The mechanism
A graph supplies nodes and edges describing task relationships. Node features are arrays, but their row order is only an indexing convention. The specimen adds self-loops, normalizes adjacency by endpoint degrees and propagates two feature coordinates through one or more layers. It uses identity feature weights and ReLU so that each output can be traced to neighbors. Renaming nodes and consistently permuting rows/edges preserves the represented result. Removing an edge changes the computation because information follows a different relationship structure. This is an executable forward mechanism; its weights are fixed, and it does not demonstrate training a graph model.
- Declare a four-node undirected graph and two features for every node; add a self-loop at every node.
- Calculate degree and the symmetric normalization coefficient for every present edge.
- Multiply normalized adjacency by the feature matrix and apply ReLU; repeat to expand the reachable neighborhood.
- Compare a consistent permutation with an actual edge removal and inspect the different invariance claims.
Science and assumptions
Graph message passing and node identity
Normalize adjacency with self-loops, aggregate feature rows and apply ReLU using identity feature weights.
Symbols and units
- H⁽ˡ⁾
- Node-feature matrix at layer l
dimensionless
- Ã
- Adjacency with self-loops
zero or one
- D̃
- Diagonal degree matrix
count
On a four-node cycle with self-loops, every node has degree 3. Node 0 initially receives features from itself, node 1 and node 3: ([1,0]+[0,1]+[0,2])/3=[1/3,1]. Removing the 0–3 edge changes both the neighborhood and endpoint normalization. A separate test permutes all node IDs and verifies equivalent outputs after undoing the permutation.
Assumptions Four-node undirected graph, self-loops, identity learned-weight placeholder and nonnegative input features. No parameter training or arbitrary graph benchmark is performed.
Engineering example
On a four-node cycle with self-loops, every node has degree 3. Node 0 initially receives features from itself, node 1 and node 3: ([1,0]+[0,1]+[0,2])/3=[1/3,1]. Removing the 0–3 edge changes both the neighborhood and endpoint normalization. A separate test permutes all node IDs and verifies equivalent outputs after undoing the permutation.
Distinctions and tradeoffs
The task graph, a computation graph, the atlas prerequisite graph and a chip netlist are different objects even when all are drawn as nodes and edges.
More aggregation steps allow information to travel farther, but representation behavior and task quality depend on the architecture, normalization, features and learning objective.
Graph learning is used where relations are part of the input. The same matrix machinery executes the aggregation, but its semantic edges are not the hardware wires carrying those numbers.
Specifications and interfaces
- Teaching specimen
- Graph computation
On a four-node cycle with self-loops, every node has degree 3. Node 0 initially receives features from itself, node 1 and node 3: ([1,0]+[0,1]+[0,2])/3=[1/3,1]. Removing the 0–3 edge changes both the neighborhood and endpoint normalization. A separate test permutes all node IDs and verifies equivalent outputs after undoing the permutation.
- Evidence boundary
- Explicit computed example
Four-node undirected graph, self-loops, identity learned-weight placeholder and nonnegative input features. No parameter training or arbitrary graph benchmark is performed.
Interfaces
- Inputs and convention Input
- Declare a four-node undirected graph and two features for every node; add a self-loop at every node.
- Computed result and scope Output
- Compute normalized neighborhood aggregation and distinguish an edge change from a harmless node renaming.
Check your understanding
Which operation should preserve the represented forward result?
- Removing an edge without changing anything else
- Changing degree normalization arbitrarily
- Renaming nodes while consistently permuting features and adjacency
Answer and reasoning
Renaming nodes while consistently permuting features and adjacency
Renaming nodes while consistently permuting features and adjacency. Four-node undirected graph, self-loops, identity learned-weight placeholder and nonnegative input features. No parameter training or arbitrary graph benchmark is performed.
Evidence and scope
The workbench executes the declared small mechanism. Numeric examples are authored and independently checked, not measurements from the cited research. Four-node undirected graph, self-loops, identity learned-weight placeholder and nonnegative input features. No parameter training or arbitrary graph benchmark is performed.
Open Mathematical foundations for the bounded executable mechanism and its record. General numerical labs are an additional reference library.