社団法人 電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS

信学技報
TECHNICAL REPORT OF IEICE.

ヒトの脳機能ネットワークに着想を得た高信頼な相互接続ネットワーク
の設計
村上

雅哉†

ライプニッツ賢治††

小南 大智††† 村田 正幸†† 大阪大学 大学院情報科学研究科 〒 565–0871 大阪府吹田市山田丘 1–5†† NICT/大阪大学 CiNet 〒 565–0871 大阪府吹田市山田丘 1–4††† 大阪大学 大学院経済学研究科 〒 560–0043 大阪府豊中市待兼山町 1–7
E-mail: †{m-murakami,murata}@ist.osaka-u.ac.jp, ††leibnitz@nict.go.jp, †††d-kominami@econ.osaka-u.ac.jp
あらまし

近年実用化が進んでいる IoT ネットワーク環境においては、ネットワークを通じて多種多様なサービスを展開するため、IoT ネットワーク同士の相互接続が必須となっている。さらに、交通、航空、電力など、主要な社会イ
ンフラもネットワーク化が進み、それらの高度利用を目的として相互接続が進められている。しかし他方では、ネッ
トワークの相互接続は、特に信頼性の観点から相互依存の状態を引き起こし、単一のネットワーク内の障害が他のネットワークへと波及するという問題も認識されている。そこで本研究では、脳機能モジュール間の相互依存性を考慮した、脳ネットワークにおけるインフルエンサーの特定手法に着目した。相互依存性の存在する相互接続ネットワークのネットワーク間接続構造を様々に変化させ、インフルエンサーを評価することにより、インフルエンサーの影響力と配置の一元化及び分散化が可能であることを示した。同時に、相互依存性の存在しない相互接続ネットワークと同等の頑健性を実現可能であることを示した。
キーワード 相互接続ネットワーク, 脳機能ネットワーク, Internet of Things, インフルエンサー

Reliable Architecture for Network of Networks with Inspiration from Brain
Networks
Masaya MURAKAMI† , Kenji LEIBNITZ†† , Daichi KOMINAMI††† , and Masayuki MURATA†† Graduate School of Information Science and Technology, Osaka University†† Center for Information and Neural Networks (CiNet), NICT and Osaka University††† Graduate School of Economics, Osaka University
E-mail: †{m-murakami,murata}@ist.osaka-u.ac.jp, ††leibnitz@nict.go.jp, †††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 even further. In the context of IoT, not only the Internet services becomes
sophisticated and diversified, the existing social infrastructure (e.g., transportation, electricity) are expected to be connected
and dependent on the Internet. It is pointed that due to the inter-dependency between networks, a partial malfunction in a
network can propagate to other interconnected networks. In this study, as an inspiration to solve this issue, we focus on amethod to detect influencers in an inter-dependent networks of the human brain. By configuring connectivity patterns between
subnetworks of an interconnected network, where inter–modular dependency exist, our evaluation shows the feasibility ofcentralization and distribution of influencers regarding its influence and deployment. At the same time, our results showed thatthe deterioration of the robustness caused by inter-dependency can be reduced as that of interconnected networks where nointer-modular dependency exists.
Key words Network of Networks, Interconnected Networks, Brain Networks, Internet of Things, Influencer—1—1. Introduction

complex and sophisticated function (e.g. cognition) in collaboration with other modules. While connections within modules justInformation networks are characterized by rapid growth and in-pass signals over nodes, i.e. neurons, around the modules, Connec-

creased complexity and has been developing toward the next gener-

tions between modules control the cooperation of different func-

ation of the Internet of Things (IoT). In the context of IoT, not only

tions. Hence, a processing halt in a module cause malfunction in

the Internet services becomes sophisticated and diversified, the ex-other modules. In the light of similarity between information net-

isting social infrastructure (e.g., transportation, electricity) or otherworks and brain networks as interdependent network systems, wevital services for everyday life are expected to be connected andapply Morone’s method to design influencers in interconnected net-

dependent on the Internet [1, 2]. These types of interdependentworks. Not only for detecting influencers or comparing interdepen-systems are often referred to as Network of Networks (NoN). The

dent and non-interdependent systems, we do also aim to configure

development toward IoT permits connecting various heterogeneous

influence and location of influencers in interconnected networks byservices in everyday life via the Internet protocol, which results inchanging connectivity between networks.

the acceleration of the emergence of NoN architectures even further.

The purpose of this study is to reveal the way to design inter-

Virtualized networks is considered as one example of interdepen-

connected networks that are robust against network malfunction

dent network systems. In the IoT scenario, virtualization of wirelesscaused by computer-virus or physical disasters, taking into accountsensor networks (WSN) has been widely studied [3,4]. A virtualized

the inter-modular dependency between subnetworks. Therefore, we

WSN consists of two network layers: infrastructure providers de-first focus on a method to detect influencers in an inter-dependent

ploy physical network resources and form infrastructure layer, andnetworks, the human brain networks. We apply this method tomultiple service layers are virtually constructed by combining thereveal the appearance pattern of influencers in interconnected net-

physical resources on the infrastructure layer. Thus, there exists in-works with various topological shape, and also to reveal the differ-

extricable interdependency via the infrastructure layer. The service

ence between the networks where inter-dependency are consideredlayers are also expected to functionally cooperate to provide further

or not. Second, by focusing on the assortativity between networks,complex and sophisticated services. Another example of interde-

we propose a method to configure the intra-modular connectivities.

pendent network system would be the functionally interconnected

Our proposed method helps to know how we can control the in-network in smart cities [5]. Toward the future IoT society, the num-

fluence and location of influencers in interconnected networks. It

ber of IoT devices and the types of services on the Internet havealso reveals how we can compensate the degradation on robustness

been showing explosive increase. Smart cities intelligently integratecaused by the inter-dependency between subnetworks. These results

information collected via the IoT devices to improving services inwill guide the future studies on the construction of interconnected

healthcare, surveillance, infrastructure, public utilities, etc., result-

networks with high reliability.

ing in the realization of smart homes or smart grids. For instance,
smart homes could contain air conditioning systems capturing tem-

2. Related Worksperature, humidity, and circulation from IoT devices in order to pro-

2. 1 Models of Network of Networks

vide best services in response to various requirements. Beside the

As shown in Table 1, in the Brain NoN model, four different

new situations we can foresee at the moment, also the number of au-

node-states are defined using three variables. The variable ni refers

tomated and interconnected service systems operating over the IoT

to the existence state of node i, the variable σi refers to the local

infrastructure is expected to drastically increase.

effectiveness of node i, and the variable ρi refers to the global ef-

While welcoming the future convenient and sophisticated tech-fectiveness of node i. The existence state is considered given, andnological society, there exists a problem behind the collaboration

local/global effectiveness of a node depends on the existence stateof those different service networks. It is pointed that due to theof itself and its neighbors. The definition for σi is given as follows:[]∏
σi = ni 1 −(1 − nj ) ,(1)

inter-dependency between networks, a partial malfunction in a network such as device failure or computer-virus infection can prop-j∈F (i)agate to other interconnected networks. Therefore, an appropriate
strategy to design interconnected networks that can deal with thewhere F(i) denotes the set of nodes connected to node i via inter-above-mentioned problem, which has not been addressed so far.

modular links. Note that F(i) takes ϕ when the node i has no inter-In this study, as an inspiration to solve this issue, we focus on a

modular links. That is, the condition for node i to become active

previous study in neuroscience field. Morone et al. [6, 7] has proposed a method to model interdependency of modularly connected
brain networks and detect influencers, i.e. a set of influential nodes

Table 1 Definition for node state
Sign
State
ni σi ρifrom a topological viewpoint, in the brain networks. The brain net-⃝

removed000work is composed of a number of modules, which provides a speci-

⃝

exist

1

0

0

⃝

locally effective

1

1

0

⃝

globally effective

1

1

1

fied function (e.g. visual processing) alone, but also provides more

—2—Module 1

Module 2#$%&&'!( ")$%&&'!( ")*!

!"#$%&'("!"#$%&')
2-.&340"#$%,3'%+-5Inter-modular link

2-.3,40"#$%,3'%+-5

Figure 2 Description of Collective Influence*+,-.'/"01"-&-.

the robustness and communication efficiency of interconnected networks with inter-modular dependency. Influencers are a set of vital
nodes in terms of the network connectivity or communication effiFigure 1 Example of node-state transition in Brain NoN model [6]ciency. Identification of those vital nodes that maximizes the influence over a network is known as NP-hard problem [8], and there

differs depending on whether it has inter-modular links. If it has one
or more inter-modular links (i) node i have to be INPUT and (ii) atleast one node connected to node i via inter-modular links have to
be INPUT. Meanwhile, if it has no inter-modular links, the condition is just (i) node i have to be INPUT. This variable σi implies
that a node that has inter-modular links cannot be useful without
support from other adjacent modules since the service provided on
this network of networks is interdependent.
In the Brain NoN model, a node is regarded as globally effective only when it belongs to the giant component of an intercon-

exist many heuristic methods have been proposed so far [9], suchas betweenness centrality or page-rank. This study focus on oneof recent works [7] which proposed the Collective Influence (CI)algorithm to identify influential nodes. Not only does the CI algorithm show superior performances for the identification of influential nodes in comparison with other conventional methods, it is also
optimized for interdependent NoN.
CI of node i represents its influence on other nodes in the same
network centered around node i. CI of node i for a single network
is defined as follows:∑

CIl (i) = (ki − 1)

nected network. Whether a node is included in the giant compo-(ki′ − 1)(3)i′ ∈∂Ball(i,l)

nent is calculated through a percolation analysis using a message
passing method. On the message passing, each node first calculates

where ki denotes the degree of node i, Ball(i, l) denotes the set ofthe probability whether the node itself belongs to the giant compo-

nodes within l hops centered around node i, and ∂Ball(i, l) denotes

nent based on the information from adjacent nodes. Then, the node

the set of nodes on the edges of Ball(i, l).

passes the resultant probability to all its adjacent nodes. The probability is defined as follows:[]∏ρi = σ i 1 −(1 − ρk→i )(2)Morone et al. [7] also define the CI algorithm for inter-dependent
modular networks as follows:∑CIl≧1 (i) = ki′i′ ∈∂Ball(i,l)

k∈S(i)+

where S(i) denotes the set of all adjacent nodes of node i, and ρk→i

∑

ki′′
[

j∈F (i):kjout =1

denotes the information from the adjacent nodes, i.e. the probabil-

kj′

∑

kj′ ′

]
(4)

j ′ ∈∂Ball(j,l)

ity for them to be included in the giant component. Starting with a

Note that the variable k considers both links within and between

set of random values for φ and ρ, nodes continue passing informa-

modules. The first term of the equation corresponds to CI of node ition until the values converge. Then, the converged probability isitself, and the second term corresponds to the sum of CI of node jregarded as the global effectiveness.

which is connected to the node i via inter-modular links. The con-

The signs in Table 1 corresponds to the node state in Fig. 2, which

dition kjout = 1 means that the node j is taken into account if itshows an example of state transition in a Network of 2 Networks (2-has only an inter-modular link. This is because if node j has more

NoN). Black lines correspond to intra-modular links and blue ones

than one inter-modular links, it is not directly affected by the state

to inter-modular links. The set of nodes circled by a red dotted lineof node i.

corresponds to the giant component detected in the perforation procedure described above.
2. 2 Identifying Influencers in Network of Networks
This study deals with identification of influencers to enhance

3. Method for Constructing Interconnected Networks
This study aims to reveal the relation of the following three ele—3—

ments: the performance of interconnected networks, the interdepen-a certain network models, which are commonly used in graph-dency between subnetworks, and the intra-modular connectivity. In

theoretical evaluation. In this study, connectivity pattern for inside

this study, we focus on the assortativity for the intra-modular con-subnetworks are considered as given, and thus, the description for

nectivity, and thus, we describe how to construct an interconnected

the network model is described in the following section.

network topology with configuration on inter-modular assortativity
for the evaluation.

Then, Linter links are assigned between the two subnetworks sothat the connectivity achieves a specified value for inter-modular

3. 1 Assortativity between Networks

assortativity of the interconnected networks. In order to obtain a

Newman was the first person who proposed measuring the assor-

appropriate connectivity pattern for the assortativity, inter-modular

tativity of a network with the assortativity coefficient [10]. Then, alinks are rewired repeatedly. The rewiring process is conductedfollowing research introduced universal assortativity coefficient tostochastically with the connectivity pattern determined by the fol-

define the assortativity between networks [11]. This coefficient re-lowing procedure:

flects the contribution of an individual edge’s to the assortativity ofthe entire network, i.e. Newman’s assortativity [10], The assortativ-（ 1 ） Linter edges are randomly assigned between the two subnet-

ity coefficient is calculated from the remaining degree distribution

works. Note that no endpoint node have multiple inter-modular

q(k) defined as follows:

links.
（ 2 ） The assortativity between the modules ρ is calculated. If ρq(k) =(k + 1)p(k + 1)∑,
j jp(j)(5)wise, the following steps are repeated.

where p(k) denotes the probability that a randomly selected node

（ 3 ） An existing link between the modules, whose assortativity isfarthest away from the target value, is deleted. If the current

has nodal degree of k.
Then, the universal assortativity coefficient ρl on a link l can be
introduced given q(k). The definition of the universal assortativity

assortativity ρ is higher than the targeted value, the most assortative inter-modular link is selected, and vice versa.
（ 4 ） A new link is created on two nodes in two different modules.

of link l is as follows:ρl =matches the targeted value, the procedure is finished. Other-(j − Uq )(k − Uq ),Lσq2

The pair of nodes are randomly selected under the condition(6)

where j and k denote the remaining degrees of the two endpointsof link l, which have the same expected value of the remaining de∑gree Uq = j jq(j). The term L denotes the number of links in(∑)2∑
the whole network, and the term σq2 = l j 2 q(j) −k kq(k)denotes the variance of the remaining degree distribution q(k). As

that the new link can move the assortativity ρ closer to the target value. Then, go back to Step 2.
The resultant topology of interconnected networks are used forthe evaluation.

4. Evaluation

seen in Fig. 3, when ρl > 0, the link is called an assortative link;

In this section, we evaluate the performance of interconnected

otherwise when ρl < 0, a disassortative link. A link with ρ = 0 has

networks. By this evaluation, we aim is to reveal topological char-no correlation.

acteristics of influencers in interconnected networks, comparing theThen, we use the term ρ =∑l∈S ρl , which reflects the sum ofcases when inter-modular dependency exists and not. We also reveal

the assortativity of all inter-modular links, for measuring the assor-

the way to enhance the robustness and efficiency by configuring thetativity between networks.

inter-modular assortativity in interconnected networks.

3. 2 Construction of Interconnected Networks

4. 1 Evaluation Environment

First, two subnetworks are prepared beforehand, that composes

First, we describe the settings for the simulation evaluation. As

an interconnected network. We deploy a set of N nodes for eachexplained in Sec. 3., we construct topologies of interconnectedsubnetwork, and connect Lintra links among the nodes based on

networks composed of two subnetworks. Each network contains
N = 100 nodes, and forms a topology with Lintra = 300 links

Disassortative

Non-assortative

based on the network models explained in Sec. 4. 1. 1. After creat-

Assortative

High-degree

ing subnetworks, we assign Linter inter-modular links between thetwo subnetwork so that the connectivity pattern achieves a specified
value of ρ. We set the value for Linter equal to N and let every
node have just an inter-modular link to simplify the problem. Al-

5067.*
806*#"

though the range of ρ ideally takes -1 to 1, it is much more limited

Low-degree

because of the topological shape of each subnetwork. To reveal the!
$ % &'()*+,-./-,,0+1-1()(12/30*44(3(*'1"deference between interconnected networks with inter-modular dependency and without, see Sec. 4. 1. 2. The metrics for evaluating

Figure 3 Connectivity between modules and the universal assortativity coefficient

influencers are described in Sec. 4. 1. 3

—4—

4. 1. 1 Intra-Modular Connectivitya)Erdös-Rényi (ER) Model

The ER model is a class of random network models [12]. The
degree distribution of the ER model follows a Gaussian distributionthat is similar to the distribution observed in information networks.
For the network construction, we repeat randomly choosing a pairof nodes and connecting them until the number of generated intramodular links reaches Lintra .
b)

Barabáshi-Albert (BA) Model

The second type of networks corresponds to the BA model [13],which is one of the most well-studied complex network models.
The BA model is characterized by that there exists extremely high(a) ER model

degree nodes (i.e. hub nodes) and a core cluster of hub nodes (i.e.
rich-club) and that its degree distribution follows a power-law.
For the topology construction, we first pick a small set of nodes tocreate an initial full-mesh topology. Then, we repeatedly add a newnode to the initial topology, and at each time we probabilisticallychoose m other nodes from the existing topology for generating alink between those nodes and the new node. At this time, the probability that node i is chosen is given by pi = ki /Σj kj , where ki
is the degree of the node i. Hence, m is chosen so that almost thesame number of intra-modular links (Lintra ) are finally generated.
4. 1. 2 Inter-Modular Dependency
As mentioned in Sec. 2. 1, we apply the rule of node-state transition in the Brain NoN model to the generated interconnected networks. In addition, we also define Pure NoN, a basic model that(b) BA model

does not consider the interdependence between subnetworks. For

Figure 4 Average hop length (AHL) for an interconnected network of no

the Pure NoN, we replace Eq. 2 with σi = ni and inter-modular

failure (black), AHL of 10%-node failed topology (red), and AHL

links works just the same as the intra-modular links. The compari-of influencers (blue)

son of Brain NoN and Pure NoN let us know what we have to consider on the construction of inter-dependent networks.

between subnetworks, and the y-axis corresponds to the hop length.

4. 1. 3 Evaluation Metrics

As seen in the x-axis, the range of assortativity is quite limited be-

We evaluate the performance based on hop length among nodes.

cause of the number of inter-modular links and the degree distribu-a ) Average Hop Length of Remaining Effective Nodestion of subnetworks. Black line shows AHL of all the possible pairs

In order to measure the effect of influencers, we remove influ-

of nodes in an original interconnected network topology, red line

ential nodes using CI algorithm on an interconnected network andshows AHL among globally effective nodes when 10% of nodes are

see the change of average hop length (AHL) of remaining networks

removed, and blue line shows AHL among influencers. Each figure

other than the failed parts. Here, AHL is defined as the average ofis the compilation of the results from 100 computer simulations.

the number of hops among all the pairs of nodes. It should be noted

Regarding the black line, it is interesting that AHL of all nodes

that only nodes with ρ = 1 are considered in the calculation of

do not change even if we configure the inter-modular assortativity

AHL. Since we focus on the most influential nodes of influencers,

between modules. Our previous research showed assortativity be-

the hop count is measured when 10% of nodes are removed.

tween modules affect performance of an interconnected networks,b ) Average Hop Length of Influencers

where inter-modular dependency are not considered. This time,

We also check the appearance patterns of influencers by focus-

however, we assign an inter-modular link to every node in each sub-ing of the hop length among influencers. After detecting influencers

network, and thus, average hop length of all nodes converge on theas 10% of all nodes after generating network failure, we count hop

same value.

count of all the pairs of influencers based on the original topology of

According to the red lines, when top 10% of influential nodes areinterconnected networks. By doing this, we confirm how sparsely

removed, the networks of Brain NoN and Pure NoN showed differ-the influencers are located.

ent behavior. Both the lines are shifted toward upper side, because

4. 2 Evaluation Resultsof the lack of influencers, i.e., high-centrality nodes. However, red-

Figures 4(a) and 4(b) shows the evaluation results for the ER anddotted line is almost horizontal because the same nodes are removed

BA model, respectively. The x-axis corresponds to the assortativity

as influencers regardless of inter-modular assortativity. It is true al—5—

though the same set of nodes are removed, inter-modular connectiv-

service modules are mutually connected via control plane or op-

ity differs. However, the effect is considered too small to change theerated by the control plane. For example, network virtualization is

performance similarly to the black line. Meanwhile, for the network

realized by the combination of virtualized service layer and physicalof Brain NoN, the hop count increases with the inter-modular assor-

control plane. Social infrastructure services such as transportation

tativity. For disassortative networks, even if highly influential nodeor electricity are also connected to and controlled by the Internet.

is removed and the other node connected via an inter-modular link is

Another untested point is regarding the connectivity within subnet-

turned down due to the inter-dependency, the latter node is of quite

works. Although we used the ER and BA model, not only the degree

low importance for sustaining connectivity. Whereas for assortative

distribution, but configuring assortativity on a given degree distribu-

networks, influential node is connected to another influential node,tion would help to control the influence and appearance pattern ofand thus, malfunction on one side becomes quite critical.

influencers.

As for the blue lines, we observe that influential nodes are basically connected close to each other, and located further closer

Acknowledgements This work was supported by a Grant-inAids for JSPS Fellows, No. JP15H01682.

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ing in the closeness of influencers. The results also show that hop
length among influencers slightly decreases when Brain NoN model
is applied. This is because a node is more likely to be chosen as
influencers if its adjacent node in the other module is also highly
influential.
When comparing Figs. 4(a) and 4(b), we can see some differenceson the performance. Both the red and blue lines showed larger gaps
to the black line. This characteristic can be attributed the degree distribution of the BA model. There exist extremely high degree nodein scale-free networks, and thus influencers are closely connected
and hop length increased when influencers are removed.

5. Conclusion and Future WorkIn this study, we applied a method to detect influencers, i.e. important node on robustness or communication, in interconnected
networks that have inter-modular dependency. We also configured
the connectivity patterns between subnetworks from the viewpointof assortativity.
It is conceivable that interdependent networks are more fragile
since a partial network failure affects the other connected module.
However, our results showed that if inter-modular connectivity corresponds to the disassortative mixing, the deterioration of the robustness can be reduced as that of interconnected networks where
no inter-modular dependency exists. At the same time, to put it
the other way, if we assortatively connect the subnetworks, unitary
management would be much easier since the influencers have higher
influence over the interconnected networks. We also confirmed thatas the inter-modular assortativity increases components of the influencers are located closer. This implies that assortative mixing between subnetworks help centralization of influencers from topological viewpoint as well, whereas disassortative mixing is preferable
to geometrically distribute the influencers. These findings would
guide to design reliable interconnected networks that match various
environmental requirements in the future IoT context.
However, there remain several issues yet to be tackled to designinterconnected and interdependent networks. In the context of information networking, we should take into consideration that different—6—