Volumes 1-5: A Song of Ice and Fire

got-tyrion-jon

[If you are new to Network Science, be sure to read A Primer on Network Analysis before continuing. If you want to know how we created the network, then check out From Book to Network.]

Let’s collect together all five books into one network. What do we learn about the saga as a whole? How do the communities and centralities for this massive network compare to the networks for individual books? Let’s find out.

The Network

The “A Song of Ice and Fire” network for the first five books has 796 nodes (characters) and 2823 weighted edges (relationships), accounting for a staggering 32,629 interactions. We limit ourselves to labeling the top 90 characters, but this is more than enough to get a sense of the network structure. In this graphic, character names are sized by PageRank centrality and nodes are sized by betweenness centrality. The thickness of an edge corresponds to its weight.

got-all-network-v2

Communities

Ordered by size, here are the 10 main communities. We list the percentage of the network contained in each community, along with their most prominent characters according to PageRank:

  • King’s Landing (25.8%): Tyrion, Jaime, Cersei, Sansa, Joffrey, Tywin
  • The Far North (14.3%): Jon Snow, Samwell, Mance
  • The Starks (12.1%): Catelyn, Robb, Bran
  • Essos (10.9%): Daenerys, Barristan, Jorah
  • The Ironborn (10.4%): Theon, Victarion, Asha, Ramsay
  • The Baratheon Host (8.3%): Stannis, Davos, Renly
  • The Game: (7.2%): Eddard, Robert, Petyr
  • Arya’s Journey (6.3%): Arya, Sandor, Gregor
  • Dorne (2.8%): Arianne Martell
  • Oldtown (1.4%): Pate

Centralities

GoT-all-centralities

It is finally time to answer the question: who is the most important character in “A Song of Ice and Fire?” There are still two volumes to go, but we can answer this question for the saga to date, and try to make some predictions given the trends that we see. Let’s deal with the centrality measures, one at a time.

Degree Centrality

First, we consider degree centrality, which counts the number of distinct connections to other characters. The top six characters are:

  • Tyrion: 122 connections
  • Jon: 114 connections
  • Jaime: 101 connections
  • Cersei: 97 connections
  • Stannis: 89 connections
  • Arya: 84 connections

So Tyrion wins this measure, with some room to spare.

Next, there is a cluster of characters with about 75 connections each: Catelyn, Sansa, Ned, Robb and Daenerys. The rest of the top 20 characters contain most of the other point-of-view characters, as well as the King’s Landing royalty: Joffrey, Theon, Robert, Bran, Tywin, Petyr, Brienne, Samwell, and Barristan Selmy.

With so many point-of-view characters, the Starks and Lannisters dominate this centrality measure. The well-travelled Daenerys only ranks 11th in this category, showing that she remains relatively isolated in spite of her expansive journeys through Essos.

Weighted Degree Centrality

Weighted degree corresponds to the number of interactions with other characters. The top finishers are

  • Tyrion: 2873 interactions
  • Jon: 2757 interactions
  • Cersei: 2232 interactions
  • Joffrey: 1762 interaction
  • Ned: 1649 interactions
  • Daenerys: 1608 interactions

The battle for the top spot is a bit closer. Tyrion earns this crown as well, though less convincingly. Daenerys comes in 6th place, behind both Joffrey and Ned. This reflects how separate her storyline is from the other characters: she is rarely mentioned in Westros, so most of her interactions are earned in the fraction of chapters set in Essos.

Our next cluster of characters is Jaime, Sansa, Bran, Robert, Arya, Robb, Stannis and Catelyn. Once again, we see the depth of the Stark and Lannister houses in the narrative focus.

Degree Centrality versus Weighted Degree Centrality

Let’s use degree centrality (connections) as a benchmark for importance. Among these top performers, which characters have more more interactions per connection? Which characters have fewer?

Compared to degree centrality, Jaime, Arya and Stannis underperform. Jaime has a lower profile than the scheming Cersei, which may explain why he does not achieve the same level as his sister. Meanwhile, Stannis’s interactions are probably suppressed because he is not a point-of-view character. As for Arya, she commands intermittent attention in Westeros, and follows a sparer plotline in Braavos.

 The overperformers include Joffrey, Bran, Ned and Robert. King Joffrey was on everyone’s lips in the second and third books. Bran has relatively fewer connections, so he ends up with a higher ratio of interactions per connection. Finally, Ned and Robert’s strong axis in the first book is self-reinforcing, and King Robert continues to be referenced in both Westeros and Essos, long after his untimely demise.

PageRank Centrality

 As you may know by now, I think that this is the most important centrality metric. The PageRank feedback loop (which spreads importance pro rated by strength of connection) does a great job at quantifying narrative tension. Scaling PageRank by the (top-scoring) bastard of Winterfell’s score, we have the following ranking:

  • Jon: 1.00
  • Tyrion: 0.97
  • Jaime: 0.82
  • Stannis: 0.72
  • Arya: 0.70
  • Cersei: 0.70
  • Daenerys: 0.68
  • Theon: 0.63
  • Eddard: 0.61
  • Catelyn: 0.59
  • Robb: 0.58
  • Sansa: 0.53

Jon edges out Tyrion by the slimmest of margins: his PageRank is 3% larger than Tyrion’s. This is such a close race, so I am tempted to declare a tie. However, looking across all five books, we see that Jon has the momentum in the category. His PageRank has slowly been building, while Tyrion’s has started to decline. We expect this trend to continue as Jon pursues his destiny in Westeros, so I award a full victory to Jon in this category.

The Kingslayer’s position is quite interesting: he distinctively earns third place, beating out the cluster of Stannis, Arya, Cersei and Daenerys. I would have expected this race to be closer. What is going on here? Taking all five books as a whole, Jaime’s connecting position  works to his advantage. Stannis is penalized for not being a point-of-view character. Arya and Daenerys are held back by their isolation from the War of the Five Kings, while Cersei’s relative confinement to King’s Landing handicaps her in this race. Like Jon Snow, Jaime’s PageRank has ascended over the second half of the books, and we expect this continue.

The righteous Stannis’s quest for the Iron Throne is an enduring counter-balance to the Stark-Lannister rivalry. It is remarkable to see Arya perform so well: “A Clash of Kings” marked her highest PageRank (3rd place), while she hovered near 10th place in the other books. She and Sansa have comparable degrees, yet Arya clearly outperforms her older sister. Both have strong and frequent connections to other important characters, so it seems that Arya’s unique community gives her the narrative boost that she needs to best her sibling. The two queens, provide a nice counterpoint to one another, with Cersei scrambling to hold onto her grip on the Iron Throne while Daenerys slowly develops the power and influence to challenge her for it.

 Theon holds eighth place after an uneven journey. It is not clear whether he will maintain this high rank, or give way to the ascendant arcs of Victarion and Asha. Next comes the triad of fallen Starks: Ned, Catelyn and Robb. Sansa follows at 12th place: as discussed above, her position seems diluted in this universal network compared to her measure in the individual books.

Eigenvector Centrality

This measure captures connections to powerful people, but not whether a character is currently taking advantage of those connections. People with high eigenvector centrality but low PageRank centrality have the potential to increase their importance in rapid fashion. So who has connections to powerful people? The eigenvector rankings are:

  • Tyrion: 1.00
  • Cersei: 0.91
  • Jaime: 0.88
  • Joffrey: 0.83
  • Sansa: 0.80
  • Robert: 0.77
  • Ned: 0.76
  • Stannis: 0.743

For eigenvector centrality, it is good to be the king, or at least, in the king’s inner circle. The Lannisters take the top spots: Tyrion wins by a wide margin, followed by the twins Cersei and Jamie, and then Joffrey. Sansa is the top-ranked Stark, thanks to her ties to the Lannisters. Thereafter, we have Robert, Eddard and Stannis. Eigenvector centrality focuses on who holds the iron throne.

We don’t encounter Jon Snow until 12th place, while the far-flung Daenerys comes in 36th place. We expect their eigenvector centralities to rise as their journeys take them closer to the main conflict for the throne.

Ultimately, Tyrion is the big winner here. He has close ties to both Lannisters and Starks. This flexibility launches him ahead of his twin siblings. In the battle for “most important character,” Jon Snow falls far behind the Imp at this point.

 

Betweenness Centrality

 Characters with high betweenness centrality help to connect different parts of the network. These characters provide the narrative glue that binds the story together. They provide efficient pathways for communication and exchange.  Jon Snow has the highest betweenness centrality. Here are the rankings for the top performers:

  • Jon Snow: 1.00
  • Tyrion: 0.84
  • Daenerys: 0.61
  • Theon: 0.58
  • Stannis: 0.57
  • Jaime: 0.52
  • Cersei: 0.46
  • Arya: 0.45
  • Ned: 0.41
  • Robert: 0.41

After five novels, Jon Snow has consistently scored well in betweenness: his lowest placing was 4th in “A Game of Thrones.” His connections to the Far North, the Night’s Watch, and Houses Stark and Baratheon keep him at the top of the heap. The well-traveled Tyrion comes in second place. He is crucial for connecting the Starks and the Lannisters. Daenerys consistently gains momentum in this category as her influence in Essos grows, and as additional denizens of Westros seek her out. Interestingly, Theon and Stannis do nearly as well in betweenness. Theon’s high ranking speaks to the ascendance of the Ironborn narrative, as well as his connections to House Stark. As for Stannis, with his longstanding claim to the Iron Throne, and his connections to Jon Snow, he forges a triangle between communities Baratheon, Lannister and Stark.

And the Winner is…

So who is the most important character in the first five volumes of “A Song of Ice and Fire?” It is a close race between Tyrion and Jon Snow

  • Degree Centrality (number of connections): Tyrion wins this handily with 122 connections compared to Jon’s 114 connections. Normalizing the score, Tyrion earns 1.0 while Jon earns 0.93.
  • Weighted Degree Centrality (number of interaction): Again, Tyrion wins with room to spare, with 2873 interactions, while Jon has 2757 interactions. The normalized score: Tyrion gets 1.0 while Jon gets 0.96.
  • PageRank Centrality (narrative importance): Jon takes the crown in this category. Tyrion earns 0.96 while Jon earns 1.
  • Eigenvector Centrality (connected to important people): Tyrion dominates this category, earning 1.0 while Jon earns 0.65
  • Betweenness Centrality (influential as a broker of information): Jon dominates this category. Tyrion earns 0.84 to Jon’s 1.0

In summary, Tyrion wins the three most straight-forward categories, while Jon wins the two subtler measures. PageRank is the best measure for narrative dominance: important things happen when influential people interact. In my opinion, Jon deserves a bonus for winning this category. Finally, betweenness centrality shows how important a character is for unifying the story as a whole. Again, Jon wins this important measure. Meanwhile, Tyrion trounces Jon in eigenvector centrality, meaning that he is connected to far more influential people. It is a tough call!

If the books ended here, then I would have to declare Tyrion the victor: he only loses PageRank by a hair’s breadth. However, there are more volumes to come, and Jon clearly has the momentum. Here is the progression of PageRank values by volume:

pagerank

We see that:

  • Jon has essentially held steady throughout the saga.
  • Tyrion, Arya and Cersei have trended downwards.
  • Jaime, Stannis and Daenerys have trended upwards

Stannis has the largest net increase in PageRank, yet he does not have the narrative momentum. Jaime’s strong showing (and inverse trajectory of his twin sister) is quite compelling. Yet one can’t help but feel that Tyrion is destined for a rebound, while Jaime will peak before the finale. Meanwhile,  Daenerys will surely continue her ascent, and Jon to continue to rise, though not at the same rapid pace. The question seems to be: will Jon’s methodical increase be enough to hold off Daenerys?

Depending on how Daenerys wields her might and influence (and dragons) when she arrives in Westeros, she could take second place, or even the top spot. Unless Tyrion finds himself on the Iron Throne, he seems destined to lose the network centrality game to Jon and Daenerys. In the same vein, Jaime remains a more distant dark horse, but the Kingslayer could surprise us– Ned did find him sitting on Aerys’s throne once before.

Finally, it is clear that Stannis deserves the next spot, especially since he is not a point-of-view character. So here is the ranking, as I see it, after the first five books

  1. Jon Snow
  2. Tyrion
  3. Jaime
  4. Cersei
  5. Daenerys
  6. Stannis

Here are the centrality measures for these six characters:

GoT-all-centralities-top

As for honorable mentions, Arya performs incredibly well when the books are taken as a whole, though her performance from book to book has been uneven. Clearly, her journey has been an enduring adventure, so she seems poised to return to the main narrative with great effect.

Theon performs exceptionally well in PageRank, especially compared to the number of his connections and interactions. Meanwhile, Sansa underperforms in this category, finishing behind the departed Ned, Catelyn and Robb, in spite of her high degree and weighted degree. So it seems that the momentum is not with the Key to the North at this point in the narrative.

Advertisements