Introduction
Multilayer networks generation models
Definitions and notations
Properties of real-world complex networks
The state of the art of generation models
-
The first may consider a set of observed properties as essential, and then sample randomly objects among the ones which have these properties. Proceeding this way, will yields a typical object with the concerned properties [28‐30]. It is then possible to determine if the retained set of properties is sufficient (do the random objects produced by the model fit well the real one? ) and to study the expected behavior of the object of interest. The relevance of the set of properties is generally checked using other known properties or behaviors of the object.
-
The second define’s a construction network generation models process inspired from the way the object is really constructed [2, 31‐33]. This construction process is generally iterated from an initial state, and eventually leads to an appropriate object. The analysis then concerns the properties induced by the construction process: do they fit real-world properties?
-
Growth: At each time \(t \ge 1\) a node with a replica node in each of the two layers is added to the multiplex. Each newly added replica node is connected to the other nodes of the same layer by m links.
-
Generalized preferential attachment: The new link in layers \(\alpha = 1, 2\) is attached to node i with probability \(\Pi _i^\alpha\) proportional to a linear combination of the degree \(k_i^{[1]}\) of node i in layer 1 and \(k_i^{[2]}\) of node i in layer 2.
Co-publications multilayer networks
Network | n | m | \({\bar{d}}\) | l | \(\gamma\) | C | T | \(\delta\) | |
---|---|---|---|---|---|---|---|---|---|
Chem | Researchers | 19794 | 80144 | 8.1 | 7.07 | 2.43 | 0.87 | 0.43 | 0.0004 |
Laboratories | 5064 | 14653 | 5.8 | 3.56 | 2.43 | 0.85 | 0.07 | 0.0011 | |
Institutions | 4563 | 32045 | 14.058 | 2.10 | 2.44 | 0.9 | 0.03 | 0.003 | |
Info | Researchers | 26492 | 485275 | 36.6 | 10.2 | 2.44 | 0.92 | 0.74 | 0.0013 |
Laboratories | 9658 | 125832 | 26.0 | 4.48 | 2.43 | 0.87 | 0.84 | 0.0026 | |
Institutions | 10463 | 163545 | 31.2 | 4.08 | 2.44 | 0.86 | 0.75 | 0.0029 | |
Math | Researchers | 9199 | 14236 | 3.09 | 13.8 | 2.40 | 0.80 | 0.54 | 0.0003 |
laboratories | 3164 | 8167 | 5.16 | 3.61 | 2.43 | 0.75 | 0.09 | 0.0016 | |
Institutions | 3558 | 12299 | 6.91 | 3.49 | 2.43 | 0.75 | 0.11 | 0.0019 | |
Phys | Researchers | 19920 | 160915 | 16.1 | 9.32 | 2.43 | 0.89 | 0.80 | 0.0008 |
Laboratories | 4793 | 18199 | 7.59 | 4.19 | 2.43 | 0.82 | 0.34 | 0.0015 | |
Institutions | 5161 | 22623 | 8.76 | 4.00 | 2.44 | 0.82 | 0.30 | 0.0016 | |
Shs | Researchers | 3955 | 6193 | 3.13 | 5.46 | 2.39 | 0.86 | 0.83 | 0.0007 |
Laboratories | 1729 | 3021 | 3.49 | 5.98 | 2.42 | 0.73 | 0.29 | 0.0020 | |
Institutions | 1903 | 4142 | 4.35 | 5.12 | 2.43 | 0.73 | 0.23 | 0.0022 | |
Sdu | Researchers | 20873 | 98593 | 9.44 | 7.49 | 2.43 | 0.87 | 0.67 | 0.0004 |
Laboratories | 7781 | 29238 | 7.51 | 3.69 | 2.43 | 0.87 | 0.12 | 0.0009 | |
Institutions | 8217 | 35408 | 8.61 | 3.63 | 2.44 | 0.86 | 0.12 | 0.0010 | |
Sdv | Researchers | 31557 | 154416 | 9.78 | 11.8 | 2.43 | 0.92 | 0.81 | 0.0003 |
Laboratories | 11010 | 46084 | 8.37 | 4.89 | 2.43 | 0.89 | 0.61 | 0.0007 | |
Institutions | 11594 | 50942 | 8.78 | 4.69 | 2.43 | 0.88 | 0.52 | 0.0007 | |
Stat | Researchers | 2867 | 5876 | 4.10 | 9.41 | 2.40 | 0.87 | 0.57 | 0.0014 |
Laboratories | 1301 | 3229 | 4.96 | 4.20 | 2.41 | 0.80 | 0.25 | 0.0038 | |
Institutions | 1541 | 5458 | 7.08 | 3.81 | 2.43 | 0.79 | 0.22 | 0.0045 |
The hierarchical network generation model
Collaboration and affiliation algorithms
-
\({\mathcal {G}} = \{G_\alpha ; \alpha \in \{0, 1, \ldots , M-1\}\}\) is a family of collaborations graphs \(G_\alpha = (X_\alpha , E_\alpha )\). This can be a collaboration networks of actors or collaboration network at each organization’s level.
-
\({\mathcal {C}} = \{E_{\alpha \beta } \subseteq X_\alpha \times X_\beta ; \alpha , \beta \in \{0, 1, \ldots , M-1 \}, \alpha \ne \beta \}\) is the set of affiliations between actors and organizations or between organizations and sub-organization \(G_\alpha.\)
Properties of the generated networks
-
\((1-\lambda )\eta\) new nodes are added at layer 0.
-
Those nodes generate \((1-\lambda _0)(1-\lambda _1)\eta\) new nodes at layer 1 by affiliating new nodes of layer 0 with the new nodes in this layer 1. Since, each new node in layer 0 has the probability \(1-\lambda _1\) (using Algorithm 3 and affiliation vector) to be affiliated to a new node in layer 1.
-
By recurrence, at layer \(\alpha\) we have \((1-\lambda _0)(1-\lambda _1)\dots (1-\lambda _\alpha )\eta\) new nodes are added as affiliation of \((1-\lambda )(1-\lambda _1)\dots (1-\lambda _{\alpha -1})\eta\) new nodes of layer \(\alpha -1\). Those \((1-\lambda _0)(1-\lambda _1)\dots (1-\lambda _\alpha )\eta\) new nodes of layer \(\alpha\) will generate \((1-\lambda _0)(1-\lambda _1)\dots (1-\lambda _\alpha )(1-\lambda _{\alpha +1})\eta\) new nodes in layer \(\alpha +1\).
-
the selection of nodes in layer \(\alpha -1\) affiliated to node of layer \(\alpha\) having degree k.
-
the selection of old nodes of degree k to affiliate new nodes of layer \(\alpha -1\) using affiliation vector.
Simulations
Designation | Description |
---|---|
M | Number of layers |
\(N_a\) | Number of collaborations to generate |
P | The distribution of the number of actors per collaboration |
\(P_i=\) probability to have i actors in a collaboration | |
\({\mathcal {V}}=\{\lambda _0, \lambda _1, \ldots , \lambda _M\}\) | Affiliation vector |
Network |
n
|
m
|
\({\bar{d}}\)
|
l
|
\(\gamma\)
|
C
|
T
|
\(\delta\)
| |
---|---|---|---|---|---|---|---|---|---|
Shs | Researchers | 3220 | 6409 | 3.98 | 4.53 | 2.41 | 0.65 | 0.19 | 0.0012 |
Laboratories | 2591 | 6290 | 4.85 | 3.95 | 2.42 | 0.62 | 0.13 | 0.0018 | |
Institutions | 2291 | 6175 | 5.39 | 3.76 | 2.43 | 0.60 | 0.12 | 0.0023 | |
Stat | Researchers | 3310 | 6879 | 4.15 | 5.53 | 2.40 | 0.80 | 0.34 | 0.0012 |
Laboratories | 2066 | 6765 | 6.55 | 3.77 | 2.43 | 0.68 | 0.14 | 0.0031 | |
Institutions | 1363 | 6534 | 9.59 | 3.21 | 2.43 | 0.63 | 0.14 | 0.0070 | |
Math | Researchers | 9137 | 15813 | 3.46 | 5.67 | 2.42 | 0.71 | 0.20 | 0.0003 |
Laboratories | 7134 | 15732 | 4.41 | 4.61 | 2.43 | 0.65 | 0.11 | 0.0006 | |
Institutions | 5710 | 15607 | 5.46 | 4.07 | 2.43 | 0.62 | 0.08 | 0.0009 | |
Sdv | Researchers | 34698 | 164777 | 9.49 | 5.45 | 2.43 | 0.90 | 0.57 | 0.0002 |
Laboratories | 20154 | 163133 | 16.1 | 3.52 | 2.44 | 0.79 | 0.16 | 0.0008 | |
Institutions | 13656 | 157545 | 23.0 | 3.07 | 2.44 | 0.75 | 0.12 | 0.0016 | |
Phys | Researchers | 29681 | 527031 | 35.5 | 3.32 | 2.44 | 0.89 | 0.32 | 0.0011 |
Laboratories | 22775 | 512941 | 45.0 | 3.05 | 2.44 | 0.86 | 0.22 | 0.0019 | |
Institutions | 17647 | 482890 | 54.7 | 2.87 | 2.44 | 0.85 | 0.20 | 0.0031 | |
Sdu | Researchers | 21878 | 117614 | 10.7 | 4.06 | 2.43 | 0.83 | 0.26 | 0.0004 |
Laboratories | 7069 | 98700 | 27.9 | 2.86 | 2.44 | 0.75 | 0.18 | 0.0039 | |
Institutions | 2430 | 62543 | 51.4 | 2.52 | 4.43 | 0.80 | 0.39 | 0.0211 | |
Chem | Researchers | 20603 | 92155 | 8.94 | 4.36 | 2.43 | 0.80 | 0.22 | 0.0004 |
Laboratories | 15326 | 91388 | 11.9 | 3.65 | 2.44 | 0.76 | 0.11 | 0.0007 | |
Institutions | 11321 | 89263 | 15.7 | 3.26 | 2.44 | 0.73 | 0.10 | 0.0013 |