Celebration of the 1818 victory of the Mahar community fighting for the British against the Peshwa, Baji Rao II, recently became a controversy in Pune, Maharashtra. While the community has traditionally seen the “victory tower” built by the British to commemorate the event as an affirmation of the lower castes, questions have been raised about the identity of the real contenders.

Another ongoing controversy is of how the 16th century Babari Majid, at Ayodhya, came to be built. While the case for a Hindu temple to be constructed at the site of the demolished mosque is with the Supreme Court, the resolution of the matter may be more through diplomacy than on the basis of facts.

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A similar, although less politically provocative, controversy is one regarding a victory of the Irish prince in 1014, which has been viewed as turning the tide against the Viking invaders. Joseph Yose, Ralph Kenna, Máirín MacCarron and Pádraig MacCarron of the Universities of Coventry, Sheffield and Oxford, write in the journal, Royal Society Open Science, that statistical analysis of the relationships described in historical sources has helped understand the political balance of the time with more clarity.

While a written, historical account is often the best source of facts that lie in the past, science and forensics have provided important supplementary information. Archaeology is an obvious example and chemical analysis or genetic profiling also has direct application. These sources become important when the written record is considered to be less than objective or misrepresenting the truth. In the case of Irish history, the traditional understanding has been that the battle of Clontarf in 1014 marks the breaking of the two centuries old Viking power in Ireland. Some modern scholarship, however, looks at the conflict as a domestic feud or civil war, rather than an international clash.

While controversy continues about how the records of a millennium ago should be interpreted, the paper in the Royal Society journal brings in mathematical analysis to present a perspective of the domestic and international issues considered in Cogadh Gaedhel re Gallaibh (“The War of the Gaedhil with the Gaill”), one of the best known acc-ounts of Viking rule in Ireland. The narrative in the Gogadh, the paper says, describes the doings of the Vikings, who came to Ireland in 795 and grew in power till the iconic battle of 1014. The text describes “multitudes of characters, alliances, conflicts, relationships and interactions of all sorts”, which have been used to model the interactions of the period in the form of a network of inter-dependent elements. The model can then be analysed using mathematics.

Mathematical methods bring objectivity and refine judgments in many fields, which are usually a part of the humanities. A ready example would be the use of mathematics in deciphering ancient writing, like the cuneiforms of Egypt or the Mohenjo Daro script. Another would be of creating a “fingerprint” of literary style, by statistical analysis of the use of words or sets of words in a writer’s work. An imitation can then be made out even it closely follows the style of the original to convince readers that a new work of their favourite author has been discovered. There are even features of numbers, themselves, that allow fake entries in bank accounts to be identified!

Relationships between groups of people, or institutions, have also been seen to form networks where the different entities connected to any one are also connected to each other and so on. These interconnections are similar to networks formed by roads connecting cities, airline routes, drainage systems, blood vessels in living things and the links between computers on the Internet. The networks have different structures, like hierarchical or organic, and the structures have different properties, which affect how fast connections between points can be or how easy alternate routes are if some routes get blocked or congested.

The mathematics of how networks function is now a subject of deep study. Methods to manage the capacity and geometry of routes for the fastest or cheapest way to transport goods, passengers or data have been developed. Analysis of networks has become important in understanding how infections spread and in controlling epidemics. Social networks are being seen as channels to spread information or for advertising.

The entities that participate in networks are known in the field as “nodes” and the interconnections as “edges”, which can be unidirectional or two-way, and the strength of the connection is called its “weight”.
The mathematical analysis of the conflict in Ireland of the 11th century consists of discovering the links, or contacts between individuals and institutions in Ireland, appearing in the Cogadh Gaedhel re Gallaibh text, and classifying them as connecting nodes with similar attributes, like Irish, Viking or other, or nodes of different types. The edges, or the connection between the nodes, were then classified as representing familial or social relationships — a “positive” edge — or conflict or animosity — a “negative” edge.

The entire narrative yielded 315 significant characters, connected by 1190 edges. While there were groups of nodes with all positive edges, there were several triads, or groups of three, predominantly with two positive and one negative edge. A triad like this represents a pair of “friends” and a common “enemy” (a friend’s enemy being mine too, or an enemy’s enemy being a friend!). And their prevalence represents a suppression of conflict between pairs, thanks to their bonding against a common foe in the third, a case of “structural balance”, the paper says.

One can see that while the participation of three kinds of nodes in positive and negative networks corresponds to the number of nodes there are, this is not the case with the kinds of edges that figure in the networks. Out of the Irish-Irish and the Viking-Viking edges, the bulk is in positive networks, representing social exchanges. But of Irish-Viking edges, there are more in negative networks, which represent hostility, compared to positive networks of the same.

“This suggests that the largest proportion of Cogadh conflict is international, but there are significant levels of intra-national hostilities too (especially Irish versus Irish)”, the paper says. The significant Irish-Irish hostility comes to light when we recognise that there is twice the number of Irish nodes as compared to Viking nodes in the network, the paper adds.

In fact, after considering the effects of the kinds of interactions possible, and the effect of “other” nodes, which are neither Irish nor Viking, the study concludes that the conflict in Ireland in 1014 was certainly more of the Irish-Viking kind, but was also Irish-Irish and Viking-Viking to a substantial degree.

The “traditional” view may have arisen from additions to the original narrative, along the way of editing and re-writing before the version that is available was created. The prose has even been described as “full of the feelings of clanship, and of the consequent partisanship of the time, disfigured also by considerable interpolations, and by a bombastic style in the worst taste…”

Mathematical analysis, based on more objective aspects of the record which would escape motivated distortion, however, has presented a picture that is more balanced and credible.

The writer can be contacted at response@simplescience.in