The original article that Jie Shan and I wrote for Math Horizons focused on “A Storm of Swords,” which corresponds to Season 3. Believe it or not, Season 3 had just ended when we started using network analysis to explore Westeros and Essos. The Math Horizons article didn’t appear until more than two years had elapsed since we started. At any rate, Season 3 holds a special place in my heart because of this.
This is also the season where the narrative fully matures. Season 1 was about Ned. Season 2 was about adjusting to the absence of Ned. In Season 3, each narrative thread has independent momentum. The system has become complex enough that the tools of network science can really add new insight to our understanding of the narrative.
So let’s see what network science has to say about Season 3, the adaptation of “A Storm of Swords.”
The Season 3 network has 125 nodes (characters) and 505 weighted edges, corresponding to 5978 interactions. Character names are sized by PageRank centrality. Character nodes are sized by betweenness centrality. Edge thickness corresponds to edge weight.
Note that this TV network is a much more streamlined version of the “A Storm of Swords” network. (The book’s network has 303 nodes , 1008 weighted edges and 8,489 interactions.) For example, Jon’s world has been drastically simplified to a small band of Wildlings. His ties to the Night’s Watch have been de-emphasized to the point that Sam and the black brotherhood are in a different community. The TV show emphasizes Jon’s exile, and keeps his allegiance to Castle Black hidden from Mance’s wildlings. This is another example of where the absence of edges contains valuable information for the informed viewer.
The Season 3 network has seven communities. Clockwise from the top, they are:
- Beyond the Wall (Jon, Ygritte)
- The North (Sam, Bran)
- The Brotherhood without Banners and Dragonstone (Stannis, Davos, Arya, Ned)
- Across the Narrow Sea (Daenerys, Robert)
- The Kingslayer’s Trek (Jaime)
- The Stark Bannermen (Robb, Catelyn)
- Theon’s imprisonment (Theon, Ramsay)
The first item we have to address is the appearance of Ned and Robert in this network. These characters met their fates in the first season. Yet we see them playing prominent roles as connectors.
Both Ned and Robert are on the lips of people across the network. So how did they end up with these particular community affiliations? Ned appears in the Brotherhood without Banners community due to an extended discussion of the Lord of Winterfell between Arya, Beric and Thoros, as the latter two try to earn her trust. Likewise, Robert is the topic of a contentious discussion between Jorah and Barristan, as each one takes a measure of the other. (Side note: Robert plays a similar connecting role in the network for the third book. However, Ned’s position is more diminished in the novel: he can be found as a far less influential node in the Stark community.)
We have already commented on the humble size of Jon Snow’s community. This Wildling band is quite tightly knit. Expanding to the rest of the Far North, we find that the book’s storyline has been drastically simplified. The second northern community has Sam strongly bound to Gilly. In addition, Sam connects two very dense sub-communities: the Night’s Watch and Bran’s small alliance. Why are these communities fused together? Because of the season finale, where Sam, Bran and Jon Snow are brought together by chance. Sam and Bran connect face-to-face, while their potential encounter with Jon is a near miss.
We encounter another conglomerate community in Westeros. This time, the pivotal character is Gendry, the bastard son of Robert Baratheon. Oblivious to his lineage, Gendry finds himself bargained from Beric Dondarion to Melisandre. This major plot pivot is enough to bind the the Brotherhood without Banners and Dragonstone into a single community, even though they only spend one episode in contact with one another. Arya anchors the Brotherhood without Banners, while Dragonstone is dominated by the unstable relationship between Stannis, Melisandre, Davos and Gendry. (Note that Stannis remains distant from Gendry to insulate himself from facing up to the full truth of his illegitimate nephew’s existence.)
Daenerys is the hub of her Essos community, as she travels Astapor and brings the Unsullied and the Second Sons into her fold.
King’s Landing is abundant with complex dynamics, including many rich narrative triangles, including (but not limited to):
- Tywin, Tyrion and Cersei
- Tywin, Cersei and Joffrey
- Cersei, Joffrey and Margaery
- Sansa, Margaery and Olenna,
- Tyrion, Sansa and Shae
- Tyrion, Sansa and Petyr
The multiple combinations and reconfigurations of the same players shows the complexity of the strategies and schemes being pursued on top of one another. Save naive Sansa and cruel prince Joffrey, these games are being played by some savvy actors.
Jaime and Ramsey helm small communities, polarized by a single adversarial relationship: noble Brienne with wily Jaime, and brutal Ramsay with poor Theon.
Finally, the Stark community is organized around Robb and Catelyn. They form a durable quartet with Edmure and Bryden. These four work together to negotiate for forces and to develop a strategy to weaken House Lannister. Robb also has a strong axis with his new bride, Talisa.
Robb, Tywin and Tyrion hold the top three spots in the first four measures. The only exception is betweenness. Here, Robb takes the connectivity crown, followed by Ned and Robert (not shown). In general, the betweenness scores seem to skew toward House Stark, so perhaps this is a less reliable measure at this point in the narrative.
So let’s look at the second tier for the first four measures:
- Degree centrality: Catelyn, Joffrey, Cersei, Jon, Sansa
- Weighted degree centrality: Sansa, Cersei, Jaime, Jon, Daenerys
- Eigenvector centrality: Joffrey, Sansa, Cersei, Catelyn, Olenna
- PageRank centrality: Sam, Catelyn, Jon, Joffrey, Cersei
We consistently find Catelyn, Jon, Sansa, Joffrey and Cersei in these rankings. Then there are two surprises: Olenna’s eigenvector centrality and Sam’s PageRank centrality; more on those scores below. Finally, Jaime and Daenerys only appear in weighted degree centrality. They have lots of interactions, but do not exert much influence or have much effect on the main narrative. Oh well, there is always next season…
And the Winner is…
Robb handily takes the top spot for Season 3. He earns first place in three categories (Degree, PageRank, Betweenness). The King-in-the-North makes solid use of his connections as he amasses an army to make a strategic play against House Lannister. Robb is the primary force behind the main action, and he connects the network together. Such a combination makes the young wolf unbeatable. (If only that invulnerability held true in Westeros as well…)
Robb comes in second to Tywin in eigenvector centrality and to Tyrion in weighted degree centrality. More broadly, the Lord of Casterly Rock and the scion of Lannister take all-around second and third place, respectively. Lord Tywin plucks the role of Hand of the King from the Imp, and lands himself in second place in the process. His top eigenvector centrality reflects the fact that he is the true center of power in King’s Landing. (Poor King Joffrey only places 7th in this category, in spite of holding the Iron Throne.) The Imp has more interactions than anyone else, but his power has been diminished, demoted to Master of Coin, and wedding planner.
Ranking the next tier of characters is more subtle, as each of them exhibits different strengths and weaknesses. For example, Sam comes in 4th in PageRank, but his middling scores in the other categories (and a terrible 52nd place showing in eigenvector centrality) take him out of the running. In the end, Catelyn wins the battle for fourth place, with strong PageRank and Degree centralities balancing out her lower weighted degree score. Indeed, the sparsity of her interactions actually makes her PageRank, eigenvector and betweenness centrality scores all the more impressive. She makes excellent use of the few encounters that she has.
Jon Snow takes fifth place, with solid performances in all categories, save eigenvector centrality. But we are already well aware that eigenvector centrality will always punish Jon Snow for being confined to the Far North. Next, we encounter a close race between Joffrey and Cersei. But Joffrey has a slight edge, only losing to Cersei in weighted degree. His boosts in eigenvector and PageRank centrality may not be deserved, but that is just one of the perks of being the King.
Now, it’s time for the honorable mentions. The first one goes to Sam, for his incredible PageRank score (4th) compared to all his other statistics. What makes him so important? He benefits from being part of two tight-knit communities: the Night’s watch and Bran’s companions. And, of course, he also connects to Jon Snow. The end-of-season encounter with Bran gives Sam a tremendous boost. This chance meeting brings together two worlds, and PageRank rewards this handsomely. However, we should take this boost with a grain of salt, since we know that this collaboration is fleeting, more happenstance than plot development.
Davos Seaworth makes impressive use of his few connections, and earns every bit of it. In spite of having few connections (39th in degree centrality), and even fewer connections to important people (63rd in eigenvector centrality), Davos snags lucky 13th place in PageRank. As the moral compass of Dragonstone, the Onion Knight shoulders a heavy burden.
Let’s talk about House Tyrell, who know a golden opportunity when they see one. Olenna deserves a shout-out for her 8th place finish in eigenvector centrality. The Queen of Thorns has quickly made a bouquet from the dead roses of Margaery’s disappointing marriage to Renly Baratheon. She deftly arranges a double wedding that would place House Tyrell into an unbreakable alliance with House Lannister, and give House Tyrell a lock on the Iron Throne. Loras gets a similar boost, but we realize that he is really benefiting from the skills of his grandmother. Meanwhile, Margaery does not capitalize on her connections: her eigenvector centrality is on par with her weighted degree centrality, and her 42nd place finish in PageRank is abysmal: Loras matches that score with half of the effort. Perhaps Margaery should spend more time with Joffrey and less time with Sansa?
And then there are the under-performers. Littlefinger is perhaps the biggest disappointment, barely cracking the top 50 in many categories. The former Master of Coin finds himself stymied this season. Or is he purposefully working by proxy, using Ros as his agent to remain unnoticed? On the other hand, Varys maintains a low profile while keeping his powerful connections alive, and is rewarded with a strong eigenvector centrality in comparison with his other statistics. The spider seems to be spinning his web properly.
Finally, we turn to Kingslayer and khaleesi. Jaime’s humiliating arc drags him into irrelevance, scoring 14th in PageRank centrality in spite of holding 6th place in weighted degree centrality. Meanwhile, Daenerys’s eigenvector score is in the more remote than the Jade Sea, and her PageRank is half as effective as her number of interactions could support. The silver queen still has a long way to go before she can turn her eyes back across the Narrow Sea.