Escalation, timing and severity of insurgent and terrorist events: Toward a unified theory of future threats
EEscalation, timing and severity of insurgent and terrorist events:Toward a unified theory of future threats
Neil F. Johnson
Physics Department, University of Miami, Coral Gables, FL 33126, U.S.A. (Dated: September 12, 2011)
Abstract
I present a unified discussion of several recently published results concerning the escalation,timing and severity of violent events in human conflicts and global terrorism, and set them inthe wider context of real-world and cyber-based collective violence and illicit activity. I pointout how the borders distinguishing between such activities are becoming increasingly blurredin practice – from insurgency, terrorism, criminal gangs and cyberwars, through to the 2011Arab Spring uprisings and London riots. I review the robust empirical patterns that have beenfound, and summarize a minimal mechanistic model which can explain these patterns. I alsoexplain why this mechanistic approach, which is inspired by non-equilibrium statistical physics,fits naturally within the framework of recent ideas within the social science literature concerninganalytical sociology. In passing, I flag the fundamental flaws in each of the recent critiques whichhave surfaced concerning the robustness of these results and the realism of the underlying modelmechanisms.
Working paper to accompany an upcoming seminar discussion. The paper will be updated sporad-ically as the research program to quantify and model collective human predation in real-world andcyberspace systems evolves. a r X i v : . [ phy s i c s . s o c - ph ] S e p . THE COMPLEX SPATIOTEMPORAL DYNAMICS OF CONFLICT Irrespective of its origin, any given conflict or terrorist campaign will play out as a highlycomplex dynamical system driven by interconnected actors whose actions are driven by awide variety of evolving information sources, myriad socioeconomic, cultural and behavioralcues, and multiple feedback processes. Furthermore, since conflicts and campaigns have abeginning and eventually an end, they will by definition exhibit non-steady state, out-of-equilibrium dynamics. To see the complications that can arise from the inherent feedbackand nontrivial temporal evolution characteristic of such complexity, one does not necessarilyhave to limit consideration to the recent conflicts in Iraq and Afghanistan, or recent globalterror campaigns. In the current year alone (i.e. 2011) three examples have arisen in whicha loose, quasi-coherent coordination between collections of humans has given rise to violentevents in response to news or rumors circulating on the Internet or electronic messagingbetween participants: The Arab Spring uprising, the London riots in the United Kingdom,and the high profile cyber-attacks on government and commercial sites by secretive hackergroups such as LulzSec. Indeed, the mixing of events in real and cyber space, togetherwith the fueling of illicit activities by the drug trade and international crime, has led to ablurring of the boundaries between terrorism, insurgency, war, so-called organized crime,and common delinquency. For example, there is a clear and present threat to the U.S.from Mexico as a result of the combination of such activities: On September 2010, U.S.Secretary of State Hillary Clinton, in remarks to the Council on Foreign Relations, said thatthe violence by the DTOs (Drug Trafficking Organizations) in Mexico may be “morphinginto, or making common cause with, what we would call an insurgency” [1]. The UnitedNations, in its report titled “The Globalization of crime: A transnational organized crimethreat assessment” [2], cites a statement by the UN Security Council from February 2010in which they highlight “.. the serious threat posed in some cases by drug trafficking andtransnational organized crime to international security in different regions of the world”.This report also states that a major stability threat will arise in the future as cartels andgangs fight for control of territory, and that the impact will be increasing violence amongcriminals and toward state officials and the public. In short, there will be more likelihood ofspillover violence, more corruption, increasing loss of territorial control, and an acceleratedundermining of both the social fabric and homeland security for nation states.2nterrelated to the situation in Mexico, is that of Colombia, where a thirty-plus year warstill continues. Though Marxist in origin, its character has been mixed up by the narcotrafficindustry, criminal gangs, mafia cartels, paramilitary groups, the presence of at least twomajor guerilla organizations (FARC and ELN), and widespread common delinquency drivenby a variety of socioeconomic factors [3]. As such, the struggle faced by state organizations tocounteradapt to ever-changing guerilla-narco-crime-cartel innovations, is immense. QuotingRef. [3], President Santos recently outlined new tactics to counteradapt to the guerrillasadoption of (i) hit-and-run raids using flexible units, (ii) mixing of rebels and criminal gangsand their use of joint activities as mutual needs arise, for example so-called Bacrims whichare organized criminal bands, (iii) dressing of insurgents as civilians to merge into the generalpopulation, (iv) carrying out small-scale attacks for maximum attention but little risk tothemselves. In response to the dynamical and fragmented nature of the insurgent cells whichthreaten the government’s control on law and order, the state’s new tactics include moreflexible units in order to mirror more closely the insurgents’ behaviors: “We have to adjustour doctrine, our operations and our procedures to the way (the rebels) are operating”(President Juan Manuel Santos [3]). These features (i)-(iv) of an insurgent Red force arenot unique to Colombia – they reflect the behaviors likely adopted by any armed group onthe Red side that is fighting to survive, whether it operate in real space or in the cyberworld,or some future hybrid mix of the two [4–6]. For this reason, these properties (i)-(iv) willplay a core mechanistic role in the generic model presented in this paper. Indeed there isan entirely parallel threat to real-space insurgencies and terrorism which is evolving on theInternet, in terms of transnational attacks in the cyber domain from both sovereign stateand non-state actors. This threat is arguably even more urgent than the real-space one,given that cyber ‘weapons’ (e.g. encounter-network worms or bots) can be assembled veryquickly, and transported in principle at the speed of light (i.e. via communications limitswithin fiber-optic networks). The advantage for Red (i.e. an insurgent or illicit organization)is that these cyber-logistics are much easier, quicker, and naturally more clandestine thanthe physical task of having to transport weapons and/or people from a point of assembly tothe place of potential attack.Future predatory threats in real and/or cyberspace, are likely to adapt to, and exploit, therapid, ongoing advances in global connectivity, and hence present clear but evolving dangersto each and every nation state, corporation or legitimate organization. The resulting arms-3ace between adaptation-counteradaptation by Red and Blue (i.e. state organizations) willlikely lead to rapid innovation of new predation methods. In addition, the backgroundcivilian population, refereed to here as Green, cannot be considered as purely passive –instead, it is a three-way struggle between Red, Blue and Green. Given this complexity, thepossibility for rapid escalation of hybrid real-world attacks, cyber attacks, and cyber-assistedattacks, therefore represents an unprecedented future risk which needs to be understood,quantified, mitigated and controlled – or at least delayed or deflated in terms of its potentialimpact. But there are so many questions that need addressing: How are these national andinternational threats likely to evolve going forward? Given their finite resources, how canstate agencies and countries be best prepared to face this challenge? What can be donein advance to prepare for the next generation of cyber–real-space attacks? Are there anylikely points of intervention that can be usefully exploited? Without quantitative modelsof such situations, solutions must be sought purely on the basis of narratives and case-studies (assuming any are available). It is clear that such narratives and case studies couldplay a crucial role, in particular where very few prior examples are known, or where strongsocioeconomic, cultural or behavioral factors play a key role. But as the amount of availabledata from such attacks, both in the real world and cyber world, increases, is there anythingadditional that can be said from a statistical viewpoint? In particular, given that humanconflicts and terror campaigns are examples of a highly complex dynamical system drivenby interconnected issues and actors, is this a potentially fruitful topic for analysis withinthe framework of the statistical physics of non-equilibrium open systems? And might such astudy then in turn shed valuable empirical light back on the emerging field of non-equilibriumstatistical physics?In this paper, I review some recent steps taken in this direction by our collaborativeteam. The results in many ways build upon earlier works within the statistical physicscommunity, e.g. Redner [7] and Rodgers [8] among others [9–17], going all the way backto the mid-20th century work of Lewis Fry Richardson [18] and Frederick Lanchester [19].In particular, we pursue a methodology which complements that of the political, socialand life science fields [4–6, 20–64], but which mirrors the approach initiated within thestatistical physics community in association with the study of financial markets [65–68].Our methodological program follows these steps: (1) Use state-of-the-art spatiotemporaldatasets with the highest available resolution, combined with current narratives from the4cademic literature, online sources, and the broader national and international media, inorder to identify systematic and anomalous behaviors in the ongoing timelines of daily,weekly and monthly events within a given domain of human predation. (2) Quantify theresulting stylized statistical facts of these multi-component time-series and hence identifystatistically significant deviations or anomalies. (3) Carry out a parallel procedure for otherpredation domains (e.g. provinces, or countries, or cultures) identifying where and whensimilar stylized facts emerge and, by contrast, where anomalies arise. (4) Develop a multi-agent mechanistic model of the underlying multi-actor dynamics for the domains of interest.Then undertake an iterative process of model modification and comparison to the datain order to obtain a minimal mechanistic model which is consistent with the most robuststylized facts which have been extracted from the data.The individual works which I summarize here using a unified perspective, have ap-peared separately in a range of different specialized journals [69–88] following our two initialpreprints [89, 90] in which we presented an initial coalescence-fragmentation model and aninitial analysis of the empirical data for the severity of events. Many others have presentedexcellent complementary or related works [7, 8, 12, 91], but I do not discuss these workshere or give a comprehensive review. Despite this progress, much remains to be done. Inparticular, our ongoing projects are focused around connecting the above listed works to thewider body of research activity within the human social, cultural and behavioral domain[41–64]. Examples that we are pursuing include adapting the model from step (4) above, toincorporate the results from studies from social psychology [92], economics, social scienceand crime science [99]. These results from other fields, while quantitative, may not be in theform of high frequency time-series, the preferred choice of the statistical physics community[67] – hence their incorporation represents a challenge to the statistical physics modelingprocess. However their inclusion is essential in order for such mechanistic analysis to betaken seriously by the other communities studying collective human predation. We are alsoactively probing time-series anomalies (e.g. see Red Queen discussion below) and interpret-ing them in terms of actor decisions, adaptations and counteradaptations. We are also usingthe multi-agent model to interpret how the underlying armed actors are themselves adapt-ing their strategies, changing their membership and reach, and counteradapting to currentgovernment and agency procedures. In parallel to earlier work in econophysics, we can thenrun these scientific models forward in time in to order to develop realistic prediction corri-5ors for where each of the indicators are likely to evolve and hence how the future threatis likely to change [85, 86]. This will allow us to change features in the parameter spaceof the models, in order to test out scenarios and identify points in the future evolution atwhich interventions might be possible, or necessary, in order to avoid particular undesirableoutcomes [85, 86].Even the simple cartoon representation in Fig. 1 demonstrates that at any one timestep,the complexity of the actors and their interactions can create a formidably complicateddynamical system. For studies of fatalities, the observable output x i ( t ) can be considereda vector whose elements describe the number of fatalities for each population type (i.e.Red, Blue, Green) at place i at timestep t . More generally, the output x i ( t ) would be atensor, showing separately the numbers of victims killed and wounded, and the differentweapon types used (e.g. Improvised Explosive Device (IED), or suicide bomb, or rocketpropelled grenade (RPG), or small arms fire). For simplicity, we will tend to refer to the‘Red’ population as ‘insurgent’, even though they may be a heterogeneous collection ofcyber-gangs, drug cartels, idealistic insurgents, rebels or rioters, and we refer to ‘Blue’ asthe ‘coalition military’ or ‘official antiterrorist organization’ even though they may be cyber-defense, police, security forces etc. Setting aside the issue of whether the data recorded hasan observational bias or not due to the way it was recorded (e.g. main street bias [83, 84]),there are many other potential complications facing a data-driven research program such asours. These include, but are not limited to, the following: (i) Heterogeneity of the insurgentforce strength (i.e. Red) which is depicted in Fig. 1 as various ‘types’ of fighter, or weapons,or assets including financing. This could also include different cultural, social and behavioraltypes within Red. Even the assumption that there is just one Red force can be misleading,as evidenced currently in Colombia (ELN, and FARC) and in Libya, with the different rebelfactions. It is not just an ‘us and them’ situation. (ii) Heterogeneity of Blue, comprisingwarfighters, equipment and money. (iii) Heterogeneity of Green, the background civilianpopulation, in terms of tribal or ethnic groups. (iv) The non-passive nature of Green dueto possible influence, sympathy, or direct recruitment to Red. For example in Fig. 1, activesupport of Red is indicated by two green figures with red heads who then get convertedin the next timestep to Red. Or it could simply be that a Green member shows an activefailure to support Blue. (v) Changing number of Red members, or Red cells. (vi) Finitelifetime of any given Red cell due to endogenous or exogenous factors, such as its implicit6 !!!"!!!" !!!" !!!"!!!" ! !! ! t ! t + t !"! FIG. 1: (Color online) Schematic of the complex spatiotemporal dynamics of multi-actor collectivepredation in real and/or cyber space. The result is a complex ecology of interactions and ob-served events, driven by some dynamically evolving but hidden network of loosely connected Redcells featuring non-local interactions aided by electronic communications [70, 89, 90]. At any onetime, there may be multiple types of actor, and these may cross different cultural and behavioralboundaries. Each population is partitioned into loose temporal cells [3–6]. Each cell may itselfsporadically combine with another cell, or simply fragment in some way – for example, as a result ofsensing danger [4–6, 16, 17, 40]. In addition to the traditional Blue population (e.g. state military,terrorist group or intelligence organization) and Red (e.g. insurgency or hacker group), there isalso a background civilian population which is labelled as Green, but which may not be passive inthe struggle. Social, cultural and human behavioral factors may play an important role for Greenand Red at the level of individual cells and its members.
II. THEORETICAL BACKGROUND
Though inspired by work in non-equilibrium statistical mechanics, it turns out that ourmechanistic approach is remarkably consistent with current thinking in the social sciencesin particular, analytical sociology as developed by Hedstrom [100]. In particular, Hedstromstates [100] “The basic idea of a mechanism-based explanation is quite simple: At its core,it implies that proper explanations should detail the cogs and wheels of the causal processthrough which the outcome to be explained was brought about.. Mechanisms consist ofentities (with their properties) and the activities that these entities engage in, either bythemselves or in concert with other entities. These activities bring about change, and thetype of change brought about depends on the properties of the entities and how the entitiesare organized spatially and temporally.” Hedstrom goes on to state that the “key challenge isto account for collective phenomena that are not definable by reference to any single memberof the collectivity. Among such properties are 1. Typical actions, beliefs, or desires amongthe members of society or a collectivity. 2. Distributions and aggregate patterns such asspatial distributions and inequalities. 3. Topologies of networks that describe relationshipsbetween members of a collectivity. 4. Informal rules or social norms that constrain theactions of the members of a collectivity.” Paraphrasing Hedstrom [100], a basic point ofthe mechanism perspective is that explanations that simply relate macro-properties to eachother are unsatisfactory. He goes on to state that these explanations do not specify thecausal mechanisms by which macro-properties are related to each other. It seems thatdeeper explanatory understanding requires opening up the black box and finding the causalmechanisms that have generated the macro-level observation [100]. According to Hedstrom,social mechanisms and mechanism-based explanations have, over the past decade, received11onsiderable attention in the social sciences as well as in the philosophy of science. As alsostated by Hedstrom, some writers have described this as a mechanism movement that issweeping the social sciences [101]. He gives the example of a car’s engine whose mechanismsand parts are quite visible when the hood is opened [100]. Hedstrom also states that “whenone appeals to mechanisms to make sense of statistical associations, one is referring tothings that are not visible in the data, but this is different from them being unobservable inprinciple”.Predator-prey systems have themselves been widely studied by many disciplines, includ-ing physics [11]. Outside the few-particle limit, mean-field mass action equations such asLotka-Volterra can provide a fair description of the average and steady-state behavior, i.e. dN R ( t ) /dt = f ( N R ( t ) , N B ( t )) and dN B ( t ) /dt = g ( N R ( t ) , N B ( t )) where N R ( t ) and N B ( t )are the Red and Blue population’s strength at time t . However, such population-leveldescriptions of living systems do not explicitly account for the well-known phenomenon ofintra-population group (e.g. cluster) formation [40], leading to intense debate concerning thebest choice of functional response terms for f ( N R ( t ) , N B ( t )) and g ( N R ( t ) , N B ( t )) in order topartially mimic such effects. Analogous mass-action equations have been used to model theinteresting non-equilibrium process of attrition (i.e. reduction in population size) as a resultof competition or conflict between two predator populations in colonies of ants, chimpanzees,birds, Internet worms, commercial companies and humans in the absence of replenishment.The term attrition just means that ‘beaten’ objects become inert (i.e. they stop being in-volved), not that they are necessarily destroyed. The combined effects of intra-populationgrouping dynamics and inter-population attrition dynamics have received surprisingly littleattention [7, 40], despite the fact that grouping and attrition are so widespread [40] andthe fact that their coexisting dynamics generate an intriguing non-equilibrium many bodyproblem. III. ESCALATION: PROGRESS CURVES AND THE DYNAMICAL RED QUEENMODEL
Before moving on to discuss specific mechanistic interactions at the level of individualcells, and hence necessarily introducing some assumptions concerning how they operate, Iwill start by taking a broad-brush view of the overall arms-race struggle between Red and12lue. This work is given in detail in Ref. [69].We consider Red (e.g. insurgents) as continually wishing to damage Blue (e.g. killcoalition military). All other things being equal, Red would like to complete successfulattacks as quickly as possible so that successive successful attacks become more frequent.Using coalition military fatality data for Afghanistan, we therefore analyzed the times forsuccessive fatal days for the coalition military, and find that they follow an approximatepower-law progress curve τ n = τ n − b [69]. Here τ n is the time between the ( n − n th fatal day, τ is the time between the first two fatal days, and b controls the escalationprocess. A fatal day is one in which the insurgent activity produces at least one death. Inparticular, we calculated the best-fit power-law progress curve parameters b and τ for eachprovince. Figure 2 shows a remarkable linear relationship which then emerges between thesebest-fit progress curve values for different provinces. Examples of the best-fit progress curvesare given in Fig. 3. This result extends to a specific weapon type (i.e. fatalities caused byimprovised explosive devices (IEDs)) and to the separate insurgent conflict in Iraq as wellas terrorist activity [69] and suggest that the insurgent (and terrorist) production process isvery similar in nature across geographical boundaries and borders. It is quite possible thatsimilar results will also be found in the future for cyberattacks. In the context of a Red-Bluestruggle where Red’s task of damaging the coalition military effectively resists completion(i.e. Blue is continually fighting back), the observed decrease in completion time may be dueto something Red is learning or doing, or Red’s increased manpower, or something Blue islearning or doing (or not doing) or a decrease in Blue’s manpower – indeed there are myriadpossibilities. The suggestion of Ref. [102] that our progress curve analysis [69] is necessarilytied to insurgent learning or experience, is false.To explain the numbers appearing in Fig. 2, in particular the observed range of b values,we have developed a dynamical version of the Red Queen evolutionary race [69] as shownschematically in Fig. 4. In essence, the theory treats the relative advantage or lead, R ,of Red over Blue as a stochastic process – and hence the exponents b are given by thescaling exponent for the standard deviation of the size of R ’s (possibly correlated) randomwalk [69]. Using this model, we can interpret the entire spectrum of observed b valuesfor different provinces, and also different terrorism domains, in an intuitive and unifiedway. Most importantly, this broad-brush Red Queen-Blue King theory does not requireknowledge of specific adaptation or counter-adaptation mechanisms, and hence bypasses13 " !"" !""" Robust linear relationship Insensitive to changes in n max ! (days) b FIG. 2: (Color online) Robust linear relationship between the escalation parameters for differentprovinces in Afghanistan shown as blue squares (from Ref. [69]). Also shown (red square) is acorresponding result for Hezbollah activity (Ref. [69]). In contrast to the claim in Footnote 13 ofRef. [102], the linear relationship is real, and does not result from the resolution of events on thedaily scale: 14 out of the 16 provinces have either zero, one or just two τ n = 1 values. For the twoprovinces with most τ n = 1 values, Helmand and Kandahar, we demonstrate the robustness byreducing n max (see Fig. 3) such that the amount of piling up of τ n = 1 events is reduced towardzero. There is no qualitative effect on the linear relationship. The green arrow shows that theirdatapoints undergo a simple to-and-fro variation along the line as n max is reduced (Helmand islight green, Kandahar is dark green). For Kandahar, for example, as n max is reduced from thesummer 2010 value of 132 down to n max = 24, for example, the values of b and τ simply move backand forth in parallel with the line, and at n max = 24 the datapoint ends up with almost exactly thesame values as for n max = 132. For Helmand, we show values down to n max = 29 as an example,as compared to the actual value of 278. For the remaining provinces, removing the very few τ n = 1values that occur also leaves the linear relationship unchanged. Several provinces have no τ n = 1values, hence the data-resolution criticism is a priori completely irrelevant. ! n n ! n n !" n max !& FIG. 3: (Color online) Successive time intervals τ n between fatal days, i.e. days in which coalitionfatalities are generated by hostile activities, for the Afghanistan province of (a) Southern borderregion and (b) Kandahar. On this log-log plot, the best-fit power-law progress curve is by definitiona straight (blue) line with slope − b (where b controls the escalation rate). In (a), as in all theprovinces other than Helmand and Kandahar, there are either no τ n = 1 values, or just one or two. issues such as changes in insurgent membership (i.e. composition, numbers or numbers ofcells), technology, learning or skill-set, as well as removing any need to know the heartsand minds of local residents. Historically, the Red Queen story features the Red Queen(e.g. insurgency or terrorist group) running as fast as she can just to stay at the sameplace, implying that her Blue King opponent (e.g. state security force or antiterrorismorganization) instantaneously and perfectly counter-adapts to her advances such that they15re always neck and neck. However, such instantaneous and perfect counter-adaptationis unrealistic – indeed, the complex adaptation-counteradaptation dynamics generated bysporadic changes in circumstances imply that the temporal evolution is likely be so complexthat it can indeed be modelled as a stochastic process. We do not need to know exactly why R changes at any specific moment, nor do the changes in R have to have the same valuesince each change is the net result of a mix of factors (e.g. changes in numbers of personnel,technology, learning or experience) for each opponent. We also find that a similar pictureto Fig. 2, showing a similarly remarkable inter-relationship between individual provinces,arises in other situations where an arms-race struggle is underway – for example, for suicidebombings in individual provinces in Pakistan (not shown) [106]. In all these cases, we stressthat a change in Red’s lead R might result from a conscious or unconscious adaptation byRed, or by Blue, or both – for example, there may be an increase in Red numbers becauseof a conscious recruitment campaign or simply due to bad press involving Blue’s activity.Likewise R may change due to a surge in Blue’s numbers or strength, or a change in itstactics or defenses. It does not matter: The precise cause for changes in R does not affectthe validity of our theory.In contrast to the claim of Ref. [102], the linear relationships that we uncover in Ref.[69] and Fig. 2 are real and do not result from any bias due to the resolution of eventson the daily scale. Of all the provinces, only Helmand and Kandahar reach an escalationsuch that events eventually occur on the daily scale. To verify the robustness of our linearrelationship, we therefore reduce n max (see Fig. 3) such that the piling up of τ n = 1 eventsis progressively reduced toward zero. This is equivalent to asking what the progress curveis up to some date prior to our end-date of summer 2010 [69]. As shown in Fig. 2, thishas no qualitative effect on the linear relationship – see the green arrow which indicates theresulting to-and-fro variations of the datapoints for Helmand and Kandahar along the lineas n max is reduced. The other provinces have very few, if any, τ n = 1 values. Of the 16provinces which define the linear relationship in Fig. 1 of Ref. [69], 14 have two or less τ n = 1 values, and none of these are piled up in a way that would induce a correlationbetween their best-fit b and τ values. Moreover, as can be checked from media and onlinereports, the τ n = 1 values that do occur in these 14 provinces can typically be tied to real,separate events on consecutive days, i.e. their one-day separation value ( τ n = 1) is realand has nothing to do with data resolution. However for the academic purpose of fully16ebutting the claim of Ref. [102], we have gone through the checking exercise of removingthese τ n = 1 values in order to test the robustness of the linear relationship between b and τ . As the low incidence of τ n = 1 values would suggest, these 14 blue squares follow thelinear relationship shown in Fig. 2, and hence also in Ref. [69], both with and withoutthe inclusion of these one or two τ n = 1 values. Going further, Zabul, Wardak, Khowstand the Southern border region have no τ n = 1 values and hence this issue is irrelevant tothem. Indeed, the fact that these four provinces’ best-fit ( b, τ ) values are widely separatedalong the linear trendline, means that this linear trend is already well-defined by these fourprovinces alone. The Hezbollah datapoint (Fig. 2, red) also has no τ n = 1 values and hencethere is again no data resolution issue. Even for the organization-wide global terrorist dataof Ref. [102], we deduced our best-fit estimates in Fig. 2 of Ref. [69] using the escalationrange within which τ n = 1 values were hardly involved (we analyzed the initial escalation inFig. 2(a) of Ref. [102]). In short, the claim in Footnote 13 of Ref. [102] that our results aredriven by data resolution, is completely false.We now make a more general point about the recent preprint Ref. [102] in relation to ourown paper which was published several weeks earlier (Ref. [69]). As stated in its title, ourpaper focuses exclusively on the escalation regime of fatal attacks against coalition militarywithin individual provinces. The approach in Ref. [102] aggregates over entire organizations,and hence has the potential to produce significant piling-up effects (i.e. densely packed τ n = 1 values) since the likelihood of a fatal attack happening on a given day somewhereacross an entire country or global organization becomes far more likely. Reference [102]treats the resulting regimes of dense τ n = 1 datapoints in an approximate ansatz-drivenmanner. However this approach [102] has several drawbacks. First, it is actually dangerousto attach too much importance to any regime where τ n = 1 values regularly occur (whichwe stress is not the case for our data). A single attack can extend over two days, as hastended to happen in the past decade with the FARC for example in Colombia. Howeverit may erroneously recorded in a terrorism or conflict database as two events separated by τ n = 1. By focusing solely on the escalation period where τ n = 1 events are rare, our paperminimizes this problem, but Ref. [102]’s aggregate approach does not. Second, if eventsare accelerating, and yet only daily data is known, then eventually the true time intervalbetween fatal attacks will be a matter of hours or minutes, thereby rendering time-serieswith strings of τ n = 1 values useless. Imagine a situation where you are monitoring the17ime between a child’s meals – having a theory (e.g. probability distribution) rounded to aninteger number of days is rather pointless since it is a forgone conclusion that each day willinvolve (at least) one meal. Third, Ref. [102]’s attempt to account for the entire durationusing a single analytic form, runs the risk of sacrificing improved accuracy and insightduring the practically important regime of initial escalation (i.e. small n ). By focusing onthis escalation regime, we were able to use a far simpler analytic form than Ref. [102] andhence were able to discover the remarkable linear dependencies shown in Fig. 2 and Ref.[69], as well as developing a potentially powerful Red Queen theory which is not tied toany specific mechanism (e.g. insurgent size). Instead, Ref. [102] forces the establishmentof a single analytic form to cover both the escalation regime with τ n (cid:29) τ n ≈
1. This is problematic, since a new conflict in its escalationphase does not ‘know’ that it is approaching the data-resolution limit τ n = 1, hence thereis no physical reason that its mathematical description should involve a smooth trajectorythrough the τ n = 1 resolution boundary. We also note that the occasional jumps that maybe observed in the time interval between successive fatal days (see for example, Fig. 3) mightbe interpreted as systematic disruptions by Red or Blue on successive days (e.g. breakingthe daily routine) as opposed to being measurement error or background fluctuations. Suchjumps may arise for any number of reasons including (but not limited to) changes in thenumber of insurgents or cells, and may even act like memory resets to the process. It isthese jumps that we believe could yield valuable insight into the effects of hidden changesin the operating landscape on both sides of a conflict, either in real or cyber space. We willreturn to this in future work.Reference [102] presents additional false criticisms of Ref. [69]. For example, it claimsthat we use learning to explain the progress-curve escalations. This is not true – there isno such statement in Ref. [69]. On the contrary, we state explicitly in Ref. [69] that ‘Ourbroad-brush theory does not require knowledge of specific adaptation or counteradaptationmechanisms, and hence bypasses issues such as changes in insurgent membership, technology,learning, or skill set, as well as a need to know the hearts and minds of local residents.’ Wealso state in Ref. [69] that ‘We do not need to know exactly why R changes at any specificmoment, nor do the changes in R have to have the same value, because each change is the netresult of a mix of factors’. Indeed, we purposely chose the term ‘progress curve’ to avoid anyexplicit connection to learning. As we acknowledge in Ref. [69], progress can arise for many18ifferent reasons (including, but not limited to, increases in insurgent numbers). Neither ourRed Queen theory nor our results depend on learning, nor do they rely on restricting thescope of possible driving mechanisms. Insurgent size increase, as promoted in Ref. [102],is simply one of the many specific mechanisms that can increase the Red Queen’s lead R and hence escalate fatal attacks. As Ref. [69] states, the power of our Red Queen theory asopposed to a specific, yet unjustified, mechanism such as insurgent size increase, is that itis not tied to one particular narrative. It properly allows for combinations of mechanismswhich may change over time. Indeed, the claim in Ref. [102] that organizational growthdrives the escalation, is highly problematic in terms of verification. No data exists – norwill any ever likely exist – for reliable insurgent numbers within the individual provincesthroughout the entire duration of our study. A similar story holds for other insurgentwars and terrorism [102]. In particular, estimates of entire organizational size (e.g. thetotal number of Taliban) are extremely crude at best, and may in fact be misleading – inparticular, it is unclear whether they bear any relationship to the actual number of activemembers who are ready to carry out attacks at a particular moment in time.The explicit explanation of escalation suggested by Ref. [102], invokes a change in the sizeof the organization. However, earlier in 2009, Ref. [70] had already introduced a detailedmodel (summarized below and in Figs. 7 and 8) in which the number of active cells N g (i.e.semi-autonomous groups [70]) can increase or decrease over time. We had then used thisto explain the change in the daily frequency distribution of fatal attacks on civilians etc.aggregated at the level of a country (see Fig. 5). This model [70] provides a quantitativeexplanation of the temporal evolution of several conflicts in terms of an increase (decrease) inthe total number of insurgent groups (i.e. cells) over time and an effective lowering of the barfor carrying out successful attacks. This finding [70] reinforces our argument above againstsize driving escalation, in that it is not enough to just focus on the changing number of cells(or total number of insurgents). For a detailed discussion, see the online SupplementaryInformation of Ref. [70], with results in Supplementary Table 2, and explicit model flow-chart in Supplementary Figure 5. Even the basic one-population version of our model (seeAppendix A) can have the number of agents increasing, or more generally changing, overtime without affecting the appearance of an approximate power-law distribution with slopenear 2.5 (Fig. 6). Going further, the following simple argument reinforces our claim thatorganizational growth cannot be the sole driver of the escalation patterns that we observe19n Fig. 3. Suppose at time t , insurgents (or terrorists) have a strength N R ( t ) while coalitiontroops (or a counterterrorism force) have strength N B ( t ), where ‘strength’ might involvemany factors, but for simplicity we take it to be the number of agents on each side. When N B ( t ) ≈ N R ( t ). Even if N B ( t ) (cid:29)
1, there will also be essentially zero fatal attacks ifthese coalition military never go out on patrol. Hence a blanket statement that insurgentsize dictates escalation is wrong. Suppose instead one tries setting the rate of fatal attacks– taken to be proportional to the Red Queen’s lead – as R ( t ) = k ( N R ( t ) − N B ( t )) where k is some conflict-specific constant. One can trivially see that increasing the size N R doesnot guarantee an increase in R and hence attack frequency, because N B may also change.Similar conclusions hold for other functional forms, since the generation of Blue fatalitiesrequires by necessity some kind of N B ( t ) dependence. Hence even before extra complicationssuch as changes in tactics or equipment on either side etc. are added, one can see that atheory based solely on N R ( t ) is not plausible.We also flag the incorrect discussion in Ref. [102] concerning our model’s ability toaccount for the lack of a strong empirical relationship between the severity of individualattacks and the escalation in the number of attacks [102]. Our full model (Fig. 8) does notin fact contradict this finding. We had already shown that the best-fit power-law exponentdoes not change much over successive epochs [76, 81, 89, 90] even though the frequencydistribution for the number of events per day is changing (Fig. 5) [70]. Turning to the basicone-population version, the severity of an attack is given by the size of the cell decidingto perform the attack, while the size of the entire insurgent or terrorist organization isgiven by N ( t ). Figure 7 shows an example where prior to fragmentation of the cell of size3 into 3 cells of size 1, N ( t ) is partitioned in such a way that n s =1 ( t ) = 0, n s =2 ( t ) = 1, n s =3 ( t ) = 2, n s =4 ( t ) = 0, n s =5 ( t ) = 1, n s ≥ ( t ) = 0. The total size is N ( t ) = (cid:80) s n s ( t ) =1 × × × N g ( t ) = 4. After fragmentation of the cellof size 3 into 3 cells of size 1, N ( t ) = 13 still, but now N g ( t ) = 6 and we have n s =1 ( t ) = 3, n s =2 ( t ) = 1, n s =3 ( t ) = 1, n s =4 ( t ) = 0, n s =5 ( t ) = 1, n s ≥ ( t ) = 0. Hence the actual values of N ( t ), N g ( t ) and n s ( t ) (the number of cells with size s ) can show appreciable variability fromeach other at any particular moment in time. In passing, we note the interesting feature(see Appendix A) that the actual functional form of the cell size distribution of individual20ells { n s ( t ) } , where n s ( t ) is the number of cells of strength s at time t , does not depend on N ( t ).Finally, we mention for clarification that our escalation study in Ref. [69] looked at thedeterministic (power-law) trend of the sequence of time intervals. We did not look at astatistical distribution of time intervals, nor did we analyze power law distributions – nordoes Ref. [69] claim that the time intervals are described by a power-law distribution. Thereis a fundamental difference between describing a deterministic overall trend in successive timeintervals as the conflict evolves – which is what we did for each province – and describing thestatistical distribution of the ensemble of time intervals for each province. Since the conflictis non-stationary, and shows escalation, it would make little sense to aggregate all the timeintervals and look for a single distribution. This would ignore completely the underlyingdynamical trend, and could lead to quite misleading conclusions. For example, the fewlarge τ n values during the early stages of the conflict would tend to dominate any statisticalanalysis of fat-tailed behavior, giving a distribution measure which is unrepresentative ofthe entire conflict. On another point related to statistics, and specifically returning tothe conventional unweighted linear least-squares approach which we used in Ref. [69] todetect the trend in log τ n versus log n , it is well-known that this method will become veryaccurate in the limit that the residuals of log τ n approach statistical independence withidentical distributions (i.i.d.). This i.i.d. criterion is not strictly met in many applicationsof least-squares in the sciences, however it turns out to be a good approximation in ourstudy [69] and likewise in Figs. 2 and 3. This is because the error in the underlying τ n values has a crudely multiplicative form whose effect decreases with increasing n . There aremany reasons why such a scatter should occur: Insurgent attacks early in the conflict maystop short of wanting to cause coalition military deaths for fear of stimulating an increasedfuture troop presence, thereby producing large variations in τ n values for small n . Also,there were physically less coalition soldiers (i.e. targets) on the ground, so their fatalitiesmay have been more clustered. Because of the subsequent transformation to logarithmicvariables, the resulting residuals for log τ n versus log n will then have distributions which arereasonably insensitive to increasing n . It also turns out that these residuals have a fairlysmall autocorrelation, a fact that can also be seen crudely by eye simply by looking at thescatter of log τ n values. The net result is that the residuals of log τ n exhibit a distributionwhich is fairly insensitive to n and they also have very little autocorrelation – in short, the21esiduals are approximately i.i.d. as indeed they should be. IV. ATTACK TIMING: NON-POISSON DISTRIBUTION
Figure 5 reproduces a result from Ref. [70], in which we showed that the empirical datafor the timing of attacks generates non-Poissonian distributions. In line with the abovediscussion, the non-Poisson nature of the distribution for the number of attacks on a givenday of an ongoing conflict, tends to increase as the conflict evolves over time. Figure 5 alsodemonstrates a common burstiness in the distribution for the number of attacks per day.As explained in the Methods and Supplementary Information of Ref. [70], we compare thedistributions over daily event counts for different epochs within four modern conflicts, againstcontrol distributions (random war) obtained by randomizing event occurrences within eachepoch. The data for each conflict (green circles) deviate from its random war (dashed curve)in a similar way: the real war exhibits an overabundance of light days (i.e. days withfew attacks) and of heavy days (i.e. days with many attacks), but a lack of medium dayscompared with the random war (see lower panel of Fig. 5). By considering subsets of days,we determined that these features are not just an artefact of a variation in attack volumeacross days of the week (for example, Fridays). The burstiness became more pronouncedover time for the wars in both Iraq and Colombia, suggesting that they became less randomas they evolved. We checked that these findings are insensitive to the precise specificationof the epochs within a given conflict.
V. SEVERITY: POWER-LAWS AND BEYOND
As mentioned earlier, Refs. [89, 90] showed that the distributions for the severity ofattacks (i.e. the histogram of the number of events with x casualties, as a function of x )generates surprisingly broad and common distributions. These distributions approximate toa power law – or strictly speaking, a power-law cannot be rejected – and have a correspondingpower-law exponent around 2.5. However in Ref. [70], we looked at features beyond a simplepower-law and found that additional information is contained in the deviations beyond thestrict power-law form. In particular, we generated successively more detailed models inwhich populations of actors interacted over time, like an ecology, and the output of these22nteractions gave the casualty distributions. The good fit between these models and theempirical deviations beyond power-law (see Fig. 6), offers new insight into subtle differencesin the rules-of-engagement for these conflicts. VI. OUR MICROSCOPIC MODEL OF MULTI-ACTOR CONFLICT
There is much work across the disciplines on how groups interact, and how these groupsbreak and form in non-violent settings – for example, in social psychology for humans,and evolutionary biology and zoology for animals, birds and fish. There is far less knownabout how human groups’ joining and breaking processes change when they are operatingin a clandestine and/or illicit way such that they do not want to get caught, and alsowhere they may have an underlying mistrust of each other which changes over time [4–6].However, clues can be gathered from a number of related areas, including animal anti-predator behaviors [17, 40] and also from the study of criminal gangs [4, 6, 99]. All theseworks – in addition to journalistic reports for situations as diverse as violence in Colombia,Iraq and Afghanistan, through to the recent riots in London – point to the idea that thereare many actors involved, and they form loose, ethereal groups which are hard to detectand whose internal structure is either continually in flux or changes sporadically. Thesefeatures inspire the relatively few, yet entirely reasonable, mechanisms that we adopt in Fig.8 in order to define a mechanistic model: namely, coalescence and fragmentation for thecell dynamics, and bounded rationality in the decision-making for the cell attack decisions.Indeed, the fact that there is only weak correlation between the severity of events and theirtiming, further supports the idea that the severity and timing mechanisms can be crudelythought of as independent processes, to a zeroth order approximation.In Ref. [70], we presented a full model which reproduces the entire severity distributionacross various insurgent conflicts (green curves in Fig. 6). In this full model, Red andBlue interact and fatalities are inflicted in Red, Blue and Green. Our results show that tofully understand the richness of the severity distribution, and in particular to understandthe deviations from power-law, the role of Red, Blue and Green must be accounted for.Doing so yields remarkably good agreement with the empirical data over the entire distri-bution (green curves in Fig. 6). However if one is simply looking for an explanation ofthe approximate power-law behavior of the distribution’s tail, and the apparent ubiquity of23ower-law exponents for casualties in the region of 2.5 for both insurgencies and terrorism,we also showed [70] that a simple, one-population version of our model will suffice (see Fig.7). Appendix A gives an explicit derivation for the resulting distribution of Red cell sizes { n s ( t ) } . Assuming that a given insurgent cell inflicts damage proportional to its size whenit attacks, the empirical observation is reproduced concerning the approximate power-lawfor casualties with exponent near 2.5.Our full model [70] (Fig. 8) combines two key human behavioral features:(1) Bounded rationality in the decision-making process of a cell when it is deciding whetherto attack on a given day. As in the famous ‘El Farol’ bar problem [68, 93], the cells arelimited by the information they have available to them, and the time they have to make theirdecisions. In the ‘El Farol’ bar problem [68, 93], a collection of boundedly rational agentsare each deciding whether to attend a potentially overcrowded bar, and hence are decidingwhether today is a good day to compete for the limited seating space. In the case of theinsurgent cells, they are each deciding whether today is a good day to attack – the limitedresource that they are competing for could be space in the international news headlines. Thisinterpretation is consistent with the behavioral feature of insurgents noted by former U.S.Senior Counterinsurgency Adviser David Kilcullen [107] that when insurgents ambush anAmerican convoy in Iraq: “...they’re not doing that because they want to reduce the numberof Humvees we have in Iraq by one. They’re doing it because they want spectacular mediafootage of a burning Humvee.” It is also consistent with small-scale laboratory experimentsstudying human groups [93]. Further support for this feature of boundedly rational decision-making, is provided by Kenney [6]: “participants in trafficking networks are only whatHerbert Simon calls boundedly rational: they face significant computational limitations intheir ability to analyze feedback from incoming stimuli”.(2)
Fragile dynamical clustering within an insurgent population (e.g. as a result of internalinteractions and/or the presence of an opposing entity such as a state army), just as schools offish or animals will go through cycles of build up and then rapid dispersal when a predatorapproaches [4–6, 17, 40]. The coalescence-fragmentation process (see Figs. 6 and 7) isconsistent with current notions of modern insurgencies as fragmented, transient, and evolving[3, 5, 14, 20, 107]. We recall the phrase of Gambetta [4] “.... contrary to widespreadbelief, criminal groups are unstable.” Further support is again provided by Kenney [6] in
From Pablo to Osama: Trafficking and Terrorist Networks, Government Bureaucracies, nd Competitive Adaptation : “To protect themselves from the police, trafficking enterprisesoften compartment their participants into loosely coupled networks and limit communicationbetween nodes”; “Trafficking networks . . . . are light on their feet. They are smaller andorganizationally flatter”; “In progressive-era New York, according to historian Alan Block,cocaine trafficking was organized by different networks of criminal entrepeneurs who formed,reformed, split, and came together again as opportunity arose and when they were able”;“loose collection of cells containing relatively small number of cell workers”; “Abu Sayyaf. . operates as a decentralized network of loosely coupled groups that conduct bombings,kidnappings, assassinations, and other acts of political violence in pursuit of a common goal. . ”. Kenney also highlights the close connection of traffickers to terrorists: “Al Qaedashare numerous similarities with drug-trafficking enterprises” [6].The coalescence process mimics the situation in which two cells (or individuals in thesecells) initiate a communications link between them of arbitrary range (for example, a mobilephone call), and hence the two cells tend to coordinate their actions from then on – albeitmaybe loosely. Indeed, the individual agents need not know each other, or be physicallypresent in the same place. The long-range nature of the coupling makes it a reasonabledescription for physical insurgencies and crime groups using modern communications in realspace, as well as cells acting in cyberspace – or any mix of the two [6]. Indeed, the languageof what is a cell and what is a group, and what is crime and what is insurgency, becomessomewhat irrelevant since the mechanistic operational details are now very similar. Furtherdetails are given in Ref. [70]. The fragmentation process (Fig. 6) may arise for a number ofsocial or situational reasons, from breakdown in trust within the cell [4] through to detectionof imminent danger [6, 40]. In addition to the quotes above concerning insurgent, drug andcriminal groups [6], it is well documented that groups of objects (e.g. animals, people) maysuddenly scatter in all directions (i.e. complete fragmentation) when its members sensedanger, simply out of fear [40] or in order to confuse a predator [40]. Or they may fragmentfollowing a clash in which the cell perceives that it is losing (see Supplementary Informationin Ref. [70] and Refs. [73, 76] for a number of variants, all of which give similar empiricaldistributions for the severity). Interactions are distance-independent as in Ref. [9] since weare interested in systems where messages can be transmitted over arbitrary distances (e.g.modern human communications). Bird calls and chimpanzee interactions in complex treecanope structures can also mimic this setup, as may the increasingly longer-range awareness25hat arises in larger animal, fish, bird and insect groups [40]. Appendix A gives an illustrationof the type of mathematical analysis which is possible, for the basic version of our model,stripped down to a simple form with no decision-making, and only one population – the Redinsurgency. Instead of having cells fragment when interacting with Blue, or when sensingimminent danger, we simply assign a probability for them to fragment. The resulting modelyields an exponentially cutoff 2.5-exponent power-law for the distribution of group sizes.Assuming that the civilian population is just some passive background that receives animpact proportional to the strength of each cell when it acts, the distribution of civiliancasualties will also have this same distribution – which is indeed what is observed for a widerange of insurgent conflicts and global terrorism, as reported in Ref. [70].Reference [103] makes a claim that our coalescence-fragmentation model falls down onthe basis that an approximate power-law severity distribution exists from the outset of theirempirical dataset for each terrorist organization, and yet the coalescence-fragmentation pro-cess surely needs some time to converge to its steady-state power-law distribution. However,this claim is false. First, the N ( t ) initial members may be coalescing and fragmenting be-fore any violent event is undertaken – indeed, there are many examples of undergroundorganizations and US-based militia who spend many years evolving without any noticeableviolent activity. No external event may be observed, but there is still a dynamical networkof groups evolving in the background. Most importantly, any such organization will un-doubtedly already have several existing clusters of contacts, hence it is not the case that thedistribution has to build up from all isolated agents. Going further, it is well known that anapproximate power-law for group sizes with slope around 2 .
5, as produced by our model, isto be expected with many different social and human activity scenarios – from the way peo-ple organize themselves in markets [114], to commerce [116], through to more casual socialsettings. A nascent insurgent, criminal, or cyber group could be created effectively instantlyfrom such an existing structure. Second, the numerical simulations show that the fat-taileddistribution develops very quickly, even if we start with isolated agents. Third, it is not thecase that starting from day 1 of a given organization, all fatal events are recorded in thedatabase. There is no guarantee that the finite time-window database of Ref. [102], whichstarted nominally in 1968, either records correctly all events since 1968, or that it capturesthe true first few events of each terrorist group. The way in which events are interpreted andrecorded has changed over time, and so in addition has the ability to name organizations26 and indeed, so have their names changed in some cases, with merging and splinter groupformation fairly common. We also note that the alternative severity model proposed in Ref.[104], is simply a combination of phenomenological broad-brush factors which happen togive a power-law, but without any specific justification for yielding the observed exponentvalue of 2.5. Instead, the parameters of this model [104] need to be cherry-picked in orderto obtain the observed power-law exponent value of 2.5. In reality, a continuum of values –including values well away from 2.5 – are just as likely within the model [104]. Nor is thereany quantitative evidence to support their proposed underlying mechanism.We note that we have also carried out preliminary investigations (and are now pursuingrigorously) the addition of heterogeneity in terms of individual character, as in Fig. 1, andits effects on team formation and kinship when both the individuals and cell are under stress.We have already presented this work for gangs and online guilds for massively parallel humanactivities involving online cyberwar games [75], as well as investigating the effect theoreticallyin a preliminary way [73]. This published work successfully uses the addition of a scalarcharacter variable to describe the empirical datasets for Long Beach street gangs and Worldof Warcraft online guilds. Enriching this structure, our preliminary work suggests thatthe inclusion of agent character may cause mixing of these divisions and initiate extremebehavior, depending on the strength of the kinship tendency (e.g. mimicking tribal andethnic tendencies). Eventually, we hope that a full character-version of our model willprovide a flexible tool which we can adapt to help address a number of issues concerningsocial and cultural intervention schemes, such as ceasefires and peace plans, and pinpointingsocial triggers that aggravate a given conflict.Another criticism of Ref. [70] which has appeared in comments on the Internet, concernsthe nature of the ‘information’ that the cells have available to them in our generalized ElFarol model for cell decision-making [70]. As in the El Farol model itself, ‘information’here simply means something which acts as a common cue – not necessarily a particularmedia source (e.g. CNN) or even type of media. Indeed, we state in Ref. [70] that:“Each group receives daily some common but limited information (for example, oppositiontroop movements, a specific religious holiday, even a shift in weather patterns). The actualcontent is relatively unimportant provided it becomes the primary input for the groupsdecision-making process.” This common information acts as a coordinating effect. Even ifit is incorrect or inaccurate, it can still act to concentrate responses in a similar way. This27rowding effect in strategy space is explained in detail, in the context of financial marketburstiness, in Ref. [68].So far we have focused on reproducing the statistical stylized facts for insurgent conflictsand terrorism. But given the development of a minimal model as described above and inFig. 7 and 8, we can also ask the practical question: Can we estimate how long a conflictwill last? To address this in a simple way, we make the assumption of applying the lawof pure attrition – or more precisely, that a conflict lasts as long as it takes to reduce oneside to a certain level of strength N ( t ). The result is shown in Fig. 9 and the detailsare given in Ref. [72]. Our results show that a minority Red population experiences alonger survival time against a majority Blue force, than it would in the case of two equallybalanced populations. This result is irrespective of whether the majority population adoptssuch internal grouping or not. Adding a third population with pre-defined group sizesallows the duration to be tailored. As shown, our findings compare favorably to real-worldobservations. We stress that these findings are not a simple consequence of either dilutionleading to reaction slow-down, or of the specific cluster selection scheme that we chose. Inour model, as in nature, opposing predator groups actively seek each other out at eachtimestep, even if their density is low, making this unlike simple chemical dilution, and henceunlike simple mass-action equations. Instead, our results emerge from the interplay betweenpopulation asymmetry, the presence of clustering, and the intentional engagement betweenthe two opposing populations. Although the specific consequences may vary by applicationarea, we believe that related phenomena lying beyond mass-action predictions will arise ina wide range of physical, chemical, biological and social systems, whenever intra-populationclustering coexists with inter-population reactions. VII. SPREADING RUMORS AND MEMES IN AN INSURGENT POPULATION
We now turn to look at the effect of spreading of a meme, or idea, or doctrine, orknowledge, within the Red population – in order to understand how such populations mightbe persuaded and even infiltrated. This seems similar to an epidemic modeling problem, andhence one might be tempted to use one of the many approaches already developed. However,just as for models of conflict discussed earlier, dynamical models of spreading tend to fallinto one of two extremes [108], neither of which is particularly realistic for modern insurgent28ars or terrorist activity – and even less so for online cyberterrorism and cyberattacks.At one extreme, they assume that the social mixing dynamics are much faster than thespreading process, and hence that mass-action models can be adopted in which a continuumapproximation can be used to generate differential equations based on calculus. This tendsto be the extreme favored by mathematicians and physicists and engineers, since it unlocksthe power of calculus and the vast spectrum of known properties of differential operators.At the other extreme, is the limit which has been favored by the social science community,in which the heterogeneity of social links is retained at the expense of assuming that thesocial network is static on the timescale of spreading-related events. Many state-of-the-artdescriptions of viral transmission processes in real populations incorporate system-specificdetails and considerations (e.g. spatial topology, differential susceptibility) [109–113]. Asindicated above, some of these focus on the well-mixed (i.e. mass-action) limit, some ofthese focus on the limit of heterogeneous networks – and some attempt to move betweenthe two by adding patch-like structure to mass-action models, or dynamical link rewiringsto network models [112, 113].Our model (Fig. 8) in which dynamical regroupings happen spontaneously and sporadi-cally over time, allows us to focus on the spreading dynamics in the realistic but less well un-derstood regime where the group-level dynamics and individual-level transmission processescan evolve on the same timescale, and hence the number and identity of a given individual’scontacts can change abruptly at any given moment in time. This is shown in Fig. 10 (see, forexample, (a) and (b)) and is explained in detail in Ref. [78]. Most importantly, the dynam-ical processes of social group or cell formation/break-up and person-to-person transmissionof information, can co-exist on comparable timescales. We adopt the simple one-populationform of our model (Appendix A). By varying the probabilities of group coalescence ( ν coal )and fragmentation ( ν frag ) relative to the standard SIR (Susceptible → Infected → Recovered)probabilities for person-to-person transmission ( p ) and individual recovery ( q ), the entirerange of relative timescales can be easily explored – from a very slowly changing insurgentcell structure (i.e. essentially a static network with infrequent rewirings) through to a rapidlychanging cell structure (i.e. essentially a well-mixed population). Our model only has fourstochastic parameters for the probabilities (and hence timescales) of the individual leveltransmission and cell dynamics, i.e. p , q for the SIR process, and ν coal and ν frag which de-scribe the probability of cells coalescing or fragmenting (Appendix A). Reference [78] shows29xplicitly that it reproduces the qualitative shapes of a wide range of empirical profilesassociated with social, financial and biological spreading, simply by varying these relativetimescales. One implication of our findings is that conventional intelligence approaches inwhich the connections and nodes are sought assuming some quasi-static network, are likelyto be unreliable at best – and dangerously wrong at worst – leading to misplaced analysesand operations and possibly ultimately endangering Blue personnel. Fuller details of thesedynamical spreading results are given in Ref. [78]. VIII. OUTLOOK
Our modeling approach described here, is characterized by two stages: First, our broad-brush dynamical Red Queen theory which describes the escalation between Red and Blue[69]. This theory and analysis does not depend on the precise mechanism which changes theRed Queen’s lead at any one time. Second, we provide a plausible microsocopic mechanisticmodel which captures more of the complexity shown in Fig. 1, with interacting populationscomprising dynamically evolving cells in some loose and sporadically-changing structure.This model accounts for both the stylized facts of the timing of events and their severity.The parameters in our model are relatively few, and the model itself allows a range ofanalytic mathematical analysis to be performed for both the severity [73] and the timing[68]. Although simple and intuitive, the mechanisms incorporated in the model mimiccertain real-world human behavioral features, such as (i) human decision-making underconditions of limited endogenous and exogenous information, (ii) the fragile and transitorynature of criminal groups, (iii) confidence levels, and desire for success, (iv) punishmentand reward [92] via the way in which strategy scores are updated, and (v) the tendencyfor human insurgents to occasionally coordinate actions using modern technology for long-distance communications, e.g. mobile phone calls and Internet use. Adding in the aspectof character to these models, which we are currently doing, will extend this work beyondthe current stage where individuals (e.g. insurgents) are just particles, and will allow us toexamine human social, cultural and behavioral issues such as reward-punishment payoffs forcell members with different social and cultural backgrounds [92].It may well turn out that other explanations of the stylized facts of human insurgencyare possible. In fact, we both hope and expect this will happen in the near future – just30s the study of financial markets has spurred the fledgling field of Econophysics to becomeso productive over the past decade [67]. As more stylized facts become available, the com-peting theories can be judged against these benchmarks. It may also occur that, as in thestudy of financial markets, certain types of stochastic time-series-generating process can alsoreproduce the observed statistical features – however, just as in the financial market field, itis well recognized that no deep understanding of market dynamics is offered by such models,other than the ability to replicate similar statistical patterns. By contrast, our goal is todeliver a model which is based on reasonable mechanisms of the dynamics of insurgencies atthe cell level, with fairly minimal assumptions, and hence open up the path to a wide rangeof uses (e.g. scenario testing, evaluation of different strategies, interpretation of the changein a war through a surge etc.).We now comment on the comparison to cybergangs and street gangs. We found thatwhen we analyzed the empirical distributions for Long Beach street gang sizes and onlineguild sizes for World of Warcraft [75], the empirical distributions were not power-law like.This can be explained by the fact that our data comprised the actual membership of onlineguilds and gangs, as well as street gangs, as opposed to the number of objects who happento be coordinated (e.g. online, or on the street) at any one time. The latter is likely to varyrapidly and spontaneously every day as members come online or onto the street, however theunderlying membership would be expected to change more slowly over timescales of months.In addition, when individuals leave a street gang or an online guild, it is unlikely that thishappens because the entire gang or guild is disbanding – hence the fragmentation processin our model is less realistic. Indeed, it is known that fission processes involving the partialdismantling of a large cell into just a few randomly chosen splinter-cells tends to generatenon-power-law distributions, as is observed for street gangs and online guilds [75]. In short,the rules for the coalescence-fragmentation in street gangs and guilds are likely to changewhen one considers longer-term membership, as compared to Figs. 7 and 8. In this case,we believe that the role of individual character will come more to the fore. This is indeedexactly what we found in Ref. [78] – by adding character, in addition to some simple rulesbased on team formation, we found that we could reproduce the size distribution results forboth cyber gangs and also street gangs on the monthly scale.This study could be opened up to other forms of collective human predation, such asthe sinister threat from online child pornography rings. Even non-human predation can be31onsidered, such as ‘battles’ involving populations of pathogens within the immune system,or even the analogy to insurgency where parts of the immune system attack itself – andwhere normal cells turn cancerous, generating primary tumors as well as secondary spreadingthrough metastasis [117]. One particular example for which there is a wealth of data forcollective human predation, is a financial market. In the fast, high-frequency regime ofintraday trading, predatory algorithms can dominate the market at particular instances.Furthermore, they operate across multiple markets on the scale of hundreds, or event tens, ofmilliseconds, without regard for geographical boundaries. This connection between marketsand predation may run even deeper, given the fact that many causes and drivers of socialunrest may ultimately be linked to individual wealth and hence to the dynamics of themarkets. Indeed, the lead article on the front page of the New York Times on Friday, July18, 2008, featured what looked like a typical picture of insurgent activity, but noted belowthat the cause was actually the successive plunges in the Pakistan stock market over a twoweek period. This new area of coupled societal risks represents a huge future modelingchallenge.
IX. ACKNOWLEDGEMENT
I am extremely grateful to the many collaborators that have made these works possible,including Mike Spagat, Brian Tivnan, Pak Ming Hui, Spencer Carran, Juan Camilo Bo-horquez, Roberto Zarama, Amith Ravindar, Juan Pablo Calderon, Guannan Zhao, ElviraRestrepo and all other co-authors on the cited papers. I also gratefully acknowledge a grantfrom the Office of Naval Research (ONR): N000141110451. The views and conclusions con-tained in this paper are those of the author and should not be interpreted as representingthe official policies, either expressed or implied, of any of the above named organizations,to include the U.S. government. Subsequent draft revisions of this paper will be posted atthe same arXiV e-print address, when updates of empirical analyses or model advances areavailable. 32 ppendix A
Here we consider the basic version of our model, stripped down to a simple form withno decision-making, and only one population – the Red insurgency. Instead of having cellsfragment when interacting with Blue, or when sensing imminent danger, we simply assigna probability for them to fragment. The resulting model yields an exponentially cutoff2.5-exponent power-law for the distribution of cell sizes. We note that generalizations ofthis model have appeared in the literature – in particular, Ref. [73] contains a number ofrelevant generalizations, including a variable number of agents in time N ( t ). A later paperprovides a different derivation of the same basic result as the one below [105], reaching similarconclusions to our earlier publication (Ref. [73]) concerning the remarkable robustness ofthe 2.5 exponent to variations in the model mechanisms.Assuming that the civilian population is just some passive background that absorbs thestrength of each cell when that cell acts, the distribution of civilian casualties should havea similar distribution to that of the insurgent cells (see main text for a fuller discussion ofthis point). Analysis of a simple version of this model was completed earlier by d’Hulstand Rodgers [8], and real-world applications have focused on financial markets – howeverthe derivation below features general values ν frag and ν coal . At each timestep, the internalcoherence of a population of N objects (which we refer to as an ‘agents’ to acknowledgeapplication to human and/or cyber systems) comprises a heterogenous soup of cells. Withineach cell, the component objects have a strong intra-cell coherence. Between cells, the inter-cell coherence is weak. An agent i is then picked at random – or equivalently, a cell israndomly selected with probability proportional to size. Let s i be the size of the cell towhich this agent belongs. With probability ν frag , the coherence of a given cell fragmentscompletely into s i cells of size one. If it doesn’t fragment, a second cell is randomly selectedwith probability again proportional to size – or equivalently, another agent j is picked atrandom. With probability ν coal , the two cells then coalesce (or develop a common ‘coherence’in terms of their thinking or activities). As discussed in the main text, Kenney provides awealth of case-study support for thinking of an insurgency as a loose soup of fragile cells [6],as do Gambetta [4] and Robb [5]. 33he Master Equation is as follows: ∂n s ∂t = ν coal N s − (cid:88) k =1 kn k ( s − k ) n s − k − ν frag sn s N − ν coal sn s N ∞ (cid:88) k =1 kn k , s ≥ , (A1) ∂n ∂t = ν frag N ∞ (cid:88) k =2 k n k − ν coal n N ∞ (cid:88) k =1 kn k . (A2)Note here we make an approximation that N → ∞ . The terms on the right hand side ofEq. (A1) represent all the ways in which n s can change. In the equilibrium state: sn s = 1 − ν frag ( ν frag + 2 ν coal ) N s − (cid:88) k =1 kn k ( s − k ) n s − k , s ≥ , (A3) n = ν frag − ν frag ) ∞ (cid:88) k =2 k n k . (A4)Consider G [ y ] = ∞ (cid:88) k =0 kn k y k = n y + ∞ (cid:88) k =2 kn k y k ≡ n y + g [ y ] , (A5)where y is a parameter and g[y] governs the cell size distribution n k for k ≥
2. MultiplyingEq. (A3) by y s and then summing over s from 2 to ∞ , yields: g [ y ] = 1 − ν frag ( ν frag + 2 ν coal ) N G [ y ] , (A6)i.e. g [ y ] − (cid:18) ν frag − ν coal ν coal N − n y (cid:19) g [ y ] + n y = 0 . (A7)From Eq. (A5), g [1] = G [1] − n . Substituting this into Eq. (A7) and setting y = 1, we cansolve for g [1] g [1] = ν coal ν frag + 2 ν coal N . (A8)Hence n = N − g [1] = ν frag + ν coal ν frag + 2 ν coal N . (A9)Substituting this into Eq. (A7) yields g [ y ] − (cid:32) ν frag + 2 ν coal ν coal N − N ( ν frag + ν coal ) ν frag + 2 ν coal y (cid:33) g [ y ] + ( N ( ν frag + ν coal )) ( ν frag + 2 ν coal ) y = 0 . (A10)We can solve this quadratic for g [ y ] g [ y ] = ( ν frag + 2 ν coal ) N ν coal − ν frag + ν coal ) ν coal ( ν frag + 2 ν coal ) y − (cid:118)(cid:117)(cid:117)(cid:116) − ν frag + ν coal ) ν coal ( ν frag + 2 ν frag ) y , (A11)34hich can be easily expanded g [ y ] = ( ν frag + 2 ν coal ) N ν coal ∞ (cid:88) k =2 (2 k − k )!! (cid:32) ν frag + ν coal ) ν coal ( ν frag + 2 ν coal ) y (cid:33) k . (A12)Comparing with the definition of g [ y ] in Eq. (A5) shows that n s = ν frag + 2 ν coal ν coal (2 s − s (2 s )!! (cid:32) ν frag + ν coal ) ν coal ( ν frag + 2 ν coal ) (cid:33) s . (A13)We now employ Stirling’s series ln [ s !] = 12 ln [2 π ] + (cid:18) s + 12 (cid:19) ln [ s ] − s + 112 s − ... . (A14)Hence for s ≥
2, we find n s ≈ (cid:32) ( ν frag + 2 ν coal ) e / √ πν coal (cid:33) (cid:32) ν frag + ν coal ) ν coal ( ν frag + 2 ν coal ) (cid:33) s ( s − s − / s s +1 N , (A15)which implies that n s ∼ (cid:32) ν s − ( ν frag + ν coal ) s ( ν frag + 2 ν coal ) s − (cid:33) s − / . (A16)In the limit s (cid:29)
1, this is formally equivalent to saying that n s ∼ exp( − s/s ) s − / (A17)where s = − (cid:34) ln (cid:32) ν frag + ν coal ) ν coal ( ν frag + 2 ν coal ) (cid:33)(cid:35) − . (A18)For large cell sizes (i.e. large s such that s ∼ O ( N )) the power law behaviour is masked bythe exponential function. The equilibrium state for the distribution of cell sizes can thereforebe considered a power-law with exponent α ∼ / .
5, together with an exponential cut-off. In the human context, the fact that the interactions are effectively distance-independentas far as Eq. (A1) is concerned, captures the fact that we wish to model systems wheremessages can be transmitted over arbitrary distances (e.g. modern human communications).Bird calls and chimpanzee interactions in complex tree canopy structures can also mimicthis setup, as may the increasingly longer-range awareness that arises in larger animal, fish,bird and insect groups. In a human/biological context, a justification for choosing a cell witha probability which is proportional to its size, is as follows: a cell with more members hasmore chances of initiating an event. It will also be more likely to find members of anothercell more frequently, and hence be able to synchronize with them – thereby synchronizing35he two cells. It is well documented that cells of living objects (e.g. animals, people) maysuddenly scatter in all directions (i.e. complete fragmentation as in Eq. (A1)) when itsmembers sense danger, simply out of fear or in order to confuse a predator. Such fleeingbehavior was discussed at length in the classic 1970 work ‘Protean Defence by Prey Animals’by D. A. Humphries and P.M. Driver, Oecologia (Berl.) , 285-302 (1970).This model also offers an alternative explanation for a variety of other complex phenomenawhich have been found to exhibit a robust 2.5 power-law. Gabaix et al. [114] found a commonpower-law distribution for individual transaction sizes with α = 2 . ± .
1, for the LondonStock Exchange, the NYSE, and the Paris Bourse. Interpreting N as the average aggregatedemand for stocks, this demand N gets shaped into a distribution of demand ‘clusters’representing potential orders of a given size s . Since it is reasonable to expect orders tobe realized at random, the distribution of individual transaction sizes is proportional tothe distribution of clusters of potential orders – hence α = 2 .
5. Similarly, Richardson [18]concluded that the distribution of approximately 10 gangs in Chicago, and in Manchoukuoin 1935, separately followed a truncated power-law with α ≈ .
3. Interpreting N as thenumber of potential gang members in each case, with each comprising a transient soup ofclusters which tend to combine or fragment over time, yields α = 2 .
5. In a similar way, therobust time-dependence of a power-law with α ≈ . N neurons[115], we can imagine a dynamical coalescence-fragmentation grouping process in whichgroups of neurons become synchronized, and then this synchronization ultimately fragments.(Members of the same group need not be physically adjacent to each other). When anentire group fires, it creates a measurable activity equal to [115] the group size s . Hence theresulting activity distribution will follow a new power-law given by s × p ( s ). The resultingpower-law exponent ( α −
1) = 1 .
5, which is exactly the famous empirical 3 / et al. [114] for markets), we know of no other singlemechanism which is simultaneously physically plausible for each application area and which36an also explain a mysterious recent finding in the field of superconductivity [74]. [1] Mexico should call in the Marines , The Washington Post, Friday 26 November (2010).[2]
The Globalization of crime: A transnational organized crime threat assessment
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69] Neil Johnson, Spencer Carran, Joel Botner, Kyle Fontaine, Nathan Laxague, Philip Nuetzel,Jessica Turnley, Brian Tivnan. Pattern in Escalations in Insurgent and Terrorist Activity,Science 333, 81 (2011)[70] Juan Camilo Bohorquez, Sean Gourley, Alex Dixon, Mike Spagat and Neil Johnson. Commonecology quantifies human insurgency, Nature (December 17, 2009)[71] For full details of earlier work and downloadable papers, see http://mathematicsofwar.com/[72] Zhenyuan Zhao, Juan Camilo Bohorquez, Alex Dixon and Neil F. Johnson. Anomalously slowattrition times for asymmetric populations with internal group dynamics, Physical ReviewLetters 103, 148701 (2009)[73] Blazej Ruszczycki, Zhenyuan Zhao, Ben Burnett and Neil F. Johnson, Relating the micro-scopic rules in coalescence-fragmentation models to the cluster-size distribution, EuropeanPhysical Journal 72, 289 (2009)[74] N. F. Johnson, J. Ashkenazi, Z. Zhao, and L. Quiroga, Equivalent dynamical complexity ina many-body quantum and collective human system, AIP Advances 1, 012114 (2011)[75] Neil F. Johnson, Chen Xu, Zhenyuan Zhao, Nic Ducheneaut, Nick Yee, George Tita, PakMing Hui. Human group formation in online guilds and offline gangs driven by a commonteam, Physical Review E 79, 066117 (2009)[76] Alex Dixon, Zhenyuan Zhao, Juan Camilo Bohorquez, Russell Denney and Neil Johnson.Statistical Physics and Modern Human Warfare, in Mathematical Modeling of CollectiveBehavior in Socio-Economic and Life Sciences, Eds. Naldi et al., Birkhauser Boston (2010)[77] Zhenyuan Zhao, Andy Kirou, Blazej Ruszczycki, and Neil F. Johnson, Dynamical Clusteringas Generator of Complex System Dynamics, Mathematical Models and Methods in AppliedSciences, 19, 1539 (2009)[78] Zhenyuan Zhao, J. P. Calderon, Chen Xu, Guannan Zhao, Dan Fenn, Didier Sornette, RileyCrane, Pak Ming Hui, and Neil F. Johnson. Effect of social group dynamics on contagion,Physical Review E 81, 056107 (2010)[79] Mark McDonald, Omer Suleman, Stacy Williams, Sam Howison, Neil F. Johnson. Impact ofUnexpected Events, Shocking News and Rumours on Foreign Exchange Market Dynamics,Phys. Rev. E 77, 046110 (2008)[80] Neil Johnson. Mathematics, physics and crime, Policing, Special Issue (2008)[81] Neil F. Johnson. The Mother (Nature) of All Wars: Conflict, Global Terrorism and Com- lexity Science, APS News, November 2006[82] Neil F. Johnson. Complexity in Human Conflict, in Managing Complexity: Insights, Con-cepts, Applications edited by Dirk Helbing (Springer, Berlin, 2008) p. 303-320[83] Jukka-Pekka Onnela, Neil F. Johnson, Sean Gourley, Gesine Reinert, Michael Spagat. Sam-pling bias due to structural heterogeneity and limited internal diffusion, Europhysics Letters85, 28001 (2009)[84] Neil F. Johnson, Michael Spagat, Sean Gourley, Jukka-Pekka Onnela, and Gesine Reinert,Bias in Epidemiological Studies of Conflict Mortality, Journal of Peace Research 45, 653(2008)[85] David M.D. Smith, Neil F. Johnson. Predictability, Risk and Online Management in a Com-plex System of Adaptive Agents, in Econophysics and Sociophysics: Trends and Perspectives,Eds. B.K. Chakrabarti, A. Chakraborti, A. Chatterjee (Wiley-VCH, Berlin, 2006)[86] D. Lamper, S. Howison, N. F. Johnson. Predictability of Large Future Changes in a Com-petitive Evolving Population, Phys. Rev. Lett. 88, 017902 (2002)[87] Neil F. Johnson, David M.D. Smith, Pak Ming Hui, Multi-Agent Complex Systems andMany-Body Physics, Europhysics Letters, 74, 923 (2006)[88] Neil F. Johnson, Pak Ming Hui, Rob Jonson, Ting Shek Lo. Self-organized segregation withinan evolving population, Phys. Rev. Lett. 82, 3360 (1999)[89] N. Johnson, M. Spagat, J. Restrepo, J. Bohorquez, N. Suarez, E. Restrepo, R. Zarama. Fromold wars to new wars and global terrorism. e-print arXiv:physics/0506213 (2005)[90] Neil F. Johnson, Mike Spagat, Jorge A. Restrepo, Oscar Becerra, Juan Camilo Bohorquez,Nicolas Suarez, Elvira Maria Restrepo, Roberto Zarama. Universal patterns underlying on-going wars and terrorism. e-print arXiv:physics/0605035 (2006)[91] M.K. Lauren. Characterising the difference between complex adaptive and conventional com-bat models arxiv.org Arxiv preprint nlin/0610035 (2006)[92] M.E. McCullough et al. Evolved Cognitive Systems for Revenge and Forgiveness, Workingpaper (2010). Also Beyond Revenge: The Evolution of the Forgiveness Instinct, McCullough,M.E. (Jossey-Bass, 2008)[93] Wang, W. et al. Heterogeneous preferences, decision-making capacity, and phase transitionsin a complex adaptive system, Proc. Natl Acad. Sci. USA 106, 8423 (2009)[94] J. Epstein, Nonlinear Dynamics, Mathematical Biology and Social Sciences (Addison-Wesley, IG. 4: (Color online) From Ref. [69], our dynamic Red Queen model for the Red-Blue struggle.Red (e.g. insurgent) advantage R is represented as a vector in a multi-dimensional space whoseaxes may represent technological, psychological, social, cultural or behavioral factors. R follows astochastic walk in this D -dimensional space. Using known results from statistical physics, exactresults can be obtained for b under different conditions of correlation etc. within the walk. Forthe simplest case of an uncorrelated walk, b = 0 .
5. For a completely correlated walk in D = 1dimensions (i.e. linear feedback), b = 1. For a mean-reverting walk in D = 1 dimensions, b ≈ number of events n per day number of events n per day number of events n per day number of events n per day p ( n ) ! ( n ) number of events n per day number of events n per day number of events n per day number of events n per day model war real war random war above random below random p ( n ) ! ( n ) FIG. 5: (Color online) From Ref. [70], the distribution of the number of violent events per day ina given conflict. Green circles show the distribution p ( n ) for the number of days with n events inthe actual conflict. Histograms below represent differences D ( n ) between real and random wars,in units of standard deviations from the mean. The average values for random wars (i.e. whereactual data is randomized over finite time window) are shown as dashed lines. Solid lines denoteaverage distributions calculated from 10,000 realizations of our model. This model is a generalizedversion of the El Farol model: Individual cells only act if they possess strategies with sufficient pastsuccess, and hence the cell surpasses a minumum confidence level [68, 70]. Error bars for randomwars, calculated as one standard deviation from the mean of 10,000 shufflings, are shown but theyare small. The error bars for the model wars demonstrate a small spread in run outcomes. c u m u l a t i v e f r equen cy P ( X ! x ) casualty event size x Afghanistan(a) dataagent modelpower law c u m u l a t i v e f r equen cy P ( X ! x ) casualty event size x Iraq (b) c u m u l a t i v e f r equen cy P ( X ! x ) casualty event size x Colombia(c) c u m u l a t i v e f r equen cy P ( X ! x ) casualty event size x Peru(d)
FIG. 6: (Color online) From Ref. [70], the log–log plot of the complementary cumulative distri-bution of event size P ( X ≥ x ) (i.e. the probability of an event of size greater than or equal to x)for four conflicts. Horizontal axis shows event size x, namely the number of casualties. Solid greencurves show the results from our model. Blue dashed line is a straight line guide to the eye, nota power-law fit. As also shown in Ref. [70], the hypothesis that these quasi-straight-line graphsfollow a power-law cannot be rejected. The exponents for the underlying best-fit power-laws, allhave values near 2 .
5. This can be explained using a simple, one-population version of our model(see Appendix A), assuming that a given insurgent cell inflicts damage proportional to its size whenit attacks. However, to fully understand the richness of the full distribution, and in particular tounderstand the deviations from power-law, the activity of the other actors (i.e. state military,police or civilians) must be accounted for. Doing this yields remarkably good agreement with theempirical data (green curves) [70]. ells join together cells fragment Population could be a real world insurgency, terrorist group, criminal gang, Internet/multimedia driven delinquency or rebellion, cyber-insurgency, cyber-terrorism group, online criminal gang or informal collection of hackers people may be recruited or converted at each timestep people may leave or be captured/killed at each timestep (cid:0)
N t ( ) : total strength at timestep tN g t ( ) : total number of cells at timestep t where ≤ N g t ( ) ≤ N t ( )
Both N g t ( ) and N t ( ) may have complex time - variation
FIG. 7: (Color online) From Ref. [70], the insurgent population comprises an overall strength N ( t ),distributed into dynamically evolving cells – with time-varying size, number and composition, andwith diverse strengths at each time-step t . The total number of cells N g ( t ) at time t can varywith time, as can the total number of composite objects (i.e. insurgent members, equipment,information) N ( t ). Since N g ( t ) is the number of cells, and N ( t ) is the total number of objects(e.g. insurgents) these two quantities are fairly independent with the only constraint being that N g ( t ) ≥ N g ( t ) ≤ N ( t ) (i.e. the largest number of cells is when every object is isolated). In this exampleshown, the number of cells of a given size s at this timestep t , prior to fragmentation of the cellof size 3 into 3 cells of size 1, is n s =1 ( t ) = 0, n s =2 ( t ) = 1, n s =3 ( t ) = 2, n s =4 ( t ) = 0, n s =5 ( t ) = 1, n s ≥ ( t ) = 0. The total number of insurgents is N ( t ) = (cid:80) s n s ( t ) = 1 × × × N g ( t ) = 4. After fragmentation, N ( t ) = 13 still, but now N g ( t ) = 6. ngoing grouping dynamics through coalescence & fragmentation YES
Attack at timestep t ? Time-series containing the numbers of events at timestep t with sizes x Update news of events up to timestep t Update strategies and confidence levels
MODEL MECHANISM AT TIMESTEP t Number of events of size x aggregated over time (Figs. 1 and 2) Broadcast news of events up to timestep t START
YES YES
Number of events at timestep t aggregated over size (Fig. 3) DATA ANALYSIS ! n x t ( ) { } ! n x t ( ) t " ! n x t ( ) x " Update news of events, number of insurgents, strategies, confidence level for attacking etc.
FIG. 8: (Color online) From Ref. [70], our full model describing the attack severity distributionand attack timings comprises two main processes: (1) The interaction between Red (e.g. insurgentforce, shown here in black) of size N ( t ) and Blue (e.g. coalition military, not shown) and Green(e.g. civilian population, not shown) which then generates the severity distribution for casualtiesper event. As a result of these interactions, the insurgent force of size N ( t ) undergoes an ongoingprocess of coalescence and fragmentation. (2) A repeated decision process in which each of theindividual N g ( t ) cells which exist at time t , assess whether to attack or not based on the informationand resources it has available to it. This is a two-step process: First the cell will either be potentiallyactive or not [70] based on its confidence level (e.g. based on the performance of of their strategiesin the past). Then if active, it will decide whether to specifically attack at that timestep or holdoff momentarily. The fact that the severity of events and their timing show a low correlation in ourdatasets, is reflected in the fact that N g ( t ) and N ( t ) can vary almost independently, with the onlyconstraint being that N g ( t ) ≥ N g ( t ) ≤ N ( t ) (i.e. the smallest number of cells is when every object belongsto this same cell). In this example, the number of cells of a given size s at this timestep t is n s =1 ( t ) = 6, n s =2 ( t ) = 2, n s =3 ( t ) = 3, n s =4 − ( t ) = 0, n s =7 ( t ) = 1, n s ≥ ( t ) = 0. The total numberof insurgents is therefore N ( t ) = 26. The number of cells N g ( t ) = 12. .0 0.2 0.4 0.6 0.8 1.001x10 Colombia(1984-present)Peru(1980-1999)EI Salvador(1979-1992)Iraq(2003-present)Afghanistan(2001-present)Algeria(1991-2002)Vietnam(1958-1975)Sierra Leone(1991-2002) US Civil War(1861-1865)World War II(1939-1945) ‘ new’ war (post-WWII) ‘ old’ war (pre-WWII) dynamical clustering model directed fire model undirected fire model D u r a t i on Population asymmetry |x|
World War I(1914-1918)Spanish Civil War(1936-1939) !" ! " !1.)("%)/’%/0*&$ FIG. 9: (Color online) From Ref. [72], our model gives an estimate of the duration of a conflict.Duration T of human conflicts, as function of asymmetry x between the two opposing populations A and B . x = | N A (0) − N B (0) | / [ N A (0) + N B (0)]. Data are up to the end of 2008, hence finaldatapoints for the three ongoing wars will lie above positions shown, as indicated by arrows. Lowertwo blue lines are the mass-action results. IG. 10: (Color online) From Ref. [78], the process of transmission through a population com-prising a loose soup of cells (groups) which follow our model’s simple dynamical rules for growthand break-up. a: Schematic of dynamical grouping of insurgents on the Internet, or Facebooketc. b: Schematic of our model, featuring spreading in the presence of dynamical grouping viacoalescence and fragmentation. Vertical axis shows number of cells (groups) of a given size at time t . c: Instantaneous network from Fig. 10b at each timestep. d: Weighted network obtained byaggregating links over time-window T ..