Subsequent research suggests that the national degree of separation is closer to five than six. The number is 4.6 for directors of Fortune 1000 companies and 3.52 for Facebook users. Of course, this is only a restricted version of closeness centrality.

https://sinews.siam.org/Details-Page/networks-through-history-from-ancient-hierarchies-to-modern-online-communities

Networks through History: From Ancient Hierarchies to Modern Online Communities

By James Case

The Square and the Tower: Networks and Power, from the Freemasons to Facebook. By Niall Ferguson. Penguin Press, New York, NY, January 2018. 592 pages, $30.00.

The Square and the Tower: Networks and Power, from the Freemasons to Facebook. By Niall Ferguson. Courtesy of Penguin Press.

The modern world is dependent on power grids, transportation and communication networks, and water and sewer systems, while facing credible threats from the National Rifle Association, the Irish Republican Army, the Islamic State of Iraq and Syria (ISIS), Al-Qaeda, and various  terrorist networks. In graph-theoretic terms, hierarchies are themselves networks—specifically trees—with a single "root node" at the top, connected by sequences of edges to a multitude of "leaf nodes" at the bottom.

Financial historian Niall Ferguson's fifteenth book, The Square and the Tower, explores the tensions between social networks and political hierarchies in human history. Ferguson asserts that his fellow historians have somewhat overlooked networks, despite extensive written records left by hierarchies like the Roman Empire, the Third Reich, and General Motors; other networks, such as the early Christian church, the Protestant Reformation, and the Populist Movement, began as loosely-connected grassroots organizations with minimal paper trails.

The book's title is inspired by the majestic Torre del Mangia, which looms allegorically over the Piazza del Campo in Siena, Italy. The piazza was designed to host voluntary interactions among relative equals, while the tower symbolizes the imposition of hierarchic secular power. Together they remind us that social networks and political hierarchies have coexisted in varying degrees of harmony since as early as the 14th century.

The early chapters of The Square and the Tower present the rudiments of network theory, which is at once an empirical science and a branch of mathematics. Ferguson is careful to introduce only mathematical principles that are easily absorbed and obviously relevant to the analysis of historical trends. Perhaps the simplest concept is the "density" (edge density) of a network, which is merely the number of edges in a particular graph on

$n$

vertices divided by the number

$n(n-1)/2$

 of potential edges. Duncan Luce and Albert Perry's "clustering coefficient"—the number of complete triangles divided by the number

$n(n-1)(n-2)/6$

 of potentially-complete triangles—is a related notion, as both measure global properties of a graph.

Ferguson also presents three measures of "centrality," a local property pertaining to a single node. The simplest is "degree centrality," which counts the number of edges on a given node or vertex. It differs from both "betweenness centrality," which counts the number of "complete paths" in the network that pass through a given node, and "closeness centrality," which computes the number of edges in the shortest path connecting the node of interest to another — averaged over all other nodes. He makes no mention of more sophisticated centrality measures, such as Katz, eigenvector, or PageRank centrality.

Though Ferguson avoids the term "graph theory," he begins his exposition with Euler's analysis of the Königsberg Bridge problem, followed by a description of social psychologist Stanley Milgram's "small world experiment." Milgram asked randomly-chosen residents of Wichita, Kan., and Omaha, Neb., to forward letters to designated recipients in or near Boston, Mass. Residents were to send the letters to their planned recipients if they knew them on a first-name basis; if not, they were to direct them to people who potentially knew the recipients, aiming to bring the letters closer to their intended targets. 44 of the original 160 letters reportedly reached their destinations, having been passed along by an average of five intermediaries. Since a path with two terminal and five intermediate nodes contains six edges, the situation came to be known as "six degrees of separation." Since that time, six degrees of Marlon Brando, Monica Lewinsky, Kevin Bacon (which later became a board game) and mathematician Paul Erdös have been catalogued. Subsequent research suggests that the national degree of separation is closer to five than six. The number is 4.6 for directors of Fortune 1000 companies and 3.52 for Facebook users. Of course, this is only a restricted version of closeness centrality.

Ferguson extracts seven network insights relevant to the analysis of social networks, two of which seem particularly worthy of mention here. One is his observation that "structure determines virality." Ferguson argues that some ideas "go viral" due to structural features of the networks through which they propagate. To illustrate the point, he invokes an example involving a graph on  nodes arranged in a circle, in which each node is connected by an edge to only its two nearest and two next-nearest neighbors. Without appreciably increasing the edge density of such a graph, the addition of a few edges connecting widely separated nodes could substantially increase the likelihood of virality.

The other noteworthy insight is that the majority of social networks are profoundly inegalitarian, in the sense that new edges are most often added to nodes that are already abundantly connected. This leads to the emergence of the type of "hubs" that populate airline route maps, and immediately distinguishes social networks from those studied by Erdös and Alfréd Rényi, in which edges are added at random to a preexisting set of vertices. The degrees of the several vertices in Erdös-Rényi networks tend toward equality as additional edges are added.

The latter chapters of The Square and the Tower consist of descriptions and analyses of historically significant networks. Of these, few have rivaled the impact of the Freemasons' network. Freemasonry began in the late middle ages as a fraternal organization of stone masons and other skilled craftsmen, ultimately joined by members of the nobility. Local Masonic lodges were among the few places where noblemen and affiliates of the bourgeoisie could freely mingle. Following the appearance of The Constitutions of the Free-Masons in 1723, the organization entered a period of rapid growth.

Leading up to the American Revolution, five associations in Boston were more or less sympathetic to the revolutionary cause. Aside from the local committee of correspondence, the Masonic Lodge that met at the Green Dragon Tavern was probably the most active. A total of 137 men belonged to one or more of the five associations, though the majority belonged to only one. Joseph Warren belonged to four, while Paul Revere, Samuel Adams, and Benjamin Church each belonged to three. Considering two members of the set

$S=\{1,...,137\}$

 of association members to be "equivalent" if they belong to exactly the same associations, Ferguson partitions  into a mere 14 equivalence classes:

$S_1\cup \cdots \cup S_{14}$

. He then constructs a graph on 14 points in which each node

$i$

 represents an equivalence class

$S_i$

, and an edge connects node

$i$

with node

$j$

only if

$S_i$

and

$S_j$

 share a member. Because the subsets containing Warren and Revere exhibit the most betweenness centrality, Ferguson concludes that (i) they were the two most important revolutionaries in Boston and (ii) the Freemasons were the primary instigators of the American Revolution. Though historians have disputed the pros and cons of the latter proposition many times, Ferguson shines new light on the long-running debate.

Freemasons, along with their kindred spirits, the Illuminati, are frequently blamed for both the French and American revolutions. Ferguson offers a concise history of the Illuminati, which began in 1748 and was dissolved by Pope Clement XIV in 1773, well before the start of the uprising in France. Still, conspiracy theorists continue to blame them for the misdeeds of the Paris mob.

Ferguson's closest citation to a practical application of network theory is—unsurprisingly—to warfare. Its post-World War II experience in the jungles of Malaya taught the British military that insurgencies are difficult to combat, in part because they are led by loosely-affiliated tribal chieftains and village elders — no one of whom is indispensable to the cause or has the authority to conclude a negotiated settlement. They tried with little success to explain this to the American generals in Vietnam, and then Afghanistan. When the U.S. Army did at last catch on, it issued a field manual entitled Counterinsurgency: FM 3-24. "To a striking extent, FM 3-24 set out to educate the U.S. military about network theory, explaining concepts such as network density, degree centrality, and betweenness," Ferguson writes. The first edition even contained an appendix entitled "Social Network Analysis."

Ultimately, it seems likely that Ferguson's revelations will prove most useful to his fellow historians, as a great deal of information concerning historically influential networks already is—or will soon be—readily available. Ferguson considers a range of networks, including the Early Christian Church, the Knights of the Round Table, the House of Saxe-Coburg-Gotha, the Protestant Reformation, the Rothschild family, the internet, the warlords of Afghanistan, ISIS, and Al-Qaeda. In almost every case, he brings fresh perspective to familiar historical episodes and sometimes challenges conventional wisdom. Though networks have been around as long as kings, princes, and military leaders, historians might have underestimated their overall significance.

James Case writes from Baltimore, Maryland.

_- Steve