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Fig. 6. Comparison of quality of the results in terms of NMI on synthetic networks, each chart represents the results for different number of memberships (OM = [1-4]) in different mixing parameters.

Figure 6 Comparison of quality of the results in terms of NMI on synthetic networks, each chart represents the results for different number of memberships (OM = [1-4]) in different mixing parameters.

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Fig. 1. Multilayer communities may form in different subset of layers.
Fig. 1. Multilayer communities may form in different subset of layers.
where x is a normalization factor, and y; is a resolution factor, s and r indicates indices of slices, i and j, are indices of nodes, A is adjacency matrix, k is degree of node, m is number of edges, C is coupling edges matrix between slices and finally 6 is Kronecker delta. A well-known generalization of the popular modularity is introduced by Mucha et al. [14]. The authors represent each layer as a slice with an adjacency matrix as well as coupling links between nodes in slices, and the node-layer tuples are assigned to a community. So the communities are detected with regards to the layers. The following formula (2) is used for multi-slice modularity:
where x is a normalization factor, and y; is a resolution factor, s and r indicates indices of slices, i and j, are indices of nodes, A is adjacency matrix, k is degree of node, m is number of edges, C is coupling edges matrix between slices and finally 6 is Kronecker delta. A well-known generalization of the popular modularity is introduced by Mucha et al. [14]. The authors represent each layer as a slice with an adjacency matrix as well as coupling links between nodes in slices, and the node-layer tuples are assigned to a community. So the communities are detected with regards to the layers. The following formula (2) is used for multi-slice modularity:
Fig. 2. Multilayer weak community (g;) and multilayer strong community (g2). Community g, is a weak community because nodes x; and x2 have 3 out-degree and 2 in-degree with respect to community g;.
Fig. 2. Multilayer weak community (g;) and multilayer strong community (g2). Community g, is a weak community because nodes x; and x2 have 3 out-degree and 2 in-degree with respect to community g;.
With these explanations, the mathematical model for the mul- tilayer community detection problem will be: The above function (12) has two parts, the first for measuring the internal density and the second part for the rate of entropy, where A is the impact factor of entropy and is normally set to 1. Maximizing the FT2(S), can detect communities with high internal density and contributions from more layers.
With these explanations, the mathematical model for the mul- tilayer community detection problem will be: The above function (12) has two parts, the first for measuring the internal density and the second part for the rate of entropy, where A is the impact factor of entropy and is normally set to 1. Maximizing the FT2(S), can detect communities with high internal density and contributions from more layers.
Fig. 4. Multilayer overlapping communities. Subset of nodes, which have dense connections in the same subset of layers and have node x; in common.
Fig. 4. Multilayer overlapping communities. Subset of nodes, which have dense connections in the same subset of layers and have node x; in common.
Fig. 3. (a) A simple 9-node network, (b) Locus-based representation, (c) Corresponding graph, (d) Detected Communities.
Fig. 3. (a) A simple 9-node network, (b) Locus-based representation, (c) Corresponding graph, (d) Detected Communities.
Fig. 5. Pareto front of the optimization process that contains pareto optimal solutions.
Fig. 5. Pareto front of the optimization process that contains pareto optimal solutions.
Algorithm 4-1- Overlapping multilayer community selection
Algorithm 4-1- Overlapping multilayer community selection
Parameters for the generation of synthetic networks. Table 1
Parameters for the generation of synthetic networks. Table 1
Fig. 7. Comparison of quality of the results in terms of NMI on synthetic networks, each chart represents the results for different number of memberships (ON [1-6]) and OM = 3 in different mixing parameters jj.
Fig. 7. Comparison of quality of the results in terms of NMI on synthetic networks, each chart represents the results for different number of memberships (ON [1-6]) and OM = 3 in different mixing parameters jj.
Fig. 10. Layers, nodes and multilayer communities in Aucs dataset. Fig. 9. NMI and optimization iterations for 10 runs and error bars. users can directly post messages and comments on other mes- sages much like in Facebook and other similar OSNs. From these, a multilayer network was retrieved, with users who their Twitter and YouTube accounts was associated to exactly one account.
Fig. 10. Layers, nodes and multilayer communities in Aucs dataset. Fig. 9. NMI and optimization iterations for 10 runs and error bars. users can directly post messages and comments on other mes- sages much like in Facebook and other similar OSNs. From these, a multilayer network was retrieved, with users who their Twitter and YouTube accounts was associated to exactly one account.
Fig. 8. The effect of w (a) and 6 (b) on NMI in mLFR benchmark networks. Error bars show the standard deviation for 10 runs of the algorithm. The parameters are N = 1000, L = 3, OM = 2, ON = 40.
Fig. 8. The effect of w (a) and 6 (b) on NMI in mLFR benchmark networks. Error bars show the standard deviation for 10 runs of the algorithm. The parameters are N = 1000, L = 3, OM = 2, ON = 40.
Structural measures and their formula for multilayer networks. Table A.1 5.4. Parameter analysis over two months. It contains three layers, which are commenting,
Structural measures and their formula for multilayer networks. Table A.1 5.4. Parameter analysis over two months. It contains three layers, which are commenting,
Table 2
Table 2
Fig. 11. This diagram represents the layers, number of communities, size of communities, multilayer communities as well as the flow of nodes’ memberships between multilayer communities in Aucs dataset.
Fig. 11. This diagram represents the layers, number of communities, size of communities, multilayer communities as well as the flow of nodes’ memberships between multilayer communities in Aucs dataset.
Main characteristics of the multilayer networks. Table A.2
Main characteristics of the multilayer networks. Table A.2
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