Our “A Game of Thrones” network for consists of 187 nodes (characters) and 684 weighted edges, accounting for 7,366 interactions. Character names are sized by PageRank centrality. Character nodes are sized by betweenness centrality. Edge thickness corresponds to edge weight.
This network divides into six main communities. The communities, and their most prominent members, are (clockwise from the top):
- the Night’s Watch (Jon Snow),
- the Riverlands (Tyrion and Catelyn),
- the Dothraki (Daenerys and Drogo),
- King’s Landing (Ned and Robert),
- the Royal Children (Sansa, Joffrey and Arya), and
- Winterfell (Robb and Bran).
The Riverlands community has a distinctive structure: each of Tyrion and Catelyn act a hub for a distinct subset of the community. Tyrion is the center for the mountain clans, while Catelyn is the center for the Tully bannerman.
There are also two three-person communities. The first consists of the Night’s Watchmen from the prologue. The second triangle contains three sons from the bounty of progeny of Walder Frey.
We also note that there are quite a few edges between communities. This first volume follows the fracturing of the Stark family, both geographically and personally. Given the focus on the Starks, we discover a network that is quite interconnected. In later books, the community structures are more pronounced and distinct.
Let’s start with the obvious. It comes as no surprise that Ned Stark tops all seven centrality measures. Ned’s values are more than 30% larger than the runner-up in four out of six measures. The exceptions are betweenness (25% larger) and eigenvector centrality (10% larger). Overall, Ned’s importance stands out by a wide margin.
Robert Baratheon takes second place in all categories, except for PageRank (where Tyrion bests him— more on the Imp below). The strong performance by Ned and Robert reflects the main narrative of the book. All other action is a direct or indirect consequence of Ned’s decision to leave Winterfell to serve as the Hand of the King. We view their strong performance across the board as a validation of the fidelity of our network.
Eigenvector Centrality versus PageRank Centrality
Let’s compare the two “feedback loop” centralities. In eigenvector centrality, you get full credit for the importance of your neighbors. In PageRank centrality, you get a pro-rated share of each neighbor’s importance, depending on the strength of your connection.
The eigenvector ranking of the top five characters is:
- Ned, Robert, Sansa, Tyrion, Joffrey.
The eigenvector ranking is clearly out of whack. The Sansa/Joffrey plot line is a secondary one, and no one would argue that Sansa’s role is comparable to Tyrion. So what has gone wrong? Enmeshed in King’s Landing and one step away from the primary action, eigenvector centrality gives Sansa and Joffrey more clout than they deserve in their present circumstance. High eigenvector centrality means that they are connected to important people. However, it does not take into account how much attention they command from those connections.
PageRank centrality does much better. This feedback loop incorporates the relative strength of your connections. The top five characters according to PageRank are
- Ned, Tyrion, Catelyn, Robert, Jon.
This list is far more compelling when squared against the book. One could certainly debate the relative order of these characters, but this is a solid list. PageRank does a nice job of capturing narrative tension. Indeed, the plot develops when important characters interact repeatedly. PageRank captures this dynamic through its amplifying feedback loop. As we continue down the list of PageRank centralities, the characters continue to make sense. Robb, Sansa and Bran take the next spots, followed by Jaime, Cersei and Joffrey. Finally, Arya comes in 12th place, as her mischievous and independent ways take her furthest from the main action.
Does this mean that we should throw out eigenvector centrality? Not necessarily. Each centrality measure captures a different dynamic. As long as you understand what is being captured in the mathematics, you can make good use of the information. High eigenvector centrality means that you have powerful resources that you can attempt to tap in the future. When a character’s eigenvector centrality is higher than their PageRank centrality, then that character has potential for development in future volumes. However, that character must exert themselves to capitalize on this potential.
This brings us to the two point of view characters from outside of House Stark: Tyrion Lannister and Daenerys Targargyen. PageRank identifies Tyrion’s unique role in the book: his score is the second largest, jumping Robert by a healthy margin. While Tyrion does not have direct power, he enjoys the strength of House Lannister, while also finding himself in direct contact and conflict with important members of House Stark. (Cool fact: Tyrion is the only non-Stark character that is connected to everyone in the Stark family.) Furthermore, Tyrion forms the hub for a motley collection of Lannister bannermen and mountain clansmen within the Riverlands community. His resulting PageRank portends the important role that lies ahead for him in the saga.
Turning to Daenerys, we see how isolated the Targaryen princess truly is. While she may be a hub in the Dothraki community, she only has three tenuous connections back to King’s Landing (Ned, Robert and Varys). Her PageRank centrality (13th) roughly matches her degree centrality (15th) and slightly underperforms with respect to her weighted degree centrality (9th). Her PageRank score is lower than every prominent Stark, Baratheon and Lannister. Meanwhile, eigenvalue centrality exacts a more severe penalty for her remoteness: she ranks 42 out of the 187 characters in the network, which is a dramatic drop compared to her degree centrality. This reminds us how distant she is from the action in Westros, and what little potential for power she currently has.
Certainly, Daenarys is more important than her 13th place ranking. Or put more simply, the ways in which Daenerys is important are not captured by the feedback loop of character interactions. What PageRank tells us is that Daenarys’s storyline is far removed from the current political intrigue of Westros. As long as you keep in mind what the measure captures (and doesn’t capture), you can make good use of the information.
What about the political schemers of King’s Landing? Some will be concerned that Littlefinger does not rank higher (15th) and that Varys does not make the top performers (he is 23rd). The PageRank model does not capture the subtle ways in which they exert themselves in the shadows. On the other hand, each of those characters might be pleased that they are under PageRank’s radar, since that means that their nuanced methods are working quietly. Interestingly, Littlefinger and Varys do well in eigenvector centrality (8th and 15th, respectively). These two savvy characters know how to leverage these connections without raising their profiles.
Characters with high betweenness are located on many shortest paths connecting pairs of characters. In other words, they are essential because they bind the various narrative threads together. Daenerys rises to 6th place in betweenness (and Drogo to 7th) since they are the two hubs of the Dothraki community. Meanwhile the royal children of King’s Landing (Sansa, Joffrey and especially Arya) are penalized for being slightly off the main narrative line. Bran and Robb hold steady, since Bran’s mysterious fall drives the Catelyn/Tyrion arc, and Robb commands the North as the Lord of Winterfell. Meanwhile, Cersei underperforms with respect to betweenness, which reflects the indirect role that she pursues for the much of the book. Likewise, Littlefinger (43rd) and Varys (31st) skulk in the shadows.
And the Winner is…
Ned, of course! It’s not even close. He tops every category. Second place goes to larger-than-life King Robert, but we do note that he underperforms in PageRank compared to his many interactions. Third place goes to Tyrion, whose misadventures bridge the narrative beyond House Stark. Fourth place is a very close race between Catelyn and Jon. I award this prize to Catelyn: she wins three out of five categories (degree, eigenvector, PageRank), and her PageRank outpaces both her degree and weighted degree. This means that she is making good use of her network in this book! So our centrality ranking is:
- Ned, Robert, Tyrion, Catelyn, Jon
Honorable mentions go to Daenerys for her betweenness score, and to Sansa for her strong showing in eigenvector and PageRank centralities. Robb also deserves a nod for hanging in there, being consistently second tier.
Meanwhile, Cersei flies a bit low. But that is actually appropriate: she spends more time scheming (and publicly deferring to Robert) than she does taking direct action in this first installment.