Introduction to the Modeling and Analysis of Complex Systems

16.3. SIMULATING DYNAMICS OF NETWORKS 355where P (k) is the probability for a node to have degree k, and −γ is the scaling exponent(γ = 3 in the Barabási-Albert model). Since the exponent is negative, this distributionmeans that most nodes have very small k, while only a few nodes have large k(Fig. 16.10). But the key implication is that such super-popular “hub” nodes do exist inthis distribution, which wouldn’t be possible if the distribution was a normal distribution,for example. This is because a power law distribution has a long tail (also called a fat tail,heavy tail, etc.), in which the probability goes down much more slowly than in the tail ofa normal distribution as k gets larger. Barabási and Albert called networks whose degreedistributions follow a power law scale-free networks, because there is no characteristic“scale” that stands out in their degree distributions.0.8100.60.01P(k)0.4P(k)10 -510 -80.210 -110.02 4 6 8 10k10 -141 10 100 1000 10 4kFigure 16.10: Plots of the power law distribution P (k) ∼ k −γ with γ = 3. Left: In linearscales. Right: In log-log scales.The network growth with preferential attachments is an interesting, fun process tosimulate. We can reuse most of Code 16.10 for this purpose as well. One critical componentwe will need to newly define is the preferential node selection function that randomlychooses a node based on node degrees. This can be accomplished by roulette selection,i.e., creating a “roulette” from the node degrees and then deciding which bin in theroulette a randomly generated number falls into. In Python, this preferential node selectionfunction can be written as follows:Code 16.11: