Table 1.
Basic properties of real-world multiplex collaboration networks.
We report the number of nodes N, the average degree 〈k〉 and the clustering coefficient C for each layer of a subset of the APS and IMDb data sets. In particular, we focus on the multiplex collaboration network of all scientists active in Nuclear, Particle, Condensed Matter I and Interdisciplinary physics, and the multiplex collaboration network of all actors starring in Action, Crime, Romance and Thriller movies. All the layers of APS have a clustering coefficient C in the range [0.24, 0.30]. Conversely, the values of C of all the IMDb layers are in the range [0.56, 0.61].
Fig 1.
Similarity of communities at the different layers of real-world collaboration networks.
In each of the two graphs nodes represent the layers of the multiplex (APS on the left and IMDb on the right) and the edges are coloured according to the value of the normalised mutual information for the community decompositions at the corresponding pairs of layers.
Fig 2.
Schematic representation of network growth with intra-layer and inter-layer triadic closure.
A newly arrived node i creates m[1] new edges on layer 1 and m[2] new edges on layer 2. The new node starts by choosing at random one of the two layers {1, 2}. We indicate the first chosen layer using the label a. a) The first link of the new node is connected to one of the nodes of layer a, chosen uniformly at random and called na (solid green line). Each of the remaining m[a] − 1 links is attached with probability p[a] to a neighbour of the previously chosen node (intra-layer triadic closure) or with probability 1 − p[a] to one of the nodes at layer a, chosen uniformly at random (dashed red lines). b) Afterwards, the new node starts connecting on the other layer b. The first link on layer b is created to node na with probability p*, or to one of the other nodes at layer at random with probability 1 − p*. We call nb the first node to which i attaches on layer b. c) Each of the m[b] − 1 remaining edges on layer b are attached with probability p[b] to one of the neighbours of nb, and with probability 1 − p[b] to one of the nodes on layer b, chosen uniformly at random.
Fig 3.
Model calibration in a simple scenario.
We show the values of p and p* extracted for the different pairs of layers of the four-layer collaboration networks of APS and IMDb. (a) The clustering coefficient C depends exclusively on the parameter p, which tunes intra-layer triadic closure. Since all the layers of those two multiplex networks have comparable clustering coefficients, we are able to determine the value of the parameter p in each of the two cases. (b) For each pair of layers, we can also determine the value of the inter-layer triadic closure parameter p* by setting it equal to the value which yields an organisation in communities characterised by a value of NMI compatible with that observed in the real network.
Fig 4.
Layers with similar or dissimilar community structures.
We show the effect of the value of the inter-layer triadic closure parameter p* on the multiplex community structure. The two top layers show two typical realisations of the simplest version of the network model with N = 50, m[1] = m[2] = 2 and p[1] = p[2] = 0.9. Nodes belonging to the same community are given the same colour and are drawn close to each other. The two layers at the bottom of each multiplex are obtained by setting, respectively, p* = 0.9 (left) and p* = 0.1 (right). The nodes maintain the same placement in space on the second layer, but are coloured according to the community they belong in that layer (colours are chosen in order to maximise the number of nodes that have the same colour in the two layers). It is evident that the community structures of the two layers on the left, corresponding to p* = 0.9, are very similar, while the partition into communities of the upper layer on the left panel is substantially different from the one observed in the bottom layer of that multiplex.
Fig 5.
Intra-layer, inter-layer and mixed assortativity in collaboration networks.
We show the intra-layer (a), inter-layer (b) and mixed (c) degree-correlations for couples of layers of the IMDb and APS collaboration networks. Real data (dots) are compared with the results of our model (solid lines) generated with the extracted values p and p*. The symbols (^) indicate that the reported quantities (both for the model and the data) have been normalised to the values observed in the corresponding configuration model. As shown, the model is in general able to correctly capture the assortative trends of the three different types of correlations. Very good agreement with the data is attained in the case of the movie actor collaboration network. Less precise results are obtained for the APS network, where we deal with a system of considerably smaller size.
Table 2.
Quantitative comparison between the curves obtained from the model and the data for the inter-layer degree correlations the and mixed degree correlations.
The curves have been fitted using a function of the form f(x) ∼ xγ; the γ parameter is reported for the corresponding curves in Fig 5.
Table 3.
Basic properties of duplex networks in APS.
We consider all the possible multiplex networks with M = 2 layers obtained from combinations of the APS collaboration networks corresponding to the four sub-fields Nuclear, Particle, Condensed Matter I and Interdisciplinary Physics. For each duplex, we report the number of nodes N, the average degree on the two layers 〈k[1]〉 and 〈k[2]〉, and the values of the clustering coefficients C[1] and C[2].
Fig 6.
In panel a) we show the dependence of the clustering coefficient C on the intra-layer triadic closure parameter p for different values of the parameter m, which sets the layer’s average degree. In the multiplex consisting of the layers Particle (P) and Condensed Matter I (CM), the average degree of each layer corresponds, respectively, to m[1] = 3 and m[2] = 4. The value of p[1] and p[2] are determined to match the clustering coefficients C[1] and C[2]. In panel b), after having determined m[1], m[2], p[1] and p[2] for all the pairs of layers in the APS dataset, we run the model with such parameters for different value of p* and infer, for each pair, the value of the inter-layer triadic closure parameter p* yielding a value of NMI compatible with that observed (see Table 1 for layers’ acronyms). In panel c) we plot a heat-map of the NMI as a function of p and m, respectively for low (0.05), intermediate (0.50) and high (0.95) values of p* in the model with m[1] = m[2] = m and p[1] = p[2] = p. An increase in the link density of the layers produces a less correlated community structure in the two layers, even if the inter- and intra-layer triadic closure strengths are high.