Weighted graph

As explained in Graph Algorithms for Data Science

A weighted network is a graph whose relationships carry numerical values representing strength, cost, frequency, or another quantity.

A weighted graph stores a numerical value on each relationship. The value can represent strength, traversal cost, frequency, or another quantity, allowing relationships to be compared rather than treated as equivalent. section 2.1.2

Interpret a weight according to the network’s purpose:

• Prefer higher values when weights represent relationship strength, such as friendship. section 2.1.2
• Prefer lower values when weights represent traversal cost, such as transportation distance or time. section 2.1.2
• Use weighted variations of degree or PageRank when analyzing projected monopartite networks. section 6.0

Strength versus cost

A connection between Jane and Ingrid can carry a higher friendship weight than the connection between Ingrid and Mark. In a transportation network, the preferred path instead minimizes the total weighted cost. section 2.1.2

Figure 2.4 (A) represents an unweighted graph; (B) represents a weighted graph.

Compare an unweighted graph, where relationships have equal strength or cost, with a weighted graph, where numerical relationship values distinguish them. section 2.1.2

How Graph Algorithms for Data Science covers it

How deep it goes

Defines numerical relationship weights and explains when higher or lower values are preferred.

Examples

Uses friendship relationships, transportation networks, and projected retweet networks as examples of weighted analysis.

What you can do after

Evaluate whether relationship values should be maximized for strength or minimized for traversal cost.

How it connects

Connects unweighted graphs, shortest paths, transportation networks, monopartite projections, degree centrality, and PageRank.

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Also explained in

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  Optimization Algorithms: AI techniques for design, planning, and control problemsA graph whose edges have numerical weights representing quantities such as distance or cost.

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Related concepts

• graphkind of · A graph is a mathematical structure consisting of vertices connected by edges that represent relationships or connections.
• unweighted graphcontrasts with · An unweighted graph is a graph whose edges have no assigned numerical weights.
• Dijkstra's algorithmused by · Dijkstra's algorithm computes shortest paths from a source vertex in a weighted graph with nonnegative edge weights.
• UMAPused by · UMAP, or Uniform Manifold Approximation and Projection, is a nonlinear manifold-learning technique that embeds high-dimensional data in a lower-dimensional space while preserving important neighborhood and structural relationships.
• weighthas part · A weight is a numerical quantity assigned to an element to represent its contribution, influence, importance, strength, or resource amount in a calculation or system.

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