Modeling State Firearm Law Adoption Using Temporal Network Models

Abstract

Policy Points.

• Promoting healthy public policies is a national priority, but state policy adoption is driven by a complex set of internal and external factors.

• This study employs new social network methods to identify underlying connections among states and to predict the likelihood of new firearm‐related policy adoption given changes to this interstate network.

• This approach could be used to assess the likelihood that a given state will adopt a specific new firearm‐related law and to identify points of influence that could either inhibit or promote wider diffusion of specific laws.

Context

US states are largely responsible for the regulation of firearms within their borders. Each state has developed a different legal environment with regard to firearms based on different values and beliefs of citizens, legislators, governors, and other stakeholders. Predicting the types of firearm laws that states may adopt is therefore challenging.

Methods

We propose a parsimonious model for this complex process and provide credible predictions of state firearm laws by estimating the likelihood they will be passed in the future. We employ a temporal exponential‐family random graph model to capture the bipartite state law–state network data over time, allowing for complex interdependencies and their temporal evolution. Using data on all state firearm laws over the period 1979–2020, we estimate these models’ parameters while controlling for factors associated with firearm law adoption, including internal and external state characteristics. Predictions of future firearm law passage are then calculated based on a number of scenarios to assess the effects of a given type of firearm law being passed in the future by a given state.

Findings

Results show that a set of internal state factors are important predictors of firearm law adoption, but the actions of neighboring states may be just as important. Analysis of scenarios provide insights into the mechanics of how adoption of laws by specific states (or groups of states) may perturb the rest of the network structure and alter the likelihood that new laws would become more (or less) likely to continue to diffuse to other states.

Conclusions

The methods used here outperform standard approaches for policy diffusion studies and afford predictions that are superior to those of an ensemble of machine learning tools. The proposed framework could have applications for the study of policy diffusion in other domains.

The us constitution provides substantial powers to states to attend to the health and safety of their citizens. ¹ The major federal gun law is the 1993 Brady Handgun Violence Prevention Act, which mandated federal background checks on firearm purchases. ² Beyond background checks, however, states are largely responsible for the regulation of firearms within their borders, and each one has developed a substantially different legal environment. Theories of the policy process suggest that legislative actions may be prompted by focusing events such as mass shootings, whereas other policy changes are thought to be the result of long‐term advocacy or lobbying campaigns. ³ , ⁴ Many proposed bills are never approved by state legislatures or signed into law, and some laws that are passed may be considered to be merely symbolic in nature. ⁵ Understanding the overall firearm legal landscape, that is, knowing how, when, and why states choose to adopt new firearm laws is therefore challenging.

Despite these challenges, a number of approaches have been developed to elucidate the process of policy diffusion from one state to another, although these methods have rarely been applied to understanding firearm laws. ⁶ Key studies in this field have demonstrated that policy diffusion is more than just geographical clustering of similar laws; it is often the result of state decisionmakers’ limited time and technical expertise coupled with the need to respond to citizens’ priorities or concerns within a fixed term. ⁶ These studies have posited a number of mechanisms for state policy diffusion, from interstate competition over welfare and minimum wage laws ⁷ to policy learning for complex systems such as health insurance, ⁸ crisis management, ⁹ or obesity ¹⁰ to the work of cross‐state policy entrepreneurs who advocate for policy change using their networks of state advocates, legislators, and staff members. ¹¹ This work has further identified conditions (such as observable experiments, time to learn and adapt, expertise and incentives) that favor the spread of more effective policies and those (such as competition and imitation) that instead favor the spread of less effective ones. ¹²

Along with a growing body of evidence related to state internal and external factors associated with policy diffusion, there has been considerable innovation in the methods used to study this phenomenon. ¹³ To date, the majority of existing studies of state policy diffusion have utilized versions of event history analysis, which considers the relevant unit of analysis as state law pairs, with outcomes coded 1 in the year that a law is adopted by a state and 0 each year before that. ¹⁴ Once the law in question is adopted, states that have the law are censored or removed from future time periods, as they are no longer “at risk” of policy adoption. One of the drawbacks of this approach is related to the small number of US states, which affects the ability of any statistical method to detect significant predictors. In response, scholars have proposed changing the unit of analysis from a single state to a directed state dyad in which state pairs are constructed for each state, both as a potential “sender” (i.e., a state that has already adopted the law) and as a potential “receiver” of the law in question (i.e., a state that has not yet adopted the law). Not only does this approach improve statistical power, but it is also useful in differentiating among characteristics of states that are more likely to be policy “senders” from those more likely to be policy adopters or “receivers.” ⁸

Despite these advantages, later studies identified limitations of the directed dyad approach regarding the independence of observations, ambiguities in attributing different motives to state actions (i.e., emulation versus learning versus independent policy adoption), and modeling the independence of different policies within the same domain, among others. ¹⁵ More recent methodological innovations use spatial autoregressive and spatially lagged X models, which account for temporal and spatial lags and can be used alongside methods to identify underlying interaction structures. ¹⁶ , ¹⁷ , ¹⁸

In this paper, we pursue a new approach to understand policy diffusion with the aim of modeling the entire system of state firearm law adoption rather than viewing each law as a realization from related but independent models of policy adoption. Our objectives are to 1) assess whether the same factors identified in studies of the diffusion of other public policies also apply to the more highly politicized realm of gun laws, 2) demonstrate a method that can be used to identify factors associated with the spread of firearm laws across states, and 3) use this information to predict the adoption of subsequent firearm laws. To do this, we apply a novel method from social network analysis to predict which states are likely to adopt an existing firearm law for the first time. We demonstrate that model predictions using this method can better account for the dependence among state legislative actions and can aid in understanding the dynamic influence of internal and external drivers of state firearm laws. Finally, we demonstrate the power of this approach by considering a series of scenarios in which we simulate the enactment of certain firearm laws in one or more states and then observe the change in response probabilities across the entire system.

Materials and Methods

Data

We used the publicly available RAND Corporation State Firearm Law Database (version 3.0), one of the most comprehensive and recently updated collections of state firearm laws obtained through original legal research methods. ¹⁹ , ²⁰ Each line in this data set is a single legislative action carried out by a single state at a single point in time. A legislative action is defined either as the enactment of a new law, or as the repeal or modification of an existing law. We restricted the data to the period from January of 1979 to January of 2020, resulting in 918 new firearm laws or modifications, all of which are included in this analysis.

RAND has categorized each firearm law (e.g., child access protection, waiting periods, background checks, stand your ground, and concealed carry) as either “permissive” (i.e., it makes firearm access or use easier, such as concealed carry laws) or “restrictive” (i.e., it limits some aspect of firearm access or use, such as waiting periods and background checks). This categorization oversimplifies many details related to these laws ²¹ , ²² and fails to identify which are considered to be effective. However, the evidence base is lacking for many existing firearm‐related laws ²³ and additional legal details (such as who exactly the law applies to or how the law might be enforced) are arguably more important when assessing laws’ impact than their diffusion. In this study, we look at the passage of every firearm law over time using this simple dichotomization (permissive versus restrictive) to assess whether different patterns hold for these two major firearm law classifications. We then include additional law categories such as child access protection (restrictive) or concealed carry (permissive) in the final models to assess whether including them improves explanatory power.

State covariates come from publicly available sources and are based on previous studies of policy diffusion across US states. ¹³ They include yearly measures of government ideology using Berry's state policy scales in which 0 represents the most conservative and 100 the most liberal, ²⁴ homicides and suicides per 10,000 from the Centers for Disease Control and Prevention (CDC) to represent the extent to which firearms may be on the state's policy agenda, firearm background checks per capita from the Federal Bureau of Investigation (FBI) in order to provide a proxy of the relative prevalence of firearms in the state, data on mass shootings in the state (operationalized as any mass shooting event in that state each year) from The Violence Project database ²⁵ to assess the effects of such a high‐profile focusing event, and the state population size and income per capita from the US Census Bureau as measures of the state's resources.

States and Firearm Laws Can Be Viewed as a Sequence of Bipartite Networks

Both theory and previous research support the contention that states do not act independently when enacting legislation, even after accounting for states’ similarities. ²⁶ As a result, we formulate a class of social network models in which each state's legislative actions can impact those of other states. We represent the legislative actions throughout the United States as a sequence of bipartite networks in which states can be connected to laws but not directly to other states. States form ties with a specific law within each main category (i.e., permissive or restrictive) when they first adopt a law from that category. States are connected in this network only if they share one or more similar laws. Thus, at each point in time, we have a network where some states are connected to a law because they have already adopted it (i.e., they are now potentially “sending” states) and some that are not connected to that law because they have not yet adopted it (i.e., they are now in a position to “receive” that law from a sending state). As there is no compelling notion of directionality between a state and a law, we consider this as an undirected network.

Temporal Exponential‐Family Random Graph Models

In our setting, we have snapshots of the state law network for each year that explicitly capture the formation and dissolution processes of edges (or links) in a network over time. Separable temporal exponential‐family random graph models (STERGMs) are a special case of temporal exponential‐family random graph models, which can be used to study situations in which there is strong dependence on prior networks as well as strong dependence of each edge in the network on other edges or even on the network configuration from a previous time period. ²⁷ , ²⁸ , ²⁹ , ³⁰ , ³¹ , ³² , ³³ In our case, we have the dates each firearm law was made effective, so we can directly model the edge formation process (i.e., the passing of laws by states). To simplify these models, we only consider the adoption and modification but not the repeal of firearm laws. See Appendix 1 for more technical details on STERGMs.

Parameters from an STERGM are estimated using a conditional maximum likelihood estimator (CMLE). ²⁹ To properly fit these models, the selection of network summary statistics is crucial and similar to the problem of selecting covariates in a standard regression analysis. Here, we do this through the inclusion of two different geometrically weighted (GW) dyad‐wise shared partners (DSP) terms. ³² The first GW state DSP term accounts for states’ tendency to adopt laws that other states have also adopted, whereas the other, the GW law DSP term, accounts for the tendency toward the adoption of some laws more often than others.

Predicting Future Law Adoption Using STERGMs

Because the interpretation of the magnitude of parameters from STERGMs can be complex, we generate a series of predicted probabilities. This approach can be used to describe the transition of the network over time steps and to simulate future steps. We compare the predictive power of our STERGM with predicted laws from a linear state law event history model (the standard approach to model state policy diffusion) and an ensemble of machine learning methods that include the super learner algorithm ³¹ with a regularized generalized linear model, random forest, generalized additive model, and XGBoost for our ensemble. We fit the predictive models using data up to 2010 and compare the predictions generated by each model with the additional restrictive and permissive state firearm laws that had been enacted by 2015. We compare these different approaches using a goodness of fit measure—area under the curve (AUC)—because the data are highly unbalanced (i.e., most laws are not enacted by most states). ³⁴

We then carry out a number of scenario analyses to consider the increase or decrease in predicted probability of any given state adopting a given law if the legal environment to be different. We identify three specific scenarios to assess how well our models predict changes in the state policy network. To operationalize these simulations, we use the complete fitted model up to 2020, then impose new law‐enactment simulations to calculate the updated change statistics for each state law dyad and their related probabilities under both the current and the hypothetical future state.

In the first scenario, we identified a large state (New York) with a long legislative history of restrictive firearm laws to assess how this state's adopting a new restrictive law that it did not already have would affect its network. This scenario represents the most common course of action for most states. The second scenario asks what might happen if such a state were to suddenly change gears and adopt a permissive law that is popular among other states. This scenario represents a highly unusual, though not impossible, situation. The third scenario predicts what might happen if several states with a history of permissive laws adopt a new permissive law at the same time. This scenario provides an illustration of how blocks of states moving together might alter the entire network configuration. Note that these predictions should not be interpreted as causal estimates. Instead, these changes can be interpreted as, “had the world turned out such that these laws were passed, other state laws would have the corresponding probabilities of being passed in the near future.” ³⁵ , ³⁶

Building and Testing STERGMs

In terms of model specification, we first tested the choice of time step because this choice is subjective but must satisfy the Markov assumption. We fitted the STERGM to the network list and compared results from 2‐yearly, 5‐yearly, and decade‐wise time steps. Appendix 6 presents the fitted parameters of different time steps, which demonstrate a consistent pattern regardless of the length of the time step. Although the 10‐year time step offers the best fit to the data (lowest Akaike Information Criterion [AIC]), a decade‐wise time step could miss changes between legislative sessions, for example. Given that the values of the estimated parameters and their statistical significance were similar, we opted to use the 5‐year window in the analyses presented here. Appendix 7 presents results for different measures of state‐level firearm‐related harms. We find that although homicides and suicides showed slightly different patterns and mass shooting events were not statistically significant predictors of new firearm legislation, firearm‐related homicides resulted in the best fit to the data (lowest AIC) and were therefore retained in the final models.

We use the AIC to compare the fit of different STERGMs (in terms of time steps and covariates). We used the R package tergm for fitting STERGMs to the network data and to generate the CMLE estimates presented here. ³⁷

Results

Figure 1 is a snapshot of part of the state law network from the STERGMs in three time periods (2005, 2010, and 2015). It shows the evolution of the network, with increasing ties between laws and states in each time period. The figure demonstrates the complexity of the network in which liberal states such as New York have mostly restrictive laws, which connects them primarily with other liberal states. However, some states, such as California and Florida, are connected to an even more diverse group of states because they have both restrictive and permissive laws on the books. We also observe an increased number of states adopting the same laws over time, resulting in increased network density by 2015. Appendix 8 provides more details about the network structure.

Figure 1.

Snapshot of Part of the Network Structure Over Three Time Periods. [Colour figure can be viewed at wileyonlinelibrary.com]

Table 1 shows the parameters of the 5‐year model after testing for inclusion of additional terms that lowered AIC. In this case, the best fitting model included dummy variables for 11 additional law categories to control for the frequency of their passage over time. The positive coefficient for restrictive laws suggests that these laws were more likely to be passed than permissive laws, all else equal. More liberal government ideology is negatively associated with permissive firearm laws. We do not find evidence for the corresponding reversal for restrictive laws. The number of homicides per capita is associated with the passing of restrictive laws, though not permissive ones. Background checks per capita (a proxy for new firearm acquisition in the state) were negatively associated with restrictive, but not permissive, law adoption. Higher income per capita was negatively associated with the passing of more permissive laws and positively associated with restrictive laws. Sharing a border with a state that has already adopted a permissive law is associated with an increased likelihood of passing the same law within the receiving state. We do not observe any evidence for the same association for restrictive laws.

Table 1.

STERGM of Firearm Law Adoption With 5‐Year Time Step, 1979–2020

Parameter	Estimate
Law characteristics
Any restrictive vs. permissive law	1.309 ***
State characteristics
More liberal government ideology, permissive	−0.339 **
More liberal government ideology, restrictive	0.162
Homicides/10,000, permissive	0.049
Homicides/10,000, restrictive	0.307 *
Background checks per capita, permissive	0.27 **
Background checks per capita, restrictive	−0.486 **
Population size, permissive	−0.077
Population size, restrictive	0.137
Income per capita, permissive	−0.397 **
Income per capita, restrictive	0.523 **
Shared border, permissive	0.312 ***
Shared border, restrictive	−0.016
Network statistics
Edges	−2.939 ***
GW law DSP	−0.035
GW state DSP	0.094 ***
AIC	1,546.81

AIC, Akaike Information Criterion; DSP, dyad‐wise shared partners; GW, geometrically weighted; STERGM, separable temporal exponential‐family random graph model.

^*** p < 0.001, ^** p < 0.01, ^* p < 0.05.

Permissive refers to the adoption of laws that make firearms easier to obtain and use; restrictive refers to laws that make firearms more difficult or costly to obtain and use. The model also includes dummy variables for 11 law types (e.g., minimum age, registration, waiting period, etc.) to account for different legislative intensities.

The terms in the final block of Table 1 provide additional information about the network structure. The “edge” term accounts for the baseline probability of laws being passed. The GW state DSP term suggests that states are more likely to adopt laws that other states have also adopted. The GW law DSP term is not statistically significant, suggesting that there was no overall tendency for certain laws to be adopted more often than others.

Table 2 shows the AUC of each method of predicting law adoption for restrictive and permissive laws, respectively. For both permissive and restrictive laws, the STERGM with law‐specific dummy variables performs best in terms of the AUC statistic. Interestingly, based on comparison of the AUC statistics, these models generally performed better at predicting restrictive rather than permissive laws.

Table 2.

Comparison of AUC Statistics for Proposed Predictive Models

Method	AUC Restrictive Laws	AUC Permissive Laws
Logistic regression (law categories only)	0.674	0.604
Multiple logistic regression	0.792	0.647
Super learner algorithm	0.657	0.686
STERGM	0.804	0.697
STERGM with law categories	0.874	0.729

AUC, area under the curve; STERGM, separable temporal exponential‐family random graph model.

The first set of output from the predictive models are shown in Appendices 9 and 6, which display, respectively, restrictive and permissive laws with the highest probability of future passage based on the network structure up to 2020. For restrictive laws, the top predictions are dominated by minimum age and prohibited possessor laws, for both conservative and liberal states. The results can be interpreted as there being a high chance that most states that do not have these very common laws will likely adopt them. For permissive laws, we see that the top predicted laws are present exclusively in more politically conservative states.

Table 3 shows the laws and states with the largest increases and decreases in predicted probability of passing laws under the scenario that New York state passed a restrictive waiting period law. The largest decreases in probability are in passing permissive laws in New York, that is, conditional on a new restrictive law, the chance of new permissive laws being enacted in that state is reduced. The largest decrease is the permissive version of the law just conditioned on. There are some small changes in other laws, for example, small decreases in probabilities of adopting restrictive laws within states that are very different from New York, which may indicate the model captures legislative divergence among states. There are also small increases in the chance of other states adopting the law, capturing the increased chance of a law diffusing.

Table 3.

State Law Dyads With Largest Decreases and Increases in Probability of Enactment, Conditioned on New York Adopting a Restrictive Waiting Period Law

				Probability
Order	State	Law	Class	Old	New	Change, %
1	AL	Waiting period	Restrictive	0.0639	0.0649	1.6814
2	SD	Waiting period	Restrictive	0.075	0.0762	1.6612
3	SC	Waiting period	Restrictive	0.0916	0.0931	1.6312
4	OH	Waiting period	Restrictive	0.1415	0.1437	1.5402
5	WI	Waiting period	Restrictive	0.103	0.1037	0.6308
6	NV	Waiting period	Restrictive	0.1044	0.105	0.6298
7	UT	Waiting period	Restrictive	0.1172	0.1179	0.6212
8	IA	Waiting period	Restrictive	0.1224	0.1232	0.617
9	MI	Waiting period	Restrictive	0.1238	0.1246	0.6165
10	NE	Waiting period	Restrictive	0.1025	0.1031	0.5823
687	ID	Sales restrictions	Restrictive	0.0136	0.0132	−2.619
688	NY	Concealed carry	Permissive	0.0029	0.0028	−2.8731
689	NY	Permit to purchase	Permissive	0.0022	0.0021	−3.9252
690	NY	Sales restrictions	Permissive	0.0026	0.0025	−4.0208
691	NY	Prohibited possessor	Permissive	0.0092	0.0083	−9.4653
692	NY	Registration	Permissive	0.0017	0.0015	−9.5304
693	NY	Castle doctrine	Permissive	0.0991	0.0851	−14.1183
694	NY	Concealed carry	Permissive	0.065	0.0533	−18.0555
695	NY	Waiting period	Permissive	0.0589	0.0476	−19.1119

Predictions from models including all covariates present in Table 1 are shown.

Table 4 presents the laws and states with the largest increases and decreases in predicted probabilities under the scenario that New York state passed a permissive Castle Doctrine law. Neighboring states without such a law, such as Vermont and Connecticut, have an increased chance of enacting the same law; however, we see few other increases in probability of legislative action. We also see net decreases in several more conservative states adopting restrictive laws.

Table 4.

State Law Dyads With Largest Decreases and Increases in Probability of Enactment, Conditioned on New York Adopting a Permissive Castle Doctrine Law

				Probability
Order	State	Law	Class	Old	New	Change, %
1	VT	Castle doctrine	Permissive	0.0569	0.0677	18.9288
2	CT	Castle doctrine	Permissive	0.0503	0.0594	18.0478
3	AR	Castle doctrine	Permissive	0.4767	0.4812	0.933
4	NC	Open carry	Restrictive	0.0118	0.0118	0.3017
5	TN	Purchase permit	Restrictive	0.0134	0.0135	0.3011
6	NC	Sales restrictions	Restrictive	0.0166	0.0166	0.3002
7	TN	Sales restrictions	Restrictive	0.0174	0.0174	0.2999
8	NJ	Purchase permit	Restrictive	0.0391	0.0392	0.2933
9	NJ	Open carry	Restrictive	0.0434	0.0436	0.292
10	MA	Purchase permit	Restrictive	0.0541	0.0542	0.2887
687	MN	Castle doctrine	Permissive	0.0995	0.0862	−13.3854
688	GA	Prohibited possessor	Restrictive	0.1188	0.1022	−14.0092
689	WA	Castle doctrine	Permissive	0.0408	0.035	−14.1346
690	HI	Castle doctrine	Permissive	0.0241	0.0206	−14.2209
691	MS	Prohibited possessor	Restrictive	0.0671	0.0572	−14.7113
692	KY	Prohibited possessor	Restrictive	0.1043	0.089	−14.7179
693	ID	Prohibited possessor	Restrictive	0.0988	0.0829	−16.0859
694	NY	Waiting period	Permissive	0.0589	0.0488	−17.0432
695	NY	Concealed weapon	Permissive	0.065	0.0523	−19.5434

Predictions from models including all covariates present in Table 1 are shown.

Table 5 displays the laws and states with the largest increases and decreases in predicted probability of passing laws under the scenario that Florida, Texas, Alabama, Georgia, and Missouri all adopt a permissive firearms sales restrictions law. We observe that, as a result, all states would have an increased chance of the adopting the law, with the largest changes coming among those geographically and ideologically closer to the conditioning block of states.

Table 5.

State Law Dyads With Largest Decreases and Increases in Probability of Enactment, Conditioned on Florida, Texas, Alabama, Georgia, and Missouri Adopting a Permissive Law on Firearms Sales

			Probability
Rank	State	Law	Old	New	Change, %
1	TN	Firearm sales restrictions, permissive	0.0139	0.0506	262.96
2	AR	Firearm sales restrictions, permissive	0.0111	0.0296	167.32
3	OK	Firearm sales restrictions, permissive	0.0096	0.0249	159.95
4	MS	Firearm sales restrictions, permissive	0.0100	0.0205	104.22
5	LA	Firearm sales restrictions, permissive	0.0146	0.0287	96.60
6	KS	Firearm sales restrictions, permissive	0.0072	0.0142	96.22
7	NC	Firearm sales restrictions, permissive	0.0158	0.0296	87.76
8	KY	Firearm sales restrictions, permissive	0.0399	0.0746	86.94
9	NE	Firearm sales restrictions, permissive	0.0054	0.0096	76.64
10	NM	Firearm sales restrictions, permissive	0.0081	0.0139	71.31
623	WY	Safety training required, restrictive	0.0133	0.0133	0.00
624	WY	Permit to purchase, restrictive	0.0115	0.0115	0.00
625	WY	Dealer license, restrictive	0.0139	0.0139	0.00
626	WY	Dealer license, permissive	0.0106	0.0106	0.00
627	WY	Minimum age, permissive	0.0460	0.0460	0.00
628	WY	Permit to purchase, permissive	0.0075	0.0075	0.00
629	WY	Open carry, permissive	0.0103	0.0103	0.00
630	WY	Prohibited possessor, permissive	0.0328	0.0328	0.00

Predictions from models including all covariates present in Table 1 are shown.

Discussion

In this paper, we applied a novel method from network analysis to the US firearm legislative environment. We framed the problem as a bipartite network in which states are connected to laws and these laws are what connect different states. We model this structure explicitly and temporally. Our approach circumvents the need to specify or detect the interaction structure, as spatial models would require, or make homogeneity or dyadic independence assumptions, as event history approaches demand. Instead, we model the interaction structure directly and parsimoniously. It is, to our knowledge, the first attempt in the policy diffusion domain to apply temporal network models to a bipartite network of states and laws.

Results show that the comparison is robust to the choice of time step in terms of the significance of most parameters. Other advantages include the fact that the model does not make restrictive independence assumption or require the identification of latent influence networks. It effectively identifies important drivers of legislative activity, mostly in line with expectations and prior literature. The model outperforms other statistical approaches on held out data, suggesting it captures the structure of the problem better. Finally, the model generates interesting information by conditioning on laws being passed and taking advantage of the underlying network structure and its evolution over time.

The effects of covariate terms are largely in line with the literature. ¹³ More liberal state ideology was negatively associated with adopting more permissive firearm laws, which is in line with expectations. It was not expected that restrictive laws would be unassociated with state ideology. This may be because by 2020, 99% of state pairs had adopted at least one restrictive law, whereas only 88% had at least one permissive law. The adoption of restrictive laws was positively associated with higher levels of homicide, but from these results, we cannot ascertain the true causal relationship between these two factors. Background checks, which we interpret as a proxy for the level of interest in the state for acquiring new firearms legally, were positively associated with permissive and negatively associated with restrictive laws. This could be because of such laws reflecting state's citizen interest in making firearms easier or more difficult to acquire. Income per capita shared similar relationships with each law category as did background checks; higher income per capita was negatively associated with permissive and positively associated with permissive laws. This result might be driven by larger, more wealthy states such as New York and California adopting more restrictive rather than permissive laws, on average. State borders were important for the adoption of permissive laws only. This may reflect regional political blocks, whereby greater number of more conservative states are neighbors than are more liberal states, many of which are located along the east and west coasts.

A benefit to the approach of directly modeling the temporal dynamics of the whole system of state firearms legislation is to use the model to generate predicted probabilities of the future passing of specific laws. The results of our three scenarios yielded several insights. In the first case, we found that if New York were to pass another restrictive firearm law, then the legislative appetite for permissive laws in the state would be greatly reduced. In the second scenario, we found that if New York had a permissive Castle Doctrine law, more conservative states would be under less pressure to adopt any new restrictive laws. Conversely, we see moderate and liberal states having a decreased chance of adopting a Castle Doctrine law. This could be interpreted as liberal states acting as a block, with the passing of a new permissive law reducing the appetite for further permissive laws, when their overall legislative agenda may tend toward more restrictive ones. In the last scenario, when multiple states with a legislative history of fewer restrictions on firearms passed a permissive law, we see that the probability of such a law passing increases significantly, with the largest effects seen in neighboring states. These scenarios reinforce the idea of viewing states and their legislative activity as linked and interdependent.

To our knowledge, there has been little work using social network models to study policy diffusion. One study used a diffusion model, coupled with thresholding on the number of states, to predict influence networks among states. ¹⁸ Results confirm a number of our findings: that states are connected by more than merely geography—political homophily was nearly as important a predictor of policy diffusion as being geographically contiguous—and that larger and more wealthy states were more often “sending” states, whether because of their greater internal capacity to innovate or the fact that such states tended to have a larger number of peers in their networks.

Limitations

We attempted to understand the highly complex process by which states pass firearms legislation with models that, by comparison, are very simple. As a result, all of the results using STERGMs are subject to some misspecification with unknown consequences. Still, compared with other statistical approaches using the same covariates, STERGMs better fit the data and additionally account for the interdependent nature of state legislative activity around firearms, so we believe they provide a less biased estimation approach.

We used a 5‐year time step model, as we believe that the legislative activity in a given 5‐year period only depends on the state of the laws 5 years ago, not the complete sequence of prior legislative environments. Other time windows are possible and may provide different results, although our results appear to be mostly robust to 2‐year, 5‐year, or 10‐year windows.

Our approach cannot account for new laws. The formulation relies on assuming that the node set is fixed. Because laws are part of the node set, they are fixed throughout the analysis. This may be problematic for prediction in practice, as the passing of new categories of laws cannot be predicted and any new laws not already included in the data set would have been unaccounted for in these analyses. Moreover, we can only model laws that are actually passed by states because there is no comprehensive database on all legislative proposals related to firearms in each state for the entire period under examination. Previous studies have found that other factors, such as mass shooting events and related media coverage, are strongly associated with legislative proposals (bills) but were only associated with the passage of one firearm arm class (permissive laws) passed in states under Republication legislative control. ³

Law categorization (permissive, restrictive) is the only feature of the law considered. Laws can vary significantly among states even if they had been categorized similarly in our data. This can happen because of differences in the law's formulation, implementation, or enforcement, which are unobserved in our, and most other, analyses. We focus on all firearm laws, but future work might focus instead only on those laws for which there is robust evidence for their effectiveness, or—depending on the hypothesis to be tested—those laws which might be considered to be merely symbolic in nature. We only consider the first enactment of a given law class for each effect and further legislative actions, such as repeals, are not considered.

Finally, because of data limitations, we are only able to assess laws that are actually passed, not bills that are proposed. Had we been able to look at proposed bills, we might have seen an association with mass shooting events, for example. ³ An area of future work would be to use the models developed here to see if they might also be effective in predicting which of a set of bills might be more likely to be passed into law within a state, conditional on other state‐level factors and the rest of that state's legal network.

Conclusions

A common objective in many public health settings is to alter the environment such that healthy decisions become the default choice for most people. ³⁸ This objective may be clearer for some realms, such as obesity prevention, in which there is good evidence for what environmental factors could be changed in order to promote healthier choices for consumers and manufacturers. ³⁹ The same cannot always be said for firearm legislation, based largely on the fact that there is still a lack of solid, high‐quality evidence on the impact of most of the laws examined in this study—whether that be evidence that the laws really affect people's behaviors in intended ways or that the cumulative impact of such behavioral changes results in reductions in population‐level firearm‐involved mortality or morbidity. ¹⁹ As evidence is accumulated about the impacts of different firearm‐related laws, policymakers, advocates, and public health practitioners will need to become increasingly savvy about the importance and the limitations of policy diffusion and to encourage approaches that facilitate real policy learning. ¹² The results of this study suggest that passing new firearm laws is infrequent, is influenced by both long‐term state characteristics (such as political ideology or state resources) as well as population‐level risk factors (such as homicide rates) that themselves have multiple and complex causes. However, for those seeking to promote the adoption of more restrictive firearm laws, there is a silver lining; these laws are more likely to pass, on average, than more permissive ones and working across state borders to advocate for policy change may be a more effective strategy—especially in the presence of strong lobbying groups—than working in isolation in a single state. ⁴⁰ , ⁴¹ , ⁴²

Funding/Support: This work was funded by a grant (CMMI‐1953135) from the National Science Foundation. The funders had no role in the study.

Appendix 1.

More detailed STERGM methodological notes

For each time‐step t we consider a fixed set of nodes {1,…,n} with fixed nodal covariates { X ^{1 t} …, X ^nt }. We define y ^t to be a graph on this fixed nodal set, in particular, a graph realized from the random variable Y ^t , taking values in the set of all graphs with n nodes, $\mathcal{Y}$ = {a ∈ R^nxn | ∀ i,j a_i,i = 0, a_i,j ∈ {0, 1}}, where a_i,j = 1 if there is an edge from i to j. For undirected networks, like ours, the additional restriction that a_i,j = a_j,i ∀ i,j is imposed. A dyad is defined to be any pair of nodes (i,j). TERGMs are discrete time dynamic network models such that y ^t is a single draw from an ERGM, conditional on the network at time‐step t‐1 for a one‐step model. The nodal covariates X ^it are permitted to vary over time, but are known.

Equation 1 gives the formulation of TERGM as the transition probability under the one‐step Markov assumption; the transition probability from y ^{t ‐1} to y ^t depends on y ^k for k < t‐1. This is similar to the ERGM model, except that the statistics at time‐step t are permitted to depend on the graph at time t‐1.

$$p\left({\mathbf{y}}^t|{\mathbf{y}}^{t-1},\theta \right)=\frac{\exp \left({\theta }^{\mathrm{T}}\mathbf{g}\left({\mathbf{y}}^t,{\mathbf{y}}^{t-1}\right)\right)}{\mathrm{Z}\left(\theta ,{\mathbf{y}}^{t-1},\mathcal{Y}\right)}$$ (1)

In equation 1, $\theta$ is a vector that encodes the parameters of the network formation and dissolution processes, g stands for network summary statistics, and $\mathrm{Z}(\theta ,{\mathbf{y}}^{t-1},\mathcal{Y})$ is a normalizing constant that ensures the numbers on the left‐hand‐side amount to probability values, namely, $\mathrm{Z}(\theta ,{\mathbf{y}}^{t-1},\mathcal{Y})=\sum _{\mathbf{y}\in \mathcal{Y}}\exp ({\theta }^{\mathrm{T}}\mathbf{g}(\mathbf{y},{\mathbf{y}}^{t-1}))$.

To formalize an STERGM, we introduce the following notation: y ⁺ is y ^{t −1} with additional ties added through the formation process (whose parameters are encoded by vector ${\theta }^{+}$), and, likewise, y ⁻ is y ^{t− 1} deprived of any ties that are removed through the dissolution process (whose parameters are encoded by vector ${\theta }^{-}$). Using this notation, equation 2 summarizes the formulation of an STERGM ²⁷ .

$$p\left({\mathbf{y}}^t|{\mathbf{y}}^{t-1},\theta \right)=p\left({\mathbf{Y}}^{+}={\mathbf{y}}^{+}|{\mathbf{y}}^{t-1},{\theta }^{+}\right)p\left({\mathbf{Y}}^{-}={\mathbf{y}}^{-}|{\mathbf{y}}^{t-1},{\theta }^{-}\right)$$ (2)

STERGMs use an ERGM for both $p({\mathbf{Y}}^{+}={\mathbf{y}}^{+}|{\mathbf{y}}^{t-1},{\theta }^{+})$ and $p({\mathbf{Y}}^{-}={\mathbf{y}}^{-}|{\mathbf{y}}^{t-1},{\theta }^{-})$. Within an ERGM, the interpretation of the formation and dissolution parameters ${\theta }^{+}$ and ${\theta }^{-}$ can be viewed similarly to the log‐odds interpretation for logistic regression, but with conditioning on the rest of the network. For the specific case of an STERGM (or TERGM in general), we add possible dependence on the previous time period's network. We define ${({\mathbf{y}}^t)}_{i,j}^c$ as the graph y^t without the dyad (i, j), so that the log‐odds ratio for an STERGM is computed from equation 1 as equation 3.

$$\mathrm{Logit}\left[p\left(Y_{i,j}^t=1|\theta ,{\left({\mathbf{y}}^t\right)}_{i,j}^c,{\mathbf{y}}^{t-1}\right)\right]={\theta }^{\mathrm{T}}\delta \left(y_{i,j}^t\right)$$ (3)

In equation 3, $\delta (y_{i,j}^t)$ is the vector of change statistics due to the inclusion of the link between i and j, that is, $\mathbf{g}(({\mathbf{y}}^t|y_{i,j}^t=1),{\mathbf{y}}^{t-1})-\mathbf{g}(({\mathbf{y}}^t|y_{i,j}^t=0),{\mathbf{y}}^{t-1})$.

Appendix 2.

STERGMs of Factors Associated With State Firearm Law Adoption With 2‐, 5‐ and 10‐year Time Steps

Parameter	Two‐year	Five‐year	Ten‐year
Government ideology, permissive	−0.54***	−0.398***	−0.458***
Government ideology, restrictive	0.243*	0.104	0.142
Homicide rate, permissive	0.393*	0.056	0.378**
Homicide rate, restrictive	0.113	0.245*	0.288*
Background checks, permissive	0.172	0.308*	0.188
Background checks, restrictive	−0.194	−0.406*	−0.211
Income per capita, permissive	−0.126	−0.429**	−0.201
Income per capita, restrictive	0.274**	0.457***	0.328**
Shared border, permissive	0.074*	0.373*	0.062
Shared border, restrictive	0.214***	0.217***	0.183*
Network statistics
Edges	−4.175***	−3.8***	−2.617***
GW state DSP	0.125***	0.136***	0.301***
GW law DSP	−0.31***	−0.194***	−0.34***
AIC	2,251.52	1,641.08	1,447.91

Note that the magnitude of parameters as well as AICs are not strictly comparable across different time steps.

Permissive refers to the adoption of laws that make firearms easier to obtain and use; restrictive refers to laws that make firearms more difficult or costly to obtain and use.

GW DSP = Geometrically weighted dyad shared partners for states and laws

Models do not control for additional laws.

Appendix 3.

Comparing Different Measures of Firearm‐related Harm

Parameter	Homicide	Suicide	Mass Shooting	All
Homicide rate, permissive	0.049	–	–	0.014
Homicides rate, restrictive	0.307*	–	–	0.252*
Suicide rate, permissive	–	0.613***	–	−0.033
Suicide rate, restrictive	–	0.015	–	0.05
Mass shooting, permissive	–	–	0.004	0.059
Mass shooting, restrictive	–	–	0.006a	0.005 a
Govt ideology, permissive	−0.398**	–0.332**	–0.393**	−0.331**
Govt ideology, restrictive	0.104	0.124	0.119	0.105
Background checks, permissive	0.308**	0.17	0.305**	0.171
Background checks, restrictive	−0.406**	−0.417*	–0.412**	−0.385*
Population, permissive	0.138	0.34**	0.15	0.334*
Population, restrictive	0.069	0.128	0.119	0.053
Income per capita, permissive	−0.429**	−0.247	–0.449**	−0.243
Income per capita, restrictive	0.523**	0.457***	0.407***	0.46***
Shared border, permissive	0.373***	0.386***	0.371***	0.383***
Shared border, restrictive	0.217***	0.213**	0.212**	0.214**
Network statistics
Edges	−2.939***	−3.90***	−3.82***	−3.893***
GW law DSP	−0.194**	−0.179*	−0.188**	−0.188*
GW state DSP	0.136***	0.141***	0.138***	0.141***
AIC	1,631.77	1,641.08	1,645.06	1,636.828

Permissive refers to the adoption of laws that make firearms easier to obtain and use; restrictive refers to laws that make firearms more difficult or costly to obtain and use.

GW DSP = Geometrically weighted dyad shared partners for states and laws

Models do not control for additional laws.

a = standard errors could not be estimated.

Appendix 4.

Network Summary Statistics From STERGM

		Mean Law Degree			State Pairs With Shared Law (%)
Year	Mean State Degree	Total	Permissive	Restrictive	Total	Restrictive	Permissive
1990	6.4	15.2	18	13.5	69.8	33.5	53.5
1995	10.98	23.2	19	26.5	91.3	79.6	60.9
2000	12.72	26.5	23.1	29.4	97.2	87.1	70.1
2005	13.44	28	24.5	30.9	98.7	90.6	75.5
2010	14.6	29.2	26.5	31.7	99.4	92.6	80.2
2015	15.84	40.4	27.2	33.7	100	95.5	87.8
2020	16.16	31.1	27.7	34.5	100	99.5	88.5

Mean state degree is the average number of law categories that states have adopted.

Mean law degree is average number of states each law has been adopted by, separated by total, and stratified by main law category (restrictive versus permissive).

State pairs with shared law represents the percentage of states that share at least one law, by total, restrictive, and permissive.

Appendix 5.

Most Likely STERGM‐predicted Restrictive Laws, Trained Using Data up to 2010, With Prediction Compared With 2015

			Model
State	Law Class	Passed?	1	2	3	4	5
Texas	firearms in college	No	0.000	0.000	0.03	0.159	0.000
New Hampshire	firearms in college	No	0.000	0.000	0.023	0.157	0.000
Alabama	child access laws	No	0.115	0.008	0.009	0.184	0.108
Alabama	prohibited possessor	Yes	0.323	0.178	0.264	0.184	0.139
Idaho	prohibited possessor	No	0.177	0.145	0.174	0.184	0.148
Alabama	minimum age	Yes	0.205	0.024	0.056	0.241	0.172
Arkansas	prohibited possessor	No	0.177	0.086	0.095	0.162	0.198
New Mexico	prohibited possessor	No	0.177	0.164	0.148	0.21	0.201
Nebraska	prohibited possessor	Yes	0.075	0.089	0.139	0.179	0.204
New York	child access laws	No	0.115	0.115	0.045	0.184	0.280
Ohio	minimum age	No	0.205	0.131	0.093	0.27	0.345
Virginia	waiting period	No	0.123	0.000	0.016	0.153	0.354
Oregon	minimum age	No	0.205	0.104	0.062	0.261	0.366
Colorado	prohibited possessor	Yes	0.110	0.337	0.175	0.257	0.382
Illinois	minimum age	No	0.205	0.164	0.09	0.358	0.421
Texas	minimum age	No	0.205	0.199	0.131	0.254	0.422
Colorado	waiting period	No	0.123	0.000	0.003	0.187	0.437
New Hampshire	minimum age	No	0.205	0.181	0.071	0.298	0.464
New York	waiting period	No	0.123	0.000	0.007	0.185	0.466
Massachusetts	waiting period	No	0.123	0.000	0.01	0.177	0.469

model 1 = law effect only, model 2 = full logistic regression, model 3 = Super learner ensemble model, model 4 = STERGM (no law effects), model 5 = full STERGM

Appendix 6.

Most Likely STERGM‐predicted Permissive Laws, Trained Using Data up to 2010, With Prediction Compared With 2015

			Model
State	Law Class	Passed?	1	2	3	4	5
Oklahoma	local preemption	No	0	0	0.008	0.191	0
Tennessee	local preemption	No	0	0	0.006	0.166	0
Kentucky	firearms in college	No	0	0	0.003	0.164	0
Alabama	firearms in college	No	0	0	0.002	0.157	0
Oklahoma	firearms in college	No	0	0	0.008	0.151	0
Missouri	local preemption	No	0	0	0.297	0.114	0
Wisconsin	local preemption	No	0	0	0.005	0.113	0
Arizona	firearms in college	No	0	0	0.008	0.111	0
Idaho	open carry	No	0.031	0	0.123	0.108	0.02
Alabama	open carry	No	0.031	0	0.002	0.166	0.022
Kentucky	open carry	No	0.031	0	0.003	0.231	0.026
Arkansas	castle doctrine	No	0.115	0.848	0.298	0.151	0.1
Utah	waiting period	No	0.1	0.198	0.055	0.115	0.115
Nebraska	castle doctrine	No	0.115	0.976	0.598	0.158	0.129
Idaho	minimum age	No	0.027	0.157	0.14	0.121	0.141
Oklahoma	minimum age	No	0.027	0.055	0.014	0.111	0.169
Alabama	minimum age	No	0.027	0.043	0.007	0.154	0.176
S. Carolina	waiting period	No	0.1	0.403	0.093	0.124	0.182
Idaho	castle doctrine	No	0.115	0.975	0.597	0.146	0.211
Alabama	concealed carry	No	0.162	0.802	0.375	0.33	0.624

model 1 = Law Effect only, model 2 = Full logistic regression, model 3 = super learner ensemble model, model 4 = STERGM (no law effects), model 5 = Full STERGM