Deterministic scale-free network

Although there are multiple deterministic models to generate scale-free networks, it is common, that they define a simple algorithm of adding nodes, which is then iteratively repeated and thus leads to a complex network. As these models are deterministic, it is possible to get analytic results about the degree distribution, clustering coefficient, average shortest path length, random walk centrality and other relevant network metrics.

Deterministic models are especially useful to explain empirically observed phenomena and demonstrate the existence of networks with certain properties. For example, the Barabási-Albert model predicts a decreasing average clustering coefficient as the number of nodes increases,[1] whereas empirical evidence suggests otherwise. Hierarchical network models can explain this phenomenon while also retaining the scale-free property.[2] Another notable example is that it is possible to generate networks deterministically, which are scale-free and linear world at the same time, showing that small-world property is not a necessary consequence of the scale-free property.[3]

The exact properties of the generated networks depend on the particular algorithm with which they are constructed, therefore there are not many common properties. The single unifying property is scale-freeness, that is, the degree distribution of the nodes always follows a power law at least asymptotically, which means that a randomly selected node has k edges with probability

${\displaystyle P(k)=ck^{-\lambda }}$

where the degree coefficient (λ) depends on the model parameters.

Also, because of the iterative construction, many of the models produce hierarchical networks with fractal-like properties. Other properties, such as network diameter, average path length, clustering coefficient vary across models depending on the construction.