The developmental dynamics of terrorist organizations

1 The dev elopmen tal dynamics of terrorist organizati ons Aaron Clauset 1 , 2 , 3 , ∗ and Kristian Skrede Gleditsc h 4 , 5 , ∗ 1 Departmen t of Computer Science, Univ ersi t y of Colorado, Boulder, CO, USA 2 BioF ron tiers Institute, Univ ersi t y of Colorado, Boulder, CO, USA 3 San ta F e Institute, San ta F e, NM, USA 4 Departmen t of Go vernmen t, Univ ersity of Essex, Wiv enho e P ark, Colchester, UK 5 Cen tre for the Study of Civil W ar, O slo, Norw a y ∗ E-mail: Corresp onding aar on. clauset@colorado.edu Abstract W e identify ro bust statistical patterns in the frequency and severity of violent attacks b y terrorist or - ganizations as they grow and age. Using group- le vel static a nd dyna mic analyses of terror ist even ts worldwide from 1968– 2008 and a simulation mo de l of or g anizational dynamics, w e show that the pro duc- tion o f violen t event s tends to accelerate with incr e asing size a nd exp er ie nce. This coupling of fr e quency , exp erience and size ar ises fro m a fundamental p ositive feedback lo op in which attac ks lead to growth which lea ds to increased pro duction of new attacks. In contrast, ev ent sev er ity is indep endent of bo th size and experience. Thus lar ger, more e x pe rienced organiza tio ns are more deadly because they attac k more frequen tly , not because their attac ks are mo re dea dly , a nd large even ts are equally likely to come from lar ge and small o rganiza tions. These res ults hold acro ss p olitical ideologies and time, suggesting that the frequency and severit y of terr orism ma y b e constrained b y fundamental process es. In tro du ction Much research on patterns in terrorism has bee n inspired by particular historic even ts and “wa ves” of sp eciﬁc forms of terror ist attacks [1, 2]. Just as the rise in in terna tio nal skyjackings in the 1970s led to a resurgenc e of studies of terrorism, the 11 September 20 0 1 attac ks renewed int er est in why groups resor t to terroris m, the sp eciﬁc choice of attack targets, and the relative eﬀectiv eness of particular counterterror ism measures. As a result, ma ny researchers hav e dev elo ped typolo g ies of s pe c iﬁc forms of terrorism and highlighted the distinctiveness of diﬀerent terrorist g r oups. By contrast, in this man uscr ipt w e examine whether there are fundamen tal pa tter ns in the frequency and severit y (num ber of deaths) o f deadly even ts carried out b y terror ist o rganiza tions and what mechanisms might generate them. Little res earch on terro rism has fo cuse d on dir ectly mo deling individual even t frequency and severit y , and the wa y these c hange ov er an organiza tion’s lifetime. When dea ths a re co nsidered, they ar e typically aggre g ated and used as a cov ariate to understand other asp ects of terr orism, e.g., tre nds over tim e [3, 4], the when, where, what, ho w and why of the resort to terrorism [5–7], diﬀerences b etw een organizatio ns [8], or the incident r ates or outcomes of even ts [3, 9]. Suc h eﬀor ts ha ve used time series analysis [3, 4, 9], qualitative mo dels or h uman exp ertise of sp eciﬁc scena rios, actor s, targ ets or attacks [10] or q uantitativ e mo dels based o n factor analy sis [11, 1 2], s o cial netw orks [13, 14] or for mal adversarial in tera ctions [6, 15, 16]. Our approach is diﬀerent and complementary to these appro aches, fo cusing on globa l trends a nd patterns in the frequenc y and s e verit y of ev ents [17–25], rather than o n event particula r s or motiv atio ns. By focusing our analysis at the global scale, the importance of individual decisions in speciﬁc contexts is in fact lessened, due to the central limit theorem a nd the roug h indep endence of individual events; as a result, the impor tance of generic non-strategic pr o cesses is enhanced a nd these proces ses, if any , may be studied. Explanations of such patterns must thus fo cus o n pro c esses or constraints that ar e independent of v ariations in context or sp eciﬁc motiv ation a nd may include physical constra int s , netw ork eﬀects and endogenous po pulation dyna mics, which are well suited to explain the b ehavior of strategically unc¨ oordinated p o pulations of a ctors [24]. This appr o ach to inv estigating the fundamental laws of ter roris m 2 has much in common with that of statistica l physics, in which the s elf-av era g ing pr op erties of indep endent even ts allows for interesting p opula tion-level prope r ties to emerge from micros copic system c haos . This statistical ph ys ic s-style approach is increasingly being applied to s tudy complex so c ial systems [2 6–28], yielding a n umber of nov el insight s . Here, we aim to shed new ligh t on the fundamen tal pro cesses gov er ning the frequency and severit y of terro rist even ts b y studying their statistical relationship with the o rganiza tio ns t ha t genera te them. Our a im is to identify global patterns in these relationships and to ex pla in their origin mec hanistically . W e employ a com bination of disag g regated data analysis, studying a la rge database of terror ist ev ents worldwide from 1968–200 7, statistical mo deling and inference, co mputational modeling and r egress ion analysis to v alidate our mechanistic h yp othese s . By shedding new light on these large- scale patterns and trends in terr orism, and on how suc h pa tterns emerge from lo cal-level b ehaviors, this large- scale statistical o r pa ttern-based approach ca n supplement formal mo dels of strategic int er actions, inf or m counter-terrorism po licy a nd cla rify our general ability to for ecast or a nticipate future terr orist even ts or trends. P atter ns in global conﬂict A pattern-based approach to studying conﬂict ow es muc h to the seminal w o rk in the ea rly 20th cen tury of L e w is F ry Richardson—a physicist and meteorologist known for collecting data on conﬂicts (“deadly quarrels ” ), modeling a rms rac e s using diﬀeren tial equations, as well as ea rly contributions to under- standing the frequencies and sev erities o f w ar s. Speciﬁc a lly , Ric hardso n [2 9, 30] identiﬁed the r emark able pattern that the frequency o f wars deca ys like the in verse power o f their se verit y . (Po wer-law distri- butions can indicate unusual under lying or endogenous pro cesses, e.g ., feedba ck lo ops, netw ork eﬀects, self-orga nization o r optimization. F ro m a purely sta tistical p ersp ective, p ow er-law distributions generate large even ts orders of magnitude more often than we would expec t under a Normal assumption. Re- cently , p ow er- law distributions hav e been identiﬁed in a wide range of so cial and biolo gical systems [31 ]. See [32], [3 3] and [34] for reviews, or App endix A of [3 5] for a gentle in tro duction.) This empirical pattern implies tha t there is no f undamental statistical diﬀerence b etw een rare but ca tastrophic wars and more common but less severe wars—the likelihoo ds of both ar e describ ed by a single mathematical function: Pr(even t with severit y x ) ∝ x − α , where x c o unts the n umber of fatalities (s e verit y) and α is the “ scaling expo nent ,” which co ntrols how quickly the frequency decreases a s severity increa ses. It also implies that the under lying s o cial a nd po litical pro cesses for b oth lar ge and small wars may be fundamentally the same, i.e., la r ge w a rs may simply be “scaled up ” versions of small wars. In g eneral, the identiﬁcation of a pow er la w implies that studying the statistically more common even ts c an shed lig ht on certain a sp ects of ex tremely rare events. (Seismologists study large earthquakes in this way: the frequencies of b oth large and small q uakes follow a power-la w distributio n, called the Guten b erg -Rich ter La w, a nd the physical pro cesses that generate bo th small and large quak es are fundamen tally the same.) Recently , Clauset et al. [20, 3 1] sho wed that this same pattern—a p ow er- law, “Ric hards o n’s Law”— also holds for the frequency of severe terr orist attacks (r ep orted fatalities) worldwide, while [23] s ug gest a similar pattern for even ts within ins urgencies. The p ower-la w pattern in terror ism is highly robust: it per sists ov er t he past 40 y ea rs despite la rge structural and political changes in the international sys tem and is independent of the type of w eap o n used (explosives, ﬁrearms, ars o n, knives, etc.), the emergence and increasing p opularity of suicide a ttacks, the demise of man y individual terro rist o rganiza tio ns, and the economic developmen t of the tar g et coun try . Thu s , fundamental regula rities in terror ism can and do emerge a t the glo bal level despite the highly contingen t and co nt ex t- sp eciﬁc nature of the individual atta cks, conﬂicts and decisions. Insights into how these patterns’ arise will likely shed new ligh t on t he underlying so cial or p olitical pro cess es that drive and constrain global trends and o n eﬀective p olicies for resp o nding to or managing those proc e s ses. 3 Metho ds W e consider t he frequency and severit y of a ttacks o ver the lifetime of individual t er roris t org anizations, and the ques tion of whether or ganizations exhibit common s tatistical patterns in thes e behaviors. W e argue that organization size (n umber of personnel) pla y s a fundamen tal role in limiting the o verall fre- quency , but not the severit y , of violent event s by a group. The key idea is that o r ganizatio n size and its ov erall production rate of even ts are linked. If ev ents lead to gr owth in an y w ay , then this link implies a p o sitive feedbac k lo op in w hich each attac k increases the pro ductio n r ate of future attacks. Th us, a terroris t organiza tion can b e view ed as a kind of fa ctory whose principal pr o duct is politica l violence, and whose proce e ds are rein vested in increa sed pro duction capacity . T o test these “dev elopmental dynamics” h yp otheses, w e present nov el statistica l analy ses o f the be- havior of nea r ly 4 00 ter r orist o rganiza tions worldwide ov er the p erio d 196 8–20 0 8. W e ﬁnd strong evidence for pre c isely this kind of gener ic acce le ration in event productio n. This suppo r ts the notion that an or- ganization’s av a ilable lab or, i.e., the size of its militan t wing, is a fundamen tal c o nstraint on the o verall frequency of its attacks. W e further show that the rate a t w hich an org anization cycles through the po sitive feedback lo op can dep end on cov ariates like its p olitical ideo logy , with religiously-mo tiv ated organiza tions accelerating (growing) the fastest. In contrast, we ﬁnd no evidence that event severity depe nds o n o rganiza tional size or exp erience . Instead, the dis tribution of attack severities follows a roug h form of Ric har dson’s La w indep endent of size, exp erience or p olitical motiv ation. These results imply that very large even ts a re equally likely to b e generated by small groups as by lar ge groups, and that larger organiza tions are indeed more deadly [8], not b ecause their individual attacks ar e systematically more spectac ula r but because they typically carry o ut many more attacks. That is, the size of the b east dir ectly determines the overall level of terror activity (frequency) but not the quality (severit y) of those actions. Recently , J ohnson et al. [25] use d a similar approach to analyze the timing of even ts in the Ira q and Afghanistan conﬂicts, which w as in turn based on an earlier v ers ion of th is ma nu s c ript [22]. Although similar statistical patterns to the ones we descr ib e here were o bserved in those c onﬂicts, a diﬀerent explanation w as oﬀered for their o rigin. W e will re v isit this compar ison a nd comment on the problems our statistical results pos e for the explanation oﬀered by [2 5]. Impact of Size on F req ue ncy H1. L ab or-c onstr aints : the ov era ll pr o duction rate of violen t ev ents by a n organiza tion dep e nds o n its size, and thus the time b etw een c o nsecutive attacks ∆ t is roughly inv er sely pr o p ortional to the size s of the organization. Mathematically , s ∝ 1 / ∆ t . In other words, the pro duction of terrorist event s cannot be automated. If this were p oss ible, org a nizations could pro duce arbitrary num b er s of even ts without needing to gr ow in size, muc h lik e a fully automated factory requir e s essentially no human p ersonnel to function. (In this light, cyb er terr orism is an interesting case: it remains unclear to what degr e e the planning and exe cution of cyb er terro rist attacks can be done automatically , by computers. Our current b elief is that cyber ter rorism is a lso not mass producea ble and thus some la bo r co nstraint will p ersis t, although it may b e s ubstantially lessened rela tive to physical terroris m.) Instead, w e a r gue that eac h terroris t even t requires signiﬁcant human involv ement, e.g., to conceive, plan and exec ute it. This requirement for human eﬀort implies that for the pro ductio n rate o f an organiza tion to decrease, it m ust add a dditional members to pr o duce them. And, the resultant increas e d rate o ccur s not b ecaus e more hands make any individual even t pro c e ed more quickly , but beca use multiple even ts ma y b e car ried out in parallel. That is, the overall pro duction rate of the organiza tion is like the pro duction rate of an entire factory; as the factory (organiza tion) adds internal independent pro duction lines (terror ist cells), the eﬀective time b etw een new even ts falls even though ea ch pr o duction line op er a tes at a constant rate. 4 It is imp ortant to reco gnize that H1 does not imply that the only w ay to increase the group-level pro duction rate of attacks is thro ugh or ganizationa l growth. Indeed, many asp ects of even t productio n surely do b eneﬁt from technology o r eﬃciency improvemen ts [36–39]. Instead, H1 implies that s uch factors ca n only mo derate, no t eliminate, the fundamental constraint that size places on pro duction. T o the extent that these factor s decr ease the time betw een an or ganization’s events, the literature on learning s uggests that the ov erall impact will be mode s t [39]. In cont r ast, increases in labo r , which allow many terro rist cells to o p e r ate in parallel, can lea d to m uch lar ger improv ements. Finally , we note that this constraint should b e stro ngest for small organiza tions, who lik ely hav e the worst access to eﬃciency-impr oving resources like sp ecialized p ers onnel, tra ining fac ilities or factories and who may r eap the la rgest b eneﬁt, e.g., media visibility , from striving to maximize their even t pro duction. Because most organizations begin small and grow ov er time, this should be mos t evidence early in the lifetime of a n o rganizatio n. (A spatial cor ollary o f H1 is that if an “orga nization” is deﬁned as those militant s within some geog raphic loca le, e.g., a pro vince o r district, then the frequency of e vents within that lo cale will b e roug hly inv ers ely pro po rtional to the num ber o f militants there. That is, the s ∝ 1 / ∆ t relationship sho uld hold when b oth s and ∆ t are deﬁned by a ge ographic b oundar y . Or ganizationa l “growth” can then b e understo o d as either immigration or recruitment of new militan ts.) Ev ents, Recruitmen t and Gro wth What role do attacks play in changing organizatio nal size? If a n even t gains the or ganizatio n wider visibility among po tential members o r sy mpa thizers, the organiza tion may grow in s ize as a result of that even t. (Decreases in size are likely dr iven by distinct so cial pro cesses (see [4 0]), which we do no t consider here.) H2. Event-r e cruitmen t : orga nizational gr owth (increased s ) is partly driven by re cruitment asso ciated with the pro duction of new even ts (increased k ), i.e., even ts lead to recruitment which leads to organiza tional gr owth. Mathematica lly , d s/ d k > 0 . H2 do es not imply that growth co mes only fr o m violence-re lated recruitment. So long as recruitmen t is partly ba sed on the productio n of v io lent ev ents, H2 implies a correlatio n betw een incr eases in s ize and increased even t pro duction F requency Acceleration T ogether, H1 and H2 imply a p o sitive feedback lo o p in which attacks lead to recruitmen t whic h leads to organiza tional g rowth a nd thus an increased gr oup-level pro ductio n of new attacks. So long as a portion of the growth is a llo cated to producing additio na l ev ents, i.e., so long as the militan t wing gr ows with the ov erall organizatio n, H1 and H2 join tly imply H3. H3. F r e qu ency-ac c eler ation : as an organiza tion ca rries out more a ttacks (increased k ), the time b etw een subsequent attacks ∆ t decr e ases. Mathematically , d∆ t/ d k < 0. That is, H1 predicts s ∝ 1 / ∆ t while H2 predicts d s/ d k > 0 . E liminating the common factor of s yields the prediction that d∆ t/ d k < 0, in which the con tinued productio n o f violent even ts produces a decreasing delay b etw een those even ts. (This dynamical relationship produces a s imilar pattern to that observed in “learning” or “pro gress curves,” in whic h co nt inued pro duction cov ar ies with lo wered production co s ts or time [36, 3 9, 4 1]. Although the pattern is similar, the mec hanism is diﬀerent.) Impact of Size on Sev erit y Increased size ma y bring greater ac c e ss to ca pita l and s killed labor , e.g., experienced professio nals, ad- v anced arms, in telligence, etc., and th us mo r e s pe ctacular attacks. 5 new event recruitment organization growth feedback + ∆ t ∝ 1 /s s → s + η x ∝ s Figure 1. A mo de l of terr ori st organizations. A schematic illustrating the feedba ck lo op relationship b etw een size s a nd the frequency and sev erity of attacks: the delay b etw een subsequent attacks ∆ t is inv ersely related to an organization’s size s while the severit y of subsequent attacks x grows with s ; new even ts lead to recruitmen t which leads to growth, which increases the size v aria ble s . H4. Severity-incr e ase : the sev erity x of a new attack increa ses with orga niz a tional size s and, via H2, the n umber of attacks k . Mathematica lly , d x/ d s > 0 and d x/ d k > 0, resp ectively . Combined with H2, H3 implies that a ttacks by exp erienced, larg er gr oups s ho uld be consistently and signiﬁcantly mor e deadly than those o f less ex pe rienced or smaller groups. H4 assumes a tangible b eneﬁt for maximizing the sev er ity of attac ks, e.g., to g ain wider visibilit y for the org anization’s cause o r to demonstrate power or res olve. Suc h incentiv es a re not foregone conclusions : severe attacks may als o attra ct harsh attention from state-level actor s, leading to re pr ession, p olice action or the destructio n of physical o r ﬁnancial reso urces. They may a lso induce co unter-pro ductive eﬀects on po tent ia l sympathizer s, e.g., due to the shockingness of sp ectacular even ts. As a result, we consider the theoretical argument supp orting the severit y- inc r ease hypo thesis to be margina l. Results Mo del of terrorist organizations T o illustra te these interactions b etw een an or ganizatio n’s size and the frequency and severit y of attacks ov er its lifetime, w e co ns truct a simple model o f a terro rist organiza tion’s developmen t (see Figure 1 for a sc hematic). Historically , terro rist organizatio ns b eg in as a sma ll collections of terr orism-inclined individuals [42]. Let this initial collection be comp osed of roughly η individuals, which de no tes the typical or characteristic size o f a terrorist cell. The particular v alue o f η is not impo rtant, but may depend p olitical ideology , so cio - economic c o ntext [43], the attack’s targ et, etc. The c e ll plans a nd conducts its ﬁrst attack, which g a ins it some visibility , via either traditiona l media cov er age or informal channels. Subseq ue nt r e cruitment yields a n umber of additional mem b ers ν (H2), and no w the organization is lar ger. Again, the particular v a lue of ν is not impo rtant, but likely dep ends on co nt ex t-sp eciﬁc factors. Each cell con tinues planning and ca rrying out new attacks, roug hly o nce ev ery τ days (H1). Newly recruited member s form new cells, of size η (H1) and new cells pla n and carry out their own attac ks in parallel. It is this parallelism that a llows the larger organiza tion to appea r to b e a cting more quickly , even though the planning time τ for any pa r ticular ev ent re ma ins ﬁxed. An attack by a ny cell leads to ov erall org anizational growth via recr uitmen t (H2), which in turn increas e s the organiza tion’s o verall pro duction r ate of attacks b y a dding new cells (H3). Finally , as the gr oup grows, the increased manp ow er also increases its a bilit y to carry out more severe ev ents (H4), e.g., beca use more supp or ting roles allow 6 10 0 10 1 10 2 10 3 10 −1 10 0 10 1 10 2 10 3 Cumulative number of events, k Time to next event, ∆ t (days) ∆ t=1 day resolution Simulation, ν / η =0.5 Simulation, ν / η =1.0 Simulation, ν / η =2.0 Model, ∆ t ∝ k −1 0 500 1000 1500 2000 2500 3000 10 0 10 1 10 2 10 3 Days from first event, t Number of cells, s/ η Figure 2. Simulated de v elopm en t of a terrorist organization. (A) Median even t delay ∆ t vs. cum ula tive num b er of even ts k , for 10,0 00 s imulated terrorist organizations and three choices of the nu mber of cells ν /η added per event. Das hed line shows the function ∆ t ∝ k − 1 , from Eq. (1). (B) Median size (num b er of terr orist “ cells” s/ν ) vs. calenda r time from the ﬁrs t even t, sho wing exponential growth with rate set by ν /η . better surv eillance, access to better equipment, etc. Co ordinating the activities o f these additional individuals, or the developmen t of non-vio lent initiatives like a political wing or the pr ovision of social ser vices, will dr aw so me members awa y fr om these militant activities. How ever, so long as rec r uitment contin ues to grow the nu mber of militan t cells, the p ositive feedback lo op remains. This simple mo del inten tionally o mits ma ny factors, such a s orga nizational structure, p olitical moti- v ation, g e o graphy , etc., that ar e likely to impact the b ehavior of any particular organizatio n. W e also int entionally omit an y p otential respons e b y state-level actors and their consequences on the o rganiza - tion’s ev olution. This last decision is made in or der to fo cus on the developmen t of the o rganiza tion, i.e., its early lifetime, where la b o r constraints are likely most profound, a lthough such pro cesses could naturally b e added. Omitting these facto r s k eep the model simple and a llows us to ma ke quan titative predictions of the gener ic relationship b etw een organiz a tion size and the frequency and severity of its attacks v ia direct numerical sim ulatio n. T o mimic the na tural v ariation b etw een particular even ts, for each new ev ent b eing planned by a cell, w e draw a de lay τ fr om a ﬁxed distribution. (In general, our r e - sults hold so long as the distr ibution o f τ is well-be hav ed and stationary with resp ect to k .) Spe c iﬁc a tion details and computer code for the simulation are given in the Supp or ting Information. Each simulated terr o rist organization gener ates a unique sequence of ev ents repre senting the collec- tive b ehavior of its cells ov er time, and we extract the g e ne r ic behavior by computing quantiles o ver v ariables of interest for many such simulated o rganiza tions. Here, we are in terested in ho w the delay betw ee n subsequent a ttacks ∆ t v ar ies with cumulativ e nu mber of ev ents k (H3), a nd ho w the size of the organiza tion, mea sured b y the num b e r of cells s/η v aries with calendar time t from the ﬁrst ev ent (H2). H4 predicts that even t severit y corr elates with or ganization size and thus no a dditional information is gained b y explicitly simulating ev ent severities. Figure 2 shows the res ults for 10,00 0 simulated organiza tions, for thr ee ch o ices of the r atio ν / η , which represents the growth rate of the org anization’s militant wing . When ν /η < 1 regime, organiza tio nal growth is s low beca us e m ultiple even ts are requir e d to establish a new terrorist cell; but, when ν /η > 1, organiza tional gr owth is fast becaus e each even t pro duce s at leas t one new cell. The gener ic behavior of our mo del is clear: (i) o rganiza tional size gr ows exp onentially with time, at rate ν /η , and (ii) the feedbac k betw een size and pro duction rate induces a str ong corr elation betw een 7 exp erience, size and the fr equency of ev ents. Finally , the model pr o duces a univ ersa l functiona l rela - tionship betw een dela y ∆ t a nd cum ulative pro ductio n k of the form ∆ t ∝ k − 1 , and this relations hip is independent of the growth rate ν /η . This latter p oint is worth reiter a ting: so long a s eac h new event leads to some marginal increase in the overall pro duction ra te (H2), a p ositive feedback lo op betw een size and even t pro duction will exist. This feedback will b e linear ∆ t ∝ k − 1 if the g rowth rate ν /η do e s not v ar y with exp er ience k . If the militant wing is a decreasing fraction of the ov erall organization ( ν /η decre a ses o ver time), the feedback will b e sub-linear a nd k − β with β < 1, while if it increases with time, the feedback will be super-linea r and β > 1. These prop erties imply that if a growing org a nization do es prov oke resp onses from state-level actors, these resp ons e s will not break the feedback lo op unless they s ucceed in both limiting the gr owth and reducing the size of the or ganizatio n, a point to which w e will return later. These quant itative predictions ca n be tested with empirical data b y examining ∆ t as a function of k across man y or g anizations. If ∆ t ∝ k − 1 holds in the data, we have strong evidence for precisely the size-mediated feedback lo op describ ed here . Empirical data Organiza tional size data were drawn from the Big Allied And Dange r ous (BAAD) data set [8], which oﬀers the currently b est a v ailable size estimates for terro rist o rganiza tions worldwide. Other sources of size data lack the breadth or temp ora l r esolution for accura te ana lysis. F o r instance, the ST AR T progr a m and the MIPT database pr eviously held a small n umber of es timates of uncertain a ccuracy , gene r ated by Detica, Inc., a British defense con tracto r, and [44] compiled a database of information o n 649 ter roris t groups that included only estimates of the maxim um size o ver a group’s entire lifetime. The BAAD data were generated b y a survey of domain exper ts at the Monterey Institute of International Studies (MI IS) who estimated the rough order o f magnitude (1–10 0, 1 00–10 00, 1000–10 ,000 and > 10 ,000 p ersonnel) of the maximum size a chiev ed b y each o f 381 groups, be t ween 1998 and 2005, iden tiﬁed in the [45] even t database. Of these, 161 organizations conducted at least one deadly attac k, and 80 conducted at leas t t wo in that perio d. T o ensure go o d compatibility with this o rganiza tion list, even t da ta w ere drawn from the MIPT T error ism Kno wledg e Bas e [45], which con tained 3 5,668 terror ism e vents, of which 13,27 4 res ulted in at least one fatality , as of 29 Jan uar y 2008. (O ther sources of ev ent data include the Global T error ism Database [46], the W orldwide Incide nt T racking System [47] and the ITERA TE data [48]. W e note that neither these nor the MIP T database provide complete and consistent worldwide coverage.) F or the p er io d 1968– 1997 , the MIPT databas e includes mainly in ternational even ts in volving actors from at least t wo countries, while for 1 9 98–2 008 it includes b oth domestic and international even ts from muc h of the world. (The MIPT data w er e originally dra wn from the RAND T errorism Chr o nology 1968–19 97, the RAN D- MIPT T err orism Inciden t database (199 8–Pr e sent), the T e r roris m Indictmen t da tabase (Universit y of Ark ansas & Univ er sity of Oklaho ma), and DFI International’s research on terror ist o rganiza tions. In 2008, how ever, the U.S. Departmen t of Homeland Security disco nt inued its funding for the maintenance of the databas e in favor of the Universit y of Mar y land’s ST AR T center’s Glo bal T e rror is m Databa se [4 6].) Each even t is deﬁned as an attack on a single tar get in a single lo cation (city) on a single day . F o r example, the Al Qaeda attacks in the United States on 11 September 2001 app ear as three even ts in the database, one for each of the New Y ork C ity , W ashington D.C. and Shanksville, Pennsylv ania lo c a tions. E ach record includes the date, ta rget, cit y (if applicable), co untry , type of weapon used, terro rist group(s) resp onsible (if kno wn), n umber of deaths (if k nown), n umber of injuries (if kno wn), a brief descr iption of the attack and the source of the information. The organiza tions iden tiﬁed in the MIPT database are a sup erset of those co ntained in the BAAD data set, and we will us e these additional data analyses that do not r equire size estimates. F o r ea ch organiza tion, we extr acted the full sequence of its attributed o r c laimed even ts. This yields 10,33 5 even ts worldwide fr om 1968– 2008 associa ted with 910 iden tiﬁable or ganizations . F or each o f the 1,204 even ts 8 worldwide with unkno wn severit y , w e assign a severit y o f x = 0 to pre s erve timing information. F urther, bec ause of the day-level tempor a l resolution of even ts in the database , multiple even ts on the same day by the sa me gr oup hav e a mbiguous “delay” (inv erse frequency). W e eliminate this a mbiguity by agg r egating such even ts in to a single “ even t da y” with sev erity eq ual to the sum of the component se verities. This slightly reduces the num b er o f even ts, mainly for the most active organizatio ns la te in their life history . As a co nsequence, the minimum res o lv able delay in the databa se for tw o even ts by the same organiza tion is ∆ t = 1 day . Regression mo dels Before analyzing the ev o lution of attacks by individual o rganiza tions we conduct static o r cross- sectional regres s ion analysis a t the lev el of individual organizations. W e examine the relationship b etw een group size and attack patterns, in particular the delay b etw een attac ks , the exper ience of a g roup in terms o f nu mber of even ts, and the sev er ity of attacks. T o r e cap, we exp ect lar ger groups to gener ate a larg e r num b er of a ttacks, hav e sho rter delays b etw een attacks (H1), and g enerate mor e severe attacks ev en a ccounting for other attack patterns (H4). W e can ev aluate H1 by comparing max imum gro up size s from BAAD and the minim um delay b etw een attacks ∆ t in MIPT. W e ca n asse s s H4 b y comparing size and the maximum severity x of a ttacks. Finally , H2 implies that larger g roups should hav e higher maximum exper ie nc e k or cumulativ e num b er of events. (H3, p o stulating a declining delay with subsequent attack, cannot b e ev aluated with static data; w e re tur n to this p oint later.) Although group size sho uld predict attac k patterns, individual measures suc h as maxim um severit y will be at least in pa rt a function o f the total num b er of attacks. Tha t is , for an y distribution of severities, an increa sed production ra te (sa mpling int ens ity) will naturally inﬂate the maxim um severit y ov er a ﬁxed time perio d, even if the distribution is s tationary . Thus, in o rder to exa mine the pa rtial relationship b etw een size and the r elated attack v ariables—o r their indep endent predictive v alue on size once we take in to a ccount the other attack pattern characteristics—it is mor e conv enient to consider to what extent we can ac count for size as function of the attack measures. W e use a n order ed logit regr ession model of size since the BAAD data g ive order- o f-magnitude es- timates of maximum s iz e. As the BAAD data p ertain to the time p erio d 19 98–2 0 05, we restrict our attack pattern measures to attac ks dur ing this same time p erio d. Since the distributions of minim um delay , maximum exp erience, and maximum severit y are all highly skew ed we take the natural lo garithm, adding 1 to se verity to preven t tak ing the log of 0 in the case of non-fatal even ts. W e rep ort the empirical estimates in T able 1. The results display a signiﬁcant nega tive r e lationship b etw een fatal attack delay and group size , consistent with our claim that larger g roups will have shorter dela ys betw een attacks (H1). W e also ﬁnd a positive r elationship betw een group s ize and experience, consistent with our claim that larger groups T able 1. Ordered logit regressio n of gr oup size, b y fatal attack patter ns V ariable ˆ β SE( ˆ β ) Delay: ln min(∆ t ) -0.351 0.119 Exp erience: ln max ( k ) 0.707 0.193 Severit y: ln max( x ) 0.150 0.159 ˆ α 0 | 1 -0.163 0.840 ˆ α 1 | 2 2.652 0.895 ˆ α 2 | 3 5.039 1.056 N = 80, LR χ 2 = 41.42, df = 3, 5 8.75% correctly classiﬁed 9 T able 2. Linear regres sion of exp er ie nc e , b y attack delay and severity F atal attacks ( F ) All attacks ( A ) V ariable ˆ β SE( ˆ β ) ˆ β SE( ˆ β ) Delay F : ln min(∆ t ) -0.119 0.042 -0.110 0.040 Delay A : ln min(∆ t ) -0.7 78 0.110 -0.795 0.105 Delay F × Dela y A 0.074 0.017 0.073 0.016 Severit y: ln max( x ) 0.190 0.059 0.150 0.056 ˆ α 3.115 0.236 3.336 0.225 N = 167, R 2 = 0.545 N = 167, R 2 = 0.565 R 2 ( ¬ severit y) = 0.515 R 2 ( ¬ severit y) = 0.546 R 2 ( ¬ delay) = 0.22 2 R 2 ( ¬ delay) = 0.18 2 generate a hig her num b er of a ttacks (H2). Finally , the maximum severit y of the attacks is no t signiﬁca ntly related to group size, once we hav e con trolled for delay and experience v ariables . This contradicts t he hypothesis that larg er gro ups ar e s ystematically more likely to genera te severe attacks (H4). Overall, the mo del places 58.75% of all the groups in the co r rect bins for group size. Only 5% of the observ ations are badly mis- classiﬁed, with predictions o ﬀ by mo re than o ne or der o f ma g nitude. By cont r ast, a nu ll mo del predicting a ll gro ups to have the mo dal siz e categ ory (100 − 1000 ) only class iﬁed 43.75 % of the observ ations cor rectly . (W e co nsidered a num b er of alter na tive speciﬁca tions. Sev er ity remains an insigniﬁcant predictor of gr oup size when we consider combinations o f dela y and exp erie nc e fo r both deadly and no n-deadly attacks. Using a linear reg ression model rather than o rdered log it do e s not change our substantiv e conclusions.) Since the BAAD data cov er only a bo ut half o f the iden tiﬁable organizations in the MIPT database ov er a restricted time span (1998–2 005), we conduct a supplemen tar y analys is with the full MIPT datas e t, where w e consider how a group’s total experie nce can b e accounted for by diﬀerenc e s in minim um de lay and maximum a ttack severity . (W e limit the analysis to MIPT org a nization that generated at least t wo even ts (frequency) and one deadly event (severit y); only 16 7 o r ganizatio ns satisfy these criter ia.) T able 2 rep ort the results for a linear r e g ressio n with logg ed v alues for a ll the terms for fatal ( F ) and all a ttacks ( A , including no n-fatal attacks) exp erience res p e c tively . The results clearly show that the minim um delay is a signiﬁcant predictor of g roup e x pe rience, and they mildly supp or t the claim ab out severit y , as the p ositive co eﬃcient for sev er ity is signiﬁcan tly diﬀerent from 0. How ever, comparing the c hange in the R 2 for estimating the mo de l with and without the severity and delay terms resp ectively indicates that dropping the severit y v aria ble leads to a rela tively small decline, while the impact of omittin g the delay v a riables is substantial. Hence, v aria tion in delay b etw een attac ks accoun ts for m uch more o f the v ariation in experience than do es severity . These static analyses provide substantial preliminary e vidence in suppo rt of H1 and H2 and little evidence to supp or t H4. W e now go beyond static analyses and test our predictions for all organizations in the MIPT database using a novel dynamical analysis to ol called a “dev elopment c ur ve.” Dev elopmen tal dynamics A developmen t curve is a statistical to ol that mea s ures the ev olution of org a nization b ehavioral v ar iables along a common quantitative timeline [22]. It is similar in structure and us e to the “exp er ience”, “learn- ing” and “progress curv es” sometimes used in managemen t science [36, 39] to quan tify the relatio nship betw ee n per -item pr o duction c ost (o r time) and “ex pe r ience” (cumulativ e item pro duction). B e cause we study b ehavioral v ariables rather than the costs of pro ductio n, and to e xplicitly avoid implying lea rning- based mechanisms, we c ho os e a distinct term. The analys is of these developmen tal curves facilitates 10 direct compariso ns of th e b ehaviors of diﬀerent g roups at similar p o ints in their life histories, which is useful for testing our h yp otheses . W e instrument a common timeline using org anizational exp erie nce k , deﬁned as the cumulativ e nu mber of even ts pro duced b y or asso cia ted with a particula r orga nization, and we co mpare the delay ∆ t b etw een the k th and ( k + 1 )th even ts, or the severity x of the k th attack, acr oss all or ganizations in our sa mple. F o r each of the 910 orga niz a tions, we ex tract from the MIPT ev ent da ta an o rdered sequence of coo rdinates { (1 , z 1 ) , (2 , z 2 ) , . . . } , which represe nt the g roup’s b ehavioral tra jector y on the v ar ia ble z o ver its lifetime. The visualiza tio n of such tra jector y is typically made using double-logarithmic axes , a s illustrated in our simulation results in Figure 2. Although the curve constructio n itself ignor es details suc h as the date of an org anization’s ﬁrst attack, its loca tion, ideology , etc., these v ar iables can b e used for s ubsequent analysis, e.g., comparing the tra jectories across co v aria tes. Constructing a development curve for an individual orga niz a tion (see Suppor ting Information) can fa- cilitate the in vestigation of speciﬁc b ehavioral dynamics of individual groups o ver their lifetimes. How ever, the spec iﬁc factor s asso cia ted with particular organizations may obscure the ge ne r ic tendency embo died by our hypothesis. T o investigate these, we ex a mine the av erag e tra jectory acro ss many o rganiza tions by tabulating the co nditional distribution Pr(∆ t | k ) of delays, for a sp eciﬁed level of exp erie nc e k . Thus, an organiza tion that has ca rried out k ∗ even ts con tributes to each of the k ≤ k ∗ conditional distributions. This approach pro vides a strong test of th e frequency-a cceleratio n (H3) and attack-severity hypotheses (H4) predictions. F requency of attac ks o v er tim e Figure 3A shows the comp osite frequency curve for all o rganiza tions in our study . T o reduce the overprin t- ing eﬀects of showing the tr a jector ie s for so man y orga nizations, we bin the v alues of k on a lo garithmic scale and plot the mean and 1st and 3 rd quartiles of the da ta within each bin. Remark ably , the o bserved empirical pattern agree s v ery close ly with our sim ulatio n model’s predictio ns (Figure 2). The pr ogres s ive decr e a se of the dela y distributions indicates a generic tendency tow ard faster pr o- duction with increased experience for all types of o rganiza tio ns, in stro ng agr eement with the frequency- acceleratio n h yp othesis (H3). But, the rela tionship betw een dela y and exp erie nc e is not deterministic: not ev ery even t occurs more quic kly than the last but the statistical tendency to ward s horter dela ys is clear. A terrorist o rganiza tion thus typically b egins in the low-frequency domain (large ∆ t ) and mov es in ﬁts and star ts tow ard the hig h-frequency domain (sma ll ∆ t ). This tr end is not subtle: the median delay after the 1st even t is ∆ t = 1 24 da ys, while by the 12th even t, it has dropp ed to 35 days a nd b y the 25 th, the next even t typically comes only 2 1 da ys later. This trans itio n to fast pro ductio n do es take considerable calendar time: for groups tha t ac hieve k = 12 ev ents, the median t o tal calendar time be tw ee n the ﬁrst and tw elfth ev ent is 4 . 4 years. Similar results hold for the timing b etw een deadly attacks. None of the s ampled orga nizations pro gressively slow ed their a ttack rate ov er time, moving fro m high- frequency to low-frequency . A few unusual gr oups, such as Al-Qaeda in the Land of Two Riv er s, begin and remain in the high-freq uency domain. But, Al-Qaeda in the Land of Two Rivers is an in ter esting case b ecause it is well-known to hav e op erated under a diﬀerent name prior to 20 04 [49]; thus, their initial high-frequency b ehavior can be interpreted as supp or t for th e labor - constraint hyp o thesis (H1) because their initial larger s ize—a hold over from their previous identit y—a llowed them to “ b egin” life ( k = 1) at a relatively high initial pro duction rate of attacks. Statistical mo del for the frequency of attac ks Quantifying the dynamical r e lationship betw een delays and exp erience allows us to go beyond our static analyses. T o do this, we statistically mo del the conditio nal dis tr ibution Pr(∆ t | k ) from whic h dela ys are drawn and how this distribution v aries with exp e r ience. 11 10 100 1000 Delay, ∆ t (days) Empirical mean 25−75% quantiles Model 1 5 10 50 100 500 1 10 100 1000 Cumulative number of events, k Num(k ≥ K) 10 0 10 1 10 2 10 3 10 4 10 5 0 0.2 0.4 0.6 0.8 1 Scaled delay, ∆ t k β Pr(T ≥ ∆ t) Events no.1−4 Events no.5−8 Events no.9−12 Events no.13−16 Events no.17−39 Events no.40−99 Events no.100−519 log−normal Figure 3. Timing of ev en ts . (A) Mea n dela y h log ∆ t i b etw een attacks, with 1st and 3rd quartiles, vs. group exper ience k . Solid line shows the exp ected mea n dela y , from the statistical model desc rib ed in the text. Low er panel sho ws the num be r of organizatio ns with at least k evens. (B) A “data collapse” showing the alignment of the re-s c aled c o nditional delay distributions Pr(∆ t · k ˆ β | k ) with the estimated underlying log-nor mal distribution, as predicted by the mo del. F or these data, a truncated lo g-norma l distribution, with the following ma thematical form Pr(∆ t | k ) ∝ exp  − (log ∆ t + β log k − µ ) 2 2 σ 2  , (1) provides an excellent ﬁt to the empirical dela y data for all o rganiza tions. Here , σ 2 is the v a riance in delays a t a given k , µ is related to the c ha racteristic delay b etw een attacks and β controls the rate at which that dela y decreases with increa sed exper ience k . That is, β go verns the strength of the feedback lo op b etw een orga niza tional ex pe rience and the pro duction of new even ts. T o include the eﬀect of the minim um timing r esolution ∆ t ≥ 1 present in the empirica l data, w e force Pr(∆ t | k ) = 0 for ∆ t < 1 da y . This mathematical structure implies that the typical delay betw een attacks g enerically decr eases according to a pow er -law function with incre a sing experience ∆ t ≈ e µ k − β . (2) (Details of this deriv ation are giv en in the Suppor ting Infor mation.) Thus, if β > 0, we will o bserve a transition toward inc r easingly fas t event pro duction, indicating supp ort for H3. In contrast, if β = 0, pro duction rates do no t v ary with organiza tional exp erience, while if β < 0, pro duction rates will decre ase (larger ∆ t ) with increasing experience. In the β > 0 reg ime predicted b y H3, the acceleration eﬀect is damp ened a s the mean delay a symptotes to the minimum timing resolution at ∆ t = 1; this pr o duces slight upw ard curv ature for large v alues of k (see Supp or ting Info r mation). The particular v alue of β has a strong eﬀect o n the material dyna mics of the feedback lo op b etw een increasing exper ie nce and increasing production. If β = 1, then the feedback loo p is linear, as in our simulation mo del, and increas es in organizatio nal exp erience lea d to prop ortio nal incre ases in even t pro duction. Line a rity implies that th e ma r ginal growth a sso ciated with an additiona l ev ent is relatively constant ov er the o rganiza tio n’s lifetime and a roughly constant fraction of new recruits are alloca ted to increase ov erall tempo of militant activ ities . In contrast, β 6 = 1 implies a non-linear feedback pro cess. No tably , non-linear feedback pro cess es are not common mo dels of so cia l pro ces ses (but see the literature on arms-r aces, particularly [17] and [50]). T raditional mo dels often fo cus on pr op ortiona l eﬀects in which incre a ses in one v a riable cause prop ortio nal 12 T able 3. F requency curve para meter s for o r ganizatio ns with similar p olitical motiv ations. Note: statistical signiﬁcance estimated via Monte Carlo simulation of a tw o-tail test against a n ull mo del with β = 0 (no frequency a c celeration), us ing the sum-o f-squared errors (SSE). V alues in paren theses indicate bo otstrap standard uncertaint y in the last digit. po litical motiv ation groups ev ents µ σ β signiﬁcance nationalist-sepa ratist 55 2959 5 . 1(5) 2 . 2(1) 0 . 9(2) p < 0 . 0 01 reactionar y 5 143 3 . 2(6) 1 . 8(2) 0 . 1(3) p < 0 . 0 01 religious 17 999 5(1) 2 . 4(5) 1 . 7(5) p < 0 . 0 0 1 revolutionary 53 2 527 5 . 7(4) 2 . 3(2) 1 . 1(2) p < 0 . 0 0 1 all secular 883 6232 5 . 2(2) 2 . 25(9) 0 . 9 (1) p < 0 . 00 1 all groups 910 72 31 5 . 1(2) 2 . 32(9) 1 . 0(1 ) p < 0 . 001 changes in other v ariables. In non-linear feedback pro ce s ses, small increases in one v aria ble ca n produce dramatic and con tinuing swings in o ther v a riables, leading to highly unpredictable dynamics [51]. When β > 1, the feedback is sup er-linea r, a nd one o r b oth of these fa ctors must increa s e with k . Tha t is, either per- even t growth in militan t activities increases o ver time or an increasing fr action of growt h is allo ca ted to militant activities. When β < 1, the feedba ck is sub-linear and the mar g inal recruitment bene ﬁts of new events decrea se ov er time or they are cons ta nt but re c ruits are increasing ly allo cated tow ard non-militan t activities. Fitting this mo del dire ctly to the empir ical data on all ev ents, w e ﬁnd that t he maxim um likelihoo d estimate is ˆ β = 1 . 0 ± 0 . 1 (std. err .), indicating linear feedback. (This approach to estimating the par a meter gives w eig ht to the event s early in organiza tion’s lifetime that is prop or tional to the num b er of such e ven ts in our data set; in con tra st, a simple r egressio n appro ach o n the mean delays would bias the estimate b y giving signiﬁcant weigh t to the rare but long-lived groups.) Using a Monte Carlo simulation aga inst a null mo del with ﬁxed β = 0 (no acceleration o ver time) and with µ , σ estimated using maximum lik eliho o d given the ﬁxed β v alue, we ﬁnd that the v alue o f ˆ β is highly s tatistically signiﬁcant ( p < 0 . 00 1). (Fitting to deadly attac ks alo ne yields a highly statistica lly signiﬁcant ˆ β = 1 . 1 ± 0 . 2, slightly in the super- linear regime, but this v a lue is statistica lly indistinguisha ble from β = 1.) A linear feedback implies that the margina l growth from even t-driven recr uitment do es not v ary muc h with organizatio nal size or exper ience. F urthermore, it implies that organizational learning in terror is t groups [25, 38], in whic h the pr o duction rate increa ses due to improved eﬃciency of a ﬁxed num ber of individuals, plays a lesser role in explaining the ov erall acce le ration of event pro duction than do the eﬀects of increasing organizatio na l size, b eca use lea rning would mimic the eﬀect of super- linear feedback by allowing a co nstant num b er of militan ts to b ehav e iden tically to an increasing num b er. A strong test of the statistical mo del’s plaus ibility is its prediction that each of the k conditional delay distributions Pr(∆ t | k ) is a sca led v ersio n of the underlying log -normal distribution L N( µ, σ 2 ). T o test this prediction, we re-sc ale the empir ical distributions by the predicted factor , i.e., w e multiply each delay v ariable ∆ t i by k ˆ β i , a nd then plot them ag ainst the estimated underlying log-no rmal distribution. A close alignment of thes e r e-scaled conditional distr ibutions, als o called a “data c o llapse” [5 2], is strong evidence for the hypothesized data mo del ov er a wide range of a lternatives. F urthermore, for a n a lternative mo del to pro duce s uch a data colla pse requires that it follows the log-no rmal form closely enough to b e eﬀectively equiv alent . Figure 3B sho ws the results of this test, illustrating a n excellen t da ta colla pse, with eac h of the re-scaled log-norma l conditional distributions closely aligning with the under lying log-norma l form. These r esults also hold when we c onsider the dev elo pment cur ves for groups with a co mmon p o litica l ideology (see Supp or ting Information). [53] divides the political motiv ations for t er roris m in to four con- ven tional categ ories: nationalist-sepa ratist, reactiona ry , religio us and revolutionary . W e co ded ac c ording to Miller’s criteria the 131 most proliﬁc groups in our sample (all with k ≥ 10 deadly e ven ts), which 13 1 5 10 50 100 250 1 10 100 1000 Cumulative number of events, k Severity of next fatal event, x Observed mean severity 25−75% quantiles Expected mean severity 10 0 10 1 10 2 10 3 10 4 10 −3 10 −2 10 −1 10 0 Severity, x Pr(X ≥ x) Events no.1−4 Events no.5−8 Events no.9−12 Events no.13−16 Events no.17−39 Events no.40−99 Events no.100−298 Richardson’s Law Figure 4. Sev eri t y of even ts. (A) Mean severit y h log x i of deadly attac ks, with 1st and 3rd quartile, vs. group exper ience k . Solid line (with slop e zero) shows the exp ected delay , fr om a simple regression mo del. (B) Conditiona l severity distributions Pr( x | k ), showing a data c ollapse on to a heavy-tailed distribution, with the maximum likelihoo d p ow er - law mo del for all severities (Richardson’s Law). accounts for 85% o f events, and ﬁtted Eq. (1) to the data within each ideolog ical catego ry . Organizations with multiple po litica l motiv ations were placed in multiple catego ries, which would only lessen any diﬀer- ences b etw een estimated parameters for diﬀerent catego ries. Within e ach of these catego ries, we obser ve the same acce leration pattern, with the strongest accelera tion (largest β ) app ear ing for relig io us groups (T able 3). Sev erity of atta ck s o v er ti me In co ntrast to the delay dev elo pment curve, we ﬁnd no statistica lly signiﬁcant relationship b etw een the severit y of attacks and incr eased exp er ience (Pearson’s r = − 0 . 024, t-test, p = 0 . 1 7), indicating no suppo rt for the severit y-increa se h yp othesis (H4). Acros s all organizations in our sa mple, the average severit y o f the ﬁrst deadly e ven t is h x i = 6 . 7 ± 0 . 9 , which is o nly slig htly lar g er than the av er a ge s e verit y of deadly events by highly experienced groups (those with k > 100) h x i = 5 . 1 ± 0 . 6. Figure 4 A shows the comp osite severit y curv e for all organiza tio ns in our study . As with the frequency curves, we ﬁnd that the conditional severity distr ibutions Pr( x | k ) roug hly collapse on to a single, underlying for m (Figure 4B), which is similar to the power law obs erved for all deadly ter rorist attac ks w orldwide from 19 68–2 008 [20, 31]. Tha t is , Richardson’s Law fo r terror ism app ears to hold for b oth inex p er ienced and highly expe r ienced groups . Co mbin ed with our sta tic analysis of orga niz a tional size, this pattern implies a hig hly counter-in tuitive fact: the severity of attacks by larger , mor e exper ie nced organiza tions, is not sig niﬁcantly gre a ter than the s everit y of attacks by small, inexp erienced orga nizations. That is, the common assumption that only exp erienced gr oups are capable of such mass destruction [54] is incorrec t: inexp er ienced o rganiza tio ns a re just as likely to pro duce extremely severe even ts as highly exper ienced orga niz a tions. How ever, although more exp erie nced or ganizatio ns ar e not systematically more letha l at the individual- even t le vel, the obs erved frequency-acce leration pattern implies that more exp er ienced groups are signiﬁ- cantly more lethal overall. This pattern was o bserved by [8] in their ana lysis o f the BAAD orga nizations. Our res ults thus clarify their results, showing that the observed correlation b etw een g reater lethality (total deaths a ttributed to an o rganiza tio n) and greater o rganiza tional siz e appear s be c ause lar ger, more exp erienced organiza tions pro duce even ts more quic kly than sma ller, less experienc e d org anizations. It is the cumulative eﬀect of the man y small ev ents tha t genera tes an increased lethality , not a systematic 14 increase in the lethalit y of individual even ts. Repe a ting this analyses on our ideolog y-co ded set o f o rganiza tions, we ﬁnd no sy s tematic dependence of severit y of attacks on or ganizationa l exp erience within a ny of the ideolog ical catego ries (see Supp o rting Information). That is, no ne of the mo del coe ﬃcie nts are s ig niﬁcant, and the a verage severity of ev ents within each ca tegory v ary only a little. In short, we ﬁnd that p olitical ideology has no systematic impact on the sev er it y of even ts o r the tra jectory that event severities tak e o ver the lifespa n of an o rganiza tion. Discussion Although details and circumstances v ary widely acro ss terro r ist or ganizatio ns, the generic nature of our results suggests general conclusio ns . In particular, we ﬁnd strong ev idence for a positive feedback lo op among orga nizational size (n umber o f per sonnel), exp erience (cumulativ e num b er of even ts) and the freq uency at whic h that orga nization launches new even ts. Small a nd inexperie nc e d org anizations tend to pro duce ev ents slowly , while larger and mor e experienced organizations tend to produce ev ents sometimes h undreds of times more frequently . Within this feedback lo op, new attacks lead to organizatio na l growth and the c orresp o nding incr ease in size leads to fas ter pro duction of new even ts beca use a la rger siz e means more terro r ist cells a re op era ting in pa r allel, no t b ecause ev ents themselves are planned more quickly . The res ult of this f ee dba ck loo p is a generic “dev elopmental” t r a jectory : as an organiza tion ag es, it tends to pro duce violent events more and more quic kly . The typical form of this rela tionship can b e mathematically mo dele d by a p ow er-law function, in which the delay ∆ t b etw een co ns ecutive even ts dec r eases r oughly like ∆ t ∝ k − β where k counts the cum ula tive num b er of events and β des crib es the strength and direction of the feedback lo op. The implication of the pow er - law pattern is that large organizations are very m uch lik e “s caled up” versions of small o rganizatio ns, and in par ticular that size a nd exp erience ar e coupled in a p ositive feedback lo op. Across all organizations in our sample, we estimate β = 1 . 0 ± 0 . 1, indica ting a linear feedback lo op, which implies that a n organiza tion’s o verall s ize is strongly correla ted with the size of its militan t wing . This pattern is strongest for small or inexperience d o rganiza tio ns, e.g., those with k ≤ 1 0 even ts, which cov ers 87% of the 910 or g anizations in our sample. In con tra s t, hig hly exp erienced organizations see m to saturate their e ven t pro duction rates a t the daily or weekly level, which may be indicative of a tendency o f large or g anizations to engage in multip le t yp es of a ctivities, e.g., the pro vis ion of so cial ser v ices, criminal activities, etc., contin uing to grow their militan t wings. The mathematical precisio n of this relatio nship is striking, as is the ability of our computer simulation to repro duce it. Except for Richardson’s Law for the frequency and severity of wars, few statistica l relationships in the study of political violence exhibit such regularity . The p ower-la w r e lation betw een organizationa l exper ie nce a nd productio n r ate is b oth conceptually and mathematically similar to the relatio nship b etw een cost and cumulative pro duction observed in manuf a cturing [3 6] or or ganizatio nal learning [37, 39], where decreases in per-item pro duction costs or time can be descr ib ed by a power law in the cumulativ e num ber o f items pro duced. That a similar patterns app ears in the pro duction of terroris t even ts is sur prising, and it may not b e sup er ﬁc ia l to descr ibe terror ist organiza tions as a special type of manufacturing ﬁrm whose principal product is p olitica l violence and whose ov erall pr o duction of violence is fundamen tally constra ined b y its size. The implication is that terr orism is inherently no n- amenable to mas s pro duction, i.e., it is not a scalable en terprise, perhaps becaus e eac h ev ent must be h umanly conceived and planned around a par- ticular target, tactic or en vir onment, and there is a limit to how m uch this pro cess can be automated. One implication of this conclusion for cyber - terror ism is that even there, despite the great p otential for automating attacks, these to o will likely not b e scala ble without adv ances in general artiﬁcia l int ellig ence. In the la nguage of economics, we say that terro rism capital and lab or are not freely substitutable with r esp ect to pro ducing new events. If the day-to-day work o f even t pr o duction do es not require 15 sp ecialized skills, then the growth potential o f a n organization b e extremely la rge because it may dr aw on the largest p ossible p o ol o f p otential recruits. This point suggests that conﬂict-level event pro duction rates sho uld ultimately b e re s p o nsive to p olicy a nd counter-terrorism eﬀorts that targ et the size and mobility of the po o l of p otential recr uits. That is, suc c essful “hear ts and minds” stra tegies [55] ar e likely to lead directly to low er incide nt r a tes by b oth restricting the growth and reducing the size of terrorist organiza tions. They ma y not, how ever, eliminate th e p os sibility of sp ectacula r attacks as these do not depe nd o n organiza tional size. Recently , following our o riginal work on progre s s curves in terro rism [2 2], Johnson et al. [25] analyzed the timing of even ts in the Iraq and Afghanistan conﬂicts, ﬁnding similar p ow er- law lik e acceler ation curves in the delay b etw een ev ents. They argue that this pattern is caused b y a kind of “ red queen” eﬀect—a concept b orr owed from ar ms races in evolutionary biology [56]—in which tw o sides of the conﬂict race through some abstrac t space, and the timing b etw een even ts is given b y how f a r “ahead” the insur gent side is in the race. In pra ctice, howev er, this explanation is diﬃcult to v alidate b ecause the c onnection is not sp eciﬁed as to how real-world ev ents and structures drive the dynamics of the abstract ra ce. In con trast, our e xplanation of t he phenomena is b oth tangible, general and testable: we argue that the size of the insurgency or the terr orist group se ts the temp o o f the co nﬂict. The more peo ple ther e are ﬁghting, the more fre quently w e will obser ve events. This explanation ma kes direct and testable pr edictions ab o ut the relationship of organizational size and frequency of ev ents, which we show are upheld by empirica l data on orga niz a tional sizes. (As a technical note, in the lang uage of physics, the “size” of a n organization or ins ur gency is an extensiv e v ariable of the conﬂict system, muc h like a rea and nu mber of particles are for ph ysica l systems [57]; this fact mak es additional testable predictions of our theory .) The implication for the Iraq a nd Afgha nistan conﬂicts is that the num ber of insurgents active in the v a rious provinces is the primary determinant of the frequency of even ts observed there. Although the acceler a tion is remark ably strong, the v ast ma jor ity of organiza tions do no t achiev e high levels of ex p er ience (only 23% o f groups a re asso ciated with k > 10 even ts) or fast pro duction rates. The pr ogres sive loss of organizations co uld b e due to high rates of organizationa l death, e.g., from counter-terrorism activities or internal conﬂicts [44, 58], shifts a wa y from violence , or a right-censoring eﬀect on y o ung and still active or ganizations. Signiﬁcant ly , the pa r ticular mo de of orga nizational demise seems not to have a strong impact on the pro duction time of ev ents, sugges ting that the transition from developmen t (growt h) to death ma y happen very quickly , s o that the experience curve do es not b end up ward but rather simply halts. F urther exploration of the death of or ganizations [44, 5 8], and how it impacts the pro duction of violence, is an interesting av enue for future w or k . Regardless of the reason, we do not exp ect the feedback lo op to co nt inue as k → ∞ . If an orga nization succeeds in becoming large enough t o pro duce new ev ents ea ch day , it may function more like a stable or mature soc ia l institution, with fundament a lly diﬀerent c o nstraints and incentiv es on the pro duction of vio lence. Larg e size a nd stability may also pose specia l risks, e.g., lea ding to larger or longer conﬂicts. On the other hand, non-violent activities, e.g., engag ement with political pro ces ses, may als o b ecome more attractive with incr eased size. Explor ing these p ossibilities is an interesting av enue for future work. Unlik e the pro duction of even ts, we ﬁnd no evidence of any r e lationship with the severit y of a t- tacks (H4). Rather, Richardson’s Law—a p ow er-law distribution in the frequency of severe ev ents— characterizes the sev erity of even ts at all levels of organizational size or exp erience, and indep endent of the organiza tion’s p olitical ideolo gy . This fact clariﬁes ongoing eﬀorts to identify the underlying so cial, p olitical or physical mechanism that generates Richardson’s La w in terro r ism. Several existing explanations assume or predict a severit y- size relationship, e.g., the aggreg ation-disintegration model of Johns on et al. [23] and [35], but these seem increasingly unlikely given our results here, because they assume the maximum severit y of an even t is prop ortiona l to the organiza tion’s size N ; th us, if N is small, the severity of even ts x will a lso be small. That is, in their e xisting form, these mo dels predict a severit y-size rela tio nship that do es not appear in the data. Of course, thes e models may b e adapted to pro duce the observed size-indep endence pattern, 16 but doing so requires additional assumptions a nd additional v alidation that ma y not be warranted. In con tra s t, tw o plausible explanations are not ruled out: (i) the explanation prop osed in [2 0], whic h po sits a c o evolutionary co mp e titio n b etw een states and ter rorists in w hich e vent planning time and severit y are str ongly r elated, and (ii) the explanation prop osed in [24], in which p opulation densities are broad-sc a led a nd terro rists preferentially ta r get high-density lo cations. Both of these explanatio ns do not assume any relatio n b etw een the severity of an attac k and the size of an o rganiza tion. T ogether, our results sugg e st tha t the total lethalit y of lar ger and more mature groups obser ved b y Asal and Rethemeyer [8] is probably b est explaine d a s a natur al consequence o f their much more fr equent activities, r ather than as a systematic incr e a se in the deadliness o f individual even ts. Policies that limit the gr owth o f an or g anization’s militant wing s hould low er the lo ng-term probability o f a severe event by that org anization. Such growth-limiting p olicies co uld b e des c r ib ed as “star ving the beas t” of the la bo r necessary to produce r are but highly s evere even ts. The mos t pr o ductive tar gets of such p olicies will b e large, es tablished o rganiza tions with long his tories of pr o ducing ter r orist attac ks. By virtue o f their size, these o rganiza tions are likely to b e well-kno wn play ers in their particular co nﬂicts and thus easy targ ets for sp eciﬁc po licies. Because small or ganizatio ns are equally likely to pro duce severe events, p olicies aimed sp eciﬁcally at larg e, well-known orga nizations may not limit the overall risk of severe even ts from all s ources. F or sma ll and p otentially unknown organiza tions, the most eﬀectiv e policies may be those aimed at prev enting their for mation in the ﬁrst place, i.e., po licies that curtail the a cquisition of the means for a nd r e sort to viole nce. Lacking this, once such a terr o rist cell car ries o ut its ﬁr st attack and b egins its developmen tal tra jectory , the b est actio n by a go vernmen t ma y be an “ overwhelming resp ons e ” to enco ur age th r ough v ar ious means the dissolution of the nascent or g anization and the t r uncation of its growt h tra jecto ry . This po licy is not without risk to the state, ho wev er , a s certain co unt er measures may serve the terr orist’s goals [59, 60]. In closing, we p oint out that the acceler ation in the frequency of terrorist even ts is independent of many commonly studied factors as so ciated with terro r ism, including g e ographic lo cation, time pe rio d, in- ternational vs. do mestic targ ets, ideological motiv ations (religio us, national-sepa ratist, reactiona ry , etc.), and politica l context. Our results th us demonstra te that some a sp ects of terror ism are not nearly as contingen t or unpredictable as is o ften assumed a nd the a ctions of terro rists may be constrained by pro cesses unrelated to strategic tradeo ﬀs among c o sts, beneﬁts and pre fer ences. Identifying a nd under - standing these proc e sses oﬀers a complemen tar y approa ch to the tra ditional rationa l-actor framework, and a new w ay to understand what regularities exist, why they exist, what they imply for long -term so cial and political s tability , e.g., lar ge-scale violen t conﬂicts like civil and int er state w a rs. 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T er roris m and Political Violence 9 : 16–2 3. 21 Supp orting Information • Section A : Supplemental A n alysis of Size, F r e quency and Severity : Additional ana lysis of the organiza tional size data, with resp ect to the frequency and s everit y of their ev ents. • Section B : Development Curves for F our Pr oliﬁc Or ganizatio ns : Individual freque nc y and severit y developmen t curves for the four most proliﬁc organizations in the MIP T dataset. • Section C : T err orist Or ganization Computer Simulation : Sp eciﬁcatio n and sim ulation co de for the computer sim ulation describ ed in the main text. • Section D : Statistic al Mo del for t he F r e quency of Attacks : Mathematical details of the statistical mo del for the g eneric pattern in ev ent freq uencies v er sus organiza tio nal exp erience . • Section E : Domestic vs. T r ansnational Events : Robustness check of the freq uency acceleration pattern b y consider ing orga nizations whose ﬁrst even t was prior to 1998 (mainly in terna tional terroris t or ganizations ) versus after (mainly domestic terr orist organizations). • Section F : Politic al Ide olo gy & F r e quency and Severity Curves : V ar iation in th e developmental tra jector ies of org anizations b y p olitical ideology , showing diﬀerent frequency acceleratio n ra tes a nd no diﬀerences in ev ent severity evolution. A Supplemen tal Analysis of Size, F requency and Sev erit y The growth hypo thesis predicts that a groups maximum size will be inv ers ely related to the minim um delay b etw een its attacks o ver the 1998– 2005 per io d. T o complement the analysis in the main text, here we show the gr aphical plots and conduct additiona l analysis. An analysis o f v ariance indicates that the average minimum delays diﬀer sig niﬁcantly betw een size categorie s ( n -wa y ANOV A, F = 9 . 98, p < 0 . 00 0013). F ur ther, w e ﬁnd that larger orga nizational size is a signiﬁcant predictor of increased attack f r equency ( r = − 0 . 4 9, t -test, p < 10 − 5 ). Fig. S1a shows the distributions within the size categories. Although the distributions do ov erlap somewhat, the do wnw ar d trend is clear. In contrast, size, like exp er ience, is not a signiﬁca nt predicto r of median attac k severit y ( n -wa y ANOV A F = 0 . 59, p = 0 . 6 2). Fig. S1b shows the distributions within the p erio d. (W e c ho o se medians b ecause they are robust to the larg e ﬂuctuations caused by small samples dr awn from hea vy-ta iled distributions.) Although there is some v ar iability b etw een size categories, the lack of a trend is c le ar. B Dev elopmen t Curv es for F our Proliﬁc Organizations As a n example of dev elo pment curv e analysis, Figure S2 shows the frequency and sev er ity development curves for the four or ganizations with the g reatest num ber of attributed event-da ys in our datase t, including b oth deadly and non- deadly even ts: the Revolutionary Armed F orces of Colo mbia (F AR C; 520 ev ents), th e T a liban (349 ev ents), Ba s que F ather land and F ree do m (ET A; 3 11 ev ents), a nd Hamas (308 even ts). Non-deadly even ts ( x = 0) incr ement the counter k for the severity curve but do not a ppe ar on the severity cur ve ﬁgures; hence, E T A, whic h carried out 26 1 (84%) non-deadly even ts, has r elatively few po ints in its sev erity curve. F or these o rganiza tions, the median dela y b etw een the k = 1 and k = 2 even ts is ∆ t = 433 days. In contrast, the median delay b etw een the m o st recent pair of even ts by these g roups is only ∆ t = 4 days, a 100 -fold increase in frequency . In ea ch case, the frequency curve begins in the upp er -left corner of the ﬁgure, representin g v ery long dela ys betw een subsequent even ts, and a s k incr eases, the cur ve mov es 22 0−100 100−1000 1000−10000 10000+ 0 0.5 1 1.5 2 2.5 3 3.5 log 10 (min ∆ t) Maximum group size a 0−100 100−1000 1000−10000 10000+ 0 0.5 1 1.5 2 2.5 3 3.5 log 10 (median x) Maximum group size b Figure S1. Box-plots o f the distributions o f a groups (a), minimum dela y log(min ∆ t ) and (b), median attack severity log(median x ) for attacks within 1998–2005 , within each o f four size categories. F or conv enience, w e connect the means of each catego r y , which are signiﬁcantly diﬀerent in the case of delays ( n -way ANOV A, F = 9 . 9 8, p < 0 . 00 0013 ), but indistinguishable in the case of severities ( n -wa y ANOV A, F = 0 . 59, p = 0 . 62). consistently , alb eit sto chastically , tow ard the b o ttom-right corne r , representing a conv ergence on v ery short delays b etw een even ts. This progressio n fr o m slo w to fast ev ent pro duction appear s to happ en quickly: each of these groups achiev es delays of ∆ t ≤ 10 da ys b y their k = 12th event. Howev er, the median c a lendar time required to achiev e this high rate o f pro ductio n is 8.5 years; thus, although these ﬁr st doze n even ts a c count for a small fraction of the lifetime pro duction of these organiz a tions (less than 4% eac h), they a ccount for a la rge fraction of the or ganizations ’ ov er all lifetimes. P ut more bluntly , these ﬁr s t few even ts play a critica l role in shaping the long- term tr a jectory of an org a nization’s pro duction curve and they illustrate a dr amatic acceleratio n in the pro duction o f even ts as the org anizations mature. This impo rtant developmen tal eﬀect is obscured b y high pro duction rates later in life. In co ntrast, the pattern for the severit y de velopment curve could not be more diﬀer ent : we o bs erve no clear trend, either up or down, b etw een ev ent severity x and exp erience k for thes e org a nizations, and the median ﬁrst a nd last severities a re x = 0 and x = 1 deaths, resp ectively . If anything, the only visual pattern we can discer n is a p o s sible incre a se in the v arianc e of x a s k inc r eases. This preliminary analysis th us a lready indicates weak supp ort for the severity-increase h yp othesis (H4) but strong supp ort for the frequency-acceleratio n hypothesis (H3). In combination with our static analy s is a b ove, this provides additional evidence supp orting lab or constraints and even t-driven re c r uitment (H1 and H2). C T errorist Organization Computer Sim ulation The toy mo del descr ibe d in the main tex t can b e fo rmalized and sim ulated explicitly . Below is computer co de that implemen ts the simulation in Matla b. In words, the simulation w orks as follows. Let η b e a constant that deno tes the n umber of individuals that ma ke up a terrorist “cell” within the organiza tion, and let ν b e the n umber of individuals the org anization as a whole gains via recruitment after each even t. Thu s, η /ν even ts are requir ed to pr o duce a single new cell; the particula r v alues of η a nd ν s e r ve only to change the scale of the dynamics, not their fundamental c hara cter. Each cell is a s signed a “clo ck” that measures the num b e r of da ys remaining befo r e that cell generates an e vent. W e denote this dela y τ a nd draw it from a log-nor mal distribution with parameter s µ and σ , i.e., Pr( τ ) ∼ LN( µ, σ ). This is the only stochastic elemen t of the sim ulation. When a cell g enerates an even t, it then draws a 23 1 10 100 1000 10000 ∆ t (days) FARC Taliban ETA Hamas 1 10 100 1 10 100 x (deaths) FARC 1 10 100 Cumulative number of events, k Taliban 1 10 100 ETA 1 10 100 Hamas Figure S2. F requency (delay ∆ t ) and severity (deaths x ) developmen t curves for the Rev olutiona ry Armed F orces of Colombia (F ARC), T aliban, Basque F a therland and F reedom (ET A), and Hamas, with generic tra jector ies estimated for all gr oups. Similar results hold for less experienced gro ups. new delay from the same distribution. As described in the main text, ea ch o rganiza tion b egins as a single cell, which has genera ted a single even t at t = 0 . Thus, initially s 1 = η . W e th en c ho ose a dela y τ for its next ev ent. The sim ulation will generate a sp eciﬁed n umber of ev ents, sp eciﬁed b y the parameter no k . F or the k th even t, the simulation then c hecks which cell has the smallest r emaining delay and adv ances a ll cells’ clo cks by that mu ch. It then generates the k th even t, r ecords its time as an ordered pair ( k, t k ), and draws a new clo ck v alue for the generating c e ll. Additionally , it increments the orga nization’s size b y ν individuals, i.e., s k = s k − 1 + ν , and adds ⌊ s k /η ⌋ new cells, each with a clo ck dr awn from Pr( τ ). A num ber of v ariations of this mo del genera te e q uiv alen t r esults. F or instance, the distribution Pr( τ ) can generate very small dela ys, e.g ., less than 1 day , which may b e co nsidered unrealistic. Impos ing a minim um v alue on the Pr( τ ) do es not change the fundamental feedback b etw een siz e and even t pro duction and thus leav es the k − 1 trend unc ha ng ed. And, the ratio η /ν only re-s cales the underlying k − 1 behavior, as seen in Figure 2. Finally , c hanging the parameters o f Pr( τ ) has no impact o n the fundamen tal behav- ior: the µ parameter sets the delay betw een the ﬁrst and second even ts, whic h app ea rs a s t he expec ted y -intercept on the resulting dev elopment curv e, a nd v a rying σ simply changes the sc atter around the underlying tr end. In fac t, the particular functiona l form of Pr( τ ) w e have chosen is no t imp or tant, and other choices lead to similar results; here, we choose the log-normal distribution d ue to its simila rity to the empirical data (Fig. 3). % --- Terrorist organizatio n simulation % --- by Aaron Clauset % --- set up simulation parameters [mu sigma] = deal (5.1,2.32); % parameter s for Pr(tau) = LN(mu,sigma) [eta nu] = deal(5 ,5); % size of cell, margina l growth after an attack nok = 1000; % number of events to generate % --- set up simulation data structure s s = zeros(nok+ 1,1); % organization size over time c = s; % number of cells over time [s(1) c(1)] = deal(eta, 1); fk = zeros(nok+1,2); fk(:,1) = (1:size(f k,1))’; % assign ids to events gr = zeros(nok+ 1,2); % holds event clocks for each cell 24 gr(:,1) = (1:size(g r,1))’; % assign ids to cells % --- initialize simulation : create the first cell t = 0; % global clock k = 1; % number of attacks to date (first attack at t=0) tau = exp(sigma* randn(1)+mu); % cho ose delay from Pr(tau) gr(1,:) = [1 tau]; % make first cell % --- generate exactly nok events while k0 % create the new cells and choose their delays tau = exp(sigma* randn(dc,1)+mu); gr(c(k-1 )+1:c(k),2) = tau; end; end; % --- done generati ng events; extract results [dt k] = deal(diff( fk(:,2)),(1:size(fk,1)-1)); % --- plot resultin g development curve figure(1 ); clf; loglog(k ,dt,’r-’,’LineWidth’,2); hold on; loglog([ 1 nok],exp(mu).* ([1 nok]).^( -1),’k--’,’LineWid t h’,3); hold off; xlabel(’ Cumulative numbe r of events, \it{k}’,’F ontSize’,16); ylabel(’ Time to next event, \Delta\it{t} \rm{(days)}’,’F ontSize’,16); set(gca, ’FontSize’,16,’YTick’,10.^(-6:4)); h1=legen d(strcat(’Simulation, \nu/\ eta=’,num2str(n u/eta,’%3.1f’)), ... ’Model, \Deltat\ propto k^-^1’,1); set(h1,’Font Size’,16); D Statistical Mo del for the F requency of A ttac ks The probabilistic model for ev ent delays used in the main text, giv en b y Eq. (1), has the precise form of Pr(∆ t | k ) =   p 2 /π σ  1 − Erf h β log k − µ σ √ 2 i   exp  − (log ∆ t + β log k − µ ) 2 2 σ 2  (3) where the leading term is the no rmalization constant and Er f ( · ) is the error function. In words, this mo del a sserts that the loga rithm of the delay ∆ t is a random v ariable distributed according to a Normal distribution N ( ν, ω ) (or eq uiv alently , the delay is log-no rmally distributed) with a low er cuto ﬀ at ∆ t = 1 day (to reﬂec t the timing resolution of the ev ent da ta), constant v aria nce ω and a distributional mean 25 ν that decrea s es systematically with increasing exper ience k . In Eq. (3), the parameter µ denotes the characteristic dela y b etw een attac ks, and in particula r the delay b etw een the ﬁr s t and se cond attacks, while σ 2 denotes the v a r iance in the expected delay . The equation giv en in the main text for the exp ected dela y a s a function of experience—the cen tra l tendency of the conditional dis tribution of delay as a function of exp er ience—can b e derived in the usual wa y . Doing so yields E[log ∆ t ] = µ − β log k +   exp h − ( β log k − µ ) 2 2 σ 2 i p 2 /π σ − 1  1 − Erf h β log k − µ σ √ 2 i   , (4) which has a simple leading form and a complicated trailing term. F or small v alues of k , the expected delay is dominated b y the leading t wo terms, i.e., the trailing term is small in rela tive magnitude, and th us the trend is well-appro xima ted by a pow er- law function ∆ t ≈ e µ k − β , where e µ represents the initial rate of attack of a gr oup. At larger v alues of k , the expected delay is dominated b y the trailing term, which makes the exp ected delay to a pproach ∆ t = 1 more slowly than a power law. When ﬁtting th is model t o the empirical data , we estimate its parameters using standard n umerical pro cedures to maximize the likelihoo d of the data (in this case, the Nelder- Mead 1 965 metho d). Sta ndard error estimates for the uncertain ty in the pa rameters are then estimated using a bo otstra p pro cedure on the organiza tions in the sample. The striking “data co llapse” sho wn in Figure 3b illus trates that the conditional probability distri- butions do indeed align closely with the estimated log-nor mal mo del for dela ys. Wh y delays should be log-nor mally distributed rema ins a mystery . Finally , we p o int out that v er y few groups (e.g., Ha mas, F a tah, L TTE, F AR C, etc.) manage to b e come highly exper ienced ( k & 100 ). This means that the ﬁt o f the mo del for lar g e- k is primarily con tro lled by the dela ys at m uch smaller v alues of k , wher e the v ast ma jority of the data lay . This fact explains the slight misﬁt of the mo del to the delays for highly exp er ienced groups. Ho wever, it also highlights the fact that the b ehavior of inexperienced gr oups early in their lifetime is highly predictive of the behavior of mature organiza tio ns. E Domestic vs . T ransnational Ev en ts F rom 1968– 1997, the MIPT event database was maintained b y RAND as part of its pro ject on transna- tional terr orism. As a result, almost no domestic terror ist attacks a re included b efor e 199 8, after which the scop e of the data base w as signiﬁca nt ly expa nded (in pa rt due to the Oklaho ma Cit y b ombing in 1995) to include purely domestic event s w or ldwide. Although orga nizations and even ts are not coded as b eing transnational or domestic, the inconsistency in databa se scop e pro vides an o ppo rtunity to test whether the frequency dynamics of domestic terro r ism or ganizations diﬀers from thos e of transnational organiza tions. By dividing ev ents in to those generated by or ganizations whose ﬁrst ev ent occurr ed 1968–1997 and those g enerated by org anizations whose ﬁrst ev ent o ccurr ed in 1998– 2008 , and then r ep eating the frequency- curve analysis from the main text, we may test whether the frequency-a c celeration phenomena appears only in o ne time perio d or the other. F urther, b eca use even ts in the 19 9 8-20 08 p er io d ar e ma inly do- mestic ev ents, while t ho s e in the 19 68–1 9 97 p e rio d are only tr ansnational even ts, the t wo time pe r io ds 26 1 5 10 50 100 500 1 10 100 1000 Cumulative number of events, k Time to next fatal event, ∆ t (days) Empirical mean, t 1 <1998 Empirical mean, t 1 ≥ 1998 Model, all Figure S3. The attack frequency development curves, plotted as the av er age delay versus exp erience, for groups whose ﬁrst attac k w a s in 1968–19 97 versus those whose ﬁrst attack was in 19 98–2 008, along with the model estimated for all even ts from the main text. serve as pro xies for transnational-only and domestic-o nly terrorism. This divisio n do es not control for non-stationar y e ﬀects. Figure S3 s hows that the developmen t curv e phenomenon is robust t o this division, indicating that the freq uency-acceler ation app ear s to hold for bo th tra nsnational and domestic terr orism. One diﬀer ence betw ee n these time perio ds does emer ge: the rate of acceleration for the 1968–199 7 data (transnational only) is ˆ β t 1 ≤ 1997 = 1 . 0 ± 0 . 2 (stderr), statistically indistinguis hable from the analysis of all organizations in the main text, while the estima ted acceler ation for the 1998–200 8 data ( ma inly domestic) is sligh tly faster, with ˆ β t 1 > 1997 = 1 . 3 ± 0 . 2. The origin of this diﬀer ence may b e related to the incre asing freq uency of religiously-motiv ated terrorism in the 199 0s a nd beyond [1, 2], who collectively exhibit a lo wer v alue of ˆ β than other types o f terrorism. An in teres ting alternative expla nation, how ever, is that some non- stationary pro c ess is having a cons istent upw a r d pressure on β over time, for all o rganiza tio ns. One candidate pro cess is the dev elo pment and spread of mo der n communications a nd digital tec hnolo gy , which may enable mo re widespread or eﬀectiv e recruiting eﬀorts and th us faster orga nizational g rowth. F P olitical Id eology & F requency and Sev erit y Curv es Our results for the developmental dynamics of ev ent frequency and severit y are g o o d descriptions of the generic behavior of ter r orist or ganizations . Howev er, we hav e so far omitted an y r ole for organizational cov aria tes, many of which are b elieved to have imp or tant impacts on org anizational b ehavior and decis ions (see [8, 12, 62], among others ). W e inv estiga te this question by studying the impact, if any , p olitical or ideologica l motiv ation ma y ha ve on the frequency curv e’s structure; we lea ve the in vestigation of other cov aria tes fo r future w or k. Miller [53] div ides the p olitica l motiv ations for terrorism or gro up ideo logies int o four conv entional categorie s: nationalist-sepa ratist, re actionary , religious a nd revolutionary . W e co ded acco rding to Miller’s criteria the 131 g r oups in our sa mple with k ≥ 1 0 deadly even ts, who together a c count for 85% of ev ents (the ma jority of our data ), and ﬁtted Eq. (1) to the data within each ideolo g ical ca tegory . Org anizations 27 1 5 10 50 100 250 1 10 100 1000 Cumulative number of events, k Time to next fatal event, ∆ t (days) National−Separatist Reactionary Religious Revolutionary 1 5 10 50 100 250 1 10 100 1000 Cumulative number of events, k Severity of k th fatal event National−Separatist Reactionary Religious Revolutionary Figure S4. (a) Estimated frequency curves for four ideolog ical categories, showing that religious groups dev elo p extremely quic kly relative to other types. (b) Estimated severit y curves for the same categorie s, showing the s ame pattern of independence as Fig. 4a. with m ultiple political motiv a tions w ere placed in m ultiple categories, whic h w ould only lessen an y dif- ferences b etw een estimated para meters for diﬀerent categ o ries. Fig. S4a sho ws the corres p o nding central tendencies, as describ ed by Eq. (2). T a ble 3 summarizes the estimated par a meters for eac h ideological category and groups o verall. W e ag ain test the statistica l signiﬁcance o f the acceleratio n eﬀect within each ideo lo gical mo del using a t wo-tail test against a null mo del with ﬁxed β = 0 (no accelera tio n ov er time). In all cases, the estimated β parameter is highly sta tistica lly sig niﬁca nt (a t the p < 0 . 001 level), indicating that the acceleration within each catego ry is rea l. Among the four ideo lo gical categor ies, we obser ve wide v ariation in the estimated v alues of β and th us in the strength of the feedback lo o p governing the frequency of attacks. Religious gro ups hav e the la rgest v alue at ˆ β = 1 . 7 ± 0 . 5 , pla c ing them ﬁrmly in the supe r -linear feedbac k regime a nd implying very strong acceleratio n in the fr equency of a ttacks ov er time. In contrast rea ctionary o rganiza tions hav e the smalle st at ˆ β = 0 . 1 ± 0 . 3, placing them str o ngly in the sub-linear regime. Revolutionary and nationalist-sepa r atist categorie s a re statistically indistinguishable from the linear-feedback regime of β = 1. The typical religious gro up, i.e., one acc e lerating along the generic pro duction tra jectory iden tiﬁed ab ov e, with k = 10 deadly attacks, attacks as fre q uently a s t he t ypical revolutionary group with k = 51 deadly attacks or the typical natio nalist-separ atist gr oup with k = 129 attacks. When view ed in terms of calendar time, this diﬀer ence is even more striking: it takes the typical religio us terrorist orga nization o nly 400 days (1.1 y ea r s) to generate its ﬁrst 10 attacks and at this point its pro duction rate is approximately one attack every 5 days. In con trast, the typical r evolutionary organization tak es 1 666 da ys (4.6 years), more than four times as lo ng, and a typical natio nalist-separ atist organiza tion takes 2103 days (5.8 years), to a chiev e an equal pro duction rate. Combining this insight with the results of our static analy sis on the role of size, the explosive accele r ation b y r eligious groups implies that they grow in size extremely quickly , which is the ultimate cause of their dramatic pro duction rates. But religious organizatio ns a re not univ ersa lly more danger ous. Compar ing the ˆ µ parameters, which gov erns the c har acteristic dela y betw een subse q uent attacks, w e obs erve a more complicated story: re- actionary groups initially attack the fastest, with the ﬁtted mo de l estimating t ypica lly ∆ t = 47 days 28 T able S1 . Severit y curve par ameters for organizations with simila r politica l motiv a tio ns. Note: statistical signiﬁcance calculated using a t -test on Pearso n’s corr elation coeﬃcient. po litical motiv ation groups ev ents h x i r sig niﬁca nce nationalist-sepa ratist 51 1003 6 . 1 0 . 0071 p = 0 . 7 5 reactionar y 5 77 7 . 1 0 . 1194 p = 0 . 2 7 religious 1 7 753 5 . 2 − 0 . 0 0 62 p = 0 . 49 revolutionary 41 725 5 . 1 − 0 . 0 109 p = 0 . 38 all groups 381 3143 7 . 3 − 0 . 0240 p = 0 . 17 betw ee n their ﬁrst and sec o nd attac ks , while all o ther groups tak e substantially lo nger (∆ t > 1 00 da ys). This diﬀerence in initial pro duction rates is quic kly eliminated b y the explosiv e accelera tion of religious groups as w ell as the more measured dev elopment o f rev olutio nary and nationalist-separatist organiza- tions, whose t ypical even t production rates o vertake that of reactiona ry g roups after b etw een 5 and 25 even ts. Much previous w or k on relig io us terror ism has argued, lar gely on theoretica l grounds, that such organiza tions a re fundamentally more dang erous than sec ula r gr oups [53, 63–65] b ecaus e they hav e fewer so cial restr ictions on their activities and are thus more free to produce a nd targe t violence than secula r organiza tions, whose victims may be p otential sympathizer s. Our results provide indirect s upp o r t fo r this a rgument, in the sense that religious o rganiza tions exhibit explosiv e ac celeration in the pr o duction of violence while secular organizations exhibit more modera te acceleration. How ever, arguments that religious organiz a tions are universally mo r e dang erous may hav e ov er- simpliﬁed o rganiza tional b ehavior b y ignoring how org anizations may change their behavior ov er time and how they v ary relativ e to other orga nizational types. W e ﬁnd that v ery ea rly in their life histories, religious gro ups are in fact less danger ous than rea ctionary gr oups, and only slightly more dang erous than national-sepa ratist or rev olutio nary g roups. It is only over the long term that the explosive accele r ation exp erienced b y r eligiously-mo tiv ated or ganizations allo ws them to cum ulatively produce so many more even ts than other types of o rganiza tio ns. That is, only if a religious org anization succeeds in reaching a more mature state do es it pose a g r eater ov era ll risk than groups with secula r motiv ations. And, it is impo rtant to note that historically sp eaking, most org anizations do no t live so long [40]: fully 5 5% of organiza tions in the MIPT database are asso ciated with only a single event. T urning brieﬂy to the question of ho w even t severit y v a ries with orga nizational ideology , we rep eat the same severit y-c ur ve analysis on the dea dly even ts pro duced b y the 131 highly proliﬁc organizations. Figure S4b shows the resulting ideology-sp eciﬁc severit y curv es and T able 4 summarizes the estimated mo del para meters, where the mo del now is a simple linear r egress ion o f severit y x against exp er ience k . As ab ov e, we ﬁnd no systematic dep endence o f severit y of attacks on o rganiza tio nal experience within any of the ideological categor ies. Tha t is, none of the mode l coeﬃcients are signiﬁcant, a nd the average severit y of even ts within ea ch c a tegory v ary only a little. Thus, w e ﬁnd that politica l ideolo g y has no systematic impa c t on the severit y of event s o r the tra jectory that even t sev erities take o ver the lifespan of an organization.

The developmental dynamics of terrorist organizations

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