Scale-free networks show specific properties that are very distinct from real networks. We often see a significant number of nodes in a network with many links and many nodes with minimal links connected to them. A very good example is Google, which is connected to literally all sites worldwide, and most of these sites have a significantly smaller number of nodes. Another example could be airline networks.

Scale-free networks are complex because their structure and dynamics are independent of the system’s size N, the number of nodes the system has. In other words, a scale-free network will have the same properties no matter the number of its nodes. Their most distinguishing characteristic is that their degree distribution follows a power-law relationship P(k) ~ k¯γ where most real networks’ coefficient γ may vary approximately from 2 to 3.

In their model, Barabási and Albert [1] suggest that the underlying physical mechanisms responsible for generating a scale-free scale behavior are growth and preferential attachment. The former is connected by adding new nodes during the system’s evolution. At the same time, the latter states that the nodes exhibit an intrinsic preference to link with a node that already has relatively more edges than others. These mechanisms lead to the formation of large clusters, the typical structure of scale-free accessible networks, and yield a power-law degree distribution, but only when introduced simultaneously. Hence, these principles define the dynamics of such networks. Their combination may be summarized by the motto “the rich become richer,” indicating the uneven growth of edges from node to node.

To read the documentation about the program, click on the following link:

1 Albert R. and Barabási A.-L., Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47–97 (2002). Barabàsi, Linked: The New Science of Networks, Perseus, Cambridge, MA, 2002.