Graphs as Data: Adjacency and Node Features

A neural network needs numbers to operate on — matrices it can multiply, add, and take gradients through. A graph, as a mental object, is nodes and connections: a network of counterparties, a supply chain, a correlation structure between assets. To make a graph usable by any neural network, including a graph neural network, it first has to be converted into two plain numeric objects: a matrix describing the connections, and a matrix describing what each node is. Every graph-ML method, no matter how sophisticated, starts from exactly these two ingredients.

The analogy: a seating chart and a set of name tags

Think of a graph as a room full of people at a mixer. The adjacency matrix is the seating chart's connection list — a record of who is standing next to whom, with nothing at all about who any individual person actually is. The node feature matrix is the set of name tags, one per person, listing whatever attributes matter — job title, years of experience, department — with nothing about the room's layout. Neither object alone describes the mixer fully: you need the seating chart to know who can talk to whom, and the name tags to know anything about the individuals doing the talking. A graph neural network's entire job is to combine both.

The two matrices, defined

For a graph with $n$ nodes, the adjacency matrix $A$ is $n \times n$, with $A_{ij} = 1$ if nodes $i$ and $j$ are connected (an edge exists) and $0$ otherwise — for a weighted graph, $A_{ij}$ can instead hold the edge's weight, e.g. exposure size between two counterparties. The node feature matrix $X$ is $n \times d$, one row per node, each row a $d$-dimensional vector of that node's own attributes — for a company in a supply-chain graph, perhaps revenue, sector code, and credit rating.

$$A \in \{0,1\}^{n \times n}, \qquad X \in \mathbb{R}^{n \times d}$$

In words: $A$ answers "who is connected to whom," entirely structural, no node attributes involved; $X$ answers "what does each node look like," entirely attribute-based, no connection information involved. A graph neural network's first layer typically combines them by having each node gather its neighbors' rows of $X$, using $A$ to know which rows count as neighbors.

Worked example 1: building both matrices for a tiny counterparty network

Four firms, $\{A, B, C, D\}$, with trading relationships $A\text{-}B$, $B\text{-}C$, $C\text{-}D$, and $A\text{-}C$. The adjacency matrix (rows/columns in that order):

$$A_{\text{adj}} = \begin{pmatrix} 0&1&1&0 \\ 1&0&1&0 \\ 1&1&0&1 \\ 0&0&1&0 \end{pmatrix}$$

Row for firm $C$ reads $(1,1,0,1)$ — connected to $A$, $B$, and $D$, matching the three edges touching $C$ above. Now suppose each firm has a 2-feature vector (leverage ratio, sector code): $X = \big[(2.1, 1), (1.4, 2), (3.0, 1), (0.9, 3)\big]$. Firm $C$'s row, $(3.0, 1)$, is its leverage and sector — completely independent information from the adjacency row just computed; you need both to know that "$C$ is a highly levered, sector-1 firm connected to three others."

Worked example 2: one step of neighbor aggregation using both matrices

A common first GNN operation averages each node's own features with its neighbors' features. For firm $B$, neighbors are $A$ and $C$ (from row $B = (1,0,1,0)$), with features $(2.1,1)$ and $(3.0,1)$; $B$'s own feature is $(1.4,2)$. Averaging all three: $\big(\frac{2.1+3.0+1.4}{3}, \frac{1+1+2}{3}\big) = (2.17, 1.33)$. Firm $D$, by contrast, has only one neighbor, $C$, with feature $(3.0,1)$; averaging $D$'s own $(0.9,3)$ with just that one neighbor: $(1.95, 2.0)$. Notice how much $D$'s updated feature shifts toward $C$'s values simply because $D$ has only one connection — a direct, numeric illustration of why a node's position in $A$ (how many and which neighbors it has) directly shapes what a GNN computes for it, on top of whatever $X$ says about the node itself.

Think of the adjacency matrix the same way you'd think of the transformation matrix above: it's a fixed matrix that reshapes what a vector of node features becomes after one round of neighbor mixing, exactly as a 2×2 matrix reshapes a vector on the plane.

What this means in practice

This split matters for what a graph-ML pipeline has to assemble: a correlation-based asset graph needs a thresholding rule to turn continuous correlations into an $A$, plus a choice of directed (payment flow) versus undirected (mutual exposure); a firm's feature vector in $X$ is chosen and normalized entirely separately from that structure.