Introduction to the Modeling and Analysis of Complex Systems

354 CHAPTER 16. DYNAMICAL NETWORKS I: MODELINGcouldn’t explain how such an appropriate p would be maintained in real social networks.Second, the small-world networks generated by their model didn’t capture one importantaspect of real-world networks: heterogeneity of connectivities. In every level of society, weoften find a few very popular people as well as many other less-popular, “ordinary” people.In other words, there is always great variation in node degrees. However, since theWatts-Strogatz model modifies an initially regular graph by a series of random rewirings,the resulting networks are still quite close to regular, where each node has more or lessthe same number of connections. This is quite different from what we usually see inreality.Just one year after the publication of Watts and Strogatz’s paper, Albert-László Barabásiand Réka Albert published another very influential paper [57] about a new model thatcould explain both the small-world property and large variations of node degrees in anetwork. The Barabási-Albert model described self-organization of networks over timecaused by a series of network growth events with preferential attachment. Their modelassumptions were as follows:1. The initial network topology is an arbitrary graph made of m 0 nodes. There is nospecific requirement for its shape, as long as each node has at least one connection(so its degree is positive).2. In each network growth event, a newcomer node is attached to the network by medges (m ≤ m 0 ). The destination of each edge is selected from the network ofexisting nodes using the following selection probabilities:p(i) =deg(i)∑j deg(j) (16.16)Here p(i) is the probability for an existing node i to be connected by a newcomernode, which is proportional to its degree (preferential attachment).This preferential attachment mechanism captures “the rich get richer” effect in the growthof systems, which is often seen in many socio-economical, ecological, and physical processes.Such cumulative advantage is also called the “Matthew effect” in sociology. WhatBarabási and Albert found was that the network growing with this simple dynamical rulewill eventually form a certain type of topology in which the distribution of the node degreesfollows a power lawP (k) ∼ k −γ , (16.17)