Modeling Radicalization Phenomena in Heterogeneous Populations
Modeling Radicalization Phenomena in Heterogeneous Populations Serge Galam∗
arXiv:1508.05269v1 [physics.soc-ph] 21 Aug 2015
CEVIPOF Centre for Political Research, CNRS and Sciences Po, Paris, France Marco Alberto Javarone† Dept. Mathematics and Computer Science, Cagliari, Italy DUMAS - Dept. of Humanities and Social Sciences, Sassari, Italy (Dated: August 24, 2015)
Abstract The phenomenon of radicalization is investigated within an heterogeneous population composed of a core subpopulation, sharing a way of life locally rooted, and a recently immigrated subpopulation of different origins with ways of life which can be partly in conflict with the local one. While core agents are embedded in the country prominent culture and identity, they are not likely to modify their way of life, which make them naturally inflexible about it. On the opposite, the new comers can either decide to live peacefully with the core people adapting their way of life, or to keep strictly on their way and oppose the core population, leading eventually to criminal activities. To study the corresponding dynamics of radicalization we introduce a 3-state agent model with a proportion of inflexible agents and a proportion of flexible ones, which can be either peaceful or opponent. Assuming agents interact via weighted pairs within a Lotka-Volterra like Ordinary Differential Equation framework, the problem is analytically solved exactly. Results shed a new light on the instrumental role core agents can play through individual activeness towards peaceful agents to either curb or inflate radicalization. Some hints are outlined at new possible public policies towards social integration.
∗
[email protected]
†
[email protected]
1
I.
INTRODUCTION
The phenomenon of radicalization [1] is of absolute interest in the context of criminality and terrorism. Despite of a great interest towards its dynamics from sociologists and socio-psychologists [1–4], this phenomenon still lacks of analytical and computational investigations. Notably, although the World Wide Web daily constitutes a huge source of data (i.e., Big Data), it is quite difficult to filter information about terrorist activities in social networks. Moreover, even in the presence of datasets characterizing terrorist dynamics, studying the evolution and the spreading of radicalization would be a rather hard task. Therefore, we suggest that an analytical approach, even if it requires a high level of abstraction from real scenarios, could constitute a first step towards a deeper comprehension of these complex social phenomena. In particular, our model of radicalization is found to provide possible study cases, and possible applications. Our approach subscribes to the modern field of sociophysics [5, 6], where scientists belonging to different communities, spanning from physics to computer science, and from social psychology to sociology [5–7], are developing various models inspired from physics to understand social complexity. Indeed, a good deal of authors investigated opinion spreading [8–10], language dynamics [6, 11], crowd behavior [6], criminal activities [12–15], and cultural dynamics [16] first using simple models as the voter model [17], and then enriching their models in order to add more particular descriptions, e.g., providing agent-based models with real-like features and behaviors as conformity, competitiveness, stubbornness, and more individual features. Most models consider two state variables when representing opinion dynamics or information spreading. This approach strongly simplifies real scenarios, but it has the advantage to afford analytical calculations and numerical simulations to investigate phenomena like phase transitions in social systems [18–20]. Moreover, agent-based models are usually implemented by embedding agents in continuous [21] (bi-dimensional) spaces and in network structures [11, 22–24]. In the proposed model, we consider a structureless heterogeneous population whose evolution is studied by using a Lotka-Volterra Ordinary Differential Equation framework. In particular, our population is composed of two subpopulations: a core subpopulation characterized by a shared way of life locally rooted, and an immigrated (two, three generations) subpopulation characterized by ways of life which can be strongly different from that of 2
the core subpopulation. Being part of the country prominent culture for a long time, core agents are not likely to modify their way of life, which naturally makes them inflexible with respect to their current way of life. On the other hand, new immigrants can either choose to live peacefully with the core people adapting to the local way of life, or choose to oppose the core population. As such, they can be considered as sensitive agents, which can shift from peaceful to opponent and vice-versa. It is worth to note that the opponent state may lead to criminal activities. Accordingly, to study the corresponding dynamics of radicalization within above heterogeneous population, we introduce a simple 3-state agent model with fixed proportions of inflexible (core subpopulation) and flexible (immigrant subpopulation) agents. In terms of real scenarios, we can think about opponent agents as a tiny minority of anti-western terrorists[25] and their passive supporters [26]. In these cases, governments put into practice social strategies to fight criminal activities and terrorism, while terrorists usually follow strong and fascinating ideals, often based on cultural and religious motivations. In the proposed model, those characters (i.e., social strategies and strength of opponents’s ideals) are represented by numerical parameters. Solving the model shows that adequate individual strategies must adjust to the degree of activeness [27] of opponent agents to control the phenomenon of radicalization in a mixed population, i.e., where both inflexible, peaceful and opponent agents co-exist. In particular, radicalization is found to emerge with different degrees of intensity depending on the equilibrium constant value reached by the opponents. Two different parameters are defined to assess the degree of radicalization invasiveness within an heterogeneous population. The results shed a new light on the instrumental role core agents can play through individual activeness towards peaceful agents to either curb or inflate radicalization. The required personal involvement is a function of both the majority or minority status of the core subpopulation and the degree of activeness of opponents. Some hints are outlined at new possible public policies towards social integration within heterogeneous neighborhoods.
II.
THE MODEL
According to the scenario we aim to study above described (i.e., radicalization and criminal activities), we consider a system with N interacting agents divided into the following categories: inflexible (I), peaceful (P ) and opponent (O). Each category refers to a different 3
behavior or feeling. Then, inflexible and opponent agents have behaviors mapped to states s = ±1, while peaceful agents have a behavior mapped to the state s = 0. As their name suggests, inflexible agents never change state (see also [28]), while peaceful and opponent agents may shift state from one to another over time. Notably, opponent agents may become peaceful and peaceful agents may become opponent. Hence, nor peaceful nor opponent agents may assume the state of inflexible agents. In this scenario, inflexible agents aim to reduce to zero the amount of opponent agents, by following social strategies. Now, be α a parameter representing one of these social strategies defined to increase the number of peaceful agents (recalling that opponent and peaceful agents never become inflexible), and be β a parameter representing the strength of the ideal or feeling, promoted by opponent agents, able to turn peaceful agents into opponent ones. In doing so and considering pairwise interactions, the associated dynamics can be described by the following system of equations dσP (t) = ασI σO (t) − βσO (t)σP (t) dt dσO (t) (1) = βσO (t)σP (t) − ασI σO (t) dt σ + σ (t) + σ (t) = 1 I P O where σI is the constant density of inflexible agents, while σO (t) and σP (t) are the respective densities of peaceful and opponent agents at time t. Dealing with densities the third equation of system 1 allows to reduce the number of ODEs to one equation. In particular, choosing the peaceful agents density σP (t) we get dσP (t) = ασI (1 − σI − σP (t)) − β(1 − σI − σP (t))σP (t) dt Remarkably, the equilibrium state of the population can be obtained from
(2) dσP (t) dt
= 0, which
writes βσP (t)2 − (ασI + β(1 − σI ))σP (t) + ασI (1 − σI )) = 0
(3)
The two solutions of equation 3 read < σP >=
ασI + β(1 − σI ) ±
p [ασI + β(1 − σI )]2 − 4βασI (1 − σI ) 2β
where < σP > is the equilibrium value of peaceful agents. Those values simplify to 1 − σ I ≡ p 1 < σP >= α σI ≡ p2 β 4
(4)
(5)
FIG. 1. Evolution of the system on varying initial conditions: a σI = 0.3, and σO = 0.3, α = 1.0, β = 1.0. b σI = 0.3, and σO = 0.3, α = 1.0, β = 2.0. c σI = 0.3, and σO = 0.3, α = 4.0, β = 2.0. d σI = 0.28, and σO = 0.02, α = 0.5, β = 0.5. e σI = 0.3, and σO = 0.3, α = 1.0, β = 5.0. f σI = 0.1, and σO = 0.4, α = 4.0, β = 2.0. g σI = 0.1, and σO = 0.4, α = 12.0, β = 2.0. h σI = 0.1, and σO = 0.4, α = 22.0, β = 2.0. i σI = 0.28, and σO = 0.7, α = 0.5, β = 0.5.
Indeed equation 2 can be solved analytically, to yield
σP (t) = p2 +
1−
p1 − p2 σP (0)−p1 β(p1 −p2 )t e σP (0)−p2
Figure 1 shows the evolution of the system on varying the initial conditions. 5
(6)
A.
Analysis of the Stability
Since we are dealing with a dynamical system, it is important to investigate its stability. Notably, we analyze the respective stability ranges for p1 and p2 : dσP dσP (σP ) ' (< σP >) + (σP − < σP >)λ dt dt where
dσP dt
(< σP >) = 0 and λ ≡
d2 σP | , dtdσP
(7)
we obtain
λ = −[ασI + β(1 − σI )] + 2βσP
(8)
Therefore, for respective values p1 , p2 we obtain λ1 = −ασI + β(1 − σI ) = β(p1 − p2 )
(9)
λ2 = ασI − β(1 − σI ) = −β(p1 − p2 ) Stability being achieved for λ < 0, equation 9 shows that p1 (p2 ) is stable when p1 < p2 (p1 > p2 ). Accordingly we get two stable regimes: p1 ≤ p2 ⇔ σI ≥ Ic ⇒ {< σP >= p1 = 1 − σI , < σO >= 0} p1 ≥ p2 ⇔ σI ≤ Ic ⇒ {< σP >= p2 = α σI , < σO >= 1 − β with Ic ≡
β . α+β
σI Ic
(10) = p1 − p2 }
The first equation of system 10 highlights that in some conditions the
amount of opponent agents is equal to zero. Hence, we perform a further investigation to study under which conditions it is possible to avoid the phenomenon of radicalization (i.e., by reaching to the equilibrium state < σO >= 0).
B.
Extinction processes
From above results, radicalization can be totally thwarted if σI ≥ Ic . Accordingly, given σI and β, the individual involvement for the inflexible population in striking up with individual opponents must be at least at a level α > β(
1 − 1) σI
(11)
Therefore, as seen from equation 11, the larger σI the less effort is required from the inflexible population. However the more active are the opponents (i.e., larger β) the more involvement 6
FIG. 2. The curve
1 σN
− 1 is shown as a function of σN . All cases for which the value of
α β
is above
the curve (yellow, clear) correspond to situations for which radicalization is totally thwarted. When the value of
α β
is below the curve (blue, dark) radicalization takes place on a permanent basis.
is required. To visualize the multiplicative factor by which α must overpass β, it is worth to draw the curve
1 σI
− 1 as a function of σI as shown in Figure 2. Equation 11 shows that
in order to prevent radicalization, the inflexible agents’s involvement must be larger than that of opponents as soon as α > β, which means when
1 σI
− 1 > 1, i.e., when σI < 12 .
Therefore, with the aim to eradicate the phenomenon of radicalization, we need to consider three different cases: 1) σI < 12 , 2) σI = 12 , and 3) σI > 21 . Case 1. For values σI < 12 , if α = β the equilibrium condition entails that < σP >= σI (and < σO >= 1 − 2σI ). If α > β, we can reach the extinction of opponent agents as
σI Ic
= 1.
Obviously, if α < β opponent agents strongly prevail in the system. Case 2. For σI = 21 , for α ≥ β opponent agents extinct. Instead, for α < β peaceful and opponent agents coexist, and the former disappear as β → ∞ (i.e., < σO >→ σI ). Case 3. For σI > 12 , opponent agents need very high values of β (compared to α) to survive in the population. In particular, opponent agents survive for values of β ≥
ασI . (1−σI )
Even in
this case, for β → ∞, the amount of peaceful agents fall to zero, although opponent agents cannot prevail due to the majority of inflexible agents. 7
C.
Degree of radicalization
In order to asses the degree of radicalization in a population, we can introduce two parameters: ζ and η. The former is defined to evaluate the fraction of opponent agents among flexible agents, while the latter (i.e., η) to evaluate the ratio between opponent and inflexible agents. Therefore, ζ represents the relative ratio of opponents among flexible agents, and η gives a measure about the real power or opponents agents in a population. A high value of ζ (i.e., close to 1) in a population with σI >> 0.5 tells that strategies to fight radicalization are too weak but, at the same time, opponents are few. Therefore, in this example, governments should take an action, even if the situation seems still under control. On the other hand, a low value of ζ (i.e., close to 0) going with a high value of η, represents an alarming situations, as although there are only few opponents among flexible agents, opponents are more than inflexible. With the aim to offer these measures, ζ and η have been defined as follows ζ =
σO 1−σI
η =
σO σI
(12)
therefore, recalling that σO = 1−σI −σP and having solved analytically σP (t) (see 6), we are able to compute values of both parameters, ζ and η, at equilibrium and on varying the initial conditions —see Figure 3. It is worth to note that the parameter ζ, as defined in 12, has a range in [0, 1]. Notably, ζ = 0 means that, at equilibrium, there are no opponent agents in the population, while ζ = 1 means that all flexible agents became opponents. On the other hand, the parameter η has potentially an unlimited range, from 0 to ∞ (in the case σI be very close to 0, and σO be close to 1). To conclude, we want to emphasize the meaningful role of the two parameters ζ and η, as they represent potentially a way to quantify in which extent radicalization phenomena are taking place in a population.
III.
DISCUSSION AND CONCLUSION
The phenomenon of radicalization is a phenomenon of central interest to social psychology [29] and policy makers, in particular because it is often related to terrorism and, more in general, to criminal activities as shown by the recent anti-western trends [25]. In the proposed model we have considered a simple scenario based on an heterogeneous population 8
FIG. 3. Radicalization degree quantified according to the parameters ζ and η, on varying initial conditions: a σI = 0.3, and σO = 0.3, α = 1.0, β = 1.0. b σI = 0.3, and σO = 0.3, α = 2.0, β = 1.0. c σI = 0.3, and σO = 0.3, α = 1.0, β = 2.0. d σI = 0.3, and σO = 0.3, α = 4.0, β = 1.0.
whose agents are provided with a state representing their behavior in relation to cultural traditions and ways of life characterizing a well defined local geographic area (e.g., a country). In particular, agents can be inflexible (i.e., those belonging to the core subpopulation), peaceful and opponent. Both peaceful and opponent agents may exchange states over time by interacting with agents from the whole population. For instance, in the context of antiwestern terrorism, core individuals can be mapped to inflexible agents. Remarkably, today the proportion of the core population in a geographical area may change more rapidly than in past as a result of an increasing illegal immigration. Therefore, it is relevant try to understand and even to predict in a general context, the evolution of a social radicalization 9
in terms of population dynamics. In order to represent the strategies of inflexible agents [30] implemented to support a peaceful coexistence among individuals of different cultures versus the strength of an opponent ideal, we introduced two numerical parameters (i.e., α and β). These parameters have a fundamental role from an analytical perspective although, in real scenarios, it may be difficult to identify and to quantify them. Nevertheless, the underlying message coming from our analytical results overcome this ‘limit’ when dealing with the real world, as it shows the risks related to a change of ratio of subpopulations within a given territory. It is worth to highlight that a de-mixing of the subpopulations may lead to higher degree of radicalization in presence of fewer opponents. Therefore, social integration strategies can really represent the best peaceful solution in order to avoid criminal scenarios. To summarize, we have identified a direct relation between on the one hand, the final state of a mixed population in terms of order or disorder phases, and on the other hand, the ratio between social strategies and the strength of opponents’s ideal. Moreover, we emphasize that the proposed model may in principle be applied also to criminal and terrorist scenarios in homogeneous populations, as it happened in the cases of Italian red brigades [31] and French revolution [32]. These two cases are concerned with homogeneous populations as both inflexible, peaceful and opponents belong to the local core population. Notably, in the former case (i.e., red brigades), inflexible agents represent individuals who respect laws and believe in institutions and governments, while individuals having a different behavior can fall in the mild category of peaceful agents or in extreme category of opponents (i.e., criminals). Instead, in the case of the French revolution, inflexibles represent the small proportion of French nobility, while the remaining part of the population is represented by peaceful and opponent agents. There, the extremely difficult life conditions fed the opponent ideals and the wide proportion of the sensible subpopulation assumed completely the state of opponent, giving rise to what is know as the revolution. Above last example allows to remark that although we mention criminal activities, we are not judging nor the motivations nor the ideals of opponent agents, as they can be considered negative (as in the case of anti-western terrorism) or positive (as in the case of the French revolution). Then, once again, our unique aim is to study the emergence and the evolution of radicalization processes. Finally, we suggest that further studies on this direction are required, in particular from a computational social science perspective, as it is should be possible to identify earlier traces (i.e., Big Data) and seeds of dangerous behaviors in social networks. Suitable tools to quantify their strength 10
are also required.
ACKNOWLEDGMENTS
MAJ would like to thank Fondazione Banco di Sardegna for supporting his work. This work was supported in part from a convention DGA-2012 60 0013 00470 75 01.
REFERENCES
[1] Borum, R.: Radicalization into Violent Extremism I: A Review of Social Science Theories. Journal of Strategic Security 4-4 7–36 (2011) [2] Thompson, R.L.: Radicalization and the Use of Social Media. Journal of Strategic Security 4-4 167–190 (2011) [3] Haines, H.H.: Black Radicalization and the Funding of Civil Rights. Social Problems 32-1 31–43 (1984) [4] Kruglanski, A.W., Gelfand, M.J., Belanger, J.J., Sheveland, A., Hetiarachchi, M., Gunaratna, R.: The Psychology of Radicalization and Deradicalization: How Significance Quest Impacts Violent Extremism. Advances in Political Psychology 35-1 (2014) [5] Galam, S.: Sociophysics: a review of Galam models. International Journal of Modern Physics C 19-3 409-440 (2008) [6] Castellano, C. and Fortunato, S. and Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81-2 591–646 (2009) [7] Buechel, B., Hellmann, T., Klobner, S.: Opinion dynamics and wisdom under conformity. Journal of Economic Dynamics and Control 52 240–257 (2015) [8] Sznajd-Weron, K. and Sznajd, J.: Opinion Evolution in Closed Community. International Journal of Modern Physics C 11-6 1157 (2000) [9] Javarone, M.A.: Social influences in opinion dynamics: the role of conformity. Physica A: Statistical Mechanics and its Applications, 2014
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[10] Javarone, M.A.: Networks strategies in election campaigns. Journal of Statistical Mechanics: Theory and Experiments, P08013 2014 [11] Javarone, M.A., Armano, G.: Emergence of Acronyms in a Community of Language Users. European Physical Journal - B, 86-11 474 2013 [12] d’Orsogna, M., and Perc, M.: Statistical physics of crime: A review. Phys. Life Rev. 12 1–21 (2015) [13] Galam S., The September 11 attack: A percolation of individual passive support. European Physical Journal B 26 Rapid Note 269-272 (2002) [14] Galam S., Global physics: from percolation to terrorism, guerilla warfare and clandestine activities. Physica A: Statistical Mechanics and its Applications 330 139-149 (2003) [15] Javarone, M.A., Galam, S.: Emergence of extreme opinions in social networks Lecture Notes on Computer Science Springer, 2015 [16] Gracia-Lazaro, C., Quijandria, F., Hernandez, L., Floria, L.M., Moreno, Y.: Co-evolutionary network approach to cultural dynamics controlled by intollerance. Physical Review E, 84-6 067101 2011 [17] Sood, V. and Redner, S.: Voter Model on Heterogeneous Graphs. Phys. Rev. Lett. 94-17 178701 (2005) [18] Castellano, C., Marsili, M., Vespignani, A.: Nonequilibrium Phase Transition in a Model for Social Influence. Physical Review Letters 85-16 3536–3539 (2000) [19] Holme, P. and Newman, M.E.J.: Nonequilibrium phase transition in the coevolution of networks and opinions. Phys. Rev. E 74-5 056108 (2006) [20] Slanina, F., Lavicka., H.: Analytical results for the Sznajd model of opinion formation. The European Physical Journal - B 35-2 279–288 (2003) [21] Antonioni, A., Tomassini, M., Buesser, P.: Random Diffusion and Cooperation in Continuous Two-Dimensional Space. Journal of Theoretical Biology 344 (2014) [22] Perc, M., Gomez-Gardenes, J., Szolnoki, A., Floria, L.M., Moreno, Y.: Evolutionary dynamics of group interactions on structured populations: a review. Journal of The Royal Society Interface, 10 20120997 2013 [23] Szolnoki, A., Perc, M.: Conformity enhances network reciprocity in evolutionary social dilemmas. J. R. Soc. Interface 12 20141299 (2015)
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[24] Wang, Z., Wang, L., Szolnoki, A., and Perc, M.: Evolutionary games on multilayer networks: A colloquium. Eur. Phys. J. B 88 124 (2015) [25] Islamic Terrorism. http://en.wikipedia.org/wiki/Islamic terrorism [26] Galam, S, Mauger, A.: On reducing terrorism power: a hint from physics. Physica A: Statistical Mechanics and its Applications 323 695–704 (2003) [27] Qian S., Liu Y., Galam S.: Activeness as a key to counter democratic balance. Physica A: Statistical Mechanics and its Applications 432 187-196 (2015) [28] Galam, S, Jacobs, F.: The role of inflexible minorities in the breaking of democratic opinion dynamics. Physica A: Statistical Mechanics and its Applications 381 366–376 (2007) [29] Aronson, E., Wilson, T.D. and Akert, R.M.: Social Psychology. Pearson Ed, (2006) [30] Kelling, G.,L., Coles, C.M.: Fixing Broken Windows: Restoring Order and Reducing Crime in Our Communities Simon and Schuster (1997) [31] Red Brigades. http://en.wikipedia.org/wiki/Red Brigades [32] French Revolution. http://en.wikipedia.org/wiki/French Revolution
13
arXiv:1508.05269v1 [physics.soc-ph] 21 Aug 2015
CEVIPOF Centre for Political Research, CNRS and Sciences Po, Paris, France Marco Alberto Javarone† Dept. Mathematics and Computer Science, Cagliari, Italy DUMAS - Dept. of Humanities and Social Sciences, Sassari, Italy (Dated: August 24, 2015)
Abstract The phenomenon of radicalization is investigated within an heterogeneous population composed of a core subpopulation, sharing a way of life locally rooted, and a recently immigrated subpopulation of different origins with ways of life which can be partly in conflict with the local one. While core agents are embedded in the country prominent culture and identity, they are not likely to modify their way of life, which make them naturally inflexible about it. On the opposite, the new comers can either decide to live peacefully with the core people adapting their way of life, or to keep strictly on their way and oppose the core population, leading eventually to criminal activities. To study the corresponding dynamics of radicalization we introduce a 3-state agent model with a proportion of inflexible agents and a proportion of flexible ones, which can be either peaceful or opponent. Assuming agents interact via weighted pairs within a Lotka-Volterra like Ordinary Differential Equation framework, the problem is analytically solved exactly. Results shed a new light on the instrumental role core agents can play through individual activeness towards peaceful agents to either curb or inflate radicalization. Some hints are outlined at new possible public policies towards social integration.
∗
[email protected]
†
[email protected]
1
I.
INTRODUCTION
The phenomenon of radicalization [1] is of absolute interest in the context of criminality and terrorism. Despite of a great interest towards its dynamics from sociologists and socio-psychologists [1–4], this phenomenon still lacks of analytical and computational investigations. Notably, although the World Wide Web daily constitutes a huge source of data (i.e., Big Data), it is quite difficult to filter information about terrorist activities in social networks. Moreover, even in the presence of datasets characterizing terrorist dynamics, studying the evolution and the spreading of radicalization would be a rather hard task. Therefore, we suggest that an analytical approach, even if it requires a high level of abstraction from real scenarios, could constitute a first step towards a deeper comprehension of these complex social phenomena. In particular, our model of radicalization is found to provide possible study cases, and possible applications. Our approach subscribes to the modern field of sociophysics [5, 6], where scientists belonging to different communities, spanning from physics to computer science, and from social psychology to sociology [5–7], are developing various models inspired from physics to understand social complexity. Indeed, a good deal of authors investigated opinion spreading [8–10], language dynamics [6, 11], crowd behavior [6], criminal activities [12–15], and cultural dynamics [16] first using simple models as the voter model [17], and then enriching their models in order to add more particular descriptions, e.g., providing agent-based models with real-like features and behaviors as conformity, competitiveness, stubbornness, and more individual features. Most models consider two state variables when representing opinion dynamics or information spreading. This approach strongly simplifies real scenarios, but it has the advantage to afford analytical calculations and numerical simulations to investigate phenomena like phase transitions in social systems [18–20]. Moreover, agent-based models are usually implemented by embedding agents in continuous [21] (bi-dimensional) spaces and in network structures [11, 22–24]. In the proposed model, we consider a structureless heterogeneous population whose evolution is studied by using a Lotka-Volterra Ordinary Differential Equation framework. In particular, our population is composed of two subpopulations: a core subpopulation characterized by a shared way of life locally rooted, and an immigrated (two, three generations) subpopulation characterized by ways of life which can be strongly different from that of 2
the core subpopulation. Being part of the country prominent culture for a long time, core agents are not likely to modify their way of life, which naturally makes them inflexible with respect to their current way of life. On the other hand, new immigrants can either choose to live peacefully with the core people adapting to the local way of life, or choose to oppose the core population. As such, they can be considered as sensitive agents, which can shift from peaceful to opponent and vice-versa. It is worth to note that the opponent state may lead to criminal activities. Accordingly, to study the corresponding dynamics of radicalization within above heterogeneous population, we introduce a simple 3-state agent model with fixed proportions of inflexible (core subpopulation) and flexible (immigrant subpopulation) agents. In terms of real scenarios, we can think about opponent agents as a tiny minority of anti-western terrorists[25] and their passive supporters [26]. In these cases, governments put into practice social strategies to fight criminal activities and terrorism, while terrorists usually follow strong and fascinating ideals, often based on cultural and religious motivations. In the proposed model, those characters (i.e., social strategies and strength of opponents’s ideals) are represented by numerical parameters. Solving the model shows that adequate individual strategies must adjust to the degree of activeness [27] of opponent agents to control the phenomenon of radicalization in a mixed population, i.e., where both inflexible, peaceful and opponent agents co-exist. In particular, radicalization is found to emerge with different degrees of intensity depending on the equilibrium constant value reached by the opponents. Two different parameters are defined to assess the degree of radicalization invasiveness within an heterogeneous population. The results shed a new light on the instrumental role core agents can play through individual activeness towards peaceful agents to either curb or inflate radicalization. The required personal involvement is a function of both the majority or minority status of the core subpopulation and the degree of activeness of opponents. Some hints are outlined at new possible public policies towards social integration within heterogeneous neighborhoods.
II.
THE MODEL
According to the scenario we aim to study above described (i.e., radicalization and criminal activities), we consider a system with N interacting agents divided into the following categories: inflexible (I), peaceful (P ) and opponent (O). Each category refers to a different 3
behavior or feeling. Then, inflexible and opponent agents have behaviors mapped to states s = ±1, while peaceful agents have a behavior mapped to the state s = 0. As their name suggests, inflexible agents never change state (see also [28]), while peaceful and opponent agents may shift state from one to another over time. Notably, opponent agents may become peaceful and peaceful agents may become opponent. Hence, nor peaceful nor opponent agents may assume the state of inflexible agents. In this scenario, inflexible agents aim to reduce to zero the amount of opponent agents, by following social strategies. Now, be α a parameter representing one of these social strategies defined to increase the number of peaceful agents (recalling that opponent and peaceful agents never become inflexible), and be β a parameter representing the strength of the ideal or feeling, promoted by opponent agents, able to turn peaceful agents into opponent ones. In doing so and considering pairwise interactions, the associated dynamics can be described by the following system of equations dσP (t) = ασI σO (t) − βσO (t)σP (t) dt dσO (t) (1) = βσO (t)σP (t) − ασI σO (t) dt σ + σ (t) + σ (t) = 1 I P O where σI is the constant density of inflexible agents, while σO (t) and σP (t) are the respective densities of peaceful and opponent agents at time t. Dealing with densities the third equation of system 1 allows to reduce the number of ODEs to one equation. In particular, choosing the peaceful agents density σP (t) we get dσP (t) = ασI (1 − σI − σP (t)) − β(1 − σI − σP (t))σP (t) dt Remarkably, the equilibrium state of the population can be obtained from
(2) dσP (t) dt
= 0, which
writes βσP (t)2 − (ασI + β(1 − σI ))σP (t) + ασI (1 − σI )) = 0
(3)
The two solutions of equation 3 read < σP >=
ασI + β(1 − σI ) ±
p [ασI + β(1 − σI )]2 − 4βασI (1 − σI ) 2β
where < σP > is the equilibrium value of peaceful agents. Those values simplify to 1 − σ I ≡ p 1 < σP >= α σI ≡ p2 β 4
(4)
(5)
FIG. 1. Evolution of the system on varying initial conditions: a σI = 0.3, and σO = 0.3, α = 1.0, β = 1.0. b σI = 0.3, and σO = 0.3, α = 1.0, β = 2.0. c σI = 0.3, and σO = 0.3, α = 4.0, β = 2.0. d σI = 0.28, and σO = 0.02, α = 0.5, β = 0.5. e σI = 0.3, and σO = 0.3, α = 1.0, β = 5.0. f σI = 0.1, and σO = 0.4, α = 4.0, β = 2.0. g σI = 0.1, and σO = 0.4, α = 12.0, β = 2.0. h σI = 0.1, and σO = 0.4, α = 22.0, β = 2.0. i σI = 0.28, and σO = 0.7, α = 0.5, β = 0.5.
Indeed equation 2 can be solved analytically, to yield
σP (t) = p2 +
1−
p1 − p2 σP (0)−p1 β(p1 −p2 )t e σP (0)−p2
Figure 1 shows the evolution of the system on varying the initial conditions. 5
(6)
A.
Analysis of the Stability
Since we are dealing with a dynamical system, it is important to investigate its stability. Notably, we analyze the respective stability ranges for p1 and p2 : dσP dσP (σP ) ' (< σP >) + (σP − < σP >)λ dt dt where
dσP dt
(< σP >) = 0 and λ ≡
d2 σP | , dtdσP
(7)
we obtain
λ = −[ασI + β(1 − σI )] + 2βσP
(8)
Therefore, for respective values p1 , p2 we obtain λ1 = −ασI + β(1 − σI ) = β(p1 − p2 )
(9)
λ2 = ασI − β(1 − σI ) = −β(p1 − p2 ) Stability being achieved for λ < 0, equation 9 shows that p1 (p2 ) is stable when p1 < p2 (p1 > p2 ). Accordingly we get two stable regimes: p1 ≤ p2 ⇔ σI ≥ Ic ⇒ {< σP >= p1 = 1 − σI , < σO >= 0} p1 ≥ p2 ⇔ σI ≤ Ic ⇒ {< σP >= p2 = α σI , < σO >= 1 − β with Ic ≡
β . α+β
σI Ic
(10) = p1 − p2 }
The first equation of system 10 highlights that in some conditions the
amount of opponent agents is equal to zero. Hence, we perform a further investigation to study under which conditions it is possible to avoid the phenomenon of radicalization (i.e., by reaching to the equilibrium state < σO >= 0).
B.
Extinction processes
From above results, radicalization can be totally thwarted if σI ≥ Ic . Accordingly, given σI and β, the individual involvement for the inflexible population in striking up with individual opponents must be at least at a level α > β(
1 − 1) σI
(11)
Therefore, as seen from equation 11, the larger σI the less effort is required from the inflexible population. However the more active are the opponents (i.e., larger β) the more involvement 6
FIG. 2. The curve
1 σN
− 1 is shown as a function of σN . All cases for which the value of
α β
is above
the curve (yellow, clear) correspond to situations for which radicalization is totally thwarted. When the value of
α β
is below the curve (blue, dark) radicalization takes place on a permanent basis.
is required. To visualize the multiplicative factor by which α must overpass β, it is worth to draw the curve
1 σI
− 1 as a function of σI as shown in Figure 2. Equation 11 shows that
in order to prevent radicalization, the inflexible agents’s involvement must be larger than that of opponents as soon as α > β, which means when
1 σI
− 1 > 1, i.e., when σI < 12 .
Therefore, with the aim to eradicate the phenomenon of radicalization, we need to consider three different cases: 1) σI < 12 , 2) σI = 12 , and 3) σI > 21 . Case 1. For values σI < 12 , if α = β the equilibrium condition entails that < σP >= σI (and < σO >= 1 − 2σI ). If α > β, we can reach the extinction of opponent agents as
σI Ic
= 1.
Obviously, if α < β opponent agents strongly prevail in the system. Case 2. For σI = 21 , for α ≥ β opponent agents extinct. Instead, for α < β peaceful and opponent agents coexist, and the former disappear as β → ∞ (i.e., < σO >→ σI ). Case 3. For σI > 12 , opponent agents need very high values of β (compared to α) to survive in the population. In particular, opponent agents survive for values of β ≥
ασI . (1−σI )
Even in
this case, for β → ∞, the amount of peaceful agents fall to zero, although opponent agents cannot prevail due to the majority of inflexible agents. 7
C.
Degree of radicalization
In order to asses the degree of radicalization in a population, we can introduce two parameters: ζ and η. The former is defined to evaluate the fraction of opponent agents among flexible agents, while the latter (i.e., η) to evaluate the ratio between opponent and inflexible agents. Therefore, ζ represents the relative ratio of opponents among flexible agents, and η gives a measure about the real power or opponents agents in a population. A high value of ζ (i.e., close to 1) in a population with σI >> 0.5 tells that strategies to fight radicalization are too weak but, at the same time, opponents are few. Therefore, in this example, governments should take an action, even if the situation seems still under control. On the other hand, a low value of ζ (i.e., close to 0) going with a high value of η, represents an alarming situations, as although there are only few opponents among flexible agents, opponents are more than inflexible. With the aim to offer these measures, ζ and η have been defined as follows ζ =
σO 1−σI
η =
σO σI
(12)
therefore, recalling that σO = 1−σI −σP and having solved analytically σP (t) (see 6), we are able to compute values of both parameters, ζ and η, at equilibrium and on varying the initial conditions —see Figure 3. It is worth to note that the parameter ζ, as defined in 12, has a range in [0, 1]. Notably, ζ = 0 means that, at equilibrium, there are no opponent agents in the population, while ζ = 1 means that all flexible agents became opponents. On the other hand, the parameter η has potentially an unlimited range, from 0 to ∞ (in the case σI be very close to 0, and σO be close to 1). To conclude, we want to emphasize the meaningful role of the two parameters ζ and η, as they represent potentially a way to quantify in which extent radicalization phenomena are taking place in a population.
III.
DISCUSSION AND CONCLUSION
The phenomenon of radicalization is a phenomenon of central interest to social psychology [29] and policy makers, in particular because it is often related to terrorism and, more in general, to criminal activities as shown by the recent anti-western trends [25]. In the proposed model we have considered a simple scenario based on an heterogeneous population 8
FIG. 3. Radicalization degree quantified according to the parameters ζ and η, on varying initial conditions: a σI = 0.3, and σO = 0.3, α = 1.0, β = 1.0. b σI = 0.3, and σO = 0.3, α = 2.0, β = 1.0. c σI = 0.3, and σO = 0.3, α = 1.0, β = 2.0. d σI = 0.3, and σO = 0.3, α = 4.0, β = 1.0.
whose agents are provided with a state representing their behavior in relation to cultural traditions and ways of life characterizing a well defined local geographic area (e.g., a country). In particular, agents can be inflexible (i.e., those belonging to the core subpopulation), peaceful and opponent. Both peaceful and opponent agents may exchange states over time by interacting with agents from the whole population. For instance, in the context of antiwestern terrorism, core individuals can be mapped to inflexible agents. Remarkably, today the proportion of the core population in a geographical area may change more rapidly than in past as a result of an increasing illegal immigration. Therefore, it is relevant try to understand and even to predict in a general context, the evolution of a social radicalization 9
in terms of population dynamics. In order to represent the strategies of inflexible agents [30] implemented to support a peaceful coexistence among individuals of different cultures versus the strength of an opponent ideal, we introduced two numerical parameters (i.e., α and β). These parameters have a fundamental role from an analytical perspective although, in real scenarios, it may be difficult to identify and to quantify them. Nevertheless, the underlying message coming from our analytical results overcome this ‘limit’ when dealing with the real world, as it shows the risks related to a change of ratio of subpopulations within a given territory. It is worth to highlight that a de-mixing of the subpopulations may lead to higher degree of radicalization in presence of fewer opponents. Therefore, social integration strategies can really represent the best peaceful solution in order to avoid criminal scenarios. To summarize, we have identified a direct relation between on the one hand, the final state of a mixed population in terms of order or disorder phases, and on the other hand, the ratio between social strategies and the strength of opponents’s ideal. Moreover, we emphasize that the proposed model may in principle be applied also to criminal and terrorist scenarios in homogeneous populations, as it happened in the cases of Italian red brigades [31] and French revolution [32]. These two cases are concerned with homogeneous populations as both inflexible, peaceful and opponents belong to the local core population. Notably, in the former case (i.e., red brigades), inflexible agents represent individuals who respect laws and believe in institutions and governments, while individuals having a different behavior can fall in the mild category of peaceful agents or in extreme category of opponents (i.e., criminals). Instead, in the case of the French revolution, inflexibles represent the small proportion of French nobility, while the remaining part of the population is represented by peaceful and opponent agents. There, the extremely difficult life conditions fed the opponent ideals and the wide proportion of the sensible subpopulation assumed completely the state of opponent, giving rise to what is know as the revolution. Above last example allows to remark that although we mention criminal activities, we are not judging nor the motivations nor the ideals of opponent agents, as they can be considered negative (as in the case of anti-western terrorism) or positive (as in the case of the French revolution). Then, once again, our unique aim is to study the emergence and the evolution of radicalization processes. Finally, we suggest that further studies on this direction are required, in particular from a computational social science perspective, as it is should be possible to identify earlier traces (i.e., Big Data) and seeds of dangerous behaviors in social networks. Suitable tools to quantify their strength 10
are also required.
ACKNOWLEDGMENTS
MAJ would like to thank Fondazione Banco di Sardegna for supporting his work. This work was supported in part from a convention DGA-2012 60 0013 00470 75 01.
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