Title Youth violence and interventions: Insights from a complex agent network model

Youth violence is a growing concern in Singapore. To address this complex social issue, we surveyed the psychology, social science, and criminology literature to identify a total of 11 intrinsic (familial, individual, school) and 2 extrinsic (peer) factors linked to youth violence, and also their interdependencies. We then developed a complex agent network model where each complex agent is represented by a complex factor network of the 13 factors along with youth violence, coupled to each other through the extrinsic factors to form a complex social network. We simulated the model using as initial conditions the results from a large-scale school-based survey of the factors and random social ties. We find factors in each complex agent evolving with time under the influences from other factors, and the social ties between agents evolving with time as a result of behavioral imitation between agents. We ran a sensitivity analysis on the model, to find that the model is most sensitive to the parameters linking (1) non-intact family, (2) delinquency in general, (3) school disengagement, (4) peer delinquency, and (5) friends in gang to gang involvement. We also ran a series of intervention scenario simulations, and our results show that it is critical to intervene early, and successful interventions work by tipping the balance between competing intrinsic and extrinsic factors. Mental health professionals and school counsellors can then apply this unique insight from the model to design more effective interventions.


Introduction
Youth violence is becoming a more serious and widespread problem around the world, 1,2 and we hear more and more reports of homicides and non-fatal assaults where the perpetrators are youths. 3,4In fact, most victims of youth delinquents are themselves adolescents. 3According to a world report prepared by Krug et al., 5 in 2000 alone there are around 199,000 youth homicides worldwide, meaning nine cases per 100,000 in the population.In the United States, the O±ce of Juvenile Justice and Delinquency Prevention (OJJDP) in the Department of Justice reported that the prevalence rate of youth gangs has increased signi¯cantly since declining from its peak in the early 2000s.In particular, data from the National Gang Center shows that gang involvement increased by 6%, and the number of gangs increased by 28% from 2002 to 2008. 6In other parts of the world, the situation is even more alarming.For every 100,000 people, the youth violence rate is 84.4 in Colombia and 50.2 in El Salvador.Meanwhile, in the Caribbean, the ¯gure is 41.8 for Puerto Rico.Apart from these generally unsafe places, the ¯gure is around 18.0 in the Russian Federation, and 28.2 in Albania of southeastern Europe. 5ne would think that youth violence is a problem that does not concern Singapore.After all, the Economist Intelligence Unit ranked Singapore as the second safest city in the world (behind Tokyo) in 2015. 7However, according to the annual crime brief released by the Singapore Police Force, 8 the number of youths arrested increased from 3,031 in 2013 to 3,094 in 2014.In particular, the number of youths arrested for rioting increased from 283 in 2013 to 322 in 2014.These increases may not seem signi¯cant to those trained in the physical sciences, but the Singapore Police Force would need to act to reassure the public, especially after a high-pro¯le case on 30 October 2010, where a 19-year-old male was fatally stabbed by four youths, aged between 18 and 21.The tertiary student su®ered 28 injuries to his head, neck, chest and limbs caused by choppers, knives and a screwdriver, according to a newspaper report. 9Following this incident, on 8 November 2010, a group of eight youths armed with parangs and metal rods assaulted another group of 20 youths.Six victims, aged between 14 and 20, were treated for injuries, according to records from the Singapore Police Force. 10,11More recently, on 1 October 2013, a 19-year-old male was slashed at a shopping mall.His attacker, believed to be in his 20s, was arrested at the scene.The victim was reportedly only quarreling with the assailant following a month-long dispute, before he was stabbed outside the mall. 12,13learly, violent youths and their behaviors impose a cost to society, not only in terms of lives lost and maimed, but also in real economic and social terms.5][16] Policing to a level where crime can be prevented is extremely costly, but prosecuting, incarcerating, and rehabilitating criminals after they ended up on the wrong side of the law is no cheaper.][19] Speci¯cally, the OJJDP Study Group on Very Young O®enders recommends integrated prevention programs at the school and community levels, combining behavior management, social competence building, channeling of energy into positive recreational activities, and mentoring. 20However, interventions are in general multimodal, [21][22][23][24] and their e®ectiveness are frequently di±cult to gauge. 25,26It is also time-intensive and labor-intensive to comprehensively evaluate programs and interventions.The complexity of this social problem makes modeling and simulation a valuable approach for testing the e±cacy of intervention strategies.
It is only very recently that we ¯nd modeling work on juvenile delinquency and juvenile crime in the literature.Understandably, the ¯rst few papers took a highly simpli¯ed approach to the problem, by treating gang membership and violent dispositions as infections, and building systems dynamics models to describe it.For example, Lee and Do developed a model whereby adolescents at risk are divided into four subpopulations: (1) susceptible (S), (2) gang members (G), (3) delinquent gang members (D), and (4) delinquents who are arrested (L). 27In their model, we ¯nd intrinsic transitions that depended only on the prevalence of the source subpopulations, for example, the intrinsic transition from D to L occurs at a rate proportional to D. We also ¯nd extrinsic transitions that depended on the interactions between the susceptible subpopulations and other subpopulations, for example, the extrinsic transition from S to G occurs at a rate proportional to SðG þ DÞ.Similar models were published by Peterson in 2008 28 and Sooknanan et al. in 2012. 29Like in all epidemic models, the behavior of such models are determined by the reproductive number R: if R > 1 the youth delinquency epidemic spreads, else if R < 1, youth violence dies out.The main drawbacks in these system dynamics models are (1) the homogeneous individuals within each subpopulation, and (2) the model parameters are phenomenological, di±cult to unpack into societal, familial, school, and personal factors a®ecting the psychological states of juvenile delinquents, and therefore the model outcomes cannot directly inform intervention strategies.
More recently, Munyo developed a multi-factor economic dynamics model of juvenile crime, where individuals weigh the expected returns from studying, working, and committing crimes. 30Whether they choose to commit crimes depend on the likelihood of being arrested, how easy it is to escape from prison, and what is the reward of crime.Crime is therefore the outcome of rational decision making.Munyo identi¯ed four major factors that explain the spate of juvenile crime in Uruguay: (1)  an anemic recovery of wages relative to total income, (2) the more lenient juvenile crime law, (3) the dramatic increase in the escape rate from juvenile correctional facilities, and (4) the outbreak of a paste cocaine epidemic.Basically, the increase in juvenile crime in Uruguay is blamed on a decrease in the cost associated with criminal activity, and the gains from crime outstripping rewards from legitimate sources.In this model, the subpopulations of agents (Study, Work, Crime þ Free, Crime þ Escape, Crime þ Sentenced) are heterogeneous.There are also internal variables like education level and crime pro¯ciency level that change with time depending on the decision made in the previous time step.However, parameters that determine which utility functions dominate decision making are still phenomenological.
In these modeling papers, the large body of work done on youth violence and juvenile delinquency in the literature has not been exploited.Therefore, in Sec. 2 of this paper, we describe a review of the psychology, social science, and criminology literature to identify familial, individual, school, and peer factors known to be associated with youth violence, the interdependencies between such factors, and complex social interactions between youths at risk.We then explain how the interactions between these factors, and their associations with youth violence can be incorporated into a network model of complex agents, i.e., a complex agent network model, or as a social network of agents whose internal states are described by a complex factor network, i.e., a complex network-of-networks model.In Sec. 3, we explain how we use results from a large-scale school survey as initial conditions for the psychological states of our simulated agents to make our study realistic.The model parameters we initialize randomly for the simulation.Because of the lack of complete incidence data for calibration, we performed a sensitivity analysis of the model instead, followed by an intervention analysis.In Sec. 4, we describe how the sensitivity analysis identi¯ed three intrinsic parameters (weighing the couplings between non-intact family, delinquency in general, and school disengagement with gang involvement) and two extrinsic parameters (weighing the couplings between peer delinquency and friends in gang with gang involvement) that the model is most sensitive to.We also describe the outcomes of various interventions targeting school engagement or gang involvement or both, for one individual or more, imposed at di®erent times.Our results suggest a critical need for interventions to be timely.We then conclude in Sec. 5.

Model
Traditionally, psychologists and criminologists working with case studies have focused on factors in°uencing violent tendencies in individuals.Because of its social nature, we argue that agent-based models are ideally suited to understanding the problems associated with youth violence. 31In the most sophisticated agent-based model we can imagine, each young individual would be represented by an arti¯cial intelligence agent.Within a setting that mimics the social environments the youths are in, agents would (1) set behavioral goals, (2) react to events and changes in the environment, and (3) interact with other agents.To build a realistic agent-based model that can do all these, we would need to ¯rst know how agents set goals, like showing up at school, meeting up with friends, join a gang, or challenge a rival gang to a ¯ght, etc.We also need to know how they react to events and changes in the environment, like comply with a new rule set by their schools, less pocket money, or having their fathers arrested by the police, etc.Finally, we need to know how they interact with other agents, like responding to a taunt by a fellow gang member, or responding to a taunt by a rival gang member, etc.There are thus too few previous studies on the numerous action items in the three categories as inputs for building such an agent-based model.
Alternatively, we can also build simple toy models, not with the goal of ¯tting to real-world data and make predictions, but to understand what fundamental features in such models can explain qualitatively what we observe in the real world.Physicists like to take this approach, because toy models are much easier to understand qualitatively and quantitatively.From this perspective, the simplest model of youth violence would be an Ising-like spin model (see Fig. 1(a)), where a normal agent would be represented by a down spin (S i ¼ À1) while a violent agent would be represented by an up spin (S i ¼ þ1).We can then couple the spins into a regular Here, a white circle represents a down spin (S i ¼ À1, normal agent) while a black circle represents an up spin (S i ¼ þ1, violent agent), and blue links represent interactions between them.(b) A complex network of complex networks.Again, a white circle represents a down spin (S i ¼ À1, normal factor) and a black circle represents an up spin (S i ¼ þ1, violent factor), but we distinguish between two types of links: red links between factors within an individual agent, and blue links between individual agents.
lattice, or more `realistically' into a complex social network.To mimic the small number of violent incidents in real life, and how violence tends to spread, we can make the coupling between spins ferromagnetic, and have the system in a symmetry broken state (hSi % À1) below the critical temperature, or have the system to be in an external downward ¯eld (so that most of the spins are down).Either way we can study how an up spin nucleate a transient up-spin domain, or one that continues growing.
Unfortunately, whether we choose to interpret S i ¼ þ1 as individual violent incidents, or a violent individual, the Ising-like model described above is unrealistic in the sense that a spin switches quickly from S i ¼ þ1 to S i ¼ À1.In the real world, it is possible to ¯nd a normal individual quickly turning violent, but most violent individuals remain violent for a long time unless they undergo counseling or psychiatric therapy.To modify the Ising-like model to make it more realistic, we therefore have to assume that individuals are not simple spins, but have complex internal structures.This can be done by replacing the Ising spin S i with a complex network of spins (see Fig. 1(b)).In this model, we can introduce heterogeneity at two levels.First, red links between factors within an individual agent are strong, while blue links between individual agents are weak.Second, the average strengths of red links can vary from individual to individual.An agent with many violent factors is violent, whereas an agent with few violent factors is normal.More importantly, a violent agent can remain violent even if it is surrounded by normal agents, if its internal links are stronger than its external links.][38][39] Because we are interested in testing interventions, we impose a deterministic di®erence equation dynamics on network-of-networks model.In Sec.2.1, we will describe the factor network model of a complex agent, explaining what each of the factors means, and providing references to studies that showed the interdependencies between factors.We will then describe in Sec.2.2, how complex agents can be coupled together to form a complex social network of factor networks, where we ¯nd links from the gang involvement and delinquency in general factors in one agent to the gang involvement factor in another agent.

The factor network model of a complex agent
Through a review of the psychology, social science, and criminology literature, we ¯nd that many factors have been suggested, but only 11 intrinsic factors and 2 extrinsic factors have been linked to youth violence at levels that are statistically signi¯cant.These are listed in Table 1 along with their symbols, and will be the variables in our model.For example, DG i ðtÞ is the level of delinquency in general for agent i at time t.In the literature, these factors are measured using a variety of scales.Selecting the appropriate scale for a factor depends on the purpose and design of the study.To keep our simulations simple to interpret, we restrict all variables to the unit interval ½0; 1.We can think of this as normalization of the di®erent scales used for di®erent variables (which is indeed what we did in Sec.3.1), or as the intensity of each variable over a cross-section of indicators, or as some fuzzy truth level for each variable.
If the factors identi¯ed in the literature were independent, we would obtain the model shown in Fig. 2(a).However, we ¯nd in the literature that these factors are interdependent.In Table 2, we list these interdependencies, and the references con¯rming the statistical signi¯cance of these connections.Once these interdependencies between factors are included, we obtain the complex factor network shown in Fig. 2(b).Now, because the factors are interdependent, it is natural to wonder whether there are factors that are dominant, because they in°uence youth violence directly as well as indirectly, and factors that are redundant, because their in°uence on youth violence can be attributed to the in°uence on them by other factors.To check this, we perform singular value decomposition on the interdependency matrix A, whose matrix elements are Að; Þ ¼ 1 if factor in°uences factor , and Að; Þ ¼ 0 otherwise.We ¯nd 10 nonzero singular values, and as shown in the left panel of Fig. 3, the ¯rst singular value is signi¯cantly larger than the rest.In the right panel of Fig. 3, we examine the ¯rst principal component comprising the most in°uential factors and also the ¯rst principal component comprising the most in°uenced factors.Judging from their amplitudes, SD, PD, and DG are the most in°uential factors, followed by SA, PC, PV, PA, NF, FG.FC, AF, DO, VI that are the least in°uential factors.On the other hand, VI, GI, DG, DO, SA are the most    in°uenced factors, followed by PC, PA, SD, PD, FG.NF, PV, FC, AF that are factors that cannot be in°uenced.Therefore, we learn from the empirical research and theory that we have summarized that while some factors may be more important than others, there is no one dominant factor that can `explain' youth violence in general.Instead, we need to consider multi-factorial explanations of the kind we demonstrate in this paper.
In this complex factor network, a directed link from factor to factor means that ðtÞ contributes to the change ÁðtÞ ¼ ðt þ 1Þ À ðtÞ.For example, in Fig. 2(b), we see that there are arrows pointing from PC, PA, GI, SA, SD, and PD to DG.This means that for agent i, PC i ðtÞ, PA i ðtÞ, GI i ðtÞ, SA i ðtÞ, SD i ðtÞ and PD i ðtÞ all contribute to ÁDG i ðtÞ.Unfortunately, previous studies in the literature were not designed to probe the exact functional forms of these dependencies, so we have to make modeling choices along the way.Let us explain how these choices are made, with the intrinsic factor PC as an example.Speci¯cally, we would like the contribution W i ðPC; DGÞ to ÁDG i ðtÞ by PC to be large when PC is large, and small when PC is small.If this is only consideration we have to take care of, then the simplest model we can write down is However, we would also like to allow W i ðPC; DGÞ to be positive (to increase DG i ðtÞ) as well as negative (to decrease DG i ðtÞ).This is possible by introducing a reference level for PC i ðtÞ, so that W i ðPC; DGÞ is proportional to the di®erence between PC i ðtÞ and its reference level.Since PC i ðtÞ is restricted to the interval ½0; 1, if we set the reference level too close to 0, PC i ðtÞ will make mostly positive contributions.On the other hand, if we set the reference level too close to 1, PC i ðtÞ will make mostly negative contributions.Assuming there is no natural bias towards mostly positive or mostly negative contributions, we set the reference level to 1 2 , so that Finally, we would like DG i ðtÞ to be restricted to the interval ½0; 1 at all times.This means that its rate of increase must slow down when DG i ðtÞ is close to 1, and its rate of decrease must slow down when DG i ðtÞ is close to 0. For the contribution W i ðPC; DGÞ by PV i ðtÞ, this can be achieved by setting where wðPC; DGÞ is the parameter that determines how strongly PC in°uences DG.The overall change to DG i ðtÞ due to the intrinsic factors PC, PA, GI, SA, and SD is thus In our model, we assume the factor-to-factor parameters wð; Þ are time-independent, and the same for all individuals.

The social network model of complex agents
Now, PD also contributes towards ÁDG i ðtÞ.Unlike PC, PA, GI, SA, and SD, PD is an extrinsic factor, which means that PD i ðtÞ is made up of contributions from agents j who are connected to agent i.For example, if agent i is connected to agents j 1 , j 2 , j 3 , j 4 , and j 5 , then PD i ðtÞ will receive contributions from DG j 1 ðtÞ, DG j 2 ðtÞ, DG j 3 ðtÞ, DG j 4 ðtÞ, and DG j 5 ðtÞ, because these are the delinquency levels of agents j 1 , j 2 , j 3 , j 4 , and j 5 , who are the peers of agent i.In Fig. 2(c), we unpack the extrinsic factors PD and FG to show how DG in agent j impacts DG, GI, SA, DO, and VI in agent i, and vice versa.Let 0 ji ðtÞ 1 denote the strength of the in°uence agent j has on agent i.Then where wðPD; DGÞ is the parameter that determines how strongly PD in°uences DG.The overall change to DG i ðtÞ due to the intrinsic factors PC, PA, GI, SA, SD and the extrinsic factor PD i ðtÞ is thus To complete our model, we need to specify how the social ties ji ðtÞ are updated.There are very few studies specifying the nature of how social ties evolve between juvenile delinquents.The most relevant publication we can ¯nd is Mo±tt's dual taxonomy theory, 16 on which recently developed a model. 67In this theory, Mo±tt observes that young children are physically and psychologically immature, and have no desire to act like adults.Therefore, they gain no real bene¯ts by going against social norms to do adult things.On the other hand, teenagers are physically and psychologically mature, but society requires them to ¯nish school and get a job before they can assume adult roles.Hence, teenagers will feel that they have gained real bene¯ts by breaking the rules to achieve ¯nancial, material, emotional, and sexual independence early through other means.We modeled this bene¯t as a function that increases linearly with age until t p , which is either 17 18 years of age depending on the country, when individuals graduate from high school and enter the job market.After t p , the bene¯t becomes constant, because the adult rights are commensurate with adult responsibilities.In the legal systems of developed economies, minors are protected against severe adult punishments when they break the law.Therefore, we model the cost of delinquency as a function that is close to 0 for young individuals, changing sharply at t p to being close to 1 for adult individuals.
The other key aspect of Mo±tt's theory is how individuals imitate each other's behavior depending on the net bene¯t (bene¯t less cost).By simulating this model, we showed the dual taxonomy of life-course-persistent and adolescence-limited delinquents emerges automatically.
In this paper, we also would like to implement this imitation dynamics in the social ties between agents.However, unlike in our recent paper 67 where the cost and bene¯t of delinquency change over a period of about 15 years, in this paper our simulations are over two years at most, before the social `coming of age' t p .Hence, we can neglect the cost and bene¯t, and focus on whether an individual choose to imitate a more delinquent neighbor, or a less delinquent neighbor.It is reasonable to assume that this imitation is facilitated by the peer factors PD i ðtÞ and FG i ðtÞ.From Fig. 2, we see that PD is in°uenced by DG and PA.Therefore, we ¯rst compute the referenced linear combination of agent i.If W i ðPDÞ > 0, agent i is inclined to imitating those more delinquent than itself.Else if W i ðPDÞ < 0, agent i is inclined to imitating those less delinquent than itself.Consider then the social tie ji ðtÞ, which determines how strongly agent i is in°uenced by agent j.If W i ðPDÞ > 0 and DG j ðtÞ > DG i ðtÞ, agent i will imitate the more delinquent agent j, and will also increase ji ðtÞ to do so better.Else if W i ðPDÞ > 0 and DG j ðtÞ < DG i ðtÞ, agent i will decrease ji ðtÞ to prevent being in°uenced by the less delinquent agent j.Conversely, if W i ðPDÞ < 0 and DG j ðtÞ < DG i ðtÞ, agent i will increase ji ðtÞ to better imitate the less delinquent agent j.Else if W i ðPDÞ < 0 and DG j ðtÞ > DG i ðtÞ, agent i will decrease ji ðtÞ to prevent being in°uenced by the more delinquent agent j.As we can see, the agency to change the social tie ji ðtÞ lies with agent i alone, so that as the net change in the social tie going from agent j to agent i.With this, we have completely speci¯ed our complex agent network model of youth violence.

Simulation
In this section, we describe in Sec.3.1 how we use the results of a large-scale schoolbased survey as the initial conditions for our simulations.Unfortunately, in this large-scale school-based survey we were not able to perform measurements to estimate the factor-to-factor parameters wð; Þ and social ties ji ðtÞ.Therefore, in Sec.3.2, we explain how these are randomly sampled from the uniform distribution Uð0; 1Þ.Thereafter, we list in Sec.3.3 simulation parameters like the time step size and the total simulation time, before going on to explain in Sec.3.4 a sensitivity analysis done in lieu of calibration against real data, and describe the intervention analysis done in Sec.3.5.

Large-scale school-based survey
To better understand juvenile delinquency in Singapore, and understand which selfreported factors contributed to this phenomenon, we surveyed 1027 adolescents (58.2% male, 41.8% female) between 12 to 19 years of age (Grades 7 through 9) from three junior high schools in Singapore. 68These three schools randomly selected for the study are public schools situated within neighborhood residential estates and these schools are the most common types of schools in Singapore.The mean age of the adolescents was 14.10 years (SD ¼ 1.15).Adolescents' self-reported ethnic identi¯cation was as follows: 65.4% Chinese, 20.8% Malay, 6.7% Indian, 1.1% Eurasian, and 6% endorsed others (including all ethnic groups not listed).Adolescents' reports of their parents' marital status were as follows: 81% married, 2.9% separated, 12.9% divorced, 1.7% never married, and 1.5% widowed.This was an anonymous survey and therefore the data are non-identi¯able.The survey consisted of four self-report scale measures of psychopathy, reactive and proactive aggression, school engagement and delinquency.These measures are well established scales with strong properties.For details please refer to Ref. 64.In addition to scale measures, we also collected demographic data, and asked adolescents about whether they have ever been involved in gang ¯ghts, whether they have ever been on probation and whether they have ever belonged to a gang.
Institutional review board approval for this study was obtained from Nanyang Technological University, Singapore.Parental consent and adolescent assent was obtained prior to data collection.Participation was strictly voluntary and at any point in time, adolescents could refuse or discontinue participation with no penalty.Permission was sought and approval obtained from the Ministry of Education, Singapore, and the respective school principals, prior to conducting the research.
In this large-scale school-based survey, all 13 factors were measured.Of these factors, PC, DG, PA, and SD were assessed on multiple items.For example, PC was assessed on 20 items, each rated from 0 to 2. We convert these to a single 40-point scale for PC by adding up responses to the 20 items.This was also done for DG, PA, and SD, so that they are measured on 18-point, 24-point, and 95-point scales.AF is self-reported from 0 years of age to 19 years of age, the maximum age in the survey.The rest were simply recorded as 0 (absence) or 1 (presence).
To make our simulations realistic, we use the outcomes of this large-scale schoolbased survey as our initial conditions.After eliminating incomplete records, we ended up with the pro¯les of 1003 individuals.This ensures that we have a population of heterogenous agents, and since our simulation is initialized using real data, there is no need for calibration and validation where this population pro¯le is concerned.However, all factors must ¯rst be normalized, so that their values are between 0 and 1.Since a ¯rst o®ence at a younger age is generally more predictive of adolescent delinquency, AF is normalized such that its value is 0 if the reported age of ¯rst o®ence is 19, and 1 if the reported age of ¯rst o®ence is 0. In the survey, school engagement is measured instead of school disengagement.Therefore, we convert a high school engagement score into a low value for SD, and a low school engagement score into a high value for SD.

Factor-to-factor parameters and social ties
In the large-scale school-based survey, social ties ji ðtÞ can in principle be measured, but they were not.The factor-to-factor parameters wð; Þ cannot be measured through self reporting.Therefore, to run the simulations, we sample random social ties ji ðtÞ from the uniform distribution Uð0; 1Þ.In this sense, all agents in our simulations are linked, but not equally strongly.More importantly, since ji 6 ¼ ij , we can have very strong in°uence by agent i on j, but only very weak in°uence the other way round.For the preliminary study reported in this paper, we set wð; Þ ¼

Simulation parameters
In our discrete time simulation, one time step Át ¼ 1 equivalent to one day, and we simulate the network of complex agents for one year, T ¼ 365.
To ensure that the factors do not change too quickly, we further divide Á obtained from (1), (2), and (3) by N ¼ 1003.

Sensitivity analysis
In principle, to calibrate wð; Þ and ji , we would need a longitudinal study tracking a cohort over time, and have psychologists decide at each point in time who in the cohort has become violent.We would then simulate our model for the same duration, starting from the same initial conditions, and vary wð; Þ and ji until the simulated life histories agree with the actual life histories as best as they can.As we can imagine, such an undertaking is forbiddingly expensive and intrusive.Therefore, instead of trying to acquire data to calibrate our model, we perform a sensitivity analysis.Basically, if the simulation outcome is insensitive to changes in a parameter, then it is not important to know the precise value of this parameter.On the other hand, if the simulation outcome is sensitive to changes in a parameter, then this sensitivity is important information, even if we ultimately cannot know the parameter value.
Leaving aside the social ties ji , which we will not test, there are altogether 49 factor-to-factor parameters between the 14 variables (11 intrinsic, 2 extrinsic, and VI; see Fig. 2).In our sensitivity analysis, we change the values of the 49 parameters one at a time, keeping the rest unchanged.We do so in four di®erent ways: (1) increase wð; Þ by a factor of 2, (2) increase wð; Þ by a factor of 1.5, (3) decrease wð; Þ to 1 2 of its value, and (4) decrease wð; Þ to 1 3 of its value.We then compare the ¯nal number of violent agents in these four scenarios with the number of violent agents in the benchmark simulation, where wð; Þ ¼ 1 7 for all and .We distinguish between positive and negative changes to the ¯nal number of violent agents when we identify the parameter that our model is most sensitive to.

Intervention analysis
After identifying parameters wð; Þ that our model is most sensitive to, we proceeded to test intervention strategies that target these sensitive parameters, against intervention strategies that do not, and ultimately against the no-intervention benchmark.Speci¯cally, we target SD and/or GI, since our sensitivity analysis tells us that wðSD; GIÞ is a very important parameter.All in all, we tested six intervention scenarios: (1) Find all potentially violent agents, but only change SD ¼ 1 to SD ¼ 0.5 for one of them after one year, and simulate for another year; ).As we can see, in some factors (including DG), the factor value decreases rapidly to zero for most agents, and increases rapidly to one for a few agents.In other factors (including SD), the factor value increases and decreases more slowly.But while the factor value decreases to zero or increases to one for most agents, for some agents the factor value tends towards some non-zero constant value over the duration of the simulation.Finally, we ¯nd in a number of factors (including VI), the factor value ¯rst increases and then decreases, or ¯rst decreases and then increases.

Sensitivity analysis
From Fig. 4, we see that after simulating the model for one year, there are ¯ve agents with VI ¼ 1 (out of 1003 agents).This number of violent agents can vary a lot, depending on our choice of the reference level in (1).Indeed, this suggests further surveys to measure correlations between the factor values and whom adolescents deem worthy of imitation.In this context, our arbitrary choice of 1 2 as our reference level can be seen as `neutral', in the sense in our model agents are not biased in choosing to imitate those more violent than themselves, or less violent than themselves.Therefore, in our sensitivity analysis, we restrict ourselves to measuring the change in the number of violent agents when each of the parameters is changed.These changes are shown as color maps in Fig. 5.
We realized that the parameters that produce the largest changes in the number of violent agents when their values are increased (wð; Þ ! 3 2 wð; Þ and wð; Þ !2wðÞÞ are not the same parameters that produce the largest changes in the number of violent agents when their values are decreased (wð; Þ ! 1 2 wðÞ and wð; Þ ! 1 3 wðÞÞ.Therefore, we identify one set of sensitive parameters (wðDG; GIÞ, wðNF; GIÞ, and wðSD; GIÞÞ for increasing parameter value, and another set of sensitive parameters (wðPD; GIÞ and wðFG; GIÞÞ for decreasing parameter value (see Fig. 5).
By running this sensitivity analysis, we have e®ectively carried out a review and analysis of the psychology, social science, and criminology literature relevant to this topic area.Naturally, our present scope is limited to wð; Þ % 1 7 for all parameters, and we intend to perform a more comprehensive survey of the parameter space in  future.Nevertheless, within the narrow scope at present we observed that the simulation outcome (number of violent agents) is sensitive only to a small number of parameters.In future, we also intend to test how sensitive the simulation outcome is to unexplored links between the factors.For example, in the literature we did not ¯nd any evidence suggesting that a non-intact family (NF) may directly a®ect the school disengagement (SD).We could add this link to the model, and simulate it with varying values for w(NF, SD) to see how strongly it a®ects the simulation outcome.More interestingly, we ¯nd from our simulations that the number of violent agents increased when w(PD, GI) and w(FG, GI) were decreased.This result runs counter to our intuition that PD and FG should be positively correlated with GI.Scrutinizing our simulation results, we found that the ¯ve agents that were violent in the benchmark simulation continued to be violent after w(PD, GI) and w(FG, GI) were decreased.We therefore focused on the four agents that became violent only after w(PD, GI) and w(FG, GI) were decreased.Examining how their VI values evolve over time, and also the DG and GI values of their neighbors, we realized that the simulation outcome for these four agents is decided by the competition between intrinsic and extrinsic factors.More precisely, these four agents are surrounded by `good' neighbors.Therefore, when wðPD; GIÞ ¼ wðFG; GIÞ ¼ 1 7 , these `good' connections drive the VI values of these four agents to zero.When w(PD, GI) and w(FG, GI) are decreased, the in°uences from their `good' neighbors are diminished, and their own `bad' intrinsic factors drive their VI values to ultimately saturate.Fig. 5. (left) Color maps of the changes in the number of violent agents when the parameters wð; Þ are changed to 1/3, 1/2, 3/2, and 2 times their original values.In this ¯gure, the 14 Â 14 color maps are organized in the same way, the factor-to-factor matrix is in Table 2. (right) The three sensitive parameters for increasing parameter values (red) and two sensitive parameters for decreasing parameter values (blue).

Intervention analysis
In scenarios (1) to (4), we ¯rst ran a year-long simulation for the ¯ve agents to become fully violent, before intervening.It is therefore not surprising that these interventions were all ine®ective (no change in the ¯nal number of violent agents).In scenarios ( 5) and ( 6), we intervened right at the start, and managed to reduce the eventual number of violent agents from 5 to 3. Scenarios (1) to (5) are targeted scenarios, i.e., we need to know beforehand which agents will eventually become violent, and apply our intervention on them directly.We might object to such a requirement as being unrealistic, but in the untargeted scenario (6), we apply our intervention on all agents with SD > 0:5 to change SD to SD ¼ 0:5.We get the same outcome, but would have to put in more resources to target more agents.This tells us that intervention must be timely to be e®ective.
Finally, instead of intervening right at the start of the simulation, we tested how late into the simulation we can intervene with strategy (6), and still obtain acceptable results.We tried to do so at t ¼ 15; 30; 60; 90 days of the simulation, and found that only the intervention applied at t ¼ 15 days was successful.From Fig. 4, we understood why this is so: after about 20 days VI would have reached VI ¼ 0 for most agents, and VI ¼ 1 for the violent agents.Beyond this point, none of our intervention strategies work.Although we have taken care with the inputs to make our simulations realistic, we have also made a number of ad hoc modeling choices.Therefore, some outcomes in our simulations will not be realistic.For example, the time scale for the evolution of VI is arti¯cially short because of our choice for wð; Þ and .Certainly, we do not believe that individuals in the real world can go from mildly violent to very violent in a matter of 20 days, leaving us with an impossibly short period of 15 days to intervene.We can easily re-run our simulations with smaller wð; Þ and , so that the average time for VI to saturate matches the average time it takes in the real world for an individual to become violent.Other simulation outcomes will remain qualitatively the same over a broad range of parameters.These are much more likely to be also true in the real world, if we have got the model mostly right.For example, however long it takes for our simulated agents to eventually become violent, there is a ¯nite time window within which we must act to prevent this.We learn therefore from our intervention simulations that early intervention works best, a conclusion that is supported by a large body of empirical research.

Conclusions
In summary, to better understand the social phenomenon of youth violence and assess the e®ectiveness of di®erent intervention measures, we developed in this paper a complex agent network model that synthesize 13 statistically signi¯cant factors linked to youth violence (Table 1), and their interdependencies (Table 2).11 of these 13 factors are intrinsic (acting within an individual), while 2 of the 13 factors are extrinsic (acting between individuals).The model con¯rms empirical research and theory in psychology and criminology suggesting that no single factor is capable of explaining youth violence, and that a multi-factorial approach is necessary to gain deeper insights.We simulated how these factors evolve with time under each other's in°uences, with random initial social ties for the agents, and using the results of a large-scale school-based survey as initial conditions for the factors.We ran a sensitivity analysis of the model, to ¯nd it most sensitive to increases in the parameters coupling non-intact family and gang involvement (NF-GI), delinquency in general and gang involvement (DG-GI), and school disengagement and gang involvement (SD-GI), and decreases in the parameters coupling peer delinquency and gang involvement (PD-GI) and friends in gang and gang involvement (FG-GI).We further simulated ¯ve targeted intervention strategies and 1 untargeted intervention strategy, with various starting times, to ¯nd that for interventions to be e®ective, they must be imposed shortly after the start of the simulation.Again, this is consistent with the large body of empirical research on interventions, giving us con¯dence that the model can be useful for the analyses of other intervention and prevention measures.Our model can also guide future studies to discover factors (like a variety of positive in°uences that can reduce violent tendencies in youths) that can potentially in°uence youth violence, but have not been systematically tested.The model can also be modi¯ed to test deterrent measures like visiting juvenile detention centers and youth gang members who land themselves in hospitals, or the e®ects of counselling and other interventions to help delinquent youths foresee the future impacts of their present actions.

Fig. 1 .
Fig. 1. (a)A complex network of simple spins.Here, a white circle represents a down spin (S i ¼ À1, normal agent) while a black circle represents an up spin (S i ¼ þ1, violent agent), and blue links represent interactions between them.(b) A complex network of complex networks.Again, a white circle represents a down spin (S i ¼ À1, normal factor) and a black circle represents an up spin (S i ¼ þ1, violent factor), but we distinguish between two types of links: red links between factors within an individual agent, and blue links between individual agents.

Fig. 2 .
Fig. 2. (a) Factor model of the 11 intrinsic factors and 2 extrinsic factors linked to youth violence, without interdependencies between factors.(b) Complex network model of the 11 intrinsic factors and 2 extrinsic factors linked to youth violence, including interdependency links between factors.In this ¯gure, familial factors are colored yellow, individual factors are colored blue, school factors are colored green, peer factors are colored red, and youth violence (VI) is colored orange.(c) Unpacking the peer interactions between intrinsic factors of two individuals.In (b), we ¯nd arrows from PD to DG, GI, SA, DO, and VI, and an arrow pointing from DG to PD.This means that DG of agent j in°uences DG, GI, SA, DO, and VI of agent i, while DG of agent i in°uences DG of agent j.Similarly, in (b), we ¯nd arrows from FG to GI and VI.This means that GI of agent j in°uences GI and VI of agent i.

5 NF 8 NF
Fig. 3. (left) Singular values of the factor network shown in Fig. 2(b).The ¯rst singular value is sig-ni¯cantly larger than the other nine nonzero singular values.(right) First-principal-component amplitudes of (a) the most in°uential factors, and (b) the most in°uenced factors.

PD i ðtÞ ¼ X j6 ¼i ji ðtÞDG j
ðtÞ; FG i ðtÞ ¼ X j6 ¼i ji ðtÞGI j ðtÞ ð 5Þ are the sums of weighted contributions of DG and GI respectively from all neighbors j of agent i.For ÁDG i ðtÞ the contribution by PD i ðtÞ is thus W i ðPD; DGÞ ¼ wðPD; DGÞPD i ðtÞ ¼ wðPD; DGÞ X j6 ¼i ji ðtÞDG j ðtÞ:

Fig. 4 .
Fig. 4. The fast time evolution of (a) DG, (b) SA, (c) VI, and slow time evolution of (d) PC, (e) PA, (f) SD for N ¼ 1003 agents in our model.

Table 1 .
Factors linked to youth violence (VI) in the psychology, social science, and criminology literature and their symbols, organized according to whether they are familial, individual, school, or peer.

Table 2 .
Interdependencies between factors linked to youth violence.In this table, references are organized into an asymmetric matrix, such that the references in the th row, th column are for factor leading to factor .
¼ DG j ðtÞ À DG i ðtÞ GI i ðtÞW i ðFGÞ < 0. Combining the two contributions, we thus have Á j ðtÞ À DG i ðtÞW i ðPDÞ < 0 facilitates the direction of change desired by agent i, and ensures also that ji ðtÞ remains within the unit interval ½0; 1.Notice there is no reference level for ji ðtÞ, because the social tie is used to both increase and decrease delinquency in agent i. ji ðtÞ ¼ wðPD; ÞÁ PD ji ðtÞ þ wðFG; ÞÁ FG ji ðtÞ ð 14Þ Youth Violence and Interventions: Insights from a Complex Agent Network Model