Fig 1.
Moffitt’s illustration of the prevalence of antisocial behavior as a function of age, redrawn from Fig 3 in Ref. [9].
The black curve gives the qualitative prevalence of antisocial behavior among youths. In Moffitt’s theory, youths below the black curve are classified into two groups, life-course-persistent youths who remain antisocial throughout their life course, and adolescence-limited youths who behave antisocially only during their adolescent period. Also shown are typical peer network structures of the same eight individuals at different life stages, where blue nodes represent pro-social individuals and red nodes represent antisocial individuals. Green nodes represent marginally pro-social individuals while yellow nodes represent marginally antisocial. Thick links between nodes indicate frequent/strong contacts between these individuals, whereas thin links between other nodes indicate infrequent/weak contacts between these other individuals. In this figure, we illustrate how their dispositions and social structure change as a result of individuals mimicking others more antisocial than themselves during the maturity gap, and individuals more pro-social than themselves after the maturity gap. This social mimicry causes the social network to change from one organized around the pro-social individuals at around age 5 to one organized around the antisocial individuals at around age 17, and thereafter back to one organized around the pro-social individuals in adulthood.
Table 1.
The list of parameters used in the simulations, their roles and their values.
For the parameters k, c, Δe, and γ, their values were increased and decreased by 50% (shown in parentheses) in the sensitivity analysis. The parameters N, T, and tp were kept fixed for our simulations.
Fig 2.
The peer network structure at age 5, age 17, and age 35 for a (a) rough network, (b) mild network, and various intervention scenarios ((c)-(f)).
In this figure, blue agents have the lowest antisocial levels while red agents have highest antisocial levels. Cyan, green, yellow, and orange agents have increasingly higher antisocial levels. In our intervention analysis, we first run the simulations repeatedly using the parameters shown in Table 1, but with different initial antisocial levels for the N = 30 agents. Depending on the initial antisocial levels, the simulations ended up with different numbers of life-course-persistent agents. We then pair (a) 50 initial antisocial levels with at least five agents having antisocial levels higher than 0.5 at the end of the simulations (rough networks) up with (b) 50 initial antisocial levels with no agent having antisocial level higher than than 0.5 at the end of the simulations (mild networks) for each of the four scenarios we analyzed. In scenario (c), we identify the least antisocial agent α1 in the rough network, and add an agent β1 with the same initial antisocial level as α1 to the mild network to simulate. In scenario (d), we identify the agent α2 ending up as marginally adolescence-limited in the rough network, and add an agent β2 with the same initial antisocial level as α2 to the mild network to simulate. In scenario (e), we identify the agent α3 ending up as marginally life-course-persistent in the rough network, and add an agent β3 with the same initial antisocial level as α3 to the mild network to simulate. Finally, in scenario (f), we identify the most antisocial agent α4 in the rough network, and add an agent β4 with the same initial antisocial level as α4 to the mild network to simulate.
Fig 3.
The (a) antisocial levels ei(t), (b) rewards ai(t), (c) costs bi(t), and (d) net rewards gi(t) of the 30 agents in a typical simulation.
In (b), the rewards increase linearly with time in the maturity gap, to reach a different constant for different agents in adulthood. In (c), we see how the costs change with time for all 30 agents. The cost for a given agent is a not a simple sigmoid because his antisocial level changes with time. We also see a gap opening up between a group of agents with nearly constant cost and a group of agents with increasing cost near the end of the simulation. This gap is also seen in (d) the net reward, but is most pronounced in (a) the antisocial level.
Fig 4.
The peer network structure at (a) t = 1 (age 7), (b) t = 100 (age 15), (c) t = 200 (age 23), and (d) t = 300 (age 32).
The boxes represent the agents, and the labels of the boxes are the indices of the agents. The agents are ordered clockwise from the least antisocial to the most antisocial, and the color of the boxes indicate the antisocial levels of the agents, according to the colorbar at the bottom. All agents are connected to each other, but here we show only the directed connections whose weights are larger than 0.3. In this figure, we see that (a) the initial connections at age 7 are strong only between pro-social agents, and agents mostly imitate agents less antisocial than themselves. During the maturity gap at (b) age 15, social mimicry amongst agents strengthens most connections, and agents mostly imitate agents more antisocial than themselves. After entering adulthood at (c) age 23, the agents split into two groups, a pro-social group where agents imitate those less antisocial than themselves, and an antisocial group where agents imitate those more antisocial than themselves. These two groups become more distinct as time goes on, as we can see in (d) at age 32.
Fig 5.
From Fig 4, we see that the pro-social group ended up with antisocial levels 0 < ei(t) < 0.5 while antisocial group ended up with antisocial levels 0.7 < ei(t) < 1.
Therefore, we can use an antisocial level between 0.5 and 0.7 as the threshold for delinquency, and count the numbers of agents with antisocial levels larger than 0.5, 0.6 and 0.7 respectively, as they evolve over time. We see that whatever threshold we use, we end up with an offending rate qualitatively similar to the one sketched by Moffitt in Fig 1.
Fig 6.
The distributions of the end-simulation adolescence-limited proportion when (a) k is increased or decreased by 50%, (b) c is increased or decreased by 50%, (c) γ is increased or decreased by 50%, and (d) Δe is increased or decreased by 50%, compared against the benchmark distribution F(p0) for the parameters shown in Table 1.
The bin size for binning the adolescence-limited fraction is 1/N where N = 30 is the number of agents in one simulation. As we can see, the simulation outcome is most sensitive to the parameter γ, which is the rate of increase of the reward for antisocial behavior. Our model is next most sensitive to Δe, the range of antisocial levels over which social mimicry can occur. Our model is very insensitive to the parameters k, which is how steeply the cost change with antisocial level and c, which is the proportionality constant that limits how fast agents can imitate each other.
Fig 7.
Intervention analysis comparing the final antisocial level of an agent in its native, rough network to his final antisocial level after he has been moved to a mild network.
For each of the following four scenarios, we ran 50 simulations. In (a), a guaranteed adolescence-limited agent is moved from the rough network to the mild network, and we see that there is little change in his final antisocial level with or without intervention. In (b), a marginally adolescence-limited agent is moved from the rough network to the mild network, and we see a statistically significant reduction in his final antisocial level. In (c), a marginally life-course-persistent agent is moved from the rough network to the mild network. The reduction in final antisocial level is the largest in this scenario, even though there are cases where the intervention fails, and the agent remains life-course-persistent. Finally, in (d) we move a guaranteed life-course-persistent agent from the rough network to the mild network. Although in some cases we see reduction in the final antisocial level, the agent remains life-course-persistent.