˜ ∝(cid:104)k(cid:105)−3/4 (grayfull
c
gree (cid:104)k(cid:105). Results of our theoretical method (small symbols)
line). The vertical red line indicates the value (cid:104)k(cid:105) (cid:39)2.4554
c
are compared with those of numerical simulations of the ob- where φ˜ = 1 (horizontal red line). (b) Height of the dis-
servability model (large symbols). We consider different val- c
continuous jump P (φ+) as a function of the average de-
ues of (cid:104)k(cid:105), (cid:104)k(cid:105) = 3.0,5.0,7.0,10.0,15.0, and 20.0. Values of ∞ c
gree (cid:104)k(cid:105) (black circles) for the same multiplex networks of
P forthevariouscasesdroptozerointhesameorderifthe
∞ panel (a). We find that P (φ+) ∝ (cid:104)k(cid:105)−7/4 (gray full line).
figure is read from right to left. ∞ c
The vertical red line indicates the value (cid:104)k(cid:105) (cid:39) 2.455 where
c
P (φ+)(cid:39)0.511 (horizontal red line).
∞ c
theanalogousonesforlayerβ,byiteration. Then,wecan
plug the obtained values of the conditional probabilities
would predict. Instead, the scaling is compatible with
for generic edges into the equations for the nodes. φ˜ k −3/4 (Fig. 2a). The height of the discontinuous
c
Figure 1 shows a comparison between the results of jum∝p(cid:104)P(cid:105) (φ+) also doesn’t go to zero as k −1 as pre-
direct numerical simulations of the observability model ∞ c (cid:104) (cid:105)
dicted in the standard percolation model, but instead as
and the numerical solutions of our equations for duplex P (φ+) k −7/4 (Fig. 2b).
networksformedbylayersobeyingPoissondegreedistri- ∞ c ∝(cid:104) (cid:105)
butions, i.e., P[α](k) = P[β](k) = (cid:104)k(cid:105)ke−(cid:104)k(cid:105). The figure
(cid:104)k(cid:105)!
provides evidence of a perfect agreement between theory IV. OBSERVABILITY TRANSITION IN REAL
and simulations. As the generating functions G[α], G[α], MULTIPLEX NETWORKS
0 1
G[β],andG[β]canbewritteninaclosedformforthePois-
0 1 In this section, we develop an analytic framework able
son distribution, we can also study the numerical solu-
to approximate the phase diagram of the observability
tionoftheequationsforinfinitelylargenetworks(Fig.2).
transitionforaduplexwithgivenadjacencymatricesfor
No transition takes place for k <2.4554, i.e., the same
(cid:104) (cid:105) the two layers. This information is encoded in the sets
valuefoundforthestandardpercolationmodel[16]. This
isexpectedfromtheequivalencebetweenbetweentheob- ∂[α] and ∂[β] containing the neighbors of node i respec-
i i
servabilitymodelandthestandardpercolationmodelfor tivelyinlayersαandβ foreverynodei. Weindicatethe
φ = 1. For k 2.4554, the behavior of the observabil- sizes of these sets respectively as k[α] and k[β]. We make
(cid:104) (cid:105) ≥ i i
itymodelbecomesdifferentfromtheoneobservedinsite use of two assumptions: (i) layers have null overlap in
percolation. There, we have that the critical occupation the sense that set of neighbors of the same node in the
probability p and the height of the discontinuous jump two layers have null intersection; (ii) neighbors of a node
c
P (p+)decaytozeroasafunctionoftheaveragedegree are not attached to each other.
∞ c
k as p P (p+) k −1 [16]. For the observability Supposeweareinterestedinestimatingtheprobability
(cid:104) (cid:105) c ∝ ∞ c ∝ (cid:104) (cid:105)
model, we find instead that the rescaled critical thresh- s that node i is part of the LMOC. This will happen if
i
old φ˜ = φ +(1 φ )[1 (1 φ )(cid:104)k(cid:105)] doesn’t decrease nodeiisreceivingatleastonemessageinlayerαandone
c c c c
as k −1 as a na−ive map−ping−between the two models message in layer β about the belonging to the LMOC.
(cid:104) (cid:105)
v
1.0 indirectly observable and attached to at least one other
node in layer α and layer β that are part of the LMOC.
Considering only layer α, for case (i), we can write
(cid:20) (cid:21)
∞0.5 qi[α] = 1− (cid:81) (1−φu[jα→]i−(1−φ)vj[α→]i) . (7)
P j∈∂i[α]
For case (ii), we have instead
γ=2.5
γ=3.0 (cid:20)
γ=3.5 r[α] = 1 (cid:81) (1 φu[α] (1 φ)z[α] )+
0.0 i −j∈∂[α] − j→i− − j→i
0.0 0.5 1.0 i (cid:21) . (8)
φ −(1−φ)ki[α](cid:2)1− (cid:81) (1−zj[α→]i)(cid:3) ×
j∈∂[α]
i
The same type of equations are valid for layer β. As the
Figure 3: Observability transition in a single random scale- node must be part of the LMOC in both layer simulta-
freemultiplexnetworks. Weconsidermultiplexnetworkswith neously, we can write
layers generated independently and given by realizations of
the configuration model with N = 10,000 nodes and degree
sequencegeneratedaccordingtoapower-lawdegreedistribu- q =φq[α]q[β] , (9)
i i i
tion P[α](k) = P[β](k) ∼ k−γ with degree exponent γ (min-
imal degree is set k = 3, so that edge overlap between
min and
layers is negligible). Results of message passing framework
without overlap (small symbols) are compared with those of
numerical simulations of the observability model (large sym- r =(1 φ)r[α]r[β] , (10)
bols). We consider different values of γ, γ =2.5,3.0 and 3.5. i − i i
Values of P for the various cases drop to zero in the same
∞
order if the figure is read from right to left. where the factors φ and 1 φ come from the fact that
−
we are considering the cases of a directly observable and
a non directly observable node, respectively. The size of
We consider average message values over the ensemble the average LMOC is given by
of random placements of observers in the network. In
particular, we define three different messages for every 1 (cid:88)
P = q +r . (11)
edge j i in layer α dependent on the states of the ∞ N i i
nodes j→and i: i
1. u[α] , that is the average value of the message that Theself-consistentequationsforthemessagesareinstead
j→i given by
node j sends to node i, when j is directly observ-
able.
(cid:20) (cid:21)
2. vnj[oα→d]ei,jthsaetndisstthoenaovdereaig,ewvhaelunejofisthneotmdeisrseacgtelythoabt- (cid:20)u[jα→]i = 1−(cid:81)k∈∂j[α]\i(1−φu[kα→]j −(1−(cid:21)φ)vk[α→]j) × ,
1 (cid:81) (1 φu[β] (1 φ)v[β] )
servable. − k∈∂j[β] − k→j − − k→j
(12)
3. z[α] , that is the average value of the message that
j→i
node j sends to node i, when neither j nor i are
directly observable.
(cid:20) (cid:21)
v[α] = 1 (cid:81) (1 φu[α] (1 φ)z[α] )
u[β] , v[β] , and z[β] represent the same quantities as j→i − k∈∂j[α]\i − k→j − − k→j ×
j→i j→i j→i (cid:20)
above but for layer β. Note that messages traveling on 1 (cid:81) (1 φu[β] (1 φ)z[β] )+
the same edge but in opposite direction are different. − − k→j − − k→j ,
k∈∂[β]
ThegenericnodeiispartoftheLMOCifoneofthese j (cid:21)
two conditions are met: (i) the node is directly observ- (1 φ)kj[β](cid:2)1 (cid:81) (1 z[β] )(cid:3)
− − − − k→j
able and attached to at least one other node in layer α k∈∂[β]
j
and layer β that are part of the LMOC; (ii) the node is (13)
vi
and real-world multiplex networks provide very different sce-
narios and degree of accuracy. In Figure 4 for example,
(cid:20)
z[α] = 1 (cid:81) (1 φu[α] (1 φ)z[α] )+ we show a comparisons between numerical simulations
j→i −k∈∂[α]\i − k→j − − k→j and numerical solutions of the frameworks when applied
j (cid:21) to a multiplex transportation network [26]. The frame-
(1 φ)kj[α]−1(cid:2)1 (cid:81) (1 z[α] )(cid:3) work that accounts for edge overlap well approximates
− − − − k→j ×
k∈∂[α]\i theground-truthresultsobtainedwithnumericalsimula-
(cid:20) j ,
1 (cid:81) (1 φu[β] (1 φ)z[β] )+ tions. The observability transitions appear smooth, and
− − k→j − − k→j the transitionpoint isveryclose to zero. Theframework
k∈∂[β]
j (cid:21) that doesn’t account for edge overlap instead shows an
(1 φ)kj[β](cid:2)1 (cid:81) (1 z[β] )(cid:3) abrupttransition,andathresholdlargerthanzero. Simi-
− − −k∈∂[β] − k→j larconsiderationsarevalidalsoforthemultiplexnetwork
j
(14) representingtheC. Elegansconnectome(Fig.5)[42,43].
for a generic edge j i belonging to layer α. Equations V. CONCLUSIONS
→
with the same structure allow to compute the probabil-
ities defined for edges in layer β. We remark that the In this paper, we extended the observability model,
equations above make use of the locally treelike approxi- originally considered on isolated networks, to multiplex
mation,hencetheproductsappearingontheirright-hand networks. In particular, we focused our attention on
sides. Further, backtracking terms are excluded in the theemergenceofthelargestclusterofmutuallyobserved
products. nodes as a function of the microscopic probability of in-
The analytic framework is now completed. To esti- dividual nodes to host observers. We developed mathe-
mate the average size of the LMOC as a function of φ, maticalframeworksabletowelldescribetheobservability
one needs to: first, solve Eqs. (12), (13), and (14) by diagram in synthetic and real-world multiplex networks.
iteration; second, plug these solutions into Eqs. (7), and Our results indicate that the features of this phase tran-
(8); third, estimate in sequence Eqs. (9), (10), and (11). sition cannot be trivially deduced from those valid for
We performed comparisons between numerical solu- the percolation transition. This statement is true both
tions of the framework and results of numerical simula- for randomly generated multiplex networks, as well as
tions for a single multiplex networks formed by random for multiplex networks representing real-world systems.
scale-free network layers which have negligible overlap Interestingly, real-world multiplex networks seem to be
(see Fig. 3). The agreement between the two approaches always in the observable regime, as long as the fraction
is remarkable. ofnodesthatisdirectlyobservedislargerthanzero. This
Westressthefactthattheframeworkpresentedabove factseemsduetothenatural,andubiquitouslyobserved,
is valid under the assumption that the two layers that presence of edge overlap among the layers that compose
compose the multiplex do not share any edge. This is real multiplex networks.
a very strong assumption, often violated by real-world
networks. For instance, edge overlap may have a dra-
matic consequence on the properties of the percolation
Acknowledgments
transition [26, 28, 38–41]. We developed a mathematical
frameworkvalidincaseofedgeoverlap. Givenitslength
andcomplexity,wepresentthedetailsonlyintheSupple- FR acknowledges support from the National Science
mental Material. Indeed, results from the application Foundation (Grant CMMI-1552487) and the US Army
of the two methods (with or without edge overlap) to Research Office (W911NF-16-1-0104).
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11..00 American–Delta 1.0 American–United 1.0 Delta–United
a b c
0.8
0.6
∞0.5 0.5 0.5
P
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Figure 4: Observability transition in the US air transportation multiplex network. (a) The system is obtained by combining
AmericanAirlinesandDeltaroutes. WeconsideronlyUSdomesticflightsoperatedinJanuary,2014,andconstructtheduplex
networkwhereairportsarenodesandconnectionsonthelayersaredeterminedbytheexistenceofatleastaflightbetweenthe
two locations. In the diagram, the gray big circles represents results of numerical simulations, the red small circles stand for
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11..00 Electric–Chem.Mon. 1.0 Electric–Chem.Pol. 1.0 Chem.Mon.–Chem.Pol.
a b c
0.8
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∞0.5 0.5 0.5
P
0.4
0.2
00..00 0.0 0.0
00..00 0.5 0.2 1.0 0.0 0.4 0.5 0.6 1.0 0.0 0.8 0.5 11..00
φ
Figure5: ObservabilitytransitionintheC.Elegansconnectomemultiplexnetwork. Edgesindifferentlayersrepresentdifferent
types of synaptic junctions among the neurons: electrical, chemical monadic, and chemical polyadic. (a) Analysis of the
multiplex obtained by combining together the layers of electrical and chemical monadic interactions. In the diagram, the gray
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accountforedgeoverlap,andtheblacksmallcirclesrepresentresultsobtainedfromthemathematicalframeworkthataccounts
for edge overlap. The inset shows a zoom of a specific part of the diagram. (b) Same as in panel (a), but for the combination
of electrical and chemical polyadic interactions. (c) Same as in panel (a), but for the combination of chemical monadic and
polydiatic interactions.
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i
Supplemental Material
Message passing with edge overlap
This section is devoted to the description of the message-passing algorithm to approximate the behaviour of the
Largest Mutually Observable Cluster (LMOC) in multiplex networks. The formalism differ from the one already
presented in the main text for the fact that the only approximation used is the locally treelike ansatz. Edge overalp
among layers is instead accounted by this framework. The method presented here is a generalization of the algorithm
proposed in Ref. [28] for standard site percolation on multiplex networks. The method relies on the definition of the
multilink m(cid:126) = (a[α],a[β]) for every pairs of nodes j and i in the duplex, where a[α] = 1 if the nodes are connected
ji ji ji ji
in layer α, and a[α] =0, otherwise. The same definition applies for layer β. Messages considered in the approach are:
ji
1. um(cid:126)ji,(cid:126)n, valid when node j is directly observable.
j→i
2. vm(cid:126)ji,(cid:126)n, valid when node j is not directly observable.
j→i
3. zm(cid:126)ji,(cid:126)n, valid when neither node i nor node j are directly observable.
j→i
In the definition of the messages, we use (cid:126)n = (n[α],n[β]) to maintain the notation as compact as possible. Different
message are indicated by different values of (cid:126)n, namely (0,0), (1,0), (0,1), and (1,1). Note that if n[α](1 a[α])+
− ji
n[β](1 a[β]) = 0, the corresponding message is automatically zero. Further, normalization implies that um(cid:126)ji,(0,0) =
− ji (cid:54) j→i
1 um(cid:126)ji,(1,0) um(cid:126)ji,(0,1) um(cid:126)ji,(1,1). It is further convenient to cumulate messages over layers as:
− j→i − j→i − j→i
u[α] =um(cid:126)ji,(1,0)+um(cid:126)ji,(1,1) ,
j→i j→i j→i
u[β] =um(cid:126)ji,(0,1)+um(cid:126)ji,(1,1) ,
j→i j→i j→i
and
u[α,β] =um(cid:126)ji,(1,0)+um(cid:126)ji,(0,1)+um(cid:126)ji,(1,1) .
j→i j→i j→i j→i
The same definitions are valid for u- and z-type messages.
The probability that node i is in the LMOC is calculated in different manner depending on whether node i is (i)
directly observable or (ii) not directly observable. For case (i), we have
(cid:34) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:35)
q =φ 1 (cid:89) (cid:0)1 φu[α] (1 φ)v[α] (cid:1) (cid:89) (cid:0)1 φu[β] (1 φ)v[β] (cid:1) + (cid:89) (cid:0)1 φu[α,β] (1 φ)v[α,β](cid:1) .
i − − j→i− − j→i − − j→i− − j→i − j→i − − j→i
j∈∂i j∈∂i j∈∂i
(SM1)
Essentially, the probability for node i to be directly observable and part of the LMOC is given by the product of the
probability to be directly observable and attached at least to another node that is part of the LMOC. The latter is
estimated as one minus the probability that none of the nodes connected to i are part of the LMOC.
Toaccountfortheoverlapamongedgesinthetwolayers,weneedtomakeadistinctionamongneighborsofanode.
Wedefinethreesetsofneighborsfornodei: ∂[α], thatissetofnodesthatareneighborsofnodeijustinlayerα; ∂[β],
i i
that is set of nodes that are neighbors of node i just in layer β; ∂[α,β], that is the set of nodes that are neighbors of
i
node i at the same time in both layers. In particular, we indicate with k[α], k[β], k[α,β] the size of the sets ∂[α], ∂[β],
i i i i i
∂[α,β], respectively. We also define degree of each node in layer α, layer β and total as follows: k[1] = k[α] +k[α,β],
i i i i
k[2] =k[β]+k[α,β], k[1,2] =k[α]+k[β]+k[α,β].
i i i i i i i
(cid:2) (cid:3)
r =(1 φ) 1 A B+C (SM2)
i
− − −
ii
(cid:20) (cid:21) (cid:20) (cid:21)
A= (cid:89) (cid:0)1 φu[α] (1 φ)z[α] (cid:1) +(1 φ)ki[1] 1 (cid:89)(1 z[α] )
− j→i− − j→i − − − j→i
j∈∂i j∈∂i
(cid:20) (cid:21) (cid:20) (cid:21)
B = (cid:89) (cid:0)1 φu[β] (1 φ)z[β] (cid:1) +(1 φ)ki[2] 1 (cid:89)(1 z[β] )
− j→i− − j→i − − − j→i
j∈∂i j∈∂i
(cid:20) (cid:21) (cid:20) (cid:21)
C = (cid:89) (cid:0)1 φu[α,β] (1 φ)z[α,β](cid:1) +(1 φ)ki[1,2] 1 (cid:89)(1 z[α,β]) +
− j→i − − j→i − − − j→i
j∈∂i j∈∂i
(cid:20) (cid:21)
(cid:89) (cid:0)1 φu[α] (1 φ)z[α] (cid:1) (1 φ)ki[α] (cid:89) (1 z[α] )
− j→i− − j→i − − − j→i ×
j∈∂[α] j∈∂[α]
i i
(cid:20) (cid:21)
(1 φ)ki[2] (cid:89) (1 z[α] ) (cid:89) (1 z[β] ) (cid:89) (1 z[α,β]) +
− − j→i − − j→i − j→i
j∈∂[α,β] j∈∂[β] j∈∂[α,β]
i i i
(cid:20) (cid:21)
(cid:89) (cid:0)1 φu[β] (1 φ)z[β] (cid:1) (1 φ)ki[β] (cid:89)(1 z[β] )
− j→i− − j→i − − − j→i ×
j∈∂[β] j∈∂β
i i
(cid:20) (cid:21)
(1 φ)ki[1] (cid:89) (1 z[β] ) (cid:89) (1 z[α] ) (cid:89) (1 z[α,β]) (SM3)
− − j→i − − j→i − j→i
j∈∂[α,β] j∈∂[α] j∈∂[α,β]
i i i
ThetermsAandB areobvious. C insteadcontainsaseriesofexceptionsthatmustbehandledtoproperlyaccount
for overlap. In particular,
1. The first term in C stands for the probability that node i is not connected to the LMOC in none of the layers.
It accounts also for the exception where all of its neighbors are not directly observable.
2. Probability for node i of being not connected to LMOC through layer α and being not observable in layer β.
Connections in layer β do not matter, but overlapping links, may connect node i to the LMOC in layer α.
3. Same as point 2, but swapping α with β.
The exceptions mentioned above are computed noting that
(cid:20) (cid:21)
(cid:89) (cid:0)1 φu[α,β] (1 φ)v[α,β](cid:1)= (cid:89) (cid:0)1 φu(1,0),(1,0) (1 φ)v(1,0),(1,0)(cid:1)
− j→i − − j→i − j→i − − j→i ×
j∈∂i j∈∂[α]
i
(cid:20) (cid:21)
(cid:89) (cid:0)1 φu(0,1),(0,1) (1 φ)v(0,1),(0,1)(cid:1)
− j→i − − j→i ×
j∈∂[β]
i
(cid:20) (cid:21)
(cid:89) (cid:0)1 φu[α,β] (1 φ)v[α,β](cid:1) . (SM4)
− j→i − − j→i
j∈∂[α,β]
i
We can now derive self-consistent equations for the messages. For u- and v-type messages, these are provided by
the following equations:
(cid:20) (cid:21)
u(1,1),(1,1) =u(1,0),(1,0) =u(0,1),(0,1) =1 (cid:89) (cid:0)1 φu[α] (1 φ)v[α] (cid:1)
j→i j→i j→i − − k→j − − k→j
k∈∂j\i
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:89) (cid:0)1 φu[β] (1 φ)v[β] (cid:1) + (cid:89) (cid:0)1 φu[α,β] (1 φ)v[α,β](cid:1) , (SM5)
− − k→j − − k→j − k→j − − k→j
k∈∂j\i k∈∂j\i
(cid:20) (cid:21) (cid:20) (cid:21)
u(1,1),(1,0) = (cid:89) (cid:0)1 φu[β] (1 φ)v[β] (cid:1) (cid:89) (cid:0)1 φu[α,β] (1 φ)v[α,β](cid:1) , (SM6)
j→i − k→j − − k→j − − k→j − − k→j
k∈∂j\i k∈∂j\i