degv

Degv stands for degree variation vector, a concept in graph theory used to describe the irregularity of a network by capturing how far each vertex’s degree deviates from the graph’s average degree. It provides a compact descriptor of how evenly or unevenly edges are distributed among the vertices.

Formally, consider a finite, simple graph G with n vertices and degrees d1, d2, ..., dn. Let d̄ be the average degree, defined as d̄ = (1/n) Σ di. The degree variation vector is then defined componentwise by degv_i = di − d̄ for i = 1,...,n. The degv is an n-dimensional real vector whose components sum to zero, reflecting that deviations above and below the mean balance out.

Key properties include that the sum of all components of degv is zero, and the L2 norm of degv is related to the degree variance: ||degv||2 = sqrt(Σ (di − d̄)²) = sqrt(n Var(d)). The sign pattern of degv identifies vertices with higher-than-average degree (positive entries) and lower-than-average degree (negative entries).

Extensions of the concept handle directed and weighted graphs. For directed graphs, one may define separate degv values for in-degree and out-degree relative to their respective averages. For weighted graphs, degv uses the weighted degree (sum of incident edge weights) in place of the simple degree.

Applications include quantifying network irregularity, informing spectral analysis, and serving as a feature in graph classification and learning tasks. Limitations include sensitivity to outliers and dependence on the chosen averaging method.

Related concepts include degree distribution, degree variance, and irregular graphs.