Modeling Radicalization Phenomena in

Heterogeneous Populations

Serge Galam∗

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 subpopula-

tion 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 peace-

ful agents to either curb or inflate radicalization. Some hints are outlined at new possible public

policies towards social integration.

∗ serge.galam@sciencespo.fr† marcojavarone@gmail.com

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I. INTRODUCTION

The phenomenon of radicalization [1] is of absolute interest in the context of criminal-

ity 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 in-

vestigations. 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 dy-

namics, 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 comprehen-

sion 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, span-

ning 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 par-

ticular 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 infor-

mation 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 im-

plemented 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 evo-

lution is studied by using a Lotka-Volterra Ordinary Differential Equation framework. In

particular, our population is composed of two subpopulations: a core subpopulation char-

acterized 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 sub-

population) 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 crim-

inal 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 be-

come 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 equationsdσP (t)dt

= ασIσO(t)− βσO(t)σP (t)

dσO(t)dt

= βσO(t)σP (t)− ασIσO(t)

σI + σP (t) + σO(t) = 1

(1)

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)

dt= ασI(1− σI − σP (t))− β(1− σI − σP (t))σP (t) (2)

Remarkably, the equilibrium state of the population can be obtained from 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)±

√[ασI + β(1− σI)]2 − 4βασI(1− σI)

2β(4)

where < σP > is the equilibrium value of peaceful agents. Those values simplify to

< σP >=

1− σI ≡ p1

αβσI ≡ p2

(5)

4

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 +p1 − p2

1− σP (0)−p1σP (0)−p2 e

β(p1−p2)t(6)

Figure 1 shows the evolution of the system on varying the initial conditions.

5

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:

dσPdt

(σP ) ' dσPdt

(< σP >) + (σP− < σP >)λ (7)

where dσPdt

(< σP >) = 0 and λ ≡ d2σPdtdσP

|<σP>, we obtain

λ = −[ασI + β(1− σI)] + 2βσP (8)

Therefore, for respective values p1, p2 we obtainλ1 = −ασI + β(1− σI) = β(p1 − p2)

λ2 = ασI − β(1− σI) = −β(p1 − p2)(9)

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 = αβσI , < σO >= 1− σI

Ic= p1 − p2}

(10)

with Ic ≡ βα+β

. 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

σI− 1) (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

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FIG. 2. The curve 1σN− 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 αβ 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 1σI− 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 1σI− 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 = 1

2, and 3) σI >

12.

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 σIIc

= 1.

Obviously, if α < β opponent agents strongly prevail in the system.

Case 2. For σI = 12, 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.

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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 = σO1−σI

η = σOσI

(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 psychol-

ogy [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 coun-

try). 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 anti-

western 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 un-

derstand 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 un-

derlying 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 al-

though 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 terror-

ism) 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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