Degreeeither

Degreeeither is a concept in graph theory and network analysis used to describe the degrees of a given vertex across multiple graphs or layers. It is particularly applied in multiplex or temporal networks where the same node participates in several layers. The idea is to capture how a node’s connectivity changes across contexts rather than within a single graph.

Formally, for a pair of simple graphs G1 = (V, E1) and G2 = (V, E2) on the same vertex set V, let deg_G1(v) and deg_G2(v) denote the degrees of vertex v in G1 and G2, respectively. Degreeeither(v) is defined as the unordered pair {deg_G1(v), deg_G2(v)}. If the two degrees are equal, degreeeither(v) reduces to a single value. This concept generalizes to k graphs, where degreeeither_k(v) is the multiset {deg_Gi(v) : i = 1, ..., k}. An alternative representation is the degree vector (deg_G1(v), deg_G2(v)).

Applications of degreeeither include comparing network structure across layers, assessing node stability or variability in connectivity, and identifying nodes with divergent roles across contexts. Derived metrics may include the maximum or minimum degree across layers or the degree disparity between layers.

Computation is straightforward: compute the degree of each vertex in each layer and assemble the degreeeither values. Limitations include the loss of cross-layer edge patterns beyond per-vertex degrees and potential ambiguity when many layers are involved. Degreeeither is commonly encountered in literature on multiplex and multi-layer networks, where it aids in concise cross-layer comparison of node connectivity. See also: graph degree, multiplex networks, multi-layer networks, degree distribution.