FREQUENTLY ASKED QUESTIONS

Last updated: February 2013

Over the past 6 months we have given many seminars, talks and interviews at academic

conferences, research institutes and for newspapers and radio, worldwide, concerning our

social-network approach to comparative mythology. There has also been enormous reaction

on twitter and the blogosphere. Certain questions frequently arise. Here we collate some of

these, together with our answers.

What inspired you to investigate mythological narratives?

Ralph Kenna is a theoretical physicist with longstanding amateur interests in mythology. He saw

network theory, which emerged within statistical physics in recent years, as a way to bring the two

disciplines together. The Leverhulme Trust supported his proposal for a PhD topic around this

interdisciplinary theme and Pádraig Mac Carron was an ideal candidate with a physics background

and a similar interest in mythology. Both authors are from Ireland, a country with an extensive

mythological literature (the known literature of the Ulster Cycle alone is estimated to occupy at least

2,000 pages of a modern volume if all repetitions were edited out), a fact which undoubtedly inspired

the project.

What has network theory been applied to previously and what are the results?

In the original 1998 paper in Nature on complex networks by Watts & Strogatz similarities

were noticed between the structures of systems as diverse as an electrical grid, Hollywood

actors and the wiring of the neural system of a nematode worm. Transport networks are

another easily conceived example. Since then, many social networks have been studied and

their properties catalogued. Examples include the networks formed by musicians in Jazz

bands, users of online forums, scientific co-authors collaborating on research papers and

directors of various companies. The collection of the various characters which have appeared

in Marvel comics and the relationships between them form an example of a social network

which is clearly fictional. The general properties of a wide variety of networks are therefore

well established and well documented.

These real-world social networks tend to share similar properties. They are highly connected

and small world. They are also assortative, and their degree distributions are often scale free,

meaning only a small number of people tend to know a large number of people.

(Thee terms are explained below.)

Slightly technical questions on networks and social networks

What is a complex network?

A complex network is a set of nodes and links between some of them – it is a graph with non-

trivial topological structure, not a simple lattice and not a random graph.

Here are schematic diagrams representing three different types of networks: random graph (left), regular lattice (middle) and a complex network (right).

What is a social network?

In a social network, the nodes represent people and links (or edges) represent relationships between

them.

What is path length and degrees of separation?

The mean path length is the average shortest number of steps between pairs of individuals in

a social network. In the above figure, for example, there are various routes by which

individuals G and D may be connected (e.g., GEHD), but the shortest of these have

only two steps (i.e., route GHD). Thus the path length between G and D is 2. We next

measure the (shortest) path lengths between all other pairs of nodes (e.g. A to B; A to C; ...; G

to H). The average of all of the resulting numbers is the mean path length of the network. The

idea of mean path length connects to the famous notion in sociology of six degrees of

separation. This is the idea that, despite the world’s population of over seven billion,

everyone is, on average, only about six steps away from any other person, in the sense that a

chain comprising of "a friend of a friend of a friend of..." connects most pairs of people,

worldwide, in only six or fewer steps (hence the term “small world”).

How does mathematics and statistics come in?

In network theory, we are interested in gathering statistics which capture how the links between nodes

are distributed. We have had the example of six degrees of separation. In a sense, the number

6 characterises the social network of the entire world. In a transport network, one could ask

how many times must one change busses (on average) to get from one point in a city to

another. If the answer is 2, say, that number characterises the public-transport network.

So networks can be characterised by numbers (6 and 2 in these examples). These numbers

can be compared to each other get an idea of how different or similar networks are in a

quantitative manner. We are mainly looking for averages – hence the statistics. Because of this, we require fairly large

systems – the larger the network, the better we can pin down its statistics.

What is clustering?

The clustering coefficient is designed to give an indication as to how cliqued a network is; if

an individual knows two other people then there is a high probability that those two people

also know each other. In social networks the average clustering coefficient tends to be very

high compared to other networks; for example in a client server network where each

computer is connected to a central sever, there is no clustering as the computers are not

directly connected to each other − there are no closed triadic loops.

An example of how to calculate the clustering coefficient of an individual node is shown here

for node B. Of the three potential relationship-triads involving it (ABC, ABD, and BCD), only one is

realised, namely BCD. The clustering coefficient of node B is therefore deemed to be one out of three

or 1/3. The clustering coefficient for the entire network is determined by simple averaging

over all nodes.

What does degree mean?

The degree of a node is the number of links it has to other nodes. In social networks it measures how

many relationships a person has - how popular they are. We measure how the degrees are distributed

throughout the entire network. This means that we are looking at links or interactions between

characters rather than characters themselves.

What does scale free mean?

People talk about scale-free networks, but really they mean scale-free degree distributions (it has

nothing to do with coarse graining). An example of a distribution with a scale is an exponential or

stretched exponential. In the former case, the probability function for degree k looks like

p(k) = A exp (-k/b).

The parameter b provides a scale for the degree distribution. A scale-free network is one with a

power-law degree distribution

p(k) = A k -γ,

with exponent where 2 < γ < 3.

On the one hand, if γ < 2, there is no finite mean. On the other hand, if γ > 3, the variance converges

and there is a scale. This is why we need 2 < γ < 3 for a scale free power-law distribution.

Preferential attachment is a possible explanation for how scale free arises.

What does small world mean?

A network is small world if it has a low average path length (i.e., a mean path length similar to that of

a random network of the same size and average degree) and a high clustering coefficient C. (large

relative to comparable random networks).

What does assortativity mean?

Assortativity is the tendency of a character of a certain degree to associate with similarly popular

characters and the opposite tendency is called dissassortativity.

Real social networks tend to be assortative (popular individuals associate with other popular

individuals).

individuals).

The network on the left represents assortativity because nodes of similar degree tend to be linked to each other. E.g., the three nodes of highest degree are linked to each other. The network on the right, which has the same distribution of nodes, represents disassortivity – e.g., the three most linked nodes are not mutually connected.

What does “vulnerable to attack” mean?

A network is vulnerable to a targeted attack if removal of the most popular or most connected

characters leads to rapid fragmentation of the network – it splits into disconnected societies. This is a

property of real social networks and we have found that it is also a feature of networks underlying the

three mythologies we looked at. In contrast, fictional networks tend to be very robust -- even after

removing all the main characters, the network can remain intact. This seems to be because, in fiction

(at least the ones we looked at), everybody tends to know everyone else. In reality, and in our three

myths, this is not the case. One can also consider random attack where nodes are removed randomly

rather than by specifically targeting the most important characters.

What does structural balance mean?

A structurally balanced social network is one in which there is unlikely to be an odd number of

hostile links in a closed triangle. i.e., if two nodes are in conflict, and if a third node enters the game,

they tend to side with one or other of the existing parties. This relates to the notion that “enemy of my

enemy is my friend”. Note that every social network we looked at (fictional or otherwise) was

structurally balanced, so this is not a good criterion to decide whether a narrative pertains to reality or

fiction.

Questions about the mythological epics we studied

Tell us a little about the epic narratives you investigated

Beowulf is an Old English heroic epic, set in

Scandinavia. A single codex survives which has

been estimated to date from between the 8th

and

11th

centuries. The story tells of the coming of

Beowulf, a Gaetish hero, to assist Hrothgar, king of

the Danes. (The Gaets were a North Germanic tribe

living in what is now Götaland in modern Sweden.)

After slaying two monsters, Beowulf returns to

Sweden to become king of the Geats and, following

another fabulous encounter many years later, is

fatally wounded. Although the poem is embellished

with obvious fictional elements and creatures,

archaeological excavations in Denmark and

Sweden offer support for the historicity associated

with some of the human characters. The character

Beowulf himself is, however, not to have existed in

reality.

The first page of Beowulf

The Iliad is an epic poem attributed to Homer and is dated to the 8th

century BC. It is set

during the final year of the war between the Trojans and a coalition of besieging Greek

forces. It relates a quarrel between Agamemnon, king of Mycenae and leader of the Greeks,

and Achilles, their greatest hero. Although debated throughout the years, recent evidence

suggests that the story may be based on a historical conflict around the 12th

century BC

interwoven with elements of fiction.

Early people and politics of

Ireland

The Táin Bó Cúailnge (Cattle Raid of Cooley) is the

most well known epic of Irish mythology. The tale

describes the invasion of Ulster by the armies of queen

Medb of Connacht and the defence by Cúchulainn,

Ireland's most famous hero. Related to The Táin itself

are a number of pre-tales and tangential tales (remscéla)

which give the backgrounds and exploits of the main

characters. The Táin has come down to us in three

recensions. The first has been reconstructed from partial

texts contained in Lebor na hUidre (the Book of the

Dun Cow, dating from the 11th

or 12th

century) and

Lebor Buide Lecáin (the Yellow Book of Lecan, a 14th

century manuscript) and other sources. The second,

later recension is found in Lebor Laignech (the Book of

Leinster, a 12th

century manuscript formerly known as

the Lebor na Nuachongbála or Book of Nuachongbáil).

A third recension comes from fragments of later

manuscripts and is incomplete.

Are there other evidences for or against historicity related to these epics?

The Táin was dated by medieval scholars to the first centuries BC, but this may have been an

attempt by Christian monks to artificially synchronise oral traditions with biblical and

classical history. Its historicity is often questioned. O’Rahilly (1946) that such tales have no

historical basis whatsoever. Jackson (1964) argues that, while such narratives corroborate

Greek and Roman accounts of the Celts and offer us a “window on the iron age”, “the

characters Conchobar and Cúchulainn, Ailill and Medb and the rest, and the events of the

Cattle Raid of Cooley, are themselves entirely legendary and purely un-historical”. On the

other hand, Lynn (2003) has claimed some evidence that the landscape reflected in the story

may be based on reality. While the analysis which we present here is based on Kinsella’s

version, we also examined Carson’s version to check that the two translations give very

similar networks. Therefore, for the purposes of this analysis, Kinsella’s translation serves as

a proxy for what is commonly understood as the Táin Bó Cúailnge.

Questions about the outcomes of our analyses

In terms of the Iliad, how confident are you that this poem is based on real life events?

As stated, we look at interactions between characters rather than characters or events themselves. In

this way we obtain measurements characterising the society’s structure. For the Iliad, the numbers we

obtain are similar to numbers pertaining to real societies and in this sense the Iliad’s society looks

believable. We can't really comment so much on the events - we're not saying that this or that

happened, or even that the individual people portrayed in this story are real, but that the overall make-

up of the society and interactions between characters seems believable. On this basis we think it's

unlikely the society portrayed in the Iliad was entirely made up.

In your analysis of the Iliad, have included divine characters in social networks?

In fact we did the analysis with and without the gods and there is little difference. They are

not the most dominant characters. Remember, we are not really interested in what a character

does - we are interested in how they interact and who they interact with.

Which of the three mythological narratives studied has the most realistic social

network? And why?

The Iliad appears most real. One reason, from a mathematical point of view, is because it is

has all the properties associated with previously studied real networks. This coincides with

the points of view of some archaeologists. Many believe the Iliad to be based on real events

and a real society.

The next most realistic is Beowulf. In fact we find that the entire society (except Beowulf

himself) looks real. Again, this is in agreement with the opinions of archaeologists.

The least realistic is the network underlying the Irish story. This coincides with mainstream

opinion amongst most archaeologists that the Táin is not real.

However, and importantly, we find that the artificiality does not pervade the entire society

depicted in the Táin. Instead, it is localised in only 6 characters. On this basis, we speculate

that the top six characters may be exaggerated versions or amalgams of characters. When we

test this by “reducing” the popularity of these characters, the society looks realistic.

But are you not simply chopping the societies in the network back until you obtain

what you want? In this sense, are the gymnastics necessary to make certain

networks assortatively mixed rather contrived?

In fact this is precisely what we are doing and indeed, we agree they are contrived. But this is

precisely the point: we are asking, what would it take to render these networks "realistic".

The point is that we only have to tweak a very tiny amount to render Beowulf and the Táin

realistic. That is one of the exciting outcomes of our analysis!

Which translations of the three epics did you use?

We base our Beowulf network analysis on Heaney’s translation.

Our network analysis of the Iliad is based upon the translation by Rieu (2003).

Two popular English translations of The Táin (Kinsella, 1969; Carson, 2007) are mainly

based on the first recension, although they each include some passages from the second. We

used Kinsella’s version.

What does it mean that there mythological stories more realistic than Harry Potter?

The three mythological epics we studied are more realistic than Harry Potter in the sense that

the social network in Harry Potter does not look like real social networks. It is too highly

connected.

Does that mean the theory of networks may be used to determine the credibility of

other myths and epic literature as the Spanish El Cid?

Yes, we are investigating this. But we are investigating not the events – rather the society. We can't

comment too much on work which has not yet been peer reviewed.

I find it a bit disturbing that you don't seem to believe Richard III is a historical figure.

If a network does not look real that does not mean a character in it is not a real historical figure,

similarly if a network does looks real that does not mean a character in it is a real historical figure.

This work looks at the society portrayed in the narrative. One has to take our evidence together with

other evidence – from archaeology and other sources.

Shakespeare was constrained by having to use actors on stage and therefore the social network is

more limited that a real one.

What surprises me even more than your finding that the ancient texts look like real social

networks is that works of modern fiction do not. Why do you think they don't?

Certainly, there are similarities between fiction and real life. Aristotle considered drama as a mimesis

of reality. Indeed, theatre is reliant upon our social cognitive capacities: the relationships between

characters must be realistic to be convincing.

In fiction, however, the networks (at least those we looked at) tend to be very highly connected and

supporting characters tend to be very close to the main character and have similar degrees. This makes

the network very robust. Minor characters in fiction tend to be linked to the main character rendering

it disassortative and the diameter of the network small. In other words everyone tends to be connected

to everyone else, otherwise the story becomes too hard to follow.

We look not at the quality of the relationships between characters (other than the fact that they are

friendly or hostile). Instead we look at the quantity (the distribution) of these relationships.

And the distribution is different depending on whether we are looking at fiction or myth.

In your EPL paper you say that the reader should keep in mind that these networks, like

many others, are necessarily of limited extent and incomplete - potentially representing

spotlights on the societies from which they are drawn. But isn't it true of any narrative, real

or fictional?

Yes – most networks are incomplete - not just those to do with narratives. (In fact one could argue

that fictional networks, by their nature, are complete.)

Another interpretation of your results is that ancient narratives are told in a similar way to

how people talk about real social networks, i.e. the sampling is similar. In this case your

results don’t necessarily mean that the ancient myths are more likely to be based on real

social networks than the modern fiction.

We think there may be an element of "over-interpretation" in what you are saying. We have to be a

little more conservative.

Our main purpose is to compare myths to (a) each other, (b) to random networks, (c) to real social

networks and (d) to fiction.

For (a), we find similarities in the three myths. This is important, and central to the paper, especially

in the Irish case. The Irish myth is considered to have the least historicity by archaeologists. Beowulf

is considered to be well grounded in reality, except for the main character. Our crucial observation is

that the Irish myth network is very similar to that of Beowulf except for the distribution associated

around the top 6 characters. This anomaly hints that these 6 may be amalgams. Tweaking them

renders the Irish myth on a similar to the Iliad, which has historicity. This is the primary comparison

of our paper.

Towards the end of the paper we compare to fiction. We do this to test whether these also look

realistic, and to compare to each other. We find they do not look realistic.

By realistic, we are talking about well-defined, real social networks, e.g., the scientific collaboration

network, the jazz musicians’ network, the movie actors' network. These are defined in a manner

which is independent of how people talk about them.

You are suggesting another investigation - about how fictional narratives compare to non-fictional

narratives. This is certainly interesting and something we have considered.

However we do not wish to speculate too much around this because we need to stick to this particular

work, as it has not yet been peer reviewed.

Someone suggested that we look at Cold Blood, by Truman Capote. “That story is based on a real

social network, but it is told very much in the style of fiction. (That's what was so innovative about it,

and it launched the contemporary style of "creative nonfiction" writing.) If a book like In Cold Blood,

or any of the later creative nonfiction books, fail the test, then that would lend strong support to the

hypothesis that story-tellers really do give themselves away when they create fictional social

networks. The alternative explanation is that the style of storytelling is different in fiction, and that

shows up in creative nonfiction as well.”

This is an interesting suggestion and one we should take up.

What you may have discovered is a marker that distinguishes group-authored

fiction like the Iliad and Beowulf from fictions that are the creations of individuals.

We can’t logically exclude this possibility. One has to take a holistic view to try to decide

whether we have found a way to investigate remnants of historicity or different narrative

styles. As we say, this is an initial foray into ongoing research and there are speculative

elements. We hope that other people will investigate the networks underlying other narratives

so that we can gather more evidence.

Why did you chose to analyse the fictional narratives that you did?

We were unaware of existing analyses of artificial or fictional networks (apart from the

Marvel Universe), so we had to have a look at something. We chose Rowling’s Harry Potter,

Tolkien’s Lord of the Rings, Shakespeare’s Richard III and Hugo’s Les Misérables. Clearly

these four, along with the Marvel Universe, are not meant to embody all of the network

features of the entire corpus of world fiction. Some are chosen because they are examples of

clear works of fiction – works involving fabulous characters, which do not attempt to emulate

reality. For others, the data were readily available, facilitating a speedy study. It is hoped that,

in the future, other scientists will analyse other works of fiction, so that a more extensive list

of typical features will ultimately be built.

We found that each of the fictional networks we analysed are small world, just like real social

networks but none of their degree distributions are well fitted to power laws. Crucially, they

are all disassortative. Moreover, unlike real social networks, the fictional ones are very robust

upon targeted removal of the most connected nodes.

Can your technique pick up the reality underlying ancient histories? For example, a

comparison with narratives from similar times that are known to be based on

reality, such as the ancient Greek histories.

Some historical works span generations and multiple cultures so you would not get one large

society in one location to be able to study the social network. Similarly most histories just

deal with the major players at a given time rather than a large society. Historical biographies

also do not work as it will just link everyone a character meets in their documented life and

not give a major indication of the overall society he played a part in.

E.g., Suetonius' Twelve Caesars gives biographies of the emperors of Rome and it mostly

tells who gave orders, where they went, etc. It is not a story with a large amount of

participating characters like a myth and it spans an individual’s entire life.

Here is the network from the first 20 pages of Suetonius' book. As this is a biography you

can see that virtually every character is just linked to Caesar, though some of them have their

father's named, this network doesn’t give much information about what the society looks like,

just how many people that it's documented that Caesar met in his adult life, as a result it will

be very disassortative. We decided not to pursue this further.

Remember, if they don't look like real social networks that doesn't mean they're artificial, it means it's not describing a society, just the major players or the elite. So some historical biographies do not seem to work, and histories spanning a large period of time will also not work. Perhaps Livy might be okay though.

What about Game of Thrones?

We are investigating this. We will report on it later.

What about the Bible?

This something that many people suggest. It is outside the remit of this project. We are not

directly interested in analysing religious texts.

It also has the problem of taking place over a large period of time and in multiple places,

therefore making it hard to get information for a full society. Also there are large ancestor

lists which again will not tell you anything about the society.

What can your results can bring to our enjoyment of these epics?

Well, the question of “enjoyment” is very subjective and qualitative. Questions such as these are far beyond our expertise. However, we can venture to say that our work helps bring to the fore the Irish

narrative which is far less well known than the other two.

What do you plan to study next?

We are currently studying other European mythologies and sagas including Arthurian, Norse,

Germanic and Slavic. In addition, colleagues in Brazil are working with us on South

American mythology.

The Referees of our EPL both interpreted our paper as "sufficient for proof-of-concept" and

an "initial foray" into the topic. We agree - this is precisely what we have in mind and why

we thought a Letter is appropriate. (EPL is a letters journal.) Our next paper will contain a

analyses of wider sets of myths, epics and fiction.

We hope that other people will investigate the networks underlying other narratives –

mythological, fictional and reality-based, so that we can gather more evidence.