The following study was originally posted by me on the Fire Emblem: Three Houses analysis page on TV Tropes.
In this study, we analyze the complex meshwork of developer-defined relationships between recruitable characters found in the 2019 strategy role-playing game Fire Emblem: Three Houses by Intelligent Systems for the Nintendo Switch.
Tools and data
For the following study, we have used the yEd free graph editor. All files discussed below can be downloaded here.
We define the nodes/vertices of our social network as the set of all named characters of Fire Emblem: Three Houses, including the post-release ones (Jeritza, "Cindered Shadows "DLC), who have support conversations ("supports") with other characters. Excepted from this definition are the main player character Byleth, because they have supports with every other character and would therefore obscure the underlying structures of the relationship network; and Sothis and Rhea, who only have supports with Byleth and would therefore be isolated by their removal.
We define the connections/edges as the set of all supports between characters, weighted/labeled by the total number of support conversations between any pair of characters. We furthermore normalize the edge weights to the interval [0,1] where the weight of 0, by convention, stands for "no connection/edge", so that the support path C›B maps to the weight 0.33; C›B›B+ and C›B›A, to 0.67; and C›B›B+›A and C›B›A›A+, to 1.00. Furthermore, because yEd cannot process undirected graphs, each support is modeled as a pair of directed edges with the same weight/label connecting character X to character Y and vice versa.
The resulting raw data can be found in fe3h_supports_raw.tgf in the downloadable archive.
Analysis
For node sizes, we used yEd's centrality algorithm, specifically Tools > Centrality Measures > Page Rank (with the "Directed" and "Get From Edge Labels" options checked), which scales the nodes according to how "important" they are in the overall network. For node colors, we used a manual coloring according to their respective House/faction's preferred color scheme. For layout, we used yEd's Organic layout algorithm with the option for Clustering set to Louvain Modularity and the "Handle Cluster as Substructure" option checked.
We observe that Jeritza's node appears to confuse the clustering algorithm, having an outsized impact on the layout due to his out-group A-supports despite his overall peripheral position in the network. We have therefore opted to exclude him from further analysis; for the interested reader, the file fe3h_supports_jeritza.graphml contains the untrimmed network. We have also opted not to use yEd's built-in node grouping function to manually cluster members of the same House/faction, because not doing so produced more interesting results. For the interested reader, however, the grouped layout can be found in fe3h_supports_groups.graphml and fe3h_supports_groups_jeritza.graphml.
The final results are thus discussed based on the contents of fe3h_supports.graphml.
Results and discussion
Predictably, members of the same House have been clustered together even when not explicitly grouped, thanks to being fully connected within the respective in-group. The only exception is the Church of Seiros faction, which is not fully connected internally (e.g. Seteth has no supports with Alois). The House leaders (Edelgard, Dimitri, Claude, Yuri) all ended up on the periphery of their respective clusters and of the network as a whole, presumably because they have much fewer out-group supports than most of their rank-and-file House members. Their right hands (Hubert, Dedue) are likewise on the periphery, but have much lower Page Ranks, presumably because, unlike their superiors, they do not have any out-group supports at all (except with the Church).
The Church is the only faction (it is technically not even a House) whose nodes are scattered all over the network, indicating that they generally have more and stronger out-group ties than in-group ones. Shamir and Cyril, in particular, are so strongly associated with the Golden Deer house, they might as well be part of that particular cluster (which makes narrative sense, as their "foreign commoner" identities would pull them towards Claude's House over all others). The remaining Church members cluster around the Blue Lions, reflecting the Church's in-story association with the Kingdom of Faerghus; Gilbert, in particular, may just as well be a Lion (and, indeed, he is only recruitable on the Azure Moon route). Likewise, Alois and Catherine's strong association with the Lions makes sense given their status as Church Knights, whereas Hanneman and Manuela, being of Adrestian descent, gravitate more towards the Black Eagles. Finally, Seteth and particularly Flayn cluster with the Lions but lean towards the Deer — or rather, away from the Eagles, which makes sense when Edelgard's intentions for Children of the Goddess are considered.
Among the Black Eagles, Caspar is a surprising outlier, as he consistently ends up closer to the Lions than to his own House due to his A-supports with Ashe and Annette and an A+-support with Catherine (who, as noted above, clusters with the Lions). This somewhat reflects his brash, combative personality that rarely gels well with the rest of his House.
The DLC-only Ashen Wolves generally tend to cluster close to, but separately from the Eagles, largely because two out of four (Yuri and Constance) have two A-supports with that House each. Only one member (Balthus) is associated with the Deer, but because all of his out-group supports are with that House, he ends up as more-or-less as its honorary member, much like Shamir and Cyril.