Hierarchical network model

Being part of the scale-free model family, the degree distribution of the hierarchical network model follows the power law meaning that a randomly selected node in the network has k edges with a probability

${\displaystyle P\left(k\right)\sim ck^{-\gamma }\,}$

where c is a constant and γ is the degree exponent. In most real world networks exhibiting scale-free properties γ lies in the interval [2,3].[4]

As a specific result for hierarchical models it has been shown that the degree exponent of the distribution function can be calculated as

${\displaystyle \gamma =1+{\frac {\ln M}{\ln(M-1)}}}$

where M represents the replication factor of the model.[5]

In contrast to the other scale-free models (Erdős–Rényi, Barabási–Albert, Watts–Strogatz) where the clustering coefficient is independent of the degree of a specific node, in hierarchical networks the clustering coefficient can be expressed as a function of the degree in the following way:

${\displaystyle C\left(k\right)\sim k^{-\beta }\,}$

It has been analytically shown that in deterministic scale-free networks the exponent β takes the value of 1.[2]

Based on the actor database available at www.IMDb.com the network is defined by Hollywood actors who are connected to each other if they both appeared in the same movie, resulting in a data set of 392,340 nodes and 15,347,957 edges. As earlier studies have shown, this network exhibits scale-free properties at least for high values of k. Moreover, the clustering coefficients seem to follow the required scaling law with the parameter -1 providing evidence for the hierarchical topology of the network. Intuitively, one-performance actors have by definition a clustering coefficient of one while actors starring in several movies are highly unlikely to work with the same crew which in general results in a decreasing clustering coefficient as the number of co-stars grows.[1]

Words can be regarded as network if one specifies the linkage criteria between them. Defining links as appearance as a synonym in the Merriam-Webster dictionary a semantic web of 182,853 nodes with 317,658 edges was constructed. As it turned out, the obtained network of words indeed follows a power law in its degree distribution while the distribution of the clustering coefficient indicates that the underlying web follows a hierarchical structure with γ=3.25 and β=1.[1]

By mapping the www.nd.edu domain a network of 325,729 nodes and 1,497,135 edges was obtained whose degree distribution followed a power law with γ_out=2.45 and γ_in=2.1 for the out- and in-degrees, respectively. The evidence for the scaling law distribution of the clustering coefficients is significantly weaker than in the previous cases although there is a clearly visible declining pattern in the distribution of C(k) indicating that the more links a domain has the less interconnected the linked/linking web pages are.[1][6]

The domain network, i.e. the internet at the autonomous system (AS) level where the administrative domains are said to be connected in case there is a router which connects them, was found to comprise 65,520 nodes and 24,412 links between them and exhibit the properties of a scale-free network. The sample distribution of the clustering coefficients was fitted by the scaling function C(k)~k^−0.75 whose exponent is (in absolute terms) somewhat smaller than the theoretical parameter for deterministic scale-free networks.[1][7]