Viewpoint Few and Far Between

At the end of the 20th century, the study of networks—systems of nodes connected by links—took off as scientists realized how ubiquitous they were. Complex networks describe interactions among proteins in a cell, coordinate communication among the neurons in our brain, and govern how individuals in a society connect. The list goes on. Many fundamental questions about networks remain unanswered, however, including several on scale-free networks, which are characterized by a powerlaw distribution in the number of connections (degree) each node has. Now, in a paper in Physical Review Letters, Charo Del Genio at the Max Planck Institute for the Physics of Complex Systems, Germany, and coauthors tell us that the only kind of scale-free network that is possible is one with an average degree that remains finite as its size becomes arbitrarily large [1]. The connection density C of a network can be interpreted as the ratio of the average degree to the network size (the number of nodes, N), and so this result implies that all realizable scale-free networks are sparse (C → 0 as N →∞). The paper should stimulate fresh activity in this area, as it appears to contradict earlier mathematical results [2] which suggested that growth by node and link duplication would give rise to dense scale-free networks. Scale-free networks [3] have become prominent recently, but power-law degree distributions have been known for some time. In the 1960s, the polymath Derek de Solla Price proposed [4] a preferential attachment scheme—now colloquially referred to as the “rich get richer” effect, first studied by George Yule—that generates power-law distributions seen in many contexts, including several in economics, such as the Pareto law for income distribution [5]. For networks, preferential attachment works as follows [6]: Start with a existing network and at subsequent time steps add new nodes that connect to existing ones preferentially, according to their degree. If the probability that a new node will connect to a node of degree k is proportional to k, then it can be shown that in the steady state the network exhibits a power-law degree distribution P (k) ∼ k−γ , where the exponent γ = 3. As many networks (the internet is one example) that evolve by accumulating nodes show scale-free degree distribution, one expects the evolutionary algorithm above to describe their growth. However, scale-free networks that are prevalent, whose exponents can span a variety of values, appear to mostly cluster around γ = 2, not 3. Why does the value of the exponent matter? For one thing, it governs the statistics of the various moments of the distribution of the degree (including the average degree and its standard deviation). This has repercussions for the dynamical processes on the networks. An example is how a contagion spreads in a population where the contact network among individuals is scale-free. The epidemic can only spread if the rate of infection exceeds a threshold, which is the ratio of the first and second moments of the degree distribution. If the distribution is a power law with an exponent between 2 and 3, the second moment diverges so that this ratio is zero and the threshold disappears. In other words, even an infection with an arbitrarily small rate of spread can result in an epidemic that spans the entire population [7]. This may appear to have ominous consequences, but there is a silver lining. A sparse scale-free network is characterized by the existence of hubs—nodes with a very high degree compared to the average (Fig. 1)—that dominate the spreading process. Therefore, identifying and selectively immunizing the few “super-spreaders” would be a workable control strategy. However, when the exponent is smaller than 2, the first moment diverges with system size. This means that the number of hubs is no longer small but rather of the order of the size of the network. Selective control of such a large number of hubs is no longer an efficient strategy. Fortunately, with a few key exceptions, most social networks do not appear to be scale-free: diseases that spread through casual social interaction may have a fi-