Statistical and Machine Learning Approaches for Network Analysis

3.2 Unified Evolving Network Model: Reproduction of Heterogeneous Connectivity, Hierarchical Modularity, and Disassortativity

As mentioned above, several remarkable structural properties have thus far been characterized in biological networks via network analysis.

A well-known feature of networks is heterogeneous connectivity, which indicates that the node degree k (the number of edges that the node has) approximately follows a power–law distribution: P(k) ∝ k^−γ. The exponent γ, called the “degree exponent,” is empirically known to be between 2 and 3. This power–law degree distribution implies that a few nodes (hubs) integrate numerous nodes while most of the remaining nodes do not; this feature is clearly different from that of the Poisson (homogeneous) distribution obtained from classical (ER) random networks. To reproduce heterogeneous connectivity, the BA model was proposed. This model has two mechanisms: the growth mechanism, in which an added node connects to existing nodes, and the preferential attachment (PA) mechanism, in which the existing node i is selected with the probability Π_i = k_i/∑ _jk_j, where k_i is the degree (the number of edges) of node i. The BA model generates heterogeneous networks with the power–law degree distribution P(k) ∝ k⁻³ [12].

The other structural property of networks is hierarchical modularity. Most real-world networks are clustered; this is characterized by the clustering coefficient C. The clustering coefficient of node i (C_i) is defined as 2 ...