Weighted Graph

A weighted graph is a graph in which each edge carries a numerical value — a weight — representing the strength, cost, capacity, or intensity of the relationship between the connected nodes. The weights transform the graph from a topological structure into a metric space, enabling analyses that depend not merely on whether two nodes are connected but on how strongly or at what cost.

In network analysis, edge weights are typically interpreted as interaction frequency, tie strength, or probability of influence. In game-theoretic networks, weights may represent payoffs or trust levels. In transportation and infrastructure networks, weights are literal costs — distance, time, or energy. The mathematical treatment is uniform: shortest-path algorithms like Dijkstra's operate on weights regardless of their interpretation, and spectral methods like the graph Laplacian generalize naturally to the weighted case.

The conceptual risk is that weights are not always comparable across edges. A weight of 5 in one relationship may not mean the same thing as a weight of 5 in another, yet most weighted-graph algorithms assume additive comparability. This is a genuine limitation: the mathematics of weighted graphs assumes a single numerical type, while real relational systems often involve qualitatively distinct dimensions of strength that cannot be reduced to a scalar without loss of structure.

The weighted graph is network theory's most useful and most abused formalism: useful because it adds expressive power, abused because every weighted edge looks the same to the algorithm even when the weights mean different things to the system.