Learning the secret of 'scale-free' networks allowed the Inquisition to stamp out heresy. Its strategy could help us fight disease, computer viruses and terrorists, argue Andrew Roach and Paul Ormerod

In 1293, a group of religious police swooped on a remote village 100 miles east of Venice. They had been sent from distant Pavia in Lombardy by the inquisitor Lanfranco da Bergamo to seize a Milanese heretic called William. Their quarry's whereabouts had already been ascertained by a spy dispatched to the area. Then a team of burly "heretic hunters" had been recruited from Pavia's house of Franciscan friars. William was captured, tried and burnt in Milan to have the maximum deterrent effect on his fellow townsfolk.

The mission was costly - in modern terms, about ?30,000 for the arrest alone. But da Bergamo judged the price to be justified. After all, more than 100 years of expertise in the elimination of religious dissent had culminated in this sort of precisely targeted, specialist-led operation. The strategy had been developed through trial and error. But similar conclusions have now been reached by mathematicians modelling the spread of ideas and viruses. It seems that the experience of the Inquisition may help us defeat al-Qaida and global terrorism.

In the late 12th century, the Catholic Church had faced its greatest challenge of the Middle Ages - the Cathar heresy. These dualists believed that the world was the creation of an evil god that accounted for all the misery in it. Catharism had become endemic in southern France and Italy, and the spread of heresy exercised many Catholic writers. The parallel with disease was common. One French writer observed: "Just as one bunch of grapes can take on a sickly colour from the aspect of its neighbour...so neighbouring towns and villages...became infected with the dreadful plague."

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But early attempts at treatment were crude. The Albigensian Crusade of 1209 made indiscriminate war on the south of France. The storming of the town of Beziers rapidly became infamous among contemporaries. One recorded: "Knowing from the confessions of these Catholics that they were mixed up with heretics, (the crusaders) said to the abbot 'What shall we do, lord? We cannot tell the good from the bad.'

The abbot "...is said to have said: 'Kill them all. For God knows his own.'

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Thus innumerable persons were killed in that city."

By the time these words were written in 1220, the strategy of trying to contain heresy by random suppression of the population was recognised as futile. In 1231, the first specialist investigators, or inquisitors, were appointed. The growing body of knowledge of heresy and how to deal with it was codified in books, the most famous written by a contemporary of da Bergamo's, the Dominican friar Bernard Gui. His Practica Inquisitionis , which was completed in 1324, drew heavily on earlier literature.

The successful strategies adopted by da Bergamo, Gui and their like were based on a better understanding of the nature of the social networks across which heresy spread and persisted despite attempts to suppress it. The popular view of medieval Europe as a stagnant society is undermined by the fact that ideas could and sometimes did travel rapidly through a relatively small nexus of educated churchmen. A few people exercised a disproportionate influence on the spread of new thinking. Most individuals had few social contacts, mainly within their own villages. But a small number, whether actual heretics or merely guides and messengers who carried news and gossip with them, travelled widely and frequently.

This type of network, in which a few people are connected to large numbers of others while most have a very small number of connections, has a very modern feel. Recent work by physicist Albert-L¨¢szl¨® Barab¨¢si and colleagues at the University of Notre Dame, Indiana, has shown that the worldwide web has similar features. Gene Stanley of Boston University, the editor of the journal Physica A , has demonstrated that the pattern of sexual contacts has the same sort of structure.

Of course, the work of Barab¨¢si, Stanley and others involves much more than simply counting the number of connections each individual, or node, has. They have demonstrated empirically that such networks are of a specific mathematical type known as "scale-free". These networks involve a particular relationship between the number of connections that any individual has and the overall frequency with which this number is observed. The exact nature of this need not concern us here, but its key feature is that it implies that only very few people have lots of connections, and most have just a few.

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New viruses occur all the time, whether they are actual diseases, computer viruses, heretical ideas or something else. Scale-free networks have distinct mathematical properties when it comes to describing their spread. Sir Robert May, the former government chief scientist who is now president of the Royal Society, is one of an increasing number who have worked on this matter.

In standard mathematical models of how epidemics spread, individuals are implicitly connected on a type of non-scale-free network. In such cases, unless a virus infects more than a critical percentage of a population, whether people or computers, it will fade and die of its own accord. The precise critical value will depend on circumstances, but we know how to calculate it. Furthermore, a strategy of inoculating the population at random will succeed in stopping the disease, provided that more than a critical percentage is protected in this way.

This is not true for scale-free networks. In principle, any virus can spread to the entire population. There is no critical threshold to pass. And random inoculation, even of a very high percentage of the population, has only a very low chance of success. To be successful, the small number of highly connected individuals has to be specifically targeted.

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Of course, we can never know the precise mathematical structure of medieval social networks in the same way we can know about the worldwide web. But its qualitative properties seem similar. For example, it is another distinguishing property of scale-free networks that viruses live in them for much longer than they do in standard models of epidemics. The same sort of persistence is seen with heresy. Catholic writers preparing reports for the 14 Council of Lyon thought the threat from Catharism was over, yet a Cathar revival led by ten perfecti (as Cathar priests were known) in southern France found ready converts and caused a big panic among Catholic churchmen in about 1300.

The Inquisition gradually evolved a successful strategy that involved targeting highly connected individuals such as guides and messengers. In his manual, Gui was not interested in the beliefs of the individual being questioned. Rather, he counselled that suspects be asked: "Whether he had any familiar association with heretics; when; how; and who was responsible for it." The physical organisation of the network was of particular interest: "Whether he received any heretical person or persons in his home; who they were; who brought them there; Iwho visited them there and escorted them thence." Those identified as heretics always faced the risk of imprisonment or burning, but increasingly the tactic of leaving contacts at liberty but forcing them to wear yellow crosses was used. In this way, other people were dissuaded from consorting with them.

Knowledge of scale-free networks may hold lessons for us now in how to best deal with al-Qaida. We do not know the precise structure of the terrorist network, and evidence would suggest that it is very loose. But there seems to be a very small number of charismatic individuals whose ideas inspire larger numbers of less connected individuals. In Iraq, the Americans issued their famous pack of cards of key people in the Saddam network to be detained. The capture of one of those was regarded as far more important than detaining thousands of Iraqi foot soldiers.

Modern maths tells us that the best strategies for containing a virus, or a group of terrorists, depends critically on the type of network that connects the relevant individuals. Medieval history gives us a dramatic illustration of why this is the case. Once they appreciated more clearly the type of social network they were dealing with, the inquisitors enjoyed great success. The challenge for modern society is to find ways of containing viruses or terrorism without resorting to the same apparatus of repression and cruelty.

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Andrew P. Roach is a lecturer in history at the University of Glasgow. His book The Devil's World: Heresy and Society, 1100-1300 will be published next year. Paul Ormerod is an economist and founding director of Volterra Consulting.