berdiameterX

berdiameterX is a generalized graph metric used in graph theory and network analysis to quantify the spatial extent of a network while accounting for edge heterogeneity and a tunable parameter X. It extends the standard graph diameter by replacing each edge length with a function of its weight and X, then taking the maximum of the resulting shortest-path distances over all pairs of vertices.

Definition and interpretation: For a weighted graph G=(V,E) with nonnegative edge weights w(e), one may define the X-scaled length of an edge as l_X(e) = w(e)^X. The berdiameterX, Diam_X(G), is the maximum, over all vertex pairs u,v ∈ V, of the shortest-path distance from u to v using edge lengths l_X. Varying X tunes sensitivity to heavier or lighter edges; X=0 yields a uniform length of 1 for all edges, while X=1 recovers the original weighted distance. In unweighted graphs (all w(e)=1), Diam_X(G) equals the ordinary diameter for any X.

Computation and applications: For fixed X, Diam_X(G) can be computed with standard single-source shortest-path algorithms from each vertex, giving an all-pairs approach with typical complexity O(n m log n) for sparse graphs. Applications appear in transportation planning, telecommunications, and social-network analysis, where understanding how heterogeneity in edge costs or priorities changes the network’s global extent is useful.

Limitations: The choice of edge-weight function and the parameter X strongly influence results, and there is no unique canonical X without domain-specific justification. Comparisons across networks require consistent weighting conventions.

See also: diameter, eccentricity, shortest-path problem, weighted graphs, complex networks.