Universal bursty behaviour in human violent conflicts

Understanding the mechanisms and processes underlying the dynamics of collective violence is of considerable current interest. Recent studies indicated the presence of robust patterns characterizing the size and timing of violent events in human conflicts. Since the size and timing of violent events arises as the result of a dynamical process, we explore the possibility of unifying these observations. By analyzing available catalogs on violent events in Iraq (2003–2005), Afghanistan (2008–2010) and Northern Ireland (1969–2001), we show that the inter-event time distributions (calculated for a range of minimum sizes) obeys approximately a simple scaling law which holds for more than three orders of magnitude. This robust pattern suggests a hierarchical organization in size and time providing a unified picture of the dynamics of violent conflicts.

Understanding the mechanisms and processes underlying the dynamics of collective violence is of considerable current interest. Recent studies indicated the presence of robust patterns characterizing the size and timing of violent events in human conflicts. Since the size and timing of violent events arises as the result of a dynamical process, we explore the possibility of unifying these observations. By analyzing available catalogs on violent events in Iraq (2003)(2004)(2005), Afghanistan (2008)(2009)(2010) and Northern Ireland , we show that the inter-event time distributions (calculated for a range of minimum sizes) obeys approximately a simple scaling law which holds for more than three orders of magnitude. This robust pattern suggests a hierarchical organization in size and time providing a unified picture of the dynamics of violent conflicts. I ndividual and collective violence has a large impact on society 1-3 -implications for health, for instance, range from psychological distress to injuries and mortality. Recent estimates of mortality from war and interpersonal violence reported values around 0.5-1 million deaths annually 1,2 , while the non-lethal impacts may affect millions more. In the light of the current status of violence and their impact on society, the knowledge of dynamical properties of violent conflicts is of interest not only for theoretical reasons.
Fundamental questions in the study of collective violence concern to the estimation of the number of conflictrelated deaths in a given attack and how often the attacks will take place. In this direction, it has been reported a robust power-law distribution describing the number of conflict-related deaths which holds for wars, insurgent conflicts and acts of terrorism [4][5][6][7] . Robust patterns have also been reported for the timing of events 7-9 , like the evidence for heterogeneous temporal behaviour characterizing violent conflicts 7 . More recently, some of these findings have been extended to include a wide range of human confrontations 10 .
The heterogeneous temporal behaviour of violent events and the statistics of event sizes are both the result of a dynamical process. Despite this fact, sizes and times of violent events have been studied separately. Here we explore the possibility that both sizes and times in human conflicts can be characterized by a common approach. By analyzing available catalogs of violent events in distinct parts of the world, we show that the inter-event time distributions calculated for a range of values of minimum sizes can be approximately described by a single master curve which holds for more than three orders of magnitude. This result signalizes the existence of a scaling law expressing a hierarchical organization of times and sizes in violent conflicts.

Results
Our findings are based in the analysis of available catalogs on violent events occurred in Iraq 7 (2003)(2004)(2005), Afghanistan 11 (2008-2010) and Northern Ireland 12  which gives the estimated number of deaths s in events occurred at time t (for details on the datasets see Methods). We defined the inter-event time t as the time interval between the beginning of two successive events, t j 5 t j 2 t j21 . Here we focus on the inter-event times t j with j labelling only the events with size s $ s min , where s min is a minimum size representing a lower threshold. Figure 1 shows the empirical raw data s versus t (Fig. 1a-1c) and their correspondent inter-event times t (Fig. 1d) for particular values of s min . The growth curves represent the cumulative number of events, n(t)~Ð t 0 r(t 0 )dt 0 where r is the activity rate defined as the number of events greater than s min per unit of time. For a given s min we also define the mean activity rate AErae which is the averaged rate over the whole time period.
We then obtained P(t), the inter-event time distribution covering events with s $ s min (see Methods). Figure 2a shows P(t) for the distinct conflicts and different values of s min . All curves display fat tails which can be viewed by the broad range of values of the inter-event times. As expected, the curves are scattered reflecting quantitative differences in the timing of events for distinct conflicts and minimum sizes. In order to compare the shape of the different curves we rescaled the data using the mean activity rate AErae and plot AErae 21 P(t) versus AEraet. The rescaling causes a s-dependent shift of the curves since AErae is a function of s min . Figure 2b shows that the different curves approximately collapsed into one master curve, which holds for more than three orders of magnitude. The data collapse signalizes the existence of a scaling law in the form P(t) 5 AEraeF(AEraet), where the scaling function F displayed distinct behaviours for small and large values of the scaled variable. For comparison, we show a power-law decay with exponent 0.75 for small times and another power-law with exponent 2.6 for larger times (Fig. 2b).

Discussion
The broad distributions shown in Fig. 2a reflect a large variability of the inter-event times characterizing heterogeneous temporal behaviour. This result is consistent with previous findings 7 , pointing to a scenario where high-activity intervals alternate with long low-activity periods. Bursty behaviour has also been reported for non-violent human activities and different explanations about the origin of these bursts have been proposed [13][14][15][16][17][18][19] .
The data collapse shown in Fig. 2b signalizes the existence of a scaling law governing the inter-event time distributions at distinct minimum sizes in violent conflicts. In the range considered, the dependence on the sizes are approximately governed by the mean   activity rate AErae. Since the mean activity rate is linked to the distribution of sizes, the collapse of the curves provides a unified picture of the size structure and temporal properties of the conflicts. Our findings suggest that the power-law type decay of the inter-event time distributions at small times (characterizing bursty behaviour) is just the short time limit of a general hierarchical scaling phenomenon which occurs in a wider range of time (and size) scales.
Our results contain some elements of similarity with previous findings in natural phenomena. For example, it has been shown that the inter-event time distributions for earthquakes whose magnitudes are greater than a lower threshold (taken over different spatial windows) collapse onto a single curve, providing a basis for the proposed unified scaling law 20 . This general scaling phenomenon captures elements of the Gutenberg-Richter law 21,22 and the Omori law 23 for earthquakes unifying these observations. A similar scaling behaviour for the interevent time distribution has been reported in fracture experiments, even in the absence of spatial degrees of freedom 24 . The data collapse shown in Fig. 2b for violent conflicts is remarkably similar those obtained in earthquakes and fracture experiments 20,24 . Specifically, the shape of the scaling function F and even the reported values of the power-law exponents for small and large times are in good agreement with the results shown in Fig. 2b for human conflicts.
Exponents around 1 characterizing the inter-event time distribution at small times arises in seismic-like phenomena as a consequence of the Omori law, while exponents around 2.5 at large inter-event times have been explained in terms of temporal variations of the activity rate 20,24 . The power-law exponent <0.75 for small times (Fig. 2b) does not necessarily imply an underlying Omori law governing the timing of violent conflicts but points to this possibility. In addition, the power law exponent <2.6 for large times (Fig. 2b) could be related to temporal variations of the activity rate of violent events-which can be seem in the non-linear behaviour of the growth curves shown in Figs. 1a-1c.
Despite the fact that human activities and natural phenomena are very different in nature, it has been suggested that both could be described by a common approach 5,25 . For example, the occurrence of earthquakes has been related to the relaxation of accumulated stress after reaching a threshold as in self-organized criticality (SOC) 20,26 . Analogously, violent events in human conflicts could be associated with a threshold mechanism. In this scenario, a description of human conflicts in terms of SOC seems plausible. Our findings are consistent with this possibility, providing quantitative support for the analogy between patterns in human conflicts and natural phenomena exhibiting SOC. We consider all deaths in each dataset, independently of the group that performed the attack (e.g. non-state groups or state security organizations).

Methods
Inter-event time distributions. For each dataset, we obtained the consecutive interevent times at a given s min and removed null time intervals (t 5 0) in order to study temporally separated events. The distributions P(t) for different conflicts and values of s min correspond to the probability density function (pdf) of the data. The number of bins was computed based on the length of data and the binning procedure was applied on log-transformed data.