The various approaches described have thus far described approaches to constructing, measures to analyse, and the dynamics describing the emergent nature of multilayer networks. To succeed in creating a model that can predict properties in multilayer networks it is necessary to understand them.
This requires in-part to understand the mechanisms responsible for the emergence of the structural properties seen.
The most basic multilayer network that could be created is to populate a set of layers with a set of nodes and to randomly assign connections between nodes as an Erdős–Rényi model. This provided
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unrealistic network structures. The Barabási and Albert model has provided a simple mechanism by which realistic traditional networks can be grown (Barabási and Albert 1999). It was by identifying hidden mechanics in growth models that allowed Barabási and Albert (1999) to identify the importance of preferential attachment. This phenomenon has guided a lot of subsequent research (Barabási and Albert 1999, Barabási and Pósfai 2016).
Several growth models have been proposed. Non-linear preferential attachments were proposed as an improvement on the Barabási-Albert model (Krapivsky, Redner et al. 2000). A link selection model that selects random existing links and connects new nodes to either node that the link is connected to, thereby producing a scale-free distribution (Dorogovtsev and Mendes 2002). A copycat model has also been proposed that copies what other nodes are doing (Kleinberg, Kumar et al. 1999).
Bianconi-Barabási models attempt to improve on preferential attachment by suggesting that there is a node ‘fitness’ that determines how attractive a node is to receive new links (Bianconi and Barabási 2001). In cases where preferential attachment is driving the growth of the network, the degree of a node can be thought of as a fitness measure. However, it also opens the possibility of late-comers becoming major players.
However, each one of these models and variations include some measure of preferential attachment (whether it is direct or indirect) to create a realistic network structure. It is for this reason that the Barabási-Albert model (the first model that was able to recreate the scale-free property) has been cited over 30,000 times (verified on Google Scholar 24/06/2018). Optimization and game-theory approaches also suggest that preferential attachment is the rational choice (Fabrikant, Koutsoupias et al. 2002, Becker 2013).
However, each one of these models relies on the growth of the network, that is to say the addition of a node at every timestep. The sequential addition of nodes is a vital part of this dynamic. Each of these models do not consider that two pre-existing nodes could create additional links between them. Link addition models have found that there is a preferential attachment that occurs on both ends of such links (Barabâsi, Jeong et al. 2002).
Equally, there are various different mechanisms that affect the overall process, such as aging (e.g. in collaboration networks, an author may have a prime) (Amaral, Scala et al. 2000), node deletion (e.g. an author switches jobs or retires) (Saavedra, Reed-Tsochas et al. 2008), and the number of links grows faster than the addition of nodes (e.g. number of scientific papers, and collaborators as shown in Chapter 5) amongst many different models.
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The rich information that has been extracted from growth and evolution models has been invaluable in understanding networks. It is therefore important to develop similar models in multilayer networks. Not many studies have been performed.
For network-of-networks, Exponential Random Graph Models (ERGMs) have been extended to model the probability of local tie structures occurring. ERGMs have traditionally been thought as useful for social networks, as these are assumed to be locally emergent (Lusher, Koskinen et al. 2013). A network-of-network implementation of ERGMs was shown to successfully generalise French cancer research elites’ collaboration network (Wang, Robins et al. 2013). Equally, ERGM was used to analyse special interest groups and identify social roles (Heaney 2014). Layer interdependency has been investigated based on degrees (Lee, Kim et al. 2012, Min, Do Yi et al. 2014), and has been extended to epidemics analysis (Funk and Jansen 2010). A similar approach has been taken in generating a multiplex network, but changing node labels to vary interlayer correlation (De Domenico, Solé-Ribalta et al. 2013).
Nicosia and Latora (2015) proposed two models based on simulated annealing to reproduce observed patterns in pairwise interlayer degree-correlations.
Nicosia, Bianconi et al. (2013) attempt to create a multiplex growth model based on linear preferential attachment. The preferential attachment is conducted layer-by-layer and the preferential attachment is based not only on the degree of the node in the layer in question, but also the degree in other layers. Two findings were given in the paper: the interdependency of the layers had a significant impact on the growth of the network and newer nodes are more affected by the interdependency. Nicosia, Bianconi et al. (2014) extends their model to include non-linear preferential attachment between the two layers. The paper succeeds in showing that altering the non-linear preferential attachment mechanisms across the layers changes the degree distribution, and layer-degree-correlations. This then provides a powerful framework to compare real networks to their growth results (e.g. if a negative degree-correlation is found in a real network, this could give an indication of how it is that the layers affect each other).
Kim and Goh (2013) provide similar approach where it is shown that the layer-pair’s degree Pearson correlation coefficient changes significantly when coevolution parameter is altered. Generally speaking, the greater the extent of the coevolution, the greater the amount of degree-correlation, whilst also altering the degree distribution.
However, given the nature of multilayer networks and how many different formats there are, there are no growth models that focus on collaborations in particular. Furthermore, none of the studies provide conclusive evidence that a multiplex network structure has been successfully simulated. Finally, none of the studies have disseminated the various components and mechanisms through which multiplex networks can be formed.
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These represent the greatest gaps in knowledge that need to be addressed in this research. These all provide the knowledge needed to understand how it is that individuals across disciplines collaborate with one another.