Nexus: Small Worlds and the Groundbreaking Science of Networks – Mark Buchanan

Thoughts: I enjoyed reading Nexus - it offers a clear and approachable introduction to discoveries in the study of networks (or rather, the state of its study two decades ago). Solid book.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Buchanan, Mark. 2002. Nexus: Small Worlds and the Groundbreaking Science of Networks. W. W. Norton.

1. Strange Connections

2. The Strength of Weak Ties

• 43: Social networks tend to have clusters - where most of the people a person is connected to are connected to each other - and bridges - connections from a person in one cluster to a person in another cluster. Mark Granovetter’s studies of social networks show that, almost without exception, bridges are formed from weak links rather than strong links.

3. Small Worlds

• 54: Networks of randomly-connected nodes have the “small world” property that one can traverse from one node to any other in a small number of steps, but they are not very clustered. But Duncan Watts and Steve Strogatz’s studies of “Small World” networks have both the small world property, and are also highly clustered (Small World networks are formed by beginning with an ordered network, where nodes are connected to all nodes nearby them, with the addition of a few random connections between nodes)

4. Brain Works

5. The Small-World Web

• 88: In many real-world networks, there exist “hubs” that are connected to an outsized number of nodes. A study by physicists Ricard Solé and Ramon Ferrer i Cancho has found that the words of the English language form a small world (if words that appear next to each other somewhere in a corpus are thought of as being connected), with words like the, a, at functioning as hubs

6. An Accidental Science

7. The Rich Get Richer

• 118-119: the small world networks Watts and Strogatz modelled did not grow and were “egalitarian”, with all nodes possessing roughly the same number of connections. Networks that grow over time, in contrast, tend to be “aristocratic”, with a few nodes acting as hubs with many connections and many hubs having very few links
  • “In contrast [to Watts and Strogatz’s networks], highly connected hubs or connecters dominate the networks of Albert and Barabási. The historical mechanism of the rich getting richer leads without fail to connectors, who, by virtue of having so many links, naturally play a role similar to that of Granovetter’s bridges”
  • “So there are, it seems, two flavours of small: egalitarian networks in which all the elements have roughly the same number of links, and aristocratic networks characterized by spectacular disparity.”

8. Costs and Consequences

• 125: “small-world networks of the egalitarian variety, à la Watts and Strogatz, are far more than mere mathematical curiosities. Like the aristocratic networks of the World Wide Web or the Internet, small-world networks of the egalitarian kind can emerge from a simple process of history and growth. Whenever limitations or costs eventually come into play to impede the richest getting still richer, then a small-world network becomes more egalitarian, as seems to be the case with the airports [Buchanan had just discussed how the growth of large airports such as Chicago’s O’Hare tends to slow once they reach a certain size and congestion and delays become a problem] and a number of other real-world networks.”
• 131: physicists Réka Albert, Hawoong Jeong and Albert-László Barabási tried modelling cyber-attacks on the World Wide Web. They found that for a randomly-connected network, if they removed nodes at random, the network quickly fell apart. In contrast, in an aristocratic small-world network (like that of the World Wide Web), the network was quite robust, withstanding many random removals without its diameter (the average number of steps required to connect a randomly-chosen pair of nodes) increasing very much.

9. The Tangled Web

10. Tipping Points

• 158: after talking about phase transitions in the behavior of networks: “Much of modern physics now is not really about matter at all, but about discovering the laws of form in networks of interacting things—not only atoms and molecules, but also bacteria and people.”

11. Breaking Out, Small-World Style

• 178: modelling the spread of diseases: whereas in a clustered, non-small-world network, an infectious disease always runs out of steam as it kills off nodes and nodes that recover become immune, the presence of random long-distance connections can cause the disease to spread to a greater proportion of the population. Once the proportion of links that are long-distance becomes great enough, the network reaches a tipping point and the disease infects the entire network. This sort of phase transition occurs no matter what value R is set to.
  • “What pushes the epidemic over the edge is not the likelihood of it moving from one person to another, but a change in the very architecture of the social network.”

12. Laws for the Living

• 185: Thomas Schelling’s simulations with “people” living on a grid - each person has an attribute (“white” or “black”). At every step in the simulation, people could get up and move to a different square on the grid if they were dissatisfied with their current location. When Schelling added stipulation that people disliked being in the extreme minority among their neighbours (i.e. if they’re the only “black” person among all their neighbours), they would move, the grid became segregated, with enclaves of all “white” people separated from enclaves with all “black” people. “The slight preference of the individual to avoid an extreme minority has the paradoxical but inexorable effect of obliterating mixed communities altogether”
• 191-192: Simulation by Jean Philippe Bouchard and Marc Mézard, where simulated people had different amounts of wealth, and people could either exchange things with each other, or invest their wealth. People with more wealth were better able to withstand the shocks of investment prices fluctuating. Each person was programmed to behave similarly (i.e. each had identical “money-making” skills), but when the simulation ran, there would always be some people who ended up with enormously more wealth than the others, and the distribution of wealth followed Pareto’s law
  • this result was robust no matter how they set the balance between the tendency to “exchange” and to “invest”
  • 192: “The discovery suggests that the temptation to find complex explanations behind the distribution of wealth may be seriously misguided.”
• 193: It’s important to note that Pareto’s law does not stricly imply an 80-20 distribution: it’s just that the distribution trails off in a particular way - it is fat-tailed - but the specific proportions can vary
• 193: observing the results of multiple runs of this simulation with different conditions: “encouraging exchange between people, with other things being equal, will tend to distribute wealth more equitably. Bouchard and Mézard found greater equality whenever they boosted the flow of wealth among the links or increased the number of such links. Alternatively, stirring up the wildness and unpredictability of investment returns worked in the opposite direction.”
  • “Of course,” Buchanan cautions, “this model is so abstract that it is not meant to provide detailed recommendations for public policy.”
  • that said, other findings: “the model reveals… that taxation will tend to erode differences in wealth as long as the money is redistributed to the society in a more or less equal way. After all, taxation corresponds to the artificial addition of some extra links into the network, along which wealth can flow from the rich toward the poor.”
    • “Taxation does not alter Pareto’s law, but the wealth will become distributed somewhat more equitably, with the rich owning a smaller fraction of the overall pie. Somewhat more surprisingly, the model suggests that a like redistribution of wealth should result from any economic measures aimed at boosting spending right across the economy. Broad taxes on the sale of luxury items, for example, may even tend to increase wealth disparities.”
• 195: “In studying their model, however, Bouchard and Mézard discovered that if the investment irregularities grow sufficiently strong, they can completely overwhelm the natural diffusion of wealth provided by transactions.” i.e. a phase transition

13. Beyond Coincidence

• 199: “what distinguishes a small-world network is not only that it has a low number of degrees of separation, but also that it remains highly clustered. We might say that the fabric of the network is densely weaved, so that any element remains comfortably and tightly enmeshed within a local web of connections. Consequently, the network overall can be viewed as a collection of clusters, within which the elements are intimately linked, as in a group of friends. A few ‘weak’ links between the clusters serve to keep the whole world small.”