Friedrich Hayek characterized complex systems with two main properties; the number of variables, and the connection between them[i]. It was these properties that predominantly resulted in emergence. Hayek was describing the properties of scale free networks.
Beginning in the 1990s, scientists revived the study of network research, at least partly after they noticed that scale free networks predominate in virtually every discipline or instance where nature constructs complex systems[ii]. Examples of networks with a scale free topology include protein networks, the Internet, genetics, traffic patterns, and social networks[iii].
A scale free network follows rather simple rules. The probability P(k) that an arbitrary element of the network is connected to exactly k other elements has the form P(k) = Ck−γ, where γ is usually called the scale-free exponent. In other words, scale free networks conform to Power Laws which stipulate that a smaller number of nodes are very highly connected while most nodes are poorly connected.
The Power Law, often conceptualized as the 80/20 rule or Pareto Principle, is characterized by a very long tail unlike the more popular and better known Bell Curve. Trends conforming to Power Laws are found everywhere including wealth distribution, the size of meteorites, research citations, stock returns, gene variation, river systems, and all the examples of scale free networks listed above. It is nature’s most prevalent topology.
The question immediately arises as to why complex systems take on a scale free topology. Closer inspection seems to indicate it provides optimal levels of both reliability and adaptability compared to alternatives[iv].
Barabási and Albert stipulate that power law generated networks are the result of two complimentary phenomena; naturally occurring networks expand by adding new vertices, and by their nature, new vertices show preference for attaching to nodes that are already well connected[v].
Reliability
Since failures occur at random and the vast majority of nodes are comparatively small in size and connection, the likelihood that a larger hub would be affected is almost negligible. Even if a central hub is destroyed, the network will not lose its connectedness since the remaining hubs guarantee connection.
The weakness is of course that multiple hubs fail at the same time, which would destroy the network and render it a series of disconnected nodes. Notice that for most instances and circumstances, the chance of this occurrence is much lower than the odds of destroying other network topologies. It is possible to construct a network without this weakness, but it is comparatively inefficient and much more costly.
Clustering implication of the Power Law
Topology of scale free network
The diagram illustrates that as the nodes become smaller, so do the clusters they enjoin (The phenomenon is also illustrated in the Leis Network introduction). The hubs by their nature are the main connection points to smaller ‘communities’ of nodes or networks.
It is the Power Law that demands this structure, much as well connected people in business or politics might connect small neighborhoods of people.
In the diagram above, there are relatively few connections between smaller nodes. In reality, it is common for smaller network nodes to have their own connections to other small nodes. Think of the Internet as an example, with links to pages as node connections. In this scenario, even destroying a central hub would introduce only temporary strain on the network.
We now also see the reason for scale free network adaptability. Failed nodes are quickly replaced by smaller connected nodes either providing the same function, or growing to meet demand, or both. It is the robustness, or redundancy of the complex system, that provides for its ability to adapt to changing circumstance.
Summary
Scale free networks are nature’s favorite form. There is a simple reason. They are the most economical (i.e., cost effective and efficient) expression of reliance and adaptivity.
End Notes
[i] Friedrich Hayek, New Studies in Philosophy, Politics, Economics and the History of Ideas (Univ of Chicago Pr (T), 1985), 25-34.
[ii] Albert-László Barabási, Linked: The New Science of Networks, 1st ed. (Basic Books, 2002).
[iii] Ibid.
[iv] Maximino Aldana, “Boolean dynamics of networks with scale-free topology,” Physica D: Nonlinear Phenomena 185, no. 1 (October 15, 2003): 45-66, http://www.sciencedirect.com/science/article/B6TVK-4B1RYJP-4/2/e5f0a84cdf2e12e95188aa14636b378b.
[v] Albert-László Barabási and Réka Albert, “Emergence of Scaling in Random Networks,” Science 286, no. 5439 (October 15, 1999): 509-512, http://www.sciencemag.org/cgi/content/abstract/286/5439/509.
Further Reading
Scale free Networks at Scholarpedia
Scale free Networks at Wikipedia Photo courtesy
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