A Crazy Belief: Predicting Outcomes from Network Graphs

September 26, 1433

Two rival factions are attempting to seize Florence’s city hall. One block’s forces arrive haphazardly with new arrivals offset by departures. In contrast, the other block immediately and decisively mobilises all of their supporters.

Centuries later, Medici is a household name; Albizzi, Peruzzi, and Strozzi are largely forgotten.

Could this outcome have been predicted?

The Medici were neither the richest, oldest, newest, largest or most popular family at the time. In fact, statistically there was no difference between Medici and oligarch “block” of families across conventional metrics. Yet the Medici won; what they had was a well-constructed network.

Florentine history is well-recorded and studied. The seminal work of Padgett and Ansell [1] describes the data and method to construct the social network of Florence’s most prominent families at the dawn of the Medici’s ascent to power.

 

Beyond usual metrics, the Medici particular skill was in their positioning within the social structure of medieval Florence. This is best illustrated by measuring the relative importance of each of the 33 families within the network, using network centrality measures: betweenness, closeness, and eigencentrality. On average, the Medici are the most central family.

Centrality measures are calculated over the network links, but some bonds are stronger than others and Padgett distinguishes nine different types of connections classified as “strong” (marriage, trade, real estate, employment, partnership) or “weak” (loan, patronage, friendship, mallevadori [2]) ties. We consider link strength by assigning a weight of 3 to strong ties and 1 to weak ties. When two families share more than one link (eg, marriage and trade), we sum the weights to obtain the total strength of the bond.

Wealth and centrality values for all families, (size and color have identical meaning). While of average wealth (among elite families), the Medici have the highest betweenness and closeness values, and the second highest eigencentrality, making them the best connected family.

With the structure of the social network known, we use loopy belief propagation to predict the likelihood a family will side with the Medici during a power struggle. Initial values are set as 1 (100% sides with Medici) for the Medici themselves, and 0 (0% sides with Medici) for the Peruzzi and Strozzi families (the most prominent oligarch families). Every other family’s belief is unknown (ie, 50%) a priori.

Belief propagation results predict that, considering families’ size, the Medici will be able to mobilize 89% of their supporters, while 65% of the other families will support the oligarch block. Historical records indicate the Medici were supported by 93% of their followers,  oligarchs by 59% of other families! This is a staggering performance considering the assumptions [3].

Left: Seeding the Medici as 1 and Strozzi and Peruzzi as 0 shows a balance of forces broken by the Medici’s greater support from their followers (dark green). Right: If another family had risen instead, here Orlandini, the model predicts a crushing defeat.

Given the predictive accuracy, we can look at the paths not taken. What if a more peripheral family would have tried to rise up against the oligarchs? They would have been crushed.

 

The Florentine problem, while small in terms of nodes and links, is difficult to solve without appropriate software. When networks are orders of magnitude larger, such as current social networks, one needs a scalable graph processing framework to enable accurate predictions and unlock the potential of network graphs.

 

[1] JF Padgett and CK Ansell, “Robust Action and the Rise of the Medici, 1400-1434”, AJoS: 1259-1319, 1993

[2] A guarantor (effectively a medieval bail bondsman)

[3] This kind of predictive performance based on network structure alone is only possible because the two blocks are statistically “evenly matched” on classical metrics.

 

Clément Fredembach is a data scientist with Teradata Australia and New Zealand Advance Analytics group. With a background in Colour Science, Computational Photography and Computer Vision, Clement has designed and built perceptual statistical experiments and models for the past 10 years.

 

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