Large-scale structure of time evolving citation networks

I. INTRODUCTION

• A citation network represents documents as vertices and citations between documents as directed edges.The direction makes the relationship between the two vertices asymmetric.Source: §I · p. 1
• Vertices and edges are added over time and remain in place, so network structure changes mainly at the current leading edge.This permanence makes citation networks a simpler and cleaner laboratory for studying network growth than the web.Source: §I · p. 1
• Because documents cite only previously created documents, citation networks are typically acyclic, with edges pointing backward in time.Rare loops can occur when a paper cites forthcoming work, but are necessarily short.Source: §I · p. 1, Figure 1 · p. 1
• Citation networks include scholarly, patent, and legal opinions; the paper uses United States Supreme Court opinions as its example.Quantitative study of scholarly citations dates to the availability of citation databases, including Price’s work on degree distributions and their origins.Source: §I · pp. 1–2
• Citation networks’ time-dependent structure calls for methods beyond standard measures such as degree distributions.The paper proposes techniques intended to reveal structure both individually and collectively.Source: §I · p. 2

• The analysis asks whether documents follow distinct citation eras or recurring timing patterns, despite noise in citation data.Source: §II · p. 2
• The example is Supreme Court opinions from 1789 onward, used illustratively while also yielding results the authors consider independently interesting.Source: §II · p. 2
• The method assigns cases to citation-profile groups while iteratively adjusting the profiles, using the expectation–maximization algorithm.Source: §II · p. 2
• Each group has a time profile for when its documents receive citations and a probability representing its expected share of documents.The model fits these parameters to observed citation patterns by maximizing their likelihood, assuming citations are random and uncorrelated draws from the group's time profile.Source: §II · p. 3
• Because group memberships are unknown, the method estimates membership probabilities and uses them to update the model parameters through numerical iteration.The authors report that, under reasonable conditions, iteration converges to a self-consistent solution.Source: §II · pp. 3–4

A. Example

• With c = 2, one group contains practically all cases before 1937 and the other practically all cases after.The breakpoint coincides with a significant constitutional crisis for the Supreme Court.Source: Figure 2 · p. 4, A. Example · p. 4
• The two groups have distinct citation profiles whose time periods closely match the group-membership periods.The opinions citing cases in each group were handed down during roughly the same eras as those cited cases; the division need not reflect legal ideology.Source: A. Example · pp. 4–5, Figure 3 · p. 5
• With c = 4, the analysis finds clear opinion-date groups, including one spanning 1897–1937, approximately the Lochner era.Larger c values fit better but risk overfitting, so the authors compare results for small c values rather than selecting an optimum.Source: A. Example · p. 5, Figure 4 · p. 5
• The c = 4 groups based on citations made are remarkably similar to those based on citations received.The authors suggest this could indicate agreement between opinion writers and later citers about opinions’ positions in the literature.Source: A. Example · p. 5, Figure 5 · p. 5

III. CLUSTERING IN CITATION NETWORKS

• Unlike EM’s grouping by similar citation-time profiles, community detection groups vertices linked by more internal edges, using modularity maximization to identify divisions.The modularity compares within-group edges with the number expected under random edge placement that respects vertex degree.Source: §III · p. 6
• The method repeatedly subdivides the network to reveal major divisions before minor ones, but applies its eigenvector calculation to an undirected network by ignoring citation direction.Repeated division can stop when no further division increases modularity.Source: §III · p. 6
• The two-way modularity split shows a significant break around 1937, broadly echoing EM, but differs in group sizes and has an early tail.The authors suggest that using both citations received and citations made may help explain these differences.Source: §III · p. 6, Figure 6 · p. 6
• Clustering classifies some cases that EM cannot because they have never received citations; about 10% of cases were never cited.The authors suggest citations made may connect these cases to earlier cases and help place them in groups.Source: §III · p. 7
• The four-way modularity split resembles EM’s overall pattern less cleanly and adds a division around 1900, possibly marking the start of the Lochner era.Figure 7 shows the groups’ decision counts over time.Source: §III · p. 7, Figure 7 · p. 7

IV. VERTEX AUTHORITY SCORE AND TIME EVOLUTION

• Ordinary eigenvector centrality based on the adjacency matrix assigns zero centrality to every vertex in an acyclic citation network.The paper therefore discusses Kleinberg’s variant, which defines separate authority and hub scores.Source: §IV · p. 7
• In the authority-and-hub method, a case’s authority score depends on the hub scores of cases citing it, while its hub score depends on authorities it cites.The authority and hub scores are leading eigenvectors of the cocitation and bibliographic coupling matrices, respectively.Source: §IV · p. 7
• The average age of top authorities tends to rise, then drops sharply around 1900, 1940, and 1970, indicating that younger cases swiftly take over; the 1970 dip may partly reflect more annual decisions.Figure 8 plots average age over time.Source: §IV · p. 8, Figure 8 · p. 8
• Maturation followed by swift renewal is clearest in the age of the top ten vertices, but a similar pattern remains visible for the top 50, 100, and 500 authorities.Source: §IV · p. 8

V. DISCUSSION

• With c = 4, the EM analysis identifies a group whose start and end dates correspond closely to the accepted dates of the Lochner era.The era is described as running from 1897 to 1937.Source: §V · p. 8
• The discussion places the break in the context of the New Deal conflict and the court’s reversal toward validating state and federal measures beginning in March 1937 after Roosevelt’s court-packing proposal.The passage notes scholarly disagreement over whether the shift reflected political pressure or a pre-existing judicial majority.Source: §V · pp. 8–9
• The EM break around 1937 aligns with the Lochner era’s end; because the analysis uses citations received, the authors infer a shift in later-cited arguments, not just case impact.They qualify this interpretation as presumptive, based on later cases citing post-1937 opinions rather than earlier ones.Source: §V · p. 9

VI. CONCLUSIONS

• The conclusion describes several methods for analyzing citation networks, characterizing those networks as acyclic and directed.The supplied conclusion passage ends after this description.Source: §VI · p. 9