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Table 1 Summary of notations and their description

From: Graph convolutional and attention models for entity classification in multilayer networks

Notations Description
G A simple (i.e., monoplex) network graph
VE Set of nodes and set of edges in G
\(G_{{\mathcal {L}}}\) A multilayer network graph
\({\mathcal {V}}\) Set of entities in \(G_{{\mathcal {L}}}\)
\({\mathcal {L}}, \ell , L_l\) Set of layers, number of layers, l-th layer of \(G_{{\mathcal {L}}}\)
\(V_{{\mathcal {L}}}, E_{{\mathcal {L}}}\) Set of nodes and set of edges in \(G_{{\mathcal {L}}}\)
i Index of node \(v_i\) in G, resp. entity \(v_i\) in \(G_{{\mathcal {L}}}\)
\(\Gamma (i)\) Neighborhood of node \(v_i\) in G
\(\Gamma (i,l)\) Within-layer neighborhood of \(v_i\) in layer \(L_l\) of \(G_{{\mathcal {L}}}\)
\(\Psi (i,l)\) Outside-layer neighborhood of \(v_i\) in layers of \(G_{{\mathcal {L}}}\) different from \(L_l\)
\({\mathbf {X}}, {\mathbf {X}}_l\) Attribute (input feature) matrix in G, resp. in the l-th layer of \(G_{{\mathcal {L}}}\)
\({\mathbf {x}}_i\), \({\mathbf {x}}_{(i,l)}\) Attribute (input feature) vector for node \(v_i\) in G, resp. entity \(v_i\) in layer \(L_l\) of \(G_{{\mathcal {L}}}\)
\({\mathbf {Z}}, {\mathbf {Z}}_{l}\) Embedding (output feature) matrix in G, resp. in the l-th layer of \(G_{{\mathcal {L}}}\)
\({\mathbf {z}}_i\) Embedding (output feature) vector for node \(v_i\)
\(\widetilde{{\mathbf {Z}}}\) Embedding (output feature) matrix for the entities in \(G_{{\mathcal {L}}}\)
\({\mathbf {h}}_i\) Hidden-layer vector for node \(v_i\)
\({\mathbf {h}}^{(k)}_{(i,l)}\) Hidden-layer vector at the k-th layer of the GNN for entity \(v_i\) in layer \(L_l\) of \(G_{{\mathcal {L}}}\)
f Number of attributes (input features)
d Size of the embedding
Kk Number of GNN layers, index of layer
Qq Number of attention heads, index of attention head
\({\mathbf {W}}, {\mathbf {W}}^{(k)}\) Weight matrix of a generic, resp. k-th, layer of a GNN
\({\mathbf {A}}, {\mathbf {A}}_l\) Adjacency matrix in G, resp. in the l-th layer of \(G_{{\mathcal {L}}}\)
\({\mathbf {A}}^{\text {sup}}\) Supra-adjacency matrix in \(G_{{\mathcal {L}}}\)
\(\widetilde{{\mathbf {A}}}, \widetilde{{\mathbf {A}}}^{\text {sup}}\) Adjacency matrix, resp. supra-adjacency matrix, with self-loops
\(\sigma (\cdot )\) Activation function
\(e_{ij}\) Attention coefficient for edge between nodes \(v_i\) and \(v_j\)
\(\alpha _{ij}\) Normalized attention coefficient for edge between nodes \(v_i\) and \(v_j\)