Table 9 Summary of notations and their description
From: Co-MLHAN: contrastive learning for multilayer heterogeneous attributed networks
Notations | Description |
|---|---|
\(G_{{\mathcal{L}}}\) | Multilayer heterogeneous attributed graph |
\({\mathcal{L}}, \ell , L, l\) | Set of layers, number of layers, set of layer indexes and layer index in \(G_{{\mathcal{L}}}\) |
\(G_{l}\) | l-th layer in \(G_{{\mathcal{L}}}\) (single-layer heterogeneous attributed graph) |
\(V_{{\mathcal{L}}}, E_{{\mathcal{L}}}\) | Set of nodes and set of edges in \(G_{{\mathcal{L}}}\) |
\({\mathcal{V}}, {\mathcal{V}}_{l}, {\mathcal{V}}_{l}^{(t)}\) | Set of entities in \(G_{{\mathcal{L}}}\), set of nodes in the l-th layer, and set of entities of type t in the l-th layer |
A, R | Set of node/entity types and set of relation types |
\(A_{l}, R_{l}, n_{l}\) | Set of node types, set of relation types, and number of nodes, in the l-th layer |
\(\phi , \varphi\) | Node/entity and edge type mapping functions |
a, t, r | Node/entity type, target entity/node type, and relation type |
\(E_{r}\) | Set of edges of type r |
d | Dimension of latent space |
\({\varvec{\mathcal{X}}}_{{\mathcal{L}}} , {\varvec{\mathcal{X}}}_{l} , {\textbf{X}}^{(a)}_{l}\) | Sets of attribute matrices and layer-specific matrices, and set of attribute matrix for entities/nodes of type a |
i, j, u | Entity indexes |
\(\langle i,l \rangle ,\langle j,l \rangle\) | Node indexes (i.e., entity-layer pairs) |
\(L_{cross}, \pi , \delta (l,l')\) | Set of layer pairing indices, pair of coupled layers, and scoring function for inter-layer links |
\(R_{\langle i,l \rangle }\) | Set of relations involving node \(\langle i, l \rangle\) of target type t |
\({\textbf{x}}_{i}^{(a)},{\textbf{x}}_{\langle i,l \rangle }^{(a)}\) | Initial feature vectors for entity i and node \(\langle i,l \rangle\) of type a |
\({\textbf{h}}_{i}^{(a)},{\textbf{h}}_{\langle i,l \rangle }^{(a)}\) | Feature embeddings for entity i and node \(\langle i,l \rangle\) of type a |
\({\textbf{W}},{\textbf{b}}\) | Learnable weight matrix and bias term |
\({\mathcal{A}}, {\textbf{A}}_{\ell }, {\textbf{A}}^{sup}\) | Set of adjacency matrices of \(G_{{\mathcal{L}}}\), adjacency matrix of the l-th layer, and supra-adjacency matrix |
\(\sigma (\cdot )\) | Activation function |
\(\textbf{a}\) | Attention vector |
\(\alpha , \beta\) | Attention coefficients |
\(N^{(r)}(i,l)\) | Set of neighbors of node \(\langle i,l \rangle\) under relation r |
\({\textbf{z}}^{N^{(r)}}_{\langle i,l \rangle }\) | Embedding of node \(\langle i, l \rangle\) under relation r |
Q, q | Number of attention heads and head index |
\({\textbf{z}}^{{{{\rm NS}}}}_{i}\), \({\textbf{z}}^{{{{\rm NS}}}}_{\langle i,l \rangle }\) | Embedding of entity i and node \(\langle i,l \rangle\) under network schema view |
\(\mathcal{M}, M_{m}, p\) | Set of meta-path types, m-th meta-path type and number of (within layer) meta-path types |
\(\mathcal{M}^{\Updownarrow }, M_{(m, \pi )}\) | Set of across-layer meta-paths and m-th across-layer meta-path type |
\(N_{m}(i,l)\) | Meta-path based neighbors of node \(\langle i,l \rangle\) for the m-th meta-path |
\(N_{m}^{\Leftrightarrow }(i,l), N_{m}^{\Updownarrow }(i,l)\) | Sets of within and across neighbors of node \(\langle i,l \rangle\) for the m-th meta-path |
\({\textbf{z}}^{(m)}_{\langle i,l \rangle }\) | Embedding of node \(\langle i,l \rangle\) for the m-th meta-path |
\({\textbf{z}}^{(m)}_{\langle i,\pi \rangle }\) | Embedding of layer-pair \(\pi\) for the m-th across-layer meta-path |
\({\textbf{z}}^{{{{\rm MP}}}}_{i}\), \({\textbf{z}}^{{{{\rm MP}}}}_{\langle i,l \rangle }\) | Embedding of entity i and node \(\langle i,l \rangle\) under meta-path view |
\(\hat{{\textbf{z}}}^{{{{\rm NS}}}}_{i}\), \(\hat{{\textbf{z}}}^{{{{\rm MP}}}}_{i}\) | Projected embedding of entity i under network schema view and under meta-path view |
\(C_{i,j}\) | Number of meta-paths between entities i and j |
\(S_{i}\) | Set of entities connected to i via a meta-path (descending order) |
\(T_{pos}\), \(\overline{{\mathcal{S}}}_{i}\) | Threshold of best positives, and set of first \(T_{pos}\)-1 entities of \(S_{i}\) |
\({\mathcal{P}}_{i}, {\mathcal{N}}_{i}\) | Sets of positives and negatives for entity i |
\(\tau\) | Temperature parameter |
\(\lambda , \lambda ^{\Updownarrow }, \eta\) | Balancing coefficients |
\(L^{{{{\rm NS}}}},L^{{{{\rm MP}}}},L_{co},L_{sup},L_{tot}\) | Loss functions |