Framework definition

Multilayer networks provide the ability to differentiate between specific disciplines. This opens a lot of avenues to analyse a co-authorship network as it provides greater distinction within the model thereby increasing the analytical resolution.

Having reviewed the various approaches possible, it is possible to define the framework adopted in this research. This framework provides the foundation for analysing multiplex collaboration networks. It outlines what data is needed to create a multilayer network.

The framework here is the type of multilayer network that is best suited to analysing a multilayer co-authorship network. A network-of-networks approach is intuitive, but requires a set of assumptions on how different layers are connected to each other. Multiplex networks are more suitable as they provide a direct mathematical representation the multiplex structure by converting the network to its supra-adjacency form, and are more amenable to its tensor format.

After exploring both formats, the multiplex network format provided an easier and more rigorous implementation. The multiplex network, 𝒢, is defined by its components.

𝒢(𝑉, 𝐸, 𝐿) (8.3)

Where V is the set of nodes that exist in every layer, 𝐸 is a set containing the sets of edges for each layer 𝛼, 𝐸𝛼∈𝐿_{, and L is the set of layers. It useful to note that the convention adopted in this research}

uses superscript to define layer, and subscript to define node. However, even within the multiplex network framework, there are different ways to represent a collaboration network. There are two major approaches that could be taken to representing the layers. Either every discipline is its own layer as shown in the following expression.

𝐿 = {𝑙0, 𝑙1, … , 𝑙𝑀} (8.4)

Where 𝐿 is the set of layers, 𝑙𝛼 _{a specific layer, 𝑀 the number of disciplines. This would imply that}

every node would have a presence in every discipline and interdisciplinary collaboration would be characterised by link overlap. That is to say that if two individuals are collaborating across different

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disciplines, they would connected in both disciplines (layers). This is a multiplex network framework using a rank-3 tensor of the form 𝑁 × 𝑁 × 𝑀.

However, it is possible to define every layer as every type of link possible. This would result in 𝑀 × 𝑀 layers, as each discipline and discipline pair would be specific layer, as shown in the following expression.

𝐿 = {𝑙00, 𝑙01, … 𝑙0𝑀, 𝑙10, … , 𝑙𝑀0, 𝑙𝑀1, … , 𝑙𝑀𝑀} (8.5)

Where a layer is denoted by 𝑙𝛼𝛽_{, 𝛼 and 𝛽 are one of 𝑀 disciplines. Where 𝛼 = 𝛽, the layer is}

disciplinary, and where 𝛼 ≠ 𝛽, the layer is interdisciplinary. Unlike the other framework, individuals would have no activity in other disciplines as these would be represented exclusively in the interdisciplinary layers.

This latter way of defining layers provides a greater resolution as it distinguishes between different layer pairs. However, the former provided a highly simple and effective model that outperformed more sophisticated models. It also highlights the importance of individuals’ presence in specific layers. For this reason, and as simplicity is the preferred approach, the former was adopted. The adjacency is therefore given by the following expression.

𝐴_𝑖𝑗𝛼 = {1, 𝑖𝑓 𝐸𝑖𝑗

𝛼_{∈ 𝐸}𝛼

0, 𝑖𝑓 𝐸𝑖𝑗𝛼 ∉ 𝐸𝛼

(8.6)

The most important observation to make is that a node will exist in every layer (although they may not be active in that layer). This means that a node will almost be treated as a separate entity in every layer, which becomes an important feature in the analysis. This research names the overall presence of an individual a ‘node’, whereas the entities in individual layers are called ‘node entities’. Therefore, a node is represented by all of its node entities.

Interdisciplinary links can be represented by the overlap. However, as most established networks measures are node-centric, it is useful to distinguish between disciplinary and interdisciplinary node entities. However, unlike in Chapter 7 where a threshold was needed for an entire node, here only node entities will be interdisciplinary and remains fairly assumption free. This simply requires a definition of which node entity represents the node’s core-discipline.

The core-discipline is the layer element, 𝛼 ∈ 𝐿, that node i belongs to according to the classification method defined in Chapter 6. This is denoted as 𝐷𝑖 (the discipline of i). Thus, a node in layer 𝐷𝑖 =

𝛼 will represent its disciplinary entity, whereas the node in 𝐷𝑖≠ 𝛼 will represent one of its

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This culminates in the aggregate network being created and then split into layers as demonstrated in Figure 8.3

Figure 8.3. A conceptual representation of a traditional network (top box) and its counterpart multiplex network (bottom box). The networks are node aligned. The colours of the nodes represent the disciplines they belong to, each discipline having a layer. Any interdisciplinary links exhibit link overlap in both layers (e.g. the blue is connected to a yellow node in both the blue and yellow layers).

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Finally, it is important to be aware that if a node has no links in a layer, this node is considered to be inactive in that layer. Node-layer activity is a vector of length M that is defined by the following expression.

𝑏_𝑖𝛼= {1, 𝑖𝑓 𝑘𝑖𝛼 > 0

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (8.7)

Multiplex node activity is the overall activity throughout the multiplex network given by the following expression.

𝐵𝑖 = ∑ 𝑏𝑖𝛼 𝑀 𝛼=1

(8.8) This provides the number layers a node is active in and can be thought of the node-layer degree.

Contribution to knowledge:

The section created a bespoke framework for multiplex collaboration networks seeking to investigate the effect of different classifications of individual such as individual’s disciplines. The framework is unique in that it identifies node classifications, but adopts a rank-3 tensor notation as opposed to a rank-4 tensor. This has a few pros and cons. The pros are that it

circumvents ‘the curse of dimensionality’, wherein the more dimensions there are, the sparser the space, and more difficult it is to make any predictions. It also creates node entities, where every person has a network representation in every layer, which becomes vital in the predictive model this research produces. The cons are that it reduces the specificity of the analysis as specific interdisciplinary links are not identified. However, due to nature of multiplex networks, this information is not lost, but is rather shifted to the link overlap. In this respect, the rank-3 tensor provides all of the benefits with no loss of information (i.e. a rank-4 tensor representation can be constructed out of the information held in a rank-3 tensor representation).

This framework is an original contribution to knowledge that layers down the foundation for multiplex collaboration networks.

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Chapter 9: Multilayer evolution in collaboration networks