degreedependent

Degree-dependent is an adjective used across disciplines to indicate that a quantity, rule, or process varies with degree. In graph theory and network science, degree typically refers to the number of incident edges of a node, though the idea extends to directed graphs (in-degree and out-degree) and to the degree of a polynomial in algebra. A degree-dependent model assigns values that depend on the degree of vertices or on the degree parameter of an object.

In probabilistic graph models, degree dependence is common. The probability of forming an edge between two nodes, p_uv, may be a function of their degrees, p_uv = g(deg(u), deg(v)). Similarly, thresholds for activation in diffusion models can be degree-dependent, meaning a node’s likelihood of becoming active scales with its degree. Degree-dependent mechanisms are used to reflect empirical observations that highly connected nodes often play different roles than low-degree nodes.

In practice, degree dependence helps capture heterogeneity in real networks, such as social, biological, or technological systems, where connectivity correlates with influence, risk, or resilience. It also affects theoretical properties, including connectivity, robustness, and the emergence of giant components in random or evolving graphs.

Formal representation is typically concise: for a vertex v, a degree-dependent quantity P(v) can be expressed as P(v) = f(deg(v)) for some function f. An edge process may be defined by p_uv = g(deg(u), deg(v)). The term is descriptive and context-specific, rather than tied to a single canonical model.

See also: degree distribution, degree sequence, percolation, preferential attachment, network model.