Network topology

Different types of networks can show different types of network topology. Here we will briefly discuss the network topologies that can be found in many observed networks. Network models which can be used as surrogate networks are discussed in the network models section.

Small-world

Almost 30 years after Travers and Milgram identified the small-world phenomenon (see first paragraph in Network theory), Watts and Strogatz described the underlying principles of small-world organisation in networks. They found that networks with a small-world organisation can be found between completely ordered networks (lattice) and completely random networks (Erdős-Rényi). A simple way of generating such a small-world topology is by randomly rewiring a small percentage of the edges of a lattice network, thereby creating "short-cuts" or long-range connections.

Small-world topology

Networks with small-world topology therefore share properties of both lattice and random networks. Similar to lattice networks their clustering coefficient is high, however, compared to small-world networks, their characteristic path-length is small (similar to a random graph). Many real networks show these properties, making the small-world topology an important aspect when analysing graphs.

Rich-club

Another organisational principle focuses on the subnetwork consisting of nodes with a large percentage of connections within the network (so called hubs), which are densely inter-connected. Nodes belonging to the rich-club are rich in terms of a specified network metric, for example the number of edges. It can be said that the rich-club forms a backbone of the network and similar to internet backbones consists of principal routes between strategically interconnected nodes.

Rich-club topology

In the example above, the rich-club (and corresponding edges) are coloured in gold. The remainder of the network can then be classified into feeder (red) and local (black). The rich-club organisation has a variety of important implications, such as functional integration of nodes, resilience to random attacks and the possibility for nodal specialisation.