Degreesyielding

Degreesyielding refers to a function that associates each possible node degree k in a network with an expected outcome of a process that unfolds on the network, such as information diffusion or influence spread. The central idea is to study how node connectivity level influences performance or outcome at the node level, abstracting away from the full degree distribution.

Let G = (V, E) be a graph and deg(v) its degree. A degreesyielding function D maps the nonnegative integers to the reals such that D(k) = E[ Y | deg(v) = k ], where Y is a measurable outcome of interest for the node v under a specified process (e.g., probability of activation, final cascade size, or time to activation).

For simple contagion with independent activation probability p per neighbor, the probability that a node with degree k becomes activated in a single step is 1 − (1 − p)^k, which is a degreesyielding function for that model.

Degreesyielding can be estimated empirically from simulated or observed data by binning nodes by degree and averaging outcomes, or derived analytically for tractable models. It helps compare networks and forecast diffusion by showing how the same degree yields different outcomes under different process parameters.

Degreesyielding depends on process assumptions, network structure, and external factors; it may not capture interactions among nodes beyond degree and may be sensitive to tails of the degree distribution.

See also related concepts such as degree distribution, diffusion of information, and network diffusion models.