Network of networks: A bibliometric analysis

Highlights

• Mapping the emerging field of Network of Networks.

• Revealing the origin of the field in the context of Neural Networks from the 1990s

• Showing Multilayer and Interdependent Networks as recent dominant research topics.

• Emergence of a distinct cluster of terms related to nanoscience and nanotechnology.

Abstract

This study explores the evolving structure of the rising field of “network of networks” (NoN). Reviewing publications dating back to 1931, we describe the evolution of major NoN research themes in different scientific disciplines and the gradual emergence of an integrated field. We analyse the co-occurrence networks of keywords used in all 7818 scientific publications in Scopus database that mention NoN and other related terms (i.e., “interconnected networks”, “multilayer networks”, “multiplex networks”, “interdependent networks”, “multinetworks”, “multilevel networks”, and “multidimensional networks”). The results show that the NoN began to form as a field mainly in the 1990s around research on neural networks. Diverse aspects of NoN research, indicated by dominant keywords such as “interconnection”, “multilayer”, and “interdependence”, gradually spread to computer and physical sciences. As of 2018, network interdependence – with its application in network resilience and prevention of cascading failure – seems to be one of the key topics attracting broad academic attention. Another noteworthy observation is the emergence of a distinct cluster of terms relevant to nanoscience and nanotechnology. It is envisaged from the analysis that NoN concepts will develop stronger ties with nanoscience with increasing understanding and data acquisition from the molecular, atomic, and subatomic levels.

Introduction

Networks are embodied features of nature’s complex systems. Our bodies are composed of numerous interacting network systems ranging from proteins to genes, neurons, and blood circulatory systems. From birth, we also form social networks, starting with our family trees. The origins of network science go back to 1736 when Euler published a paper titled “Solution of a problem relating to the geometry of position” [1] about seven bridges of the Königsberg city [2]. Since then, the science of networks has been developed and has played a significant role in global socio-technical development. Over the last few decades, information technology has progressed hand in hand with virtual social connectivity and human mobility world-wide. Consequently, topics in network science related to interactions of multiple networks have received increasing attention, especially in relation to communication networks, energy grids and other cases of large and complex infrastructure. They have been also discussed from sustainability perspective [3].

Kivela et al. [4] classified the types of diverse formations of multiple networks (see Table 1 in [4]), using the term “multilayer network” as the overarching keyword. We have identified eight terms used in the research of interactions between multiple networks that are increasing in use: “interconnected networks”, “multilayer networks”, “multiplex networks”, “interdependent networks”, “network of networks”, “multinetworks”, “multilevel networks” and “multidimensional networks”. We refer to the concept represented by these terms as “network of networks” (NoN). The process of arriving at this list is explained in the data and methods section. Fig. 1 shows the annual trend of publications containing these eight terms in the Scopus database until the end of 2018. As evident, the first publication in the selection is from 1932. The rapid growth of the field, beginning in the 1990s and accelerating over the last two decades, is evident from the figure. This study explores the changes in the research into NoN in the last 88 years since the first publication appearance.

To gain a more in-depth view, Fig. 2 provides the annual publication profiles for each of these keywords. Here, we first look at the literature and investigate the original use of each of these keywords. We then conduct a detailed network analysis of the published literature containing these terms to assess the relations between the keywords used together and the evolutionary trends of this research field.

Interconnected network: As evident from Fig. 2, this term is the most common keyword among the eight discussed. There were over three thousand scientific articles about interconnected networks by the end of 2018 (see Fig. 2). The term was first used by Purcell and Powel [5], [6]. Purcell, from Duquesne Light Co., and Powel, from Westinghouse Elec. & Mfg. Co., both companies in Pittsburgh, wrote a short letter titled “Interconnected Networks” to the Electrical Engineering journal in 1931 [5], describing the successful interconnection between the electricity networks of the two companies. In 1932, they wrote a full paper with the same title and provided details of their tie-line load regulators which facilitated the interconnection [6]. The term was not used in the scientific literature for the following 25 years until Matthaei [7] published a circuit analysis paper in 1957. He described a method for finding the characteristic polynomials of interconnected networks using the properties of their component networks. Again, we found no incidence of “interconnected network” usage for the following decade, until in 1966, Rieke, Karsten, and Bross [8] discussed the complexity of simulating the water network expansion of Berlin city due to the complex nature of the interconnected networks involved. This term has become increasingly used since that time, especially over the last decade (see Fig. 2).

Multilayer network: The term “multilayer network” seems to have been coined first by Weston in 1974 [9], describing an attempt to develop a multilayer neural network model of the auditory system for the study of lateral inhibition. A decade later, Coden [10] discussed the importance of multilayer networks versus disparate networks in the planning and design of fibre optic networks. In the late 1970s, the International Organization for Standardization (ISO) proposed the Open Systems Interconnection model (OSI model) as a conceptual model that characterizes and standardizes the communication functions of a telecommunication network. This was done in response to the growth of different networks that were developing from different equipment vendors and companies that were not originally compatible with each other, and appropriate common interfaces and standards had to be developed to interconnect them. The model partitions a communication system into seven layers of abstraction, where each layer serves the layer above it and is served by the layer below it. The OSI multilayer model was published in 1984 by both the ISO, as standard ISO 7498, and the CCITT (International Telegraph and Telephone Consultative Committee), as standard X.200 [11]. The TCP/IP (Transmission Control Protocol–Internet Protocol) protocol stack was developed in parallel for the same reason, in 1970s [12]. The use of the multilayer network term started to increase mainly from 1988, after which it grew gradually over the 1990s and 2000s. A steep increase in the use of this term has further been recorded since 2015 (see Fig. 2).

Multiplex network: This is a relatively newer term. It was initially used in 1980 by Ballinger [13] regarding high-frequency telecommunication cables and the use of frequency-division multiplex (FDM) networks to address the increasing traffic requirements of the time and the need of different users to share communication channels. FDM itself is a technique from the 1860s as harmonic telegraphy [14]. The initial use of multiplex network term was mostly for communication data multiplexing in various applications such as modem design [15], vehicle local area networks [16], [17] or ATM networks [18]. However, the term did not become popular until recent years, and its use has displayed an accelerating pace from around 2012 (Fig. 2)

Interdependent network: The first recorded use of this term in the scientific literature is by Johnson [19] in 1977, discussing the benefits of interdependent networks of kin relationships in multicultural communities. However, the term was not used much until the late 2000s when it became more widely used. Its use has accelerated since 2010, but overall this term (together with “interdependence network”) is less commonly used than multilayer or interconnected networks.

Multinetwork: This term seems to have been first used by Neubert [20], who in 1966 introduced a multinetwork for studying the elastic behaviour of anisotropic rubber by superposition of independent Gaussian networks. As evident from Fig. 2, this term was rarely used for the next two decades. Its common use began in the mid-2000s and is accelerating.

Multilevel network: The origin of this term has been attributed to Haring [21] in his introduction of multi-threshold threshold elements for electronics and computer networks in 1966. As displayed in Fig. 2, its regular use began in the late 1980s and accelerated from the late 2000s.

Multidimensional networks: This term seems to have been coined by Korn in 1962 [22] in his attempt to extend the model of a one-dimensional electrical network to multidimensional networks. The next record is again from Korn in 1966 [23] relating to multidimensional electrical networks. However, this term did not become popular until the last decade (see Fig. 2).

Network of networks: The term “network of networks” seems to have been coined by Wilson in 1986 in an article titled “Fiberway” [24]. He proposed Fiberway as a new digital broadband network which enabled simultaneous transportation of multiple independent protocols over the same fibre, calling it a network of networks [25]. In the following year, Coviello [26] introduced the “network of networks” concept for distributed communication across several communities as a network redundancy tool for survivability. Coviello argued that the establishment of a virtual network of networks utilizing user terminals as the switches and existing networks as the linkages could increase the resilience of the overall system compared to any of the constituent networks. The use of this term has grown steadily since then. The Internet itself is often described as a network of networks, each running its own protocols or having a different administrator, and interconnected at special nodes called gateways or routers, depending on the layer at which they are interconnected.

In this paper, we use the term “network of networks” (NoN) as an overarching term to address all related concepts including “interconnected networks”, “multilayer networks”, “multiplex networks”, “interdependent networks”, “multinetworks”, “multilevel networks”, and “multidimensional networks”. We explore all available literature in Scopus database on these topics to the end of 2018 and longitudinally investigate the interactivity of related keywords in the development of the field.