Читать книгу Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic - Страница 78

The Graph Network (GN) framework [37] generalizes and extends various GNN, MPNN, and NLNN approaches. A graph is defined as a 3‐tuple G = (u, H, E) (H is used instead of V for notational consistency). u is a global attribute, is the set of nodes (of cardinality N^v), where each h_i is a node’s attribute. is the set of edges (of cardinality N^e), where each e_k is the edge’s attribute, r_k is the index of the receiver node, and s_k is the index of the sender node.

GN block contains three “update” functions, φ, and three “aggregation” functions, ρ,

(5.32)

where , and . The ρ functions must be invariant to permutations of their inputs and should take variable numbers of arguments.

The computation steps of a GN block:

1 φe is applied per edge, with arguments (, ,u), and returns . The set of resulting per‐edge outputs for each node i is, = , and is the set of all per‐edge outputs.

2 ρe → h is applied to , and aggregates the edge updates for edges that project to vertex i, into which will be used in the next step’s node update.

3 φh is applied to each node i, to compute an updated node attribute, . The set of resulting per‐node outputs is .

4 ρe → u is applied to E′, and aggregates all edge updates, into , which will then be used in the next step’s global update.

5 ρh → u is applied to H′, and aggregates all node updates, into , which will then be used in the next step’s global update.

6 φu is applied once per graph and computes an update for the global attribute, u′.