In this section, we detail the operationalization of open-source data to construct weighted monoplex network layers and, in turn, a multiplex network. Unitary states are modeled as nodes with consensus data from established databases used to quantify edges in the form of edge weights.Footnote 2 These are assigned in proportion to the intensity of the connections in the different network layers (Pastor-Satorras and Vespignani 2004). That is, the connections between the nodes (edges) have a value associated with them (weight) based on the “strength” of the dyadic relationship between nodes within that layer. For example, edge weights are higher between the United States and North Korea during the thrust of the Korean War, but these weights decrease in value at times where there are less overt actions and threats between the two states. The links are also directed, in that the value associated with the relationship from one node to another is not necessarily the same in the opposite direction. For example, the United States may make an explicit threat to Russia, but the Russian Federation remain silent. In the directed network, this would represent an increased edge weight from the United States to Russia while the edge weight from Russia to the United States would be lower in value.
Each of the four layers in the multiplex network corresponds to a variable theorized to affect nuclear proliferation—conflict, alliances, NCAs, and trade. The edges are built upon historical data from the period 1951–1990.
Conflict variable
The conflict variable engages with existing work that finds that the presence of disputes increases the likelihood of proliferation. To operationalize this, we use the Dyadic Militarized Interstate Dispute (MID) database that builds upon the Correlates of War (COW) MID dataset to construct the conflict variable (Ghosn et al. 2004). Specifically, we use the variables, HiactA and HiactB, denoting the highest action taken by the respective sides in any given dispute. The variable reflects a low ranking of conflict if states are solely the recipient of conflict action and do not respond (coded as 1). The ranking increases as states threaten (2), display (3), or use force (4). The highest ranking represents a declaration of war (5).
Edges in the monolayer conflict network indicate the presence of a conflict between two states, i and j, and are used to quantify a conflict metric for each dyad:
$$ C_{ij} = I_{ij}. $$
(1)
The measure of conflict, Cij, is equal to the directed conflict intensity, Iij, as defined by the MID Database, where i is the recipient of conflict action (Jones et al. 1996). For example, for two states i and j, where j has issued a display of force to i and i does not respond, Cij = 3 and Cji = 1. As a real-world example, in 1961, the United States is coded as having used force against North Vietnam, but North Vietnam is coded as only displaying its force against the United States. In this case, CNV,US = 4 and CUS,NV = 3. As this work focuses on interactions between states rather than the properties of individual states, factors such as the relative military strength are not included when assessing the conflict metric.
Alliances
To examine the effect of formal agreements between states on nuclear proliferation, the alliance value, a, defined in terms of the strength of the alliance commitment, is adapted from Moaz’s Relative Commitment variable constructed from the Leeds’ Alliance and Treaty Obligation Project dataset (Leeds et al. 2002; Maoz 2009). The variable is coded into five categories: consultation (1), nonaggression (2), neutrality (3), offense (4), and defense pacts (5).
It is possible for states to share multiple concurrent alliance commitments between them. To account for simultaneous alliances between two states, an alliance commitment variable is used to represent directed edge weights in the alliance layer of the network. As alliances are dyadic, the alliance commitment, Bij, of state i from j is the sum of the strength of the alliance commitments issued by j to i:
$$ B_{ij}=\sum a_{ij}. $$
(2)
Using the ranking of alliance strength, for example, for two states i and j, where j has issued a defense commitment to i and a consultation pact is shared, the alliance commitment score is Bij=1+5=6. Conversely, as i has no defense commitment to j, Bji=1. This example is meant only to be illustrative and in the overwhelming majority of cases, alliance commitments are mutual with symmetrical edge weights. For example, in 1971, China and North Korea maintained a mutual defense pact coded as 5, in addition to a mutual non-aggression pact coded as 2, resulting in symmetric directed links with an alliance commitment score (i.e., edge weight) of 7.
Nuclear cooperation agreements
As noted above, NCAs are formal agreements between states to cooperate on one or more matters of nuclear technologies, safety, materials, and knowledge. To address NCAs, the model uses Fuhrmann’s Nuclear Cooperation Agreement Dataset and, specifically, the nca type variable adapted from Keeley’s compilation of NCAs (Fuhrmann 2012; Keeley 2009). Fuhrmann’s seven measure scale accounts for nuclear safety, cooperation in research and training, the transfer of nuclear materials, development towards a research program, development towards a nuclear electricity program, an agreement with no restrictions, and a military assistance agreement. We use this variable to construct a 3-measure ordinal scale of NCAs based on the relevance of the type of nuclear assistance to a nuclear weapons capability from those exclusively concerned with safety-related agreements (1), non-safety-related agreements (2), and finally to agreements dealing with sensitive nuclear assistance (3).Footnote 3
Safety-related agreements cover only authorized cooperation in the realm of nuclear safety. Non-safety-related NCAs are more significant from the standpoint of enabling the technologies, facilities, materials, and expertise necessary for the development of a nuclear weapons program. These activities cover cooperation in research and development, training, transfer of nuclear-related materials (e.g., uranium, heavy water, or plutonium), development of a nuclear research program (including export of a research reactor), and development of a nuclear program for electricity production. The final coding considers sensitive nuclear assistance (e.g., enrichment, reprocessing, etc.) independent of other non-safety-related NCAs due to the increased proliferation risk associated with these technologies (Fuhrmann 2009a; Kroenig 2009).
Each NCA can last for a specific number of years or for an indefinite period of time, depending on the terms of the agreement. Since the duration of many NCAs is confidential or unknown, NCA edges are treated as indefinite in the model for the sake of consistency across cases. Further, as additional NCAs may be signed between states over time, the value of each additional NCA is added to the states’ previous NCA metric to account for the accumulation of nuclear materials, expertise, and latent technological capabilities. This coding scheme provides a measure of the amount of nuclear cooperation that occurs between states over time and reflects the accrual of weapons-related information and technology. The NCA value, nij, is quantified based on the nature of nuclear assistance state i receives from state j.
As with alliances, states can have multiple NCAs with the same partner. The accumulated nuclear cooperation, Nij, that state i receives from state j is calculated as the sum of each individual NCA coding, nij:
$$ N_{ij} = \sum n_{ij}. $$
(3)
For two states i and j, where i has been the recipient of two non-safety-related NCAs from j, but i has never supplied an NCA to j, Nij=2+2=4 while Nji=0. For example, France supplied a safety-related agreement (coded as 1) and a non-safety-related agreement (coded as 2) to the United States in 1958, yielding an accumulated nuclear cooperation value of 3. In addition, a previous safety-related agreement and a non-safety-related agreement by France to the United States was supplied in 1956. As past NCAs are treated as persistent in the model, the directed edge weight in the NCA monolayer for France to the United States was valued at Nij=3+3=6 in 1958.
Trade dependence
Trade is included in the model to test its effect on proliferation likelihood, where an edge in the trade monolayer network represents an international exchange of goods and services. While there are many ways to calculate trade dependence and considerable literature in disagreement over the most appropriate method (Gleditsch 2002; Mansfield et al. 2002; Gartzke 2007), the model uses the Russett and Oneal formulation as it reflects the economic integration of states into the global economy by measuring the proportion of their economies devoted to bilateral trade (Oneal and Russet 1997; Oneal and Russett 1999; Russett and Oneal 1999).
Trade dependence, Dij, measures the total trade between two states as a fraction of each state’s gross domestic product (GDP). For two states, i and j, the trade dependence, Dij, of state i on j equals its exports to j, Exij, plus its imports from j, Imji, divided by its GDP:
$$ D_{ij} = \frac {Ex_{ij}+Im_{ji}} {GDP_{i}}. $$
(4)
Trade dependence is asymmetric as it is shaped by the relative strength of the state’s economy and the degree of dependence on a particular partner. For example, in 1990, the United States’ trade dependence upon China was 0.003 while the Chinese trade dependence on the United States was 0.12, indicating that China’s GDP was more dependent on trade with the United States than U.S. GDP was dependent on trade with China.