Hierarchical network model

Hierarchical network models are iterative algorithms for creating networks which are able to reproduce the unique properties of the scale-free topology and the high clustering of the nodes at the same time. These characteristics are widely observed in nature, from biology to language to some social networks. Hierarchical network models are iterative algorithms for creating networks which are able to reproduce the unique properties of the scale-free topology and the high clustering of the nodes at the same time. These characteristics are widely observed in nature, from biology to language to some social networks. The hierarchical network model is part of the scale-free model family sharing their main property of having proportionally more hubs among the nodes than by random generation; however, it significantly differs from the other similar models (Barabási–Albert, Watts–Strogatz) in the distribution of the nodes' clustering coefficients: as other models would predict a constant clustering coefficient as the function of the degree of the node, in hierarchical models nodes with more links are expected to have a lower clustering coefficient. Moreover, while the Barabási-Albert model predicts a decreasing average clustering coefficient as the number of nodes increases, in the case of the hierarchical models there is no relationship between the size of the network and its average clustering coefficient. The development of hierarchical network models was mainly motivated by the failure of the other scale-free models in incorporating the scale-free topology and high clustering into one single model. Since several real-life networks (metabolic networks, the protein interaction network, the World Wide Web or some social networks) exhibit such properties, different hierarchical topologies were introduced in order to account for these various characteristics. Hierarchical network models are usually derived in an iterative way by replicating the initial cluster of the network according to a certain rule. For instance, consider an initial network of five fully interconnected nodes (N=5). As a next step, create four replicas of this cluster and connect the peripheral nodes of each replica to the central node of the original cluster (N=25). This step can be repeated indefinitely, thereby for any k steps the number of nodes in the system can be derived by N=5k+1. Of course there have been several different ways for creating hierarchical systems proposed in the literature. These systems generally differ in the structure of the initial cluster as well as in the degree of expansion which is often referred to as the replication factor of the 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 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 .