Table 1 Summary of notations and their description

G

A simple (i.e., monoplex) network graph

V, E

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

K, k

Number of GNN layers, index of layer

Q, q

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 \(v_j\)