Krugman meets Barabasi: scale-free complex networks, hubs and increasing returns in economics

Studies in complexity gained momentum in Economics after Brian Arthur’s work (Arthur (2015) and Foster (2005)) as the head of New Mexico’s Santa Fe Institute in the late 1980s. With applications on various fronts, complex dynamical systems approaches have been applied to different fields of research in Economics and other sciences. Applications are used, for example, in Game Theory, Political Science, Biology and Physics. Original applications in Economics were on modeling of financial markets, individual agents’ decision-making rules in various contexts and studies on path-dependence and technological dynamics with increasing returns. The Atlas of Economic Complexity presented in the previous section advances the discussion of complexity combining it with Big Data techniques to create what is perhaps one of today’s most relevant economic databases for world trade analysis. The term Big Data has been widely used in various contexts to describe the explosive growth of data available from the digital world. At its roots, Big Data deals with a large volume and variety of high-velocity data.
In a compilation of his works on and the history of scale-free complex networks, Barabasi (2002) provides a detailed explanation of the concepts and recent contributions to network science within the context of Big Data in different fields of knowledge; some practical examples of which include the internet itself, the network of Hollywood actors and films, biological and linguistic networks, among many more. The simple case of the US airlines network (see figure 1 below) as presented by Barabasi (2002) explains in a clear manner the concept of scale-free complex networks. The first network is that of the US highway system with many connection nodes (each city is a node) and no relevant hubs. The airlines network in the same graph is the opposite case: a complex network with hubs (that is, large nodes with many connections), therefore a non-random network. A few hubs exist that concentrate the majority of connections (Chicago, New York, Houston, LA, etc.). In such complex, non-random networks, a few hubs hold the majority of connections and many other nodes have very few connections. A new city that tries to compete in terms of “receiving” and “sending” flights will face great difficulty when competing with the mega hubs. Its status as an “ordinary hub” in the network makes entry into this “space” far too difficult. The network is considered to be scale-free because the number of links connecting to the nodes does not follow a well-behaved pattern, but rather a power-law distribution.

Figure 1: Complex scale-free and random networks


Source: Barabasi (2002)

Nodes in a random network have a random number of links. In a scale-free complex network, a few nodes have the majority of the links (the hubs) and the great majority of other nodes have very few links. A Gaussian distribution characterizes the former kind of network, while the latter is characterized by a power-law distribution. Non-random networks show a hierarchy where the hubs prevail because they have far more access to links than “ordinary” nodes: a “topocracy” reigns (Borondo et al 2014). Competition inside these networks is uneven in the sense that, over time, certain nodes collect large numbers of links to become hubs with greater access to other nodes of the network. An “ordinary” node faces great difficulty when competing with a hub because it starts out from a poor position in terms of its stock of accumulated links.

Barabasi and his team created a simplified model that reproduces with remarkable accuracy this kind of real-world network dynamics; the model has three pillars: i) a network that grows with new nodes being incorporated to other nodes by means of links at every point in time; ii) a preferential attachment rule according to which each new node prefers to connect to an existing node with lots of links; and, iii) fitness: some nodes are more competent link-accumulators than others, which may help a new node to overcome the difficulty of lacking links when it enters the network.

Barabasi and his team use these three simple rules to formally replicate the characteristics of such networks in the real world, including the appearances of power-law distributions as indicated above in the case of the US airlines network. Barabasi’s “preferential attachment” mechanism is nothing more than the familiar dynamics of increasing returns illustrated in a single urn Polya process or in a generalized several urns Yules process. H. Simon showed that power laws may emerge as consequences of Yule-type processes (Newman 2010). These findings are crucially important for economists because they formalize and add analytical content for already known insights and empirical regularities; particularly for discussions of the new economic geography and trade theory (as previously noted by A. Marshall, Krugman et al (1999) among others). This kind of Barabasi network dynamics clearly illustrates the increasing returns and path-dependent processes that Arthur (2015) demonstrated in his works on economic complexity and technological dynamics.


Arthur, W., B. (2015), Complexity and the Economy, Oxford University press, New York.

Foster, J., (2005) “From simplistic to complex systems in economics”, Cambridge Journal of Economics 29, 873–892, doi:10.1093/cje/bei083

Barabasi, A.L. (2002) Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life, Basic Books ed. NY

Barabasi A.L., (2016) Network Science,

Borondo, J.,F. Borondo,C. Rodriguez-Sickert,C. A. Hidalgo (2014), “To Each According to its Degree: The Meritocracy and Topocracy of Embedded Markets”, Scientific Reports4, Article number: 3784 doi:10.1038/srep03784

Krugman, p., Fujita, and Venables (1999) The Spatial Economy, cities regions and international trade, Mit Press

Newman, M.E.J., (2010) Networks, an introduction, Santa Fe Institute, Oxford University Press

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