Scaling property and opinion model for interevent time of terrorism attack
aa r X i v : . [ phy s i c s . s o c - ph ] N ov Scaling property and opinion model for interevent time of terrorism attack
Jun-Fang Zhu, ∗ Xiao-Pu Han, † and Bing-Hong Wang ‡ Department of Modern Physics, University of the Science and Technology of China, Hefei 230026, China
The interevent time of terrorism attack events is investigated by empirical data and model analysis. Empiricalevidence shows it follows a scale-free property. In order to understand the dynamic mechanism of such statisticfeature, an opinion dynamic model with memory effect is proposed on a two-dimension lattice network. Themodel mainly highlights the role of individual social conformity and self-affirmation psychology. An attackevent occurs when the order parameter of the system reaches a critical value. Ultimately, the model reproducesthe same statistical property as the empirical data and gives a good understanding of terrorism attack.
PACS: 89.75.Da, 89.65.Ef, 89.75,-k.Key words: Interevent time distribution, Terrorism event, Scaling law, Zipf’s law, Opinion dynamic.
I. INTRODUCTION
In recent years, terrorism attack events have occurred frequently. It has attracted interests of many scientists such as historians,politicians, physicists and so on. Many people might think terrorism attack is random and unpredictable. But, indeed there aresome general, common statistic property, just like the result from other fields where the non-poison distribution is captured forthe interevent time of the e-mail [1], surface mail[2], short message lending [3], web browsing [4,5], and rating of movies [6]etc.As early as 1948, Richardson found the number of casualty follows a power-law distribution in interstate wars [7]. Recently thesame statistic property is revealed for the casualty numbers in the global terrorism events [8]. Actually the terrorism events can bethought as wars in war. However, so far, the fundamental reason of terrorism events still remain unclear in view of its complexityand diversity. At present, there are mainly three kinds of viewpoints to explain the burst of terrorism events. The first kind is theself-organized critical notion. At first, Cederman [9] gave a possible interpretation for the finding of Richardson[7] by an agent-based model. After 9/11 event, The analysis for Iraq, Colombia, Afghanistan[10-12] displays the power law distribution with ascaling exponent α = 2 . which conforms with the non-G7 countries, the ones except the major industrialized nation: Canada,France, Germany, Italy, Japan, the United Kingdom and the United States, in old war [8]. Meanwhile the author proposed theself-organized critical model of interaction among terrorist groups who make decision of coalescence or fragmentation randomlyor with some one probability. This model produces the results coinciding with the statistic data and give insight in term of theconception of complex system. Further, it is generalized and perfected by Clauset[13], and the solution of the steady-statebehavior is obtained analytically under the conditions of constant number of terrorism-inclined individuals and the proportionalrelation between the severity and the size of the attacking cell. The second one proposed by Galam[14-17] is the terrorism modelof percolation theory based on individual passive supporters. In this model one territory is under the terrorist threat if the densityof the passive supporters exceeds percolation threshold in this territory. Further one clue is given to curb terrorism threat withoutharming the passive supporters. The specific scheme is increasing the value of terrorism percolation threshold by decreasing thespace dimension but not the number of nearest neighbors. This interesting model describes the state of terrorism and gives anorient to fight against terrorism. The third one is the competition, selection viewpoint. Clauset et al. [18] found the scale-freeproperty of frequency-severity ditribution has evident robustness on burst means and stability over time since 1968. they havedeveloped a toy model to explain this kind of behavior by the mechanism of competition between states and the non-state actorssuccessfully. In another reference [19], from the strategic selection, they illuminate the substitution and the competition in theIsrael-Palestine conflict is the reason whether an organization resort to terrorism where the public standing is first thought aorigin of attack occurrence.From the aspect of fighting against terrorism, besides the model of percolation theory, the conditions of promoting a vio-lence[20] are also referetial. Lim et al.[20] think that a violence arises at boundaries between groups with culture diffentiationwhen the group size achieves a critical scale and point out the violence might be prevented or minimized if appropriate bound-aries is created for current geocultural regions. However that is not indisputable[27]. Recently, the variation of the intereventtime over time have been concerned, and the research of Clauset et al.[21] shows the organizational growth leads to the decreaseof the interevent time. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] (corresponding author) t FIG. 1: The succession of terrorism events in Iraq (upper panel) and Afghanistan (lower panel). Each vertical line represents a single event.
In this paper we focus on the interevent time of terrorism events in Iraq and Afghanistan from 2003 to 2007. It is found toobey Zipf’s law corresponding with the power-law distribution with scaling exponents α = 2 . and α = 2 . for Iraq andAfghanistan respectively. Considering the importance of memory[22-24], we propose an opinion dynamic model with memoryeffect to understand such a statistic property. Due to individual social conformity and self-affirmation psychology, the publicopinion is formed and varies with time under the coaction of influence between individuals [25] and individual history memory[26]. Ultimately, under some certain social circumstance, an attack event occurs when the public consensus reaches some degree.We have found that this model can reproduce the same intervent time distribution of terrorism events. II. THE EMPIRICAL DATA p ( τ ) = τ − α , the Zipf’sexponent α ′ should satisfy the relation [28,29] R − /α ′ = Z ∞ R p ( τ ) dτ ∝ R − ( α − . (1)Here R represents the rank. We find the Zipf’s exponent α ′ = 0 . for Iraq and α ′ = 0 . for Afghanistan. In term of theabove relation, we have α = 1 + 1 /α ′ . So the interevent time distribution has power exponent α = 2 . and α = 2 . for Iraqand Afghanistan respectively. For Afghanistan, the smaller exponent means the interevent time is more heterogenous than Iraq,namely, the occurrence of terrorism events in Afghanistan shows more burstiness property. -2 -1 (a)slope = -0.70 i n t e r e v en t t i m e ( da y ) rank -1 (b)slope = -0.73 rank FIG. 2: The interevent time distribution of terrorism attack events in Zipf’s plot for Iraq (a) and Afghanistan (b) from 2003 to 2007. The circlesrepresent the empirical data, and the solid line is the linear fitting.
III. THE MODEL
To understand the underlying mechanism of the scaling property in terrorism events, we introduce an opinion dynamic modelwith memory effect to explain this statistic property. In the present agents model, every node is connected with its four adjacentneighbors inwardly and outwardly on a 2-dimension lattice network. Here nodes and links represent the individuals in a terrorismsocial system and the interaction between nodes respectively. Before an attack event occurs every individual has own viewpoint,support or opposition which are denoted by σ i = 1 and σ i = − . Individual opinion is determined by two factors. One is theinfluence of its adjacent neighbors, it is described by W ( σ i,t ) = σ i,t − X j =1 σ j,t − ( j = 1 , , , . (2)the other is the individual history memory effect measured by W ( σ i,t ) = (cid:26) , σ i,t − σ i,t − > , σ i,t − σ i,t − < . (3)Individual opinion turnovers with time by the above factors in terms of next rule: W > means individual has consis-tent opinion with the majority of its adjacent neighbors and change his/her own opinion with the probability [ exp ( − aW ) + exp ( − bW )] /T . W < corresponds with the opposite case. In this case history memory effect dominates and individualchanges his/her opinion with the probability exp [ − bW ] . So the overturning probability is P ( σ i ) = (cid:26) [ exp ( − aW ) + exp ( − bW )] /T, W > exp ( − bW ) , W < . (4)Here the parameters a and b are the main factors indicating the social conformity psychology and self-affirm psychology respec-tively. T ≥ . is an index to describe the social chaotic degree. By this rule the system undergoes self-organization evolutionin the non-equilibrium state.Now we measure the order degree of the public opinion by an order parameter m ( ≤ m ≤ ) on a lattice network with size L × L and periodic boundary, m = | L L X j =1 σ i | , σ i ∈ (+1 , − (5)Generally the total population of a terrorism social system is approximately invariant. So let L = 10 and the initial states aregiven randomly. No matter how complicated that the practical reasons may be, we think an attack event is triggered when m reaches a critical value m c . Next we investigate the influence of different parameters on the interevent time statistic property bysimulation. -7 -6 -5 -4 -3 -2 -1 s l ope = - . s l ope = - . a (a) a=0.5 a=1.2 a=4.0 p r obab ili t y -7 -6 -5 -4 -3 -2 -1 (b) b=0.0 b=0.5 b=1.0 b=1.5 -7 -6 -5 -4 -3 -2 -1 s l ope = - . s l ope = - . T (c) T=4.0 T=5.5 T=7.0 p r obab ili t y interevent time -7 -6 -5 -4 -3 -2 -1 (d) m c =0.50 m c =0.70 m c =0.80 m c =0.85 interevent time FIG. 3: (Color online) The normalized distribution of interevent time by simulation under the influence of one parameter among a = 1 . , b =0 . , T = 5 . , m c = 0 . . (a) The distribution curves when the parameter a is changed. The two solid lines are the fitting of power-law withthe exponent α = 1 . and α = 2 . respectively. The top inset: relationship between a and α . (b) The variation of distribution curves whenthe paramter b is changed. (c) the distribution curves when the parameter T is changed. Two curves fit by two solid lines has scaling exponent α = 1 . and α = 2 . respectively. The top inset: α as a function of T . (d) The variation of distribution curves when the parameter m c ischanged. Each data is obtained by averaging over 100 independent runs. IV. SIMULATION AND ANALYSIS
First, the social conformity will impact on the whole social order degree and smaller a indicates individual has larger will-ingness to follow social public opinion. In Fig. 3 (a) the interevent time distribution is plotted for different values of a . It isshown that the distribution transits to power-law style from power-law-like style with a tail gradually when a increases. Forpower-law-like style at a = 0 . a natural cutoff of tail is executed, then the power exponent α is obtained by linear fitting.Obviously, the power exponent becomes larger with the increase of a , which means the distribution of terrorism events is moreinhomogeneous when individual inclines to follow public opinion more easily. From the inset in Fig. 3 (a), one can see that α increases as a increases and converges to a steady value when a is large enough. This is because the effect of the public opinionis so weak for large a that the dynamic evolution of terrorism attack is hardly affected.In the evolution of terrorism events, self-affirmation psychology is also very important for social order degree. Together withthe social conformity, they countermine wether an individual overturns his/her opinion when individual opinion agrees withthe major opinion. However, in converse case, the self-affirmation play a decisive role in individual opinion selection. At themoment, individual will make a decision according to history selection in memory. Now let us pay attention to the effect ofself-affirmation factor b on time statistic property of terrorism event. As shown in Fig. 3 (b), the distribution of interevent timeexceeds a power law when the self-affirmation is inadequate at large b value. The distribution curve becomes a power-law andthen tends to a stretched exponent style with the decrease of b in that the strengthening of self-affirmation reduces the timeinterval between two consecutive terrorism events. Meanwhile it makes terrorism event with short interevent time occur withsmaller probability. Indeed social conformity and self-affirmation are two factors who are complementary to each other andmutual restraint to reach balance.The third tunable parameter is T , which implies the chaotic degree of a social circumstance. It will be effective as individualopinion keeps consistent with the public opinion. Similar with Fig. 3 (a), the increase of T also leads to the increase of powerexponent and reduce of interevent time of terrorism events from Fig. 3 (c). But, compared with the a , it is different that the large T still has obvious influence on power exponent from the inset.The above three factors determine the order degree of public opinion. We need to set a critical value of order parameter m c to -7 -6 -5 -4 -3 -2 -1 (a)a=1.31b=0.50T=5.50m c =0.70 p r obab ili t y interevent time slope = -2.43 -7 -6 -5 -4 -3 -2 -1 (b)a=1.20b=0.50T=5.50m c =0.70 interevent time slope = -2.35 FIG. 4: The circles represent simulation results for interevent time distribution of the terrorism attack events in (a) Iraq and (b) Afghanistanand the solid line is the linear fitting. Each data is obtained by averaging over 100 independent runs. judge whether a terrorism event burst. Fig. 3 (d) displays the influence of different critical value m c on the curve style. It is veryeasy to achieve consensus when m c is small. So the terrorism event bursts frequently. Contrarily time interval of the terrorismevent becomes longer in a way. The curve style transits to the stretched exponent because the proportion of the middling timeinterval is prominent relatively.From Fig. 3, one can find that the style of distribution curve is decided by the self-affirmation psychology and the critical orderparameter. The social conformity and the chaotic degree of a social circumstance decide the power exponent. So the specificpower-law style with some exponent will be got if only we choose appropriate parameter values. According to the propertyabove, we choose a = 1 . , b = 0 . , T = 5 . , m c = 0 . and a = 1 . , b = 0 . , T = 5 . , m c = 0 . to simulate theterrorism events in Iraq and Afghanistan respectively. The simulation results are shown in Fig. 4. The present model generatesthe power-law distribution with exponent α = 2 . and α = 2 . which accord with the empirical data. It is noted that thestrong self-affirmation and weak social conformity are the significant character for Iraq. Nevertheless these two factors arealmost equivalent for Afghanistan. V. CONCLUSION
In conclusion, from real data we find the scale-free feature of interevent time distribution for terrorism event in Iraq andAfghanistan from 2003 to 2007. Here we consider the assumption that the burst of a terrorism event is closely relative to theformation of opinions. This formation process depends on not only the social influence but also the individual memory. Previousterrorism models have noted the former but ignored the later. So, to understand the observed statistic property from empiricaldata, we proposed an opinion dynamic model with memory effect in this paper. In the model, the order degree of public opiniondetermines the burst of a terrorism event. In certain social circumstance, the formation of public opinion depends individualpsychology character of social conformity and self-affirmation. So individual psychology factor is the crucial reason whether aterrorism burst in a given social circumstance. This also alert us to poll is important in some wars. Winning morale to strengthenthe social conformity is a possible means of reduce terrorism events. These results obtained by this model are coincide with thereality intuitively and it can reproduce the same power-law interevent time distribution of terrorism attack as the empirical datain Iraq and Afghanistan. It confirms the rationality of our assumption and provides a better understanding of the terrorism attack.In addition, terrorism events can be treated as a kind of collective behaviors of human. Our studies show that the memory andsocial effect could be an origin of the power-law properties in many collective behaviors of human.The authors would like to thank Dr. Han-Xin Yang for helpful conversation. This work is supported by the National NaturalScience Foundation of China (Grant No. 60744003, 10975126, and 70871082), the Specialized Research Fund for the DoctoralProgram of Higher Education of China (Grant No. 20060358065),and the National Basic Research Program of China (973Program No.2006CB705500). [1] A.-L. Barab ´ a si, Nature 435 (2005) 207.[2] J.G. Oliveira, A.-L. Barab ´ a si, Nature 437 (2005) 1251.[3] W. Hong, X.-P. Han, T. Zhou, B.-H. Wang, Chin. Phys. Lett. 26 (2009) 028902.[4] Z. Dezs ¨ o , E. Almaas, A. Luk ´ a cs, B. R ´ a cz, I. Szakad ´ a t, A.-L. Barab ´ a si, Phys. Rev. E 73 (2006) 066132.[5] B. Goncalves, J. J. Ramasco, Phys. Rev. E 78 (2008) 026123.[6] T. Zhou, H. A. T. Kiet, B. J. Kim, B.-H. Wang, P. Holme, Europhys. Lett., 82 (2008) 28002.[7] L.-F. Richardson, Amer. Stat. Assoc. 43 (1948) 523.[8] A. Clauset, Y. Maxwell, 2005. arXiv:physics/0502014.[9] L. Cederman, Amer. Pol. Sci. Rev. 97 (2003) 135.[10] N. Johnson, M. Spagat, J. Restrepo, J. Bohorquez, N. Suarez, E. Restrepo, R. Zarama, 2005. arXiv:physics/0506213.[11] N. Johnson, S. Mike, J. A. Restrepo, B. Oscar, C.-B. Juan, S. Nicolas, M. R. Elvira, Z. Roberto, 2006. arXiv:physics/0605035v1.[12] N. Johnson, Complexity in Humuan Conflict, Springer Berlin / Heidelberg, New York, 2008.[13] A. Clauset, F.-W. Wiegel, 2009. arXiv:0902.0724v1.[14] S. Galam, Eur.Phys.J.B 26 (2002) 269.[15] S. Galam, A. Mauger, Physica A 323 (2003) 695.[16] S. Galam, Physica A 330 (2003) 139.[17] S. Galam, Int. J. Mod. Phys. C 19 (2008) 409.[18] A. Clauset, M. Young, K. S. Gleditsch, Journal of Conflict Resolution 51 (2007) 58.[19] A. Clauset, L. Heger, M. Young, K. S Gleditsch, Cooperation & Conflict (2009), to appear.[20] M. Lim, R. Metzler, Y. Bar Yam, Science, 317 (2007) 1540.[21] A. Clauset, S.-G. Kristian, 2009. arXiv:0906.3287v1.[22] S. M. Cai, Z. Q. Fu, T. Zhou, J. Gu, P. L. Zhou, EPL, 87 (2009) 68001.[23] L. Telesca, M. Lovallo, Physica A 362 (2006) 480.[24] J. Alvarez-Ramirez, E. Rodriguez, R. Urrea, Physica A 377 (2007) 291.[25] L.-L. Jiang, D.-Y. Hua, J.-F. Zhu, B.-H. Wang, T. Zhou, Eur.Phys.J.B 65 (2008) 251.[26] K.-I. Goh, A.-L. Bar ´ a ♯♯