Network structure reveals patterns of legal complexity in human society: The case of the Constitutional legal network

Complexity in nature has been broadly found not only in physical and biological systems but also in social and economic systems. Although many studies have examined complex systems and helped us understand real-world complexity, the investigation to the legal complexity has not been thoroughly investigated. Here we introduce a novel approach to studying complex legal systems using complex network approaches. On the basis of the bipartite relations among Constitution articles and Court decisions, we built a complex legal network and found the system shows the heterogeneous structure as generally observed in many complex social systems. By treating legal networks as unique political regimes, we examine whether structural properties of the systems have been influenced as the society changes, or not. On one hand, there is a core structure in all legal networks regardless of any social circumstances. On the other hand, with relative comparison among different regimes’ networks, we could identify characteristic structural properties that reveal their identity. Our analysis would contribute to provide a better understanding of legal complexity and practical guidelines for use in various legal and social applications.


S1 Korean Legislative System
The Korean legislative system consists of the Constitution, Acts and administrative legislations such as Presidential Decree [1]. There is hierarchy in the legislative system. The Constitution is the paramount law. Acts are subordinate to the Constitution. Acts should not be in conflicts with the Constitution. Administrative legislations are subordinate to Acts. The Constitution is amended by national referendum. Acts are determined by voting among members of the National Assembly. Administrative legislations are set by the government [1].

S1.1 Constitution of Korea
Since the Korean government established in 1948, the Constitution has been revised 9 times, and the latest version was amended by Oct. 29, 1987 [2]. The Constitution stipulates fundamental issues such as rights and obligations of people, structure of government, economic system, election management and so on. The Constitution consists of following 12 categories [2].
Duties of members of National Assembly, Session of National Assembly, Speaker of National Assembly, Majority rule of National Assembly, Open to public of National Assembly, Not deliberating the same bill, Introduction of bills, Process of bills, Budget bills, Continuing disbursement, Revised budget bill, Increase of expenditure in the budget, National bonds, Type of taxes, Powers of National Assembly to treaties, Inspection of National Assembly, Attending in National Assembly, Removal of State Council members, Internal rules of National Assembly, Impeachment of National Assembly.

S3 Construction of the CLN
We start our research with collecting all historical Constitutional decisions data for 27 years. In civil society, each individual or organization request for adjudication of certain legal activities ( Figure A(A)). When a certain legal activitie were identified as unconstitutional, the Constitutional court make a decision by citing some Constitution articles which provide a basis for the decision. Since only the unconstitutional cases quote articles, we choose 1057 decisions and produce the bipartitle network through mapping the relations among decisions and articles ( Figure A(B)). The bipartite network consists of two sets of nodes; articles and decisions. We can generate two directions of projections for each set as well known examples of previous researches like collaboration relations among actors [3] and scientists [4], human disease [5], organization [6]. The projection of the bipartite relations onto the articles shapes the Constitutional legal network (CLN) of 132 nodes which has undirected and weighted network structure ( Figure A(C)). The weight of each link corresponds the number of co-cited decisions by same decisions. The full CLN network has 132 nodes and 496 edges. Since only 71 nodes have non-zero degrees in the CLN, the effective mean degree of the network becomes 13.9 so the network becomes so dense ( Figure B). Therefore, for the better visualization, we perform the backbone extraction algorithm [7] in main manuscript. With the simple visualization, we could identify the modular structure and core articles which stands central position of the CLN. The backbone extraction algorithm, however, only used for better visualization and all the network analysis of our research was performed on the full network ( Figure B).  Table S2: Basic network properties of each regime.
In Section 2, we describe characteristics of Korean historical regimes since 1989. As stated in main manuscript, the Constitution articles of the Republic of Korea has not changed during the last 30 years but the society has experienced dramatic changes. Therefore, we construct CLN for each political regime and compare their structural properties.
In Table S2, we summarize the basic network properties of each regime's CLN. For all CLNs, only about 30% of nodes have non-zero degrees and networks are so dense in the sense of high mean degrees k and strenghts s . Furthermore, all the CLNs are highly clustered ( C ) and the considerable rate of linkweights (≈ 70%) are concentrated into the articles of "Rights and Duties of Citizens" category. These observations imply that the strengths of CLN networks are highly concentrated to smalll number of articles rather than equally distributed to all ones.

S4 Strength distributions of each regime's CLN
We observe the cumulative strength distribution P c (s) of each regime's CLN and compare the distribution with the null model counterparts ( Figure C). As we found in main manuscript, the strength distributions of the CLN of each regime are all highly skewed. This implies that there exist a number of highly-cited hub articles in all regimes.

S5 Scaling relations between degrees and strengths in each regime
For all regimes, the scaling relations between degrees and strengths are nonlinear (β > 1.0) ( Figure D). This shows that the macroscopic structural properties of CLN are robust regardless of the social (political) environments changes.

S6 Strengths of articles in each category for the different regime's CLN
In main manuscript, we observe the statistical descriptions of different categories in total CLN. Here we perform the same analysis for each different regime ( Figure E). As found in the main article, the strengths of the "Rights and Duties of Citizens" articles were much larger than the other categories. Furthermore, the distribution of the categories except for the "Rights and Duties of Citizens" would show the regime's specific characteristics. In regime 1, the categories "Preface" and "Election Management" have a relative high strengths comparing with other regimes. The relatively high strengths of "Economy" in regime 2 and "Courts" in regime 3 would also reflect the high interest about those issues at that time.

S7 Share of link weights between categories of each regime's CLN
In Figure 2E of the main menuscript, we present the chord diagram which shows the net flow between categories of the total CLN. Likewise, we aggregate all articles into their categories and observe the share of linkweights between categories for different regimes' CLNs. This can be interpreted as the blockmodel representation [8]. We visualize these information onto the heatmap ( Figure F) and found the dominant concentration of linkweights to "Rights and Duties of Citizens" for all regimes.
S8 Prevalence measures of each regime

S8.1 Nodes' prevalance
A high prevalence of an article in a given regime r means that the article was more predominant than in any other regimes. In the main menuscript, we present this information as the word clouds for better visualizations. In this section, we provide the full information as scatter plots of each regime ( Figure G). For all figures, top 3 articles of high prevalence in each regime are presented and these would coincide with the articles with big font sizes in Figure 4 of the main menuscript.

S8.2 Links' prevalence
As nodes' prevalence, we can also obtain each link's prevalence of each regime. In the main menuscript, we show two adjacent regimes (regimes 4 and 5) with markedly different structures within their networks. In this section, we present all the core structure of each regime's network identified by link prevalence ( Figure H).
S9 Time-dependent analysis of the evolution of the network

S9.1 Giant component evolution
The giant component analysis has been a representative approach to study the evolution of the network [9,10]. Recently, the approach has been found useful to analyze the complex legal system [11]. We observe the giant component size as the event time goes on ( Figure I). The event time is defined by one singular legal decision so it corresponds to the link addition to the system. The giant component emerges at the early stage and grows slowly as time goes on. As we found in the main result, only a few of the preexisting (central) articles were preferentially cited then the cumulative strength distribution becomes fat-tailed. In the middle of the period, the giant component size grows fast again and the period is corresponds to the start of regime 4. As we described in the manuscript, the high prevalence of "Education" and "Family, mother and health" articles shows the distinctive tendency of the regime, and the characteristics can be also captured in the growth of the giant component size ( Figure I).                 After the increase of the giant component saturated for a while, it grows fast again when regime 4 starts. In regime 4, several articles like "Education" and "Family, mother and health", which were not paid attention to other regimes, entered the spotlight.