Designing Interconnected Networks
for Improving Robustness and Efficiency
Masaya Murakami∗ , Kenji Leibnitz†, Daichi Kominami‡ and Masayuki Murata∗∗ Graduate School of Information Science and Technology, Osaka University, Japan

Email: {m-murakami,murata}@ist.osaka-u.ac.jp† Center for Information and Neural Networks, NICT & Osaka University, Japan

Email: leibnitz@nict.go.jp‡ Graduate School of Economics, Osaka University, Japan

Email: d-kominami@econ.osaka-u.ac.jp

Abstract—The Internet is rapidly developing toward the next
generation of the Internet of Things (IoT), which accelerates
the emergence of interconnected network architectures evenfurther. However, the way to design interconnected networks thatcan meet various changes in environment and service demands
remains an important issue that has not been appropriately
addressed yet. These interconnected networks should be robust to
suppress or prevent diffusion of malicious information, whereas
they should also be efficient to enhance diffusion of urgent
information throughout the entire network. In this study, we
propose a Network of Networks (NoN) model inspired by themodularly interconnected networks in the brain, and show thatthe model has two prominent characteristics. First, unintentional
information diffusion from one subnetwork to another can be prevented. Second, speed of information diffusion can be controlledeven when the diffusion starts from interconnecting links across
two subnetworks. For the second point, we further investigate
a strategy for influencing the information diffusion speed byappropriately configuring the connectivity within and between
subnetworks. Simulation experiments confirm that by adapting
the brain-inspired activation rules on the interconnecting links,
our model is able to switch from the unhindered and fast
information diffusion between subnetworks to the slow casewhere all interconnecting links are inoperative.
Index Terms—Network of Networks, Brain networks, Internetof Things, assortativity, centrality

I. I NTRODUCTION
Modular interconnected networks, often referred to as Network of Networks (NoN), have been frequently observedin many complex systems in biology, society, science andtechnology, as well as the Internet [1], [2]. In contrast to theother rather static kinds of complex systems, the Internet itself
is rapidly developing toward the next generation of the Internetof Things (IoT), which permits connecting various types ofinterconnected devices from everyday life via the Internet
protocol and which is expected to accelerate the emergenceof modular network architectures even further.
An example of the modular architecture in the future Internet is the functionally interconnected network in smart cities[3]. In the future IoT society, the number of IoT devices connected to the Internet as well as the types of services provided
through the Internet are expected to show an explosive andcontinuous increase. Smart cities automatically collect data
978-1-5386-0728-2/17/$31.00 c 2017 IEEE

from those IoT devices and intelligently integrate them for
improving services in healthcare, surveillance, infrastructure,public utilities, etc., resulting in the realization of smart homes
or smart grids. For example, smart homes could contain air
conditioning systems capturing temperature, humidity, andcirculation from IoT devices in order to provide best services
in response to a variety of situations. Beside the new situations
we can foresee at the moment, also the number of automatedand interconnected service systems operating over the IoT
infrastructure is expected to drastically increase in future smart
cities.
When providing services, these interconnected networks
will be required to meet various changes in environment andservice demands: robustness to suppress or prevent diffusionof malicious information or efficiency to enhance diffusion ofurgent information. However, the best way to design an NoN
architecture remains an important issue that has not been fully
addressed yet. Therefore, we first focus in this paper on models
for NoN that have been studied for modularly interconnected
networks [4], [5]. It has been shown that especially biologicalsystems have high robustness against cascading failures. For
example, Morone et al. [5] proposed an NoN model fromthe perspective of neuroscience, i.e., networks of neurons in
the brain. This NoN model, termed as Brain NoN hereafter,
considers the characteristics of an activation rule of neuralfirings in brain networks, which is well-known for its highrobustness [6], [7].
Although the activation rule in Brain NoN can be a clue toimprove robustness of emerging interconnected Internet services, its application has not been considered so far. Moreover,there are two important questions that need to be answered forinterconnected networks regarding their structural connectivity: (i) how is the connectivity of nodes within modules? and(ii) how is the connectivity between modules? In this study,
we attempt to answer those questions from the viewpoint ofinfluential nodes and node correlations. Influential nodes in
networks are defined as the fraction of nodes that have alarge influence over the whole network [8]–[12], whereas nodecorrelation [13], [14] is formulated based on the correlationof degrees of two nodes and termed as assortativity.
The aim of our work is to design an NoN architecture

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Figure 1. Activation rule in the Brain NoN model [5]for information networking that meets environmental changes
and service demands, which can be summarized as highrobustness against malicious information and efficiency forurgent information. For this goal, we first propose an NoN
model inspired by Brain NoN that matches situations in
information networking with service interdependency. Second, by taking the node influence and node correlations into
account, we propose a method to configure the intra- andinter-modular connectivities and evaluate the performance of
the NoN. Results from simulation experiments reveal that
unlike conventional NoN models without the activation ruleon interconnecting links, our proposed NoN model is able toswitch from the unhindered and fast information diffusion to
the slow case where all interconnecting links are inoperativeby configuring its connectivity.
This paper is organized as follows. Section II describes
related works already done in the field of graph theory.
Section III introduces an NoN model inspired by the humanbrain, and Section IV describes the method to configure connectivity of interconnected networks. Section V describes theresults from evaluations of interconnected networks. Finally,
Section VI provides the conclusion and future work.

input and the input of at least one node in the other modules
exists. The value of σ is defined asY(1 − nj ) ,(1)
σi = ni 1 −j∈F (i)where F (i) denotes the set of all nodes connected to node i
via inter-modular links.
B. Identification of Influence in Networks
Our study also focuses on the vital nodes in order tocontrol acceleration and suppression of information diffusion
in interconnected networks. The identification of the set ofnodes that maximizes the influence over a network is known
as NP-hard problem [8] and several heuristic solutions havebeen proposed to solve this problem [12].
We focus on [11] which proposed the Collective Influence(CI) algorithm to identify influential nodes. CI of node irepresents its influence on the other nodes in the same network
centered around node i, e.g., betweenness centrality, pagerank,
or k-core. The CI algorithm shows superior performances forthe identification of influential nodes compared to other methods using conventional centrality measurements by finding thesmallest set of nodes that totally collapses the connectivity of
the networks. CI of node i is defined asX(kj − 1),
CIl (i) = (ki − 1)j∈∂Ball(i,l)

where ki denotes the degree of node i, Ball(i, l) denotes the setof nodes within l hops centered around node i, and ∂Ball(i, l)denotes the set of nodes on the edges of Ball(i, l).
C. Universal Assortativity
Assortativity, i.e., the correlation of nodal degrees, was first
proposed by Newman with the assortativity coefficient [15].
The assortativity coefficient is calculated from the remaining
degree distribution q(k) defined as follows:

II. R ELATED W ORK
A. Models of Network of Networks
In the Brain NoN model [5], nodes can have three different
states: active, input, and no-input. Each node can be active
only when its own and its neighbors’ input satisfy certain
conditions. These three states of node i are determined by
two variables, input variable n and activation variable σ, asfollows.
:

active(ni = 1, σi = 1):
input
(ni = 1, σi = 0)
: no-input (ni = 0, σi = 0)
The patterns of each circle represent nodal states corresponding to Fig. 1, which shows an example of the statetransition in a Network of 2 Networks (2-NoN) of the Brain
NoN model. The values for the input variable n are assumed
as given and they sequentially determine the values for theactivation variables σ. Node i can be active only when its own

q(k) =(k + 1)p(k + 1)
P,
j jp(j)where p(k) denotes the probability that a randomly selected
node has nodal degree of k.
Furthermore, the universal assortativity coefficient ρl on alink l was introduced to analyze the assortativity of any part ofa network in [14] and is introduced given q(k). The definitionof the universal assortativity of link l is as follows:ρl =(j − Uq )(k − Uq )
,
M σq2

(2)

where j and k denote the remaining degrees of the two
endpoints of link l, which have
P the same expected value ofthe remaining degree Uq = j jq(j). The term M denotes
the number of edges
network and the termP in the whole2P 22
σq = l j q(j) −denotes the variance of thek kq(k)remaining degree distribution q(k). When ρl > 0, the linkis called an assortative link; otherwise when ρl < 0, it is adisassortative link.

4,5,(,).%*

Table IM APPING VARIABLES FROM B RAIN N O N TO INFORMATION NETWORKS
variables
σ=0
σ=1

Brain NoN
node is inactive
node is active

IC NoN
outer-interface is inactive
outer-interface is active

III. I NFORMATION D IFFUSION M ODEL FOR
I NTERCONNECTED N ETWORKS
Although input to nodes and activation as the result of this
input were considered in the Brain NoN model [5], effects ofnode activation on neighbor nodes have not been considered sofar. We expand the activation rule of the Brain NoN model toexpress the communication flow in interconnected networks.
First, we change the interpretation of the node states in
the Brain NoN model to states of node interfaces (network
devices) in information communication NoN, called IC NoN.
In the Brain NoN model, the activation of interconnecting links
is coupled with the activation of endpoint nodes of the interconnecting links. In information networks, however, even if
one endpoint node is deactivated and thus the interconnecting
link is also deactivated, the other endpoint node can maintainits process within the module to which the node belongs.
For this reason, the meaning of the states defined by σin Brain NoN is reinterpreted as shown in Table. I, where
the activation of nodes in Brain NoN now corresponds to theactivation of outer-interfaces. In this context, the input variablen in Brain NoN represents the input state of information.
It should be noted that inner-interfaces are always active
independent of the value of σ or n. Adding to IC NoN, wenote a basic model that does not consider the interdependence
between modules as Pure NoN in Table I. Pure NoN always
diffuses at the maximum speed that the topological connectivity can produce.
Second, in order to express the flow of information, IC NoN
adopts the notion of time-scale. In this model, the value ofvariables n and σ at current time step t is given by the previous
states at time step t − 1. We then introduce a probability
function pt for nodes to decide whether to have input or
depend on the states of neighbor nodes. Here, we supposethat each node can pass information with probability δ throughactive outer- and inner-interfaces whenever they have inputs.
Therefore, the probability function pt (i) for node i to judge
whether to have input is written as follows:YY(1 − δσit−1 σkt−1 nt−1
(1 − δnt−1
pt (i) = 1 −
j )×
k ),
j∈S(i)k∈F (i)where S(i) denotes the set of neighbors of node i within thesame module and F (i) denotes the set of neighbor nodes in
the other modules. It should be noted that all inner-interfaces
are always active, while outer-interfaces are active only whenσ = 1. An important point this equation expresses is that when
δ < 0, node i can behave differently depending on the numberof neighboring input nodes: the more input neighbors node ihas, the more likely node i will have input.

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Figure 2. Network topologies with various connectivity

Then, activation state of node i is rewritten based on therule in Eq. (1) of the Brain NoN model as follows:


Y
t
t
t
(1 − nj ) .
(3)
σi = ni 1 −j∈F (i)Equation (3) shows that the inter-modular interface of node i
becomes active only when node i and at least one neighbornode on an inter-modular link has input.
IV. M ETHOD FOR C ONFIGURING C ONNECTIVITY OF
I NTERCONNECTED N ETWORKS
In our strategy, we configure intra- and inter-modular connectivity from the perspectives of node influence mentioned
in Section II-B, since node influence and diffusion speed areclosely related.
A. Configuring Connectivity within Subnetworks
In order to increase/decrease the power of influential nodes
in terms of information diffusion speed in a subnetwork, we
extend the conventional preferential attachment method andgenerate topologies by controlling parameter γ. Given a seed
network, we add nodes one by one where each new node isconnected to m existing nodes. The probability for each linkof new node to be connected to an existing node i is defined
as follows:kγp(i) = P i γ ,j kj

where ki denotes the degree of node i. This process finishes
when all N nodes are added to the network. Figure 2 shows
the topological structure of networks with changes of theparameter γ. When γ decreases, the topology approaches auniform degree distribution with average degree k̄ ≃ 2m
and variance of degree approaches zero, and thus node centrality is also distributed. However, when γ increases, more
highly influential nodes emerge and the number of influencers
decreases. As a result, topologies will have a node degree
distribution following a power-law p(k) ∼ k −δ .

B. Configuring Connectivity between Subnetworks
When adding an interconnecting link to an NoN, we consider two points: (i) dependency on the centrality of both
endpoint nodes within each subnetwork, and (ii) dependency
on the correlation of centrality of both endpoint nodes. All
possible pairs of nodes with a certain centrality value can be
expressed by changing these two dependencies respectively.
Based on this idea, we investigate which nodes should be
preferentially selected as endpoint nodes of interconnecting
links for achieving an NoN topology with fast/slow information diffusion. In the following, we will formulate eachdependency as Dependency Coefficient (DC).
1) Coefficient for Node Centrality: First, we define the DCof centrality itself as DCcnt . Here, we consider the dependency on centrality of each endpoint node of interconnecting
links independently, DCcnt is simply defined as the sum ofcentrality of each endpoint node as follows:
DCcnt (h, i) = ch + ci ,where ch denotes any centrality value of node h withineach subnetwork the node belongs to. The values of ch andci , respectively, vary in the range of [0, 0.5]: a high value
represents high centrality, and vice versa.
2) Coefficient for Correlation of Node Centrality: We measure the correlation of node centrality based on the ideasof universal assortativity as mentioned in Section II-C. The
universal assortativity is introduced to measure the correlationof node degree centralities between networks
as shown inP
Eq. 2. Here, the expected value Uq = j jq(j) is based on
the remaining degree. We assume that interconnecting links are
generated between two different subnetworks independently of
the connectivity within each subnetwork. The probability forselecting any node in each subnetwork as an endpoint node isequal. Setting p(c) as any kind of node centrality of a subnetwork, the expected value of the centrality on an endpoint nodeof an interconnecting link is also expressed as p(c). Therefore,
we define another generalized universal assortativity ρ′l of aninterconnecting link l between subnetworks 1 and 2, modifying
Eq. (2), as follows
ρ′l =



cl1 − Up1 cl2 − Up1,
σp1 σp2(4)

where cl1 and cl2 denote centralities of endpoint nodes inthesubnetworks 1 and 2, respectively. Up1 and Up2 denoteP
expected values of node centrality, defined as Up = j jp(j),σp21 and σp22 denote the variances of node centrality distribution
P
2
P
mp(m)
p(c), given as σp2 = l l2 p(l) −. Particularly,m
if subnetworks 1 and 2 have the same node centrality distribution p(c), Eq. (4) can be rewritten as:
ρ′l =



c l1 − U p c l2 − U p
.
σp2

(5)

Assortative

Disassortative!"#$%&'()*+,")-!"#$%&'()*+,")-! " #$% &'()

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Figure 3. Various patterns of connectivity between subnetworks

Finally, we define DCcor of the correlation of node centralities of the two endpoint nodes by slightly changing thegeneralized universal assortativity as follows:ch − U p ci − U p
DCcor (h, i) =,σp2
where h and i are node indices.
3) Coefficient for Varying Connectivity between Subnetworks: To configure the connectivity between subnetworks,
we consider two aspects as mentioned above: (i) dependency
on centrality of both endpoint nodes, and (ii) dependency
on correlation of centrality among the two endpoint nodes.
That is, we combine DCcnt and DCcor and express various
interconnectivities between subnetworks using DC defined asfollows:#r cos θ"min
DCcnt (h, i) − DCcnt+1
DC(h, i) =min
DC cnt − DCcnt(6)#r sin θ"min
DCcor (h, i) − DCcor+1+min
DC cor − DCcor
where θ ∈ [−1, 1], r ∈ {0, 1} with r = 0 for random
connectivity and r > 0 for various connectivity. Each dependency coefficient is normalized so that the effect of bothcoefficients becomes the same on average. We then add 1 toboth coefficients so that the minimum dependency coefficientamong all pairs of nodes always stays 1 as a standard value
independent of θ.
For r > 0 and θ ∈ (0, π), interconnecting links become
assortative; otherwise when θ ∈ (π, 2π), the links become
disassortative. When θ ∈ (3π/2, π/2), high centrality nodes
tend to be selected as endpoints of interconnecting links, while
when θ ∈ (π/2, 3π/2), low centrality nodes are preferred.
These cases are depicted in Fig. 3.
V. S IMULATION E VALUATION
A. Simulation Settings
We evaluate the performance of the introduced NoN models
by changing their topological connectivity. In particular, weconsider the IC NoN model as our proposal and Pure NoN



































Description
information passing probability
parameter γ for preferential attachment
parameter m for preferential attachment
parameter r for connectivity between subnetworks
parameter θ for connectivity between subnetworks
number of nodes in each subnetwork
number of inter-modular links
maximum nodal degree of intra-modular links
maximum nodal degree of inter-modular linksValues0.5[-50,20]21[0,2π)
100
25
25
1







Variables
δ
γ
m
r
θ
N
E
max
kin
max
kout





Table IIPARAMETER SETTINGS

as a reference for comparison. Both models are described in
Sect. III.
To conduct the evaluation, we configure the parametersettings on NoN models and topologies according to Table II.
We measure the required time-steps for information to diffuse
over the entire NoN topologies to evaluate whether the NoN
diffuses information quickly or slowly. Two alternatives are
considered as origin of the diffusion: (i) the highest loadedinter-modular links, and (ii) randomly selected inter-modular
links. Starting the diffusion from an interconnecting link
matches both our research objective and the natural behavior
of information networking. Although it is the original behaviorof the IC NoN model that nodes become empty after passing
their information, we designate the source inter-modular link,i.e., the source endpoint nodes, to continuously send theinformation. This is because the diffusion is following a probabilistic method and it is therefore possible for the diffusion
to disappear from the network in the first few iteration steps.
B. Basic Properties of Independent Subnetwork
Before we start with evaluating the information diffusion
efficiency, we first investigate the basic properties of eachsubnetwork of interconnected networks, which will allow fora deeper understanding on how to evaluate interconnected
networks in the following section. Figure 4 shows the maximum collective influence (left axis) and the required steps
for information diffusion (right axis) in the subnetwork. We
can confirm that as the parameter γ increases, the maximum
collective influence monotonously grows and the number ofrequired steps decreases. This result implies that the impact
of influential nodes can be summarized and distributed bychanging the parameter γ. However, we can also find that
there is a limitation on the feasible values of maximum
collective influence and required steps. We can also find astraightforward tendency of the speed of information diffusion
increasing when γ increases.
C. Simulation of Information Diffusion
In this subsection, we investigate the performance of both
NoN models, IC NoN and Pure NoN, through simulation of
information diffusion. Information diffusion selects influential
links as the source of the diffusion based on the average
cumulative influence of both endpoint nodes.
In Fig. 5, the required time for information to completely
diffuse over the entire network is shown. Styles of the lines in




























Figure 4. Collective influence and required steps for diffusion

Fig. 5 basically correspond to the blue lines in Fig. 4 for thecase of information diffusion in a subnetwork. As γ increases,
influential nodes gradually appear and they minimize thediameter of each subnetwork in the interconnected network.
However, the behavior of lines differ among each other,
depending on the types of NoN models and the parameter θ.
Regarding the parameter θ, we pick results of representativevalues in the range of θ ∈ [0, 2π), which sufficiently describes
the characteristics of connectivity of interconnected networks.
The first striking point is that the diffusion speed greatly
slows down when γ is small. This is because each subnetwork
becomes uniformly connected as we confirme in Sec. IV-A. In
such stretched subnetworks, the endpoints of interconnecting
links in each network are located far away. The activation rulefor outer-interfaces of the IC NoN model requires that both
endpoint nodes of an interconnecting link have input when
the outer-interfaces need to be activated. Therefore, the outerinterfaces tend to be turned off in interconnected networks
composed of such stretched subnetworks. From those reasons,the solid lines of IC NoN marked much higher values than thedotted lines of Pure NoN. IC NoN achieves almost the same
speed of information diffusion as interconnected networks
whose interconnecting links are all inoperative, which can beinterpreted from comparing Figs. 4 and 5.
On the other hand, IC NoN can achieve almost the same
speed of information diffusion as Pure NoN. This can be seen
when γ ≥ 2 and θ ≈ 0. In this range of parameters, the interconnecting link of the diffusion source is assumed to connect
the highest centrality nodes in each subnetwork. At the sametime, both subnetworks have quite small diameter centeredaround those high centrality nodes. Therefore, strong diffusion
sources enable quick information diffusion for IC NoN. It isnotable that with this parameter settings, IC NoN can diffuse
information as fast as Pure NoN, where interconnecting links
are always active.






























































































(a) Diffusion from interconnecting links with high centrality endpoint nodes

ACKNOWLEDGMENT













































This research was supported by a Grant-in-Aid for Scientific
Research (A) (No. JP15H01682) from the Japan Society for
the Promotion of Science (JSPS), Japan.
























R EFERENCES














tion diffusion between modules and thus proposes maximum
diffusion speed with a given topology. We also found thateven if malicious information spreads out from interconnecting
links, we can reduce the diffusion speed to become as slow asinterconnected networks with only inoperative interconnecting
links, and thus IC NoN realizes robustness. Therefore, we canconclude that IC NoN can achieve both efficiency and robustness on information diffusion in interconnected networks.
In the evaluation of our proposal, we focused on theinformation diffusion starting from interconnecting links andwe did not discuss the diffusion starting from a node or a link
within a subnetwork, or interconnected networks composed ofthree or more subnetworks. Therefore, our future work would
be to modify IC NoN or to configure a more complicatedsettings so that we can evaluate other types of information
communication in interconnected networks.













(b) Diffusion from randomly selected interconnecting links
Figure 5. Required steps for complete diffusion with changes in connectivity

VI. C ONCLUSION AND F UTURE W ORK
In this study, we proposed an NoN model called IC NoN
inspired by the Brain NoN model, which reproduces theactivation rule of neurons of different modules that share interconnecting control links. As a basic characteristic, IC NoN
does not allow unaccepted information to pass through interconnecting links, and thus prevents interconnected networks
from unintended behavior. However, it is conceivable thatinterconnecting links themselves become sources of malicious
information diffusion. Otherwise, in case of emergency, we
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In the evaluation part, we simulated information diffusion
starting at interconnecting links, while changing connectivity
within and between subnetworks. The results showed thatIC NoN can efficiently diffuse information as fast as Pure
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