Random and scale-free graph models

At the end of the Fifties, different researches on graph and probability theory joined together into a new field of interests: random graphs (Gilbert, 1959, Erdös and Rényi, 1959). The efforts culminated in the monumental paper by Erdös and Rényi (1960), which represents a milestone for all the further studies. An Erdös-Rényi random graph is the mathematical object obtained adding, progressively and at random, successive edges between a set of isolated vertices. In a more mathematical language, let n a positive integer and 0 ≤ pER ≤ 1. The

random graph G (n, pER), which can be written ERn(pER), is a probability space over the

set of graphs on the vertex set V = {1, . . . , n} determined by

Pr [{i, j} ∈ G] = pER (2.3)

with these events mutually independent. Many properties of such a mathematical structure were highlighted. For example, Erdös-Rényi random graph exhibits a phase transition in the size of the maximal component when pERvaries. Because of its simplicity, which is repre-

sented by the use of probabilistic rules, the random graph was initially used for describing real networks, which are large in size and difficult to model deterministically (Van Der Hofstad, 2009). The most important issue of random networks, which is relevant for the understanding of the following discussion, is that the distributions of vertex degrees follows a Poisson dis- tribution (Erdös and Rényi, 1960, Bollobás, 2001). That is, the probability that a vertex has k edges is

Pr (k) = e−λλ

k

k! (2.4)

and the small generating unit at Swift was also tripped. Subsequently, when the reactive power output of the MacNary generation units was at about 480 MVAR to provide reactive support, the protective relays started tripping the McNary units one by one because of faulty relay operation. As the McNary units went out of service, the inter-area oscillations grew in magnitude, and the damping of the 0.25 COI inter-area mode appeared to change from positive damping values to negative damping values. At 15:48:51, within 75 seconds after the initial fault on the Ross-Lexington line, the COI lines were tripped which resulted in system separation and the blackout (Venkatasubramanian and Li, 2004).

where λ =n − 1 k  pkER 1 − p k ER
n−1−k . (2.5)

At the beginning, mathematicians were enthusiasts and thought to have found a good representation of the reality, but they were wrong. The ability of random networks to model the real world was put to the test few years later. de Solla Price (1965), analysing a catalogue of journal references, found that, for a scientific paper in a specific field of research, half of the references are to a research front of recent papers and the other half are to papers scattered uniformly through the literature. In particular, he plotted the percentage of papers containing a certain number of bibliographic references, see Figure 2.7, and noted that the average number of references per paper is about 15, but the distribution is far from being Gaussian. In particular, 50 percent of the references came from the 85 percent of the papers, which contain 25 or fewer references apiece. Within this category, the percentage of papers with 3 to 10 references was around 5 percent (per each class of number of references). On the contrary, considering the percentage of papers with many references each, de Solla Price observed that a 25 percent of the references came from the 5 percent of all papers containing 45 or more references, while 12 percent of the references came from one percent of papers (the ones having 84 or more references). Finally, he noted that the number of papers cited n times in a year followed an inverse power law (a Zipf Law) with the exponent in the range 2.5 – 3.0.

The heavy-tailed distribution found by de Solla Price is a symptom of scale-free trend. The explanation of this particular shape found in the network of references would come later under the denomination “Cumulative Advantage Distribution”: the success of a publication fall equally on the heads of previous successes (de Solla Price, 1976). In other words, the most cited papers tend to be referenced more than the ones less mentioned.

In the Nineties, research interests in scale-free networks rose after the topological map- ping of the World Wide Web by Barabási and colleagues (Albert et al., 1999, Barabási and Albert, 1999). They mapped the complete nd.edu domain (Notre Dame University, where they worked) containing n = 325729 documents and e = 1469680 links. In doing so, they de- termined the probabilities pout(k)and pin(k)that a document has k outgoing and incoming

link, respectively and found that these follow a power law over several orders of magnitude, see Figure 2.8. In particular the tail of the distribution follows

Pr (k) ∼ k−γ, (2.6)

with γout = 2.45and γin= 2.1. Similar behaviours were recorded for the network of actors

playing together, in this case with Kevin Bacon, with γin = 2.3. Other examples can be

found in Van Der Hofstad (2009). As a result, large networks self-organize into a scale-free state, a feature unpredicted by random network models. The way this is done, in the idea of Barabási and Albert, is that networks continuously grow by the addition of new vertices and new vertices connect preferentially to highly connected ones: this is the “preferential

Chapter 2 - A connected world - 25

Figure 2.7: Percentage (relative to the total number of papers published in 1961) of papers published in 1961 which contain various number (n) of bibliographic references, after the original paper by de Solla Price (1965).

Figure 2.8: Distribution of links on the World-Wide Web obtained the complete map of the nd.edudomain. Left-hand side plot: Outgoing links (URLs found on an HTML document). Right-hand side plot: incoming links (URLs pointing to a certain HTML document). Dotted lines represent analytical fits used as input distributions in constructing the topological model of the web, after the original paper by Albert et al. (1999).

attachment”, as de Solla Price (1976) imagined, finally modelled by Dorogovtsev et al. (2000), Krapivsky et al. (2000) and Bollobás (2001).