Administration Statistics

This document describes the dynamics of multistate litigation by the presidential administrations of Obama and Biden. We first compare the number of attorney generals and cases filed over the period. We see a gradual increase in the number of AGs and cases filed. Then, we compare the titles of attorneys across administrations. We find only small differences in the numbers of distinct titles used.

Next, we compare network statistics between networks, such as the centrality, density, and assortativity by degree and AG political affiliation.

Finally, we construct a multilayer network of the state graphs. This graph combines all the connections between states across administrations where the layers represent the various administrations and the edges in the layers connect states who participate together on multistate litigation. Using a multiple dimensional network is a significant contribution to the field of network methodology in legal academia.

Descriptive Docket Statistics

Figure 1 shows the number of AGs that participate on a case by filing date. We see some gradual increase over the period with greater variation in the Biden administration.

Figure 2: Number of Cases by Filing Year and Administration

Figure 2 categorizes the number of cases filed in each year by administration. For the Obama administration, we see both an increase through 2015 followed by an decrease through 2018. The Biden administration starts off at a peak of 33 cases in a 2021 and continues to decrease.

Table 1: Number of Distinct Titles By Administration
Administration	Distinct Titles	Distinct Titles (cleaned)
Obama	63	58
Biden	69	55
^a Title (cleaned) refers to titles that have been harmonized referring to the same position and removing phrases such as Of Legal Policy, For Legal Strategy, To the Attorney General, Special Litigation Unit, Civil Division, etc. For a full list, see the Appendix ...

Table 1 describes the distinct number of titles for attorneys participating in multistate litigation. It appears that the number has increased from 63 to 69 in the Biden administration. However, after cleaning, we see that the number of unique titles was less than in the Obama administration. However, this may be due to the Biden administration having a fewer number of cases than the Obama administration and has not yet finished its course.

Network Statistics

Table 2: Network Statistics by Graph and Administration
	Obama	Biden
States Network
Eigen Centrality	0.38	0.53
Density	0.52	0.47
Degree Assortativity	0.20	1.00
Political Assortativity	0.15	0.61
Attorney Network
Eigen Centrality	0.71	0.73
Density	0.13	0.16
Degree Assortativity	0.26	0.93
Political Assortativity	0.05	-0.08

Table 2 shows the network statistics computed at the network level for the State and Attorney networks in each administration.

A centralization score is a graph level centrality measure based on the node-level centrality scores to show how central a graph’s most central node is in relation to how central all other nodes are.¹ It is computed as \(C(G) = \sum_{v} (\max_{w}c_{w} - c_{v}),\) where \(c_{v}\) is the centrality of vertex \(v\). The most centralized structure in eigenvector centrality is the graph with a single edge and potentially many isolates. Values closer to one represent greater centralization or that information flows disproportionaly through one or more members of the organization rather than being equally distributed.²

We find that the Biden State and Attorney networks are more centralized than the Obama networks and that both State networks are less centralized than the Attorney networks.

The graph density is the ratio of the actual number of edges present in a graph and the maximum number of edges that a graph may contain. Values of density range from 0 to 1.³ A highly connected network has a greater density.

We find that the States Network is much more dense than the Attorney Network in both administrations. There are fewer states than attorneys involved, so it’s probable that more of them would be connected to each other than the large number of attorneys and judges connected to each other. The Obama State Network is more dense than the Biden State Network while the Biden Attorney Network is more dense than the Obama Attorney Network.

The assortativity coefficient measures the homophyly of the graph based on labeling or values of the verticies. High assortativity means that connected verticies are labeled with similarly assigned values. Nominal assortativity is defined as \(r = \frac{\sum_{i} e_{ij} - \sum_{i} a_{i}b_{i}}{1 - \sum_{i}a_{i}b_{i}},\) where \(e_{ij}\) is the fraction of edges connecting verticies of type \(i\) and \(j\), \(a_{i} = \sum_{j} e_{ij}\), and \(b_{j} = \sum_{i} e_{ij}\).⁴ Assortativity values range from -1, disassortative in which nodes mix of opposite types, to 1, assortative where nodes connect with the same type. We calculate assortativity by degree and by political association.

We find that the Biden State network is completely assortative by degree– nodes of high degree centrality connect with other connected nodes. The Biden Attorney network is also quite assortative in degree. While still assortative, the Obama networks are much lower than the Biden networks in degree assortativity. Both State networks are assortative by political association, meaning that Republican States are likely to connect with other Republican states and Democrat states are likely to connect with other Democrat states. In contrast, the Biden Attorney network is slightly politically disassortative and the Obama Attorney network is marginally politically assortative.

Multidimensional Network Graphs

We next describe the multidimensional structure of these networks. Multidimensional networks are increasingly common to explain real-world phenomenon.⁵ Multiplex and multilayer networks have unique structural properties.⁶ The multilayer structure allows us to see how states connect through different administrations. At the same time, we can also separate the layers to view the network of each administration separately. We consider each administration as layer of the network, where nodes of the layer are connected to each other via edges in multistate litigation. Using a multilayer network is a great contribution to the legal academic literature, because most papers have focused on employing a single network, but this paper employs a temporal dimension across networks.

Figure 3: Multidimensional Plot of Networks across Administrations

Figure 3 shows the multidimensional network with edges labeled according to Presidential administration. We observe that the Biden network is fairly separated and differs in states connected than in the Obama administration.

Figure 4 breaks down the multidimensional plot of the network structure into the two separate networks for the Obama and Biden administrations while still using the same layout on each graph.

Figure 5: Multilayer Network Graph of Administrations (Edges) and AG Political Party Affiliation (Verticies)

Figure 5 shows another format of the multidimensional network of states across presidential administrations.

Figure 6 separates the multilayer network into its two distinctive administrations, Obama and Biden.

Figure 7: Interactive State Network Plot of Multiplex Network Graph of Administrations

Figure 7 shows an interactive version of the multilayer graph.

Footnotes

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1. Ethan S. Bernstein, Jesse C. Shore & Alice J. Jang, Network Centralization and Collective Adaptability to a Shifting Environment, 34 ORGANIZATION SCIENCE 2064 (2023), https://pubsonline.informs.org/doi/10.1287/orsc.2022.1584 (last visited Feb 8, 2024).↩︎
Subham Datta, Graph Density | Baeldung on Computer Science, (2021), https://www.baeldung.com/cs/graph-density (last visited Feb 8, 2024).↩︎
M. E. J. Newman, Assortative Mixing in Networks, 89 PHYS. REV. LETT. 208701 (2002), http://arxiv.org/abs/cond-mat/0205405 (last visited Feb 8, 2024); M. E. J. Newman, Mixing Patterns in Networks, 67 PHYS. REV. E 026126 (2003), http://arxiv.org/abs/cond-mat/0209450 (last visited Feb 8, 2024).↩︎
Federico Battiston, Vincenzo Nicosia & Vito Latora, The New Challenges of Multiplex Networks: Measures and Models, 226 EUR. PHYS. J. SPEC. TOP. 401 (2017), https://doi.org/10.1140/epjst/e2016-60274-8 (last visited Feb 9, 2024); Amy C. Kinsley et al., Multilayer and Multiplex Networks: An Introduction to Their Use in Veterinary Epidemiology, 7 FRONT VET SCI 596 (2020), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7500177/ (last visited Feb 9, 2024); Manlio De Domenico, More Is Different in Real-World Multilayer Networks, 19 Nat. Phys. 1247 (2023), https://www.nature.com/articles/s41567-023-02132-1 (last visited Feb 9, 2024).↩︎
Federico Battiston, Vincenzo Nicosia & Vito Latora, Structural Measures for Multiplex Networks, 89 PHYS. REV. E 032804 (2014), https://link.aps.org/doi/10.1103/PhysRevE.89.032804 (last visited Feb 9, 2024); S. Boccaletti et al., The Structure and Dynamics of Multilayer Networks, 544 PHYS REP 1 (2014), https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7332224/ (last visited Feb 9, 2024).↩︎