Adjacency matrix

As explained in Graph Algorithms for Data Science

An adjacency matrix is a square matrix whose elements indicate whether pairs of graph nodes are connected.

An adjacency matrix represents a graph as a square table. Each row and column represents a node, and each cell indicates whether the corresponding pair of nodes is connected. section 9.0

Four-node graph

For nodes A, B, C, and D, a cell containing 1 means the corresponding relationship exists; a cell containing 0 means it does not. The cell at column C and row D is 1 when C and D are connected. section 9.0

The representation has important consequences for graph machine learning:

• A graph with one million nodes gives every node a vector with one million elements. section 9.0
• Using the matrix as model input creates many features, makes models depend on graph size, and can increase overfitting. section 9.0

Figure 9.2 Adjacency matrix

View the four-node adjacency matrix and connect its cell values to graph relationships. section 9.0

How Graph Algorithms for Data Science covers it

How deep it goes

Defines 0/1 connectivity and scales it to million-node graphs, then identifies feature-count and overfitting problems.

Examples

Uses a four-node graph with A, B, C, and D, including a C–D cell containing 1.

What you can do after

Evaluate/understand adjacency-matrix features and their scaling and overfitting implications for graph machine learning.

How it connects

Connects graph structure with node embeddings and machine learning models.

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Also explained in

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  Optimization Algorithms: AI techniques for design, planning, and control problemsAn adjacency matrix is a matrix whose entries indicate whether pairs of graph nodes are connected, and possibly encode connection weights.
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  Graph Neural Networks in ActionAn adjacency matrix is a matrix representation of a graph in which entries encode whether or how strongly pairs of nodes are connected.
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  Data Without LabelsAn adjacency matrix is a matrix representing which pairs of nodes in a graph are connected and, when applicable, the connection weights.
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  D3.js in Action, Third EditionAn adjacency matrix is a square grid in which rows and columns represent nodes and cells indicate the presence and optionally the weight of connections between them.
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  Machine Learning Algorithms in DepthAn adjacency matrix is a matrix that records which pairs of graph vertices are connected by edges.
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  Advanced Algorithms and Data StructuresAn adjacency matrix is a matrix representation of a graph in which entries indicate whether pairs of vertices are connected, and may encode edge information.
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  Financial AI in PracticeA matrix whose entries represent whether or how pairs of graph vertices are connected.

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Related concepts

• adjacency listalternative to · An adjacency list is a graph representation that associates each vertex with a collection of its neighboring vertices or incident edges.
• graphimplements · A graph is a mathematical structure consisting of vertices connected by edges that represent relationships or connections.
• graph Laplacianused by · A graph Laplacian is a matrix formed from a graph’s degree and adjacency matrices that encodes connectivity and supports spectral analysis.
• encoderused by · An encoder is a component that transforms input data into an encoded representation for subsequent processing, reconstruction, comparison, or transmission.

This explainer was generated by Manning's knowledge graph from the text of Graph Algorithms for Data Science. The section links point to the parts of the book it draws on.