Keywords: character networks, assortativity, epic narratives, Vikings, Ireland, complexity science

2.1. Context: the war of the Gaedhil with the Gaill

Cogadh Gaedhel re Gallaibh comes down to us in three manuscripts. The oldest is in the twelfth-centuryBook of Leinster which contains part of the tale. The second (also incomplete) is theDublin Manuscript, dated to the fourteenth century. The third and only complete text is theBrussels Manuscript. This was transcribed from an earlier (now lost) manuscript by the famous Franciscan friar Mícheál Ó Cléirigh who in the seventeenth century was sent from Louvain in Belgium to Ireland to collect and preserve Ireland’s ancient heritage. The Brussels and Dublin manuscripts are close but not identical. Máire Ní Mhaonaigh gives a detailed textual history ofCogadh Gaedhel re Gallaibh in [29,30]. As a proxy for the originals, we use the nineteenth-century translation into English by Todd [4]. Todd’s edition, which was 150 years old in 2017, is accompanied by an extensive introduction and by detailed explanatory footnotes. It serves as a source for some scholars wishing to access the narrative today [38]. Todd consideredCogadh Gaedhel re Gallaibh as divisible into two parts. The first recounts the arrival and deeds of the Vikings in Ireland in a rough chronological fashion. The second part concerns Brian Boru and his Munster dynasty whose powerbase was on the banks of the river Shannon. The lives and politics of his family are outlined along with numerous encounters with the Vikings, all leading to the events at Clontarf.

Brian Boru was king of the Dál Cais in the northern part of the province of Munster (a map of Ireland during the Viking Age is provided in appendix A). After various battles at provincial level, Brian and the Dál Cais consolidated rule of Munster, defeating their Irish and Norse challengers. Brian then turned his attention to the easterly province of Leinster and the westerly province of Connacht. This brought him into contest with Máel Sechnaill mac Domnaill, king of Meath and most powerful king in Ireland, but in 997, Brian and Máel Sechnaill agreed a truce, whereby the former would rule over the (approximate) southern half of Ireland, while the latter kept the (approximate) northern half. By these means, Brian came to control Munster, the area immediately north of Dál Cais territory in southern Connacht, and Leinster as well as the Hiberno-Norse cities within, while Máel Sechnaill held the province of Meath, part of Connacht with at least a notional claim of authority over the northern part of Ireland.

In 998, Brian and Máel Sechnaill worked together against the Dublin Norse. The Vikings had established a settlement in Dublin in 838 and during the following century they developed a kingdom comprising large areas surrounding the town and controlling parts of the Irish Sea. Viking Dublin was politically linked at various times to the Isle of Man and the Hebrides, as well as to Viking settlements in Britain and Scandinavia. Dublin was joined by Leinster under a new king, Máel Morda mac Murchada, in opposing Brian and Máel Sechnaill. Leinster traditionally rejected the rule of both Munster and Meath and the Hiberno-Norse city of Dublin was ruled by Máel Morda’s nephew, Sigtrygg Silkbeard. The two sides met at Glenmama in late December 999. The Irish annals agree that the combined forces of Munster and Meath decisively defeated those of Leinster and Dublin.

The river Shannon presented a barrier to Meath receiving support from his ally Cathal mac Conchobar mac Taidg, king of Connacht, when Máel Sechnaill came under attack by Brian in the year 1000. By 1002, Máel Sechnaill had submitted to Brian at Athlone [6]. The next target for Brian was the northern kingdoms. It took 10 years, a combination of forces and coordinated use of sea and land attacks, and support from the Church in Armagh for the Northern Uí Néill and regional kings of modern-day Ulster to submit to Brian. By 1011, Brian had achieved his aim of bringing all the regional rulers of Ireland under his control.

In 1012, Máel Mórda mac Murchada of Leinster rose in rebellion. Allied with Flaithbertach Ua Néill, regional king of Ailech in the north-west, he again attacked Meath. Máel Sechnaill sought Brian’s help and the following year Brian and his son led a combined force from Munster and Connacht into Leinster, reaching Dublin in September. Out of supplies near the end of the year, they abandoned their siege of the walled city, with an intention to return.

Thus was the background to the famous Battle of Clontarf. In 1014, Máel Morda’s cousin, Sigtrygg, journeyed to Orkney and the Isle of Man seeking Viking support. These Norsemen came under Sigurd Hlodvirsson (Earl of Orkney, known as Sigurd the Stout) and Brodir, reputedly of the Isle of Man. Brian’s forces came from Munster and southern Connacht possibly supported, at least initially, by Máel Sechnaill’s Meathmen (the precise role of Meath in the battle itself is a matter of some contention [5,7,36]). The Battle of Clontarf is believed to have taken place on Good Friday, 23 April 1014 [4] (see, however, [13,57]). According to theCogadh, after a day’s fighting, the battle ended with the routing of the Viking and Leinster armies. The account tells us that their retreat was cut off by the high tide. Many of the nobles died. Brodir killed Brian, having found the old man in his tent.Njáls Saga informs us that Brodir in turn was killed by Úlf Hreða (possibly Cuduiligh in theCogadh [58], meaning Wolf the Quarrelsome), a relative of Brian Boru. Sigurd the Stout of Orkney was also killed, as was the Leinster king Máel Morda mac Murchada. Sigtrygg Silkbeard survived and remained king of Dublin, and the king of Meath, Máel Sechnaill mac Domnaill, resumed his claim to high kingship of Ireland,⁶ supported by Flaithbertach Ua Néill.

2.2. Authenticity and deficiencies ofCogadh Gaedhel re Gallaibh

It is nowadays widely accepted that one of the main aims ofCogadh Gaedhel re Gallaibh was to document the achievements of the Dál Cais and eulogise Brian Boru ‘… to create an illustrious past for his dynasty and to underline thereby later Uí Brian claims to political power’ [29]. Although it is a valuable resource for studies of the Viking Age in Ireland, it is considered a biased one. The question of its reliability has been the topic of a very long-standing debate [4–7,30,48]. Besides some clear interpolation (described in §3.3), much of its bias appears in the descriptive detail of the narrative. Ours, however, is a statistical analysis and, as such, is rather concerned with the totality of the interactions between characters rather than rhetorical levels of detail. As with any statistical analysis, what it delivers is a summary which captures aggregate characteristics, largely insensitive to individual elements. In this sense, one may hope that it delivers useful statistical information on the Viking Age in Ireland.

Estimates for the date ofCogadh Gaedhel re Gallaibh are various. Todd stated its author ‘was a contemporary and strong partizan of King Brian’ [4]. Robin Flower also considered the chronicle ‘almost contemporary’ [25]. Albertus Goedheer gives a date as late as 1160 [23] but John Ryan argues thatCogadh Gaedhel re Gallaibh ‘might have been composed about 1130 or earlier’ [24]. In [6], Donnchadh Ó Corráin refers to it as ‘written in the twelfth century’. He also describes the hypothesized text known asBrian’s saga as written about 1100 in response toCogadh Gaedhel re Gallaibh, a suggestion that implies a date before 1100 for the creation of the latter [6]. More recent scholarship by Ní Mhaonaigh gives the likely composition date ofCogadh Gaedhel re Gallaibh as between the years 1103 and 1113 [29]. (She dates the common source for the Dublin/Brussels recension as the 1120s or 1130s [27,29].) Denis Casey also reviews dating estimates in [41] and argues that there may have been multiple versions of theCogadh (see also [48,46]). Seán Duffy believes it may be ‘based on contemporary annals and, no doubt, local memory’ [7]. He suggests thatCogadh Gaedhel re Gallaibh gives ‘a vivid picture of what happened at Clontarf as related perhaps to the writer of the Cogadh by a veteran’ and gives the possibility that it ‘was written by someone who may well have lived through these last years of Brian’s life’. This bringing us back to Todd’s original estimate [4].

The interpretation ofCogadh Gaedhel re Gallaibh as propagandistic is linked to the question of the date of its composition because ‘Heroic stature presupposes nurturing by time’ [27]. Thus, its propagandistic nature ‘implied that it could no longer be considered contemporary with any of the events it describes’ [27]. The greater the distance between the events of Clontarf and the setting down ofCogadh Gaedhel re Gallaibh, the more room there is for a distorted view to take hold. This is the reason why a good estimate date for the composition of theCogadh is important in the present context. Ryan writes: ‘In the course of the eleventh century, … the view seems to have gained universal acceptance that the Battle of Clontarf was par excellence the great decisive struggle of Irish history. Brian in the retrospect was everywhere acclaimed as a national hero’ [5]. The claim is that time distorted reality; ‘The Norse were a substantial section of the opposing force, and in the mellow haze of popular imagination the battle tended to be transformed into a clear-cut issue, Irish versus Norse, with the former victorious. Even in the Northern countries the battle passed rapidly from history into saga’ [5]. The above estimates for the interval between Clontarf and composition ofCogadh Gaedhel re Gallaibh range between contemporary and about 150 years. Our approach cannot deliver an independent estimate for the date of composition and the above estimates should be kept in mind. While the above considerations suggest that theCogadh may distort in favour of an overly international picture of conflict (and, indeed, the contemporary name of the tale itself emphasizes the Viking–Irish conflict), on the other hand it should also be kept in mind that, in places, it identifies Leinster as the principal enemies of Brian [7,13].

In his Introduction toCogadh Gaedhel re Gallaibh, Todd acknowledges the defects of the work and expresses regret that it is ‘so 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 …’. In chronicle literature, an interpolation of the type mentioned by Todd is a later addition not written by the original author. We address this issue in §3.3.

Ó Corráin states that the author ofCogadh Gaedhel re Gallaibh ‘drew his material from the extant annals but he telescoped events, omitted references to other Viking leaders and concocted a super-Viking, Turgesius, whose wholesale raiding and, particularly, whose attack on Armagh was intended to demonstrate the inefficiency of the Uí Néill as defenders of the church and of the country in contrast of the achievements of the great Brian’ [6]. (Turgesius is elsewhere referred to as ‘exaggerated’ rather than ‘concocted’ [43].) Clare Downham states that throughout theCogadh, ‘records of alliances between Vikings and Irish rulers are neglected; a number of victories won by rulers other than Uí Bhriain are omitted’. Moreover, ‘paired names of Vikings rhyme or alliterate and do not transfer easily into Old Norse equivalents …. These names look as if they have been invented by the author … or drawn from a poetic source’ [48]. Downham further suggests that since ‘historical accuracy, according to the modern definitions, was not the priority’ inCogadh Gaedhel re Gallaibh, ‘the material which is unique to that narrative deserves to be treated with some caution’ [48].

Duffy, on the other hand argues that, whatever about the detail ofCogadh Gaedhel re Gallaibh ‘and its slightly cavalier approach to chronology’, the gist of the account ‘seems sound’ [7]. Duffy also discusses difficulties in using the annals to check the historicity ofCogadh Gaedhel re Gallaibh. By his reckoning, although some of the names of individuals drafted in from beyond Ireland are indeed suspicious, ‘up to half of them appear to be real and their presence at Clontarf is historically credible, if not corroborated by some other source’ [7]. In [30], Ní Mhaonaigh shows that genuine annals underlieCogadh Gaedhel re Gallaibh and that the compiler ofCogadh Gaedhel re Gallaibh ‘remained fairly true to his exemplar’. ‘Provided, therefore, that we keep the redactor’s political purpose firmly in view, we may tentatively add the annalistic material preserved inCogadh Gaedhel re Gallaibh to our list of sources for information on the history of Ireland in the Viking Age’ [30].

Todd himself also reports what he considers to be ‘curious incidental evidence’ for reliability of at least some of theCogadh account in that it ‘was compiled from contemporary materials’ [4]. ‘It is stated in the account given of the Battle of Clontarf, that the full tide in Dublin Bay on the day of the battle (23rd April, 1014), coincided with sunrise’ [4]. In a piece of ‘mathematical detective-work’ [7] that precedes our own by 150 years, Todd’s colleague established that the full tide that morning occurred at 05.30 and indeed coincided with sunrise. For Todd, this ‘proves that our author, if not himself an eye-witness, must have derived his information from those who were’ [4]. We have already seen the importance of the time of the evening tide; calculated to have been at 17.55, consistent with the account inCogadh Gaedhel re Gallaibh; it prevented the escape of the Viking forces and considerably aided Brian’s victory. (See [57] for a recent discussion on this topic.)

This is certainly among the most striking evidence in support of the account ofCogadh Gaedhel re Gallaibh. Duffy provides multiple other instances where theCogadh may be reliable [7]. Certainly bombastic statements that are not backed up by the annals have to be treated warily. But notwithstanding this, he considers the narrative as having ‘some credibility’, although ‘unreliable in its precise detail’ [7]. (For criticism of Duffy’s counter-revisionist views, see e.g. [13].)

To summarize, there is a vast amount of humanities scholarship concerningCogadh Gaedhel re Gallaibh. Although some dispute its reliability, others consider its version of events mainly credible and largely consistent with other sources and evidence. As stated by Duffy, ‘even though it is exaggerated and biased’,Cogadh Gaedhel re Gallaibh can be useful ‘if we use it judiciously’ and ‘make allowance for its propagandist tendency’. The composer surely did not think in terms of network science but, in recording a cast of hundreds connected with well over a thousand links between them, he nevertheless imprinted networks in the narrative. (We explain how we harvest these data in §3.1.) Thus, we may expect that the bulks of the networks contained inCogadh Gaedhel re Gallaibh might not be too far away from the reality of the networks of the Viking Age in Ireland. Many of the objections listed above are largely irrelevant to our approach as static networks are immune to ‘bombastic’ descriptions, ‘telescoping’ of events and ‘cavalier’ attitudes to chronology. We will see that the aggregate approach is even resistant to isolated cases of interpolation. It is with this perspective that we interrogate the narrative with a networks-science methodology. To recap, our primary aim is to determine whether the character networks inCogadh Gaedhel re Gallaibh are implicative of an ‘international contest’ or ‘local quarrel’ [12].

2.3. International contest or local quarrel?

O’Connor [2] in the eighteenth century, with Ryan [5] and Ó Corráin [6], in the twentieth, are considered early debunkers of the traditional myth of Clontarf [29,7]. O’Connor describes the conflict as a ‘civil war’ in which ‘the whole province of Leinster revolted, and called the Normans from all quarters to its assistance’ [2]. Ryan’s main claim is that ‘In the series of events that led to Clontarf it was not … the Norse but the Leinstermen, who played the predominant part’ [5]. His thesis is that the conflict is not a ‘clear-cut’ one between Irish and Viking. Firstly, Brian’s army was not a national one, but one of Munstermen supported by two small Connacht states. Secondly, the opposition ‘was not an army of Norse, but an army composed of Leinster and Norse troops, in which the former were certainly the predominant element and may have constituted two-thirds of the whole’ [5]. The battle, then, was not a contest for the sovereignty of Ireland—it was not a clear-cut issue of Irish versus Norse. Instead, the issue at hand was ‘the determination of the Leinstermen to maintain their independence against the High-King’ [5].

It was in the course of the eleventh century, Ryan argues, that the picture of a decisive struggle of Irish history gained ‘universal acceptance’ in the popular imagination. This came about because of the parts played by forces from the Isle of Man and the Orkney Islands together with the partisan nature ofCogadh Gaedhel re Gallaibh. It was only in this retrospect that Brian was acclaimed as a national hero. Ó Corráin’s view is similar [6]: ‘The battle of Clontarf was not a struggle between the Irish and the Norse for the sovereignty of Ireland …. [It] was part of the internal struggle for sovereignty and was essentially the revolt of the Leinstermen against the dominance of Brian, a revolt in which their Norse allies played an important but secondary role’.

Duffy points out that this revisionist interpretation is not supported by the other ancient annals. For example, the Annals of Inisfallen gives a short but reliable account ‘reflective of contemporary reaction to what occurred’ [7]. It is stated that ‘the Foreigners of Dublin gave battle to Brian’ and Leinstermen are also slain. According to Duffy, ‘Whereas some modern historians see the Leinstermen as Brian’s primary enemy at Clontarf, the annalist was in no doubt that the enemy was the Norse of Dublin. In fact he has the same black-and-white picture of the opposing sides that we tend to think of as later legend …’. ‘The entry in the Annals of Ulster also echoes the Annals of Inisfallen in emphasizing the primacy of the Norse as Brian’s adversaries’. Duffy states that the Annals of Ulster suggest ‘it was fundamentally a contest between the Irish and Norse (although the latter too had Irish allies)’.

Duffy provides multiple items of evidence in support of his view that ‘Brian’s principle opponents were the Hiberno-Norse allied to Leinster’ and that the Battle of Clontarf ‘was notable in particular for the great numbers of overseas Norse forces present, and for the huge losses they incurred by fighting and drowning’. ‘Implicitly, for the Cogadh’s author, two centuries of Irish opposition to Viking invasion, spearheaded by Brian’s dynasty, reached a climax at Clontarf. That picture was imprinted too, with remarkable correspondences, on the minds of … thirteenth-century Icelandic writers. Those who did battle with Brian came from the Norse world seeking a kingdom for themselves in Ireland’.

Thus, the debate about Clontarf has spanned the centuries and frames our present investigation. Here, we broaden the question to how conflictual and social relationships are presented inCogadh Gaedhel re Gallaibh.

3.1. Constructing theCogadh network

As with previous studies [49–51,60,61], we considerCogadh Gaedhel re Gallaibh as playing out on a complex network comprisingN nodes andM edges. The edges link the nodes through relationships or interactions. We distinguish between three categories—Irish, Viking and other—identifying to which group each node belongs from the text itself. We obviously cannot directly access the reality behind the text to determine any gradation between the groups. For example, we cannot know how Sigtrygg Silkbeard, who had a Viking father and an Irish mother, might have self-identified in reality; we can only take our lead from theCogadh itself and since the Hiberno-Norse of Dublin are presented there as Vikings, they are placed in that category. Nodes classified as ‘other’ are those that are not readily assigned to either camp.

Our approach to constructing the networks follows the methodology of [49–51] in that nodes and links are identified by carefully and manually reading the texts with multiple passes through all of the material by multiple readers. In our experience, such an approach is required to minimize errors and omissions as well as to reduce levels of subjectivity.Cogadh Gaedhel re Gallaibh is a very dense text and meticulous care is required to interpret extremely subtle tracts containing large amounts of explicit and implicit information. It is currently beyond technological capabilities to extract such information automatically owing to the inherent complexity of such texts (see, e.g. [62]). Establishing the technology for such an approach is another active area of research.

Figure 1 contains a network visualization of the full set of interactions recorded in theCogadh. Green nodes represent Irish characters and green edges represent interactions between them. The counterpart set of Viking nodes and their interlinks are in blue. Brown edges represent interactions between Irish and Viking nodes. Any remaining nodes and edges are in grey.

We distinguish between two types of edge: positive and negative. Positive edges are established when any two characters are related, communicate directly with each another, or speak about one another, or are present together when it is clear that they know each other. So positive edges ordinarily represent familial or social relationships. Negative links, on the other hand, are formed when two characters meet in physical conflict or when animosity is explicitly declared by one character against another and it is clear they know each other (such as a declaration of war). So negative edges typically represent actual or intended physical hostility. It is possible that two characters are linked by both positive and negative edges as relationships between characters may change over time.

Ours is a static analysis, capturing the temporal totality of theCogadh narrative. ‘Making the past just as visible as the present’, as Moretti puts it [63], is a benefit of this networks approach and one which has been used elsewhere [49,50]. Nonetheless, it should be noted that the study of dynamical properties of networks constitutes an active, broad and developing area of research and such an approach would be of interest in the future [61]. We focus primarily on thetopology of the networks underlyingCogadh Gaedhel re Gallaibh, considering undirected, unweighted networks. This means that (i) the features which connect the various nodes are not oriented and (ii) the statistics we report upon do not take into account varying levels of intensity of interactions between nodes. To account for (i), one would have to introduce a level of detail which is finer that just positivity or negativity. However, what one gains in refining details, one loses in statistical power. To account for (ii), one may place higher weight on more intense interactions, but, besides using the number of interactions between characters in the narrative, there is no established standard mode of weighting edges in character networks. Moreover, we are primarily interested in the presence or absence of conflict, not on the details of varying intensity of such hostility. Therefore, we defer consideration of directed, weighted and temporal networks for future studies and restrict the current study to network topology and related matters.

3.2. Network methodology: basic statistics

We identifiedN=315 individual interacting characters in Todd’s translation ofCogadh Gaedhel re Gallaibh.⁷ These nodes are interconnected byM=1190 edges and we refer to the corresponding assemblage as theentire network. We can also consider the positive and negative sub-networks, formed only of positive or negative edges, respectively. Examination of these allows us to gain more insight into the social and conflictual statistics contained in the narrative. Indeed, it is long known from sociology that societies exhibithomophily, the tendency of individuals to associate with others who are similar to themselves [64–66]. In the field of social network analysis, this is known as assortativity. In previous studies of epic literature [49–51,60], we studieddegree assortativity, the tendency (or otherwise) of nodes to attach to other nodes with similar numbers of links. We found some positive sub-networks exhibit degree assortativity, or are uncorrelated, while the opposite feature—degree disassortativity—is characteristic of negative sub-networks. This means that positive social networks give a ‘cleaner’ picture (relative to full networks) of the non-conflictual societies underlying such narratives, making it valuable to study them in isolation [67]. A new feature of the current study is our additional focus on the negative sub-network to statistically measure levels of hostility.

We use the termunsigned to refer to networks containing both positive and negative edges. Networks comprising only positive (or only negative) edges are then themselves termedpositive (ornegative, respectively). We use the termfull-cast to refer to networks containing the full cast of characters, Irish, Viking and others. Networks containing only Irish (or only Viking) characters are themselves referred to asIrish (orViking, respectively). This terminology is summarized intable 1. Statistics for the entire network and various sub-networks are collected intable 2.

Table 1.

Full-cast networks comprise Irish, Viking and other nodes together with interactions between them. Unsigned networks comprise positive and negative edges as well as the nodes they connect. Thus, for example, the positive, full-cast network comprises all nodes but only positive links. The unsigned, Irish network comprises only Irish nodes but both positive and negative links between them. Theentire network comprises all interacting nodes and all links.

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The average number of edges per node for the entire network is 〈k〉=2M/N≈7.6. The actual number of edges associated with theith node is denoted byk_i. This is a number which varies between 1 for the least connected characters (nodes withk_i=0 have no links and are not attached to the network at all) andk_max for the most connected (in a sense, the most important) character. For the entire network, the most connected character is Brian himself who, withk_max=105 edges, is linked to 33% of the other characters in the narrative. Besides Brian’s degree, we are also interested in the connectedness of other characters and we rank the first few characters according to their individual degrees, and according to other measures of importance, in appendix C.

Cogadh Gaedhel re Gallaibh hasN⁺=287 interacting characters in its positive sub-network, interconnected byM⁺=957 edges, corresponding to a mean degree of 〈k〉⁺≈6.7.⁸ Here and henceforth, we use the superscripts ‘+’ and ‘−’ to identify statistics associated with the positive and negative networks, respectively. (We omit such a superscript from statistics for the unsigned networks. These are distinguished from generic symbols by context.) The counterpart figures for the negative network areN⁻=180,M⁻=264 and 〈k〉⁻≈2.9, respectively. (The total number of positive and negative linksM⁺+M⁻=957+264=1221 exceeds the numberM=1190 which we previously identified for the entire network because some relationships involve both positive and negative aspects.) As for the entire network, Brian has the highest degrees in both positive and negative subgraphs, with the former measured atk⁺_max=53 and the latter atk⁻_max=63.

The adage that ‘the enemy of an enemy is a friend’ is related to the notion ofstructural balance in network science [68–70]. The maxim suggests thattriads (sets of three mutually connected nodes) with one positive and two negative edges are commonplace. More generally, triads with odd numbers of positive edges are considered structurally balanced. One way to quantify the extent to which it holds in a character network is through the statistic Δ, defined as the percentage of triads that contain an odd number of positive links. A large value of Δ means that hostility between two characters is suppressed if they have a common foe. Clearly Δ is only meaningful for the unsigned network; on the positive sub-network it is 1 by definition, while in the negative sub-network it is necessarily zero. We find that the entire network underlyingCogadh Gaedhel re Gallaibh (which has 3041 triads) is indeed structurally balanced with Δ≈93%.

As mentioned above, assortativity (disassortativity) is the tendency for the nodes of a network to attach to other nodes that are similar (different) in some way. Network theorists frequently measuredegree assortativity—the extent to which nodes of similar degree tend to link up. As with other character networks, we find that the negative full-cast network is disassortative by degreer=−0.25(3).⁹ This means that high-degree characters are hubs and their negative links preferentially attach to low-degree ones. This appears to be a generic feature of heroic tales in particular, where the hero or heroes encounter multitudes of lesser characters and defeat them in battle. The positive full-cast network, on the other hand, is uncorrelated within errors (r=−0.00(4), meaning it is neither assortative nor disassortative). These features are typical of social networks and of character networks with positive interactions [67,49].

Besides the networks comprising the full cast of characters, we can also consider the networks containing only Irish or only Viking nodes and these are also listed intable 2.¹⁰ We observe the following average properties of the various networks. In the Irish and Viking networks (as in the full-cast cases), the mean degrees are maximal for the unsigned networks and minimal for the negative sub-networks. The unsigned Viking network is more structurally balanced than its Irish counterpart. Structural balance for the Irish network, which has 830 triads, is 93% whereas the 881 Viking triads all contain odd numbers of positive links.

3.3. Effect of interpolation on network statistics

In his Introduction toCogadh Gaedhel re Gallaibh, Todd acknowledges the defects of the work and expresses regret that it is ‘so 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 …’. In chronicle literature, an interpolation of the type mentioned by Todd is a later addition not written by the original author. As scribes copied ancient material by hand, extraneous material frequently came to be inserted for a variety of reasons [71]. These may have been for bona fide intentions, perhaps as explanations; for subjective purposes; or they may simply have crept in through errors and inaccuracies arising from manual copying or, indeed, as attempts ‘to enhance the appeal of the narrative’ [27]. One way to detect such interpolation is through comparing different manuscripts.

Perhaps the most famous interpolation in the narrative is a passage which occurs in the Dublin version describing the actions of Fergal Ua Ruairc of Bréifne and associate chieftains [5,27]. (For the location of Bréifne, seefigure 3.) The Brussels manuscript, by contrast, ‘omits everything connected with Fergal and his presence in the battle’ [4]. As stated by Todd, ‘the whole story bears internal evidence of fabrication, for Fergal O’Ruairc was slain AD 966 …, and our author had already set him down among Brian’s enemies’. Ryan [5], Duffy [7] and others also identify Ua Ruairc as an interpolation and Ní Mhaonaigh gives a detailed account of Bréifne bias inCogadh Gaedhel re Gallaibh [27]. She states ‘one of the main aims of the interpolator was to portray Fergal Ua Ruairc and his followers in as favourable a light as possible, sometimes regardless of the effect this had on his text’. The point is that a pro-Ua Ruairc reviser of the narrative may have deemed it politically expedient to alter the record of relations between the Uí Ruairc and the Dál Cais by demonstrating assistance given by the former to Brian at Clontarf. Ní Mhaonaigh estimates the period when the Uí Ruairc were likely to have gained maximum advantage from such an association to have been the mid- to late 1140s, over a hundred years after Clontarf [27].

We are interested in what insight the networks methodology can give on such matters. We have already seen that 93% of the 3041 triads in the unsigned network are structurally balanced as are 93% of the 830 triads in the Irish network. The triad formed by Ua Ruairc’s enmity to Máel Sechnaill, the latter’s alliance with Brian, and the interpolated support of Ua Ruairc for Brian is one of two positive edges and one negative one, which is structurallyimbalanced. Since the vast majority of triads inCogadh Gaedhel re Gallaibh are balanced, this makes the Ua Ruairc episode stand out as relatively unusual. We removed Ua Ruairc and his three associates (Gilla-na-Naomh, Mac an Trin and Domhnall mac Raghallach [4]) from the networks to test the effects on the statistics. Besides reducing the number of edges (e.g.M reduces from 1190 to 1146 in the entire network), the effects of this removal are minimal. For example, the degree assortativies are unchanged within error estimates for the unsigned, positive and negative networks.

The possibility of interpolation applies not only to Ua Ruairc and allies. Ryan claims that ‘Many of the names mentioned are names only, for nothing is known of the persons who bear them. Some of the levies in important positions were certainly absent. In a word, no effort is made to distinguish between the genuine and the spurious, to criticise suspect sources, and to reconcile contradictions’ [5]. Given the minor effect of the most famous and easily identified, Ua Ruairc, interpolation, we do not attempt to remove other interpolations from our analysis. Besides, any attempt to do so would be incomplete because we cannot be certain that all interpolations have been identified. Indeed, as we have repeatedly emphasized, ours is a network study ofCogadh Gaedhel re Gallaibh as represented by Todd in [4] and therefore we present it in its entirety. However, we attempt to simulate the effects of interpolation by randomly removing up to 15% of nodes or edges. The process is repeated 1000 times and the averages deliver no appreciable difference to the statistics given intable 2, indicative of their robustness (see appendix C for a network-robustness analysis). For example, removal of 15% of the vertices alters the assortativity fromr=−0.09 tor=−0.08 (imperceptible change within errors). Removal of 15% of the edges leavesr unchanged within this level of precision. A more systematic and targeted quantitative study of the effects of interpolation would be interesting for future study.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

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Data Availability Statement

Data are available fromhttps://github.com/ralphkenna/CGG.