Evidence of strategic periodicities in collective conflict dynamics
aa r X i v : . [ q - b i o . P E ] J a n Evidence of strategic periodicities in collective conflict dynamics
Simon DeDeo , ∗ , David Krakauer , Jessica Flack , ∗ E-mail: [email protected]
We analyze the timescales of conflict decision-making in a primate society. We present evidence for multi-ple, periodic timescales associated with social decision-making and behavioral patterns. We demonstratethe existence of periodicities that are not directly coupled to environmental cycles or known ultrarid-ian mechanisms. Among specific biological and socially-defined demographic classes, periodicities spantimescales between hours and days, and many are not driven by exogenous or internal regularities. Ourresults indicate that they are instead driven by strategic responses to social interaction patterns. Anal-yses also reveal that a class of individuals, playing a critical functional role, policing, have a signaturetimescale on the order of one hour. We propose a classification of behavioral timescales analogous to thoseof the nervous system, with high-frequency, or α -scale, behavior occurring on hour-long scales, throughto multi-hour, or β -scale, behavior, and, finally γ periodicities observed on a timescale of days. Variability on multiple timescales is a fundamental feature of complex systems [4, 33]. Minimally, mul-tiple timescales are critical for feedback, and without them there would be no memory, regulation, oradaptation. Adaptation, for example, requires timescales fast relative to the environment. Memory,on the other hand, arises from slow variables that average over the underlying fast dynamics. Theseslow variables can serve as a reference for decision-making when the lower-level dynamics are rapidlyfluctuating [1, 10, 14, 19].In the brain, multiple timescales, or characteristic frequencies of oscillation [2], enable populations ofneurons to efficiently represent different kinds of statistical information about the environment. Timescaleshave been hypothesized to play a role in the emergence of a unitary consciousness by binding the activityof large populations of cells [29] and to provide, by increasing the combinatorial space, new means ofstoring complex temporal patterns [19] and optimizing synaptic weights.Timescale variability has also been observed in behavioral dynamics and social systems. The dynamicsof learning ( e.g. , [27]) and decision-making ( e.g. , [31]) occur at timescales from seconds through to monthsand years. Social systems are comprised of emergent, hierarchically organized social networks that changeover a broad range different timescales from hours to years [1, 14]. These observations raise interestingquestions including, how new timescales emerge, and what the optimal coupling is between time constantsgiven functional requirements at the individual and collective levels. To answer these questions we mustfirst quantitatively characterize the range of time constants in our study systems. Whereas much is knownabout the range of time constants in neural systems, there is has been less quantitative characterizationof time constants and their implications in social phenomena.Here we show that conflict decision-making behavior – specifically, the decision to join fights – ina primate society is characterized by multiple, periodic timescales. We report the range of timescalesdetected, and propose a broad classification of behavioral time scales into α waves on hour scales, β waveson multi-hour scales, and γ waves from six hours up to days. We find that the timescales we detect areproperties of demographic classes defined by biological properties, like age and sex, or social properties,like power and social roles.Our analyses take as input a well-studied conflict timeseries [8] collected from a large, socially-housed,1rimate group (pigtailed macaques, Macaca nemestrina ) at the Yerkes National Primate Center inLawrenceville, Georgia (see Sec. 4). To characterize the range of timescales in our study system, weadapt a technique developed to study irregularly-sampled astronomical phenomena, the Lomb-Scargleperiodogram. As described in Sec. 3, the Lomb-Scargle periodogram can be used to detect a very im-portant class of timescales generated by regular periodicities in the dynamics. Examples of phenomenawith these kinds of periodicities include the ultraridian waves of physiology, circadian rhythms correlatedwith the photoperiod, and seasonal and reproductive infraridian periodicities [15].Using the Lomb-Scargle periodogram, we extract signatures of broad-band variation from the conflicttime-series. We consider three alternative hypotheses to account for the time constants we observe. Theseinclude two null models intended to determine whether the timescales are the consequence of exogenousor endogenous drivers of behavior, and a third hypothesis: the timescales are generated by strategicallytimed decisions to join or avoid fights. By “strategic” we mean the decision to join or avoid fights istimed in response to the pattern of social interactions rather to external cues or physiological clocks (seeSec. 5.2 for an operational definition on what is meant by “strategy”).
The quantitative study of timescale variation falls under the heading of spectral analysis . A tool ideallysuited to the spectral analysis of sparsely and irregularly sampled data is the Lomb-Scargle periodogram[23, 28, 30]. One of the advantages of irregularly sampled data is the great reduction in windowing andaliasing effects [28].The Lomb Periodogram for a time-series, { h j } , j = 1 . . . N , sampled at times { t j } , is defined by P ( ω ) ≡ Z hP j ( h j − ¯ h ) cos ω ( t j − τ ) i P j cos ω ( t j − τ ) + hP j ( h j − ¯ h ) sin ω ( t j − τ ) i P j sin ω ( t j − τ ) , (1)where Z is a normalization and τ is defined bytan 2 ωτ = P j sin 2 ωt j P j cos 2 ωt j . (2)As discussed in Ref. [28], if we write h ( t ) = A sin ωt + B cos ωt, (3)the definition of τ amounts to setting P ( ω ) proportional to A + B , where the coefficients are set by alinear least-squares fit. Choosing the normalization, Z , to be the variance of { h j } makes the estimator theLomb-Scargle normalized periodogram. A nice feature of the periodogram is that, under a null hypothesisof i.i.d. Gaussian variables, the distribution of P ( ω ) is an exponential distribution with mean unity [30].The output of the Lomb-Scargle is a (normalized) strength-of-signal as a function of frequency, P ( ω ).The frequency, ω , measured in Hz, is simply the inverse period (times 2 π ). In our case, the different { h j } for which we measure P ( ω ) are associated with conflicts involving different individuals and demographicclasses, as described in the following section. Our study group contained 48 socially-mature individuals and 84 individuals in total. Conflicts, or ‘fights,’in this system involve two or more individuals and are separated by peaceful periods – defined as the2bsence of fights among any of the group members. Operational definitions, and additional details onthe data set and data collection protocol appear in the Materials and Methods.Briefly, a “fight” was operationally defined to include any interaction in which one individual threatensor aggresses a second individual. A conflict was considered terminated if no aggression or withdrawalresponses (fleeing, crouching, screaming, running away, submission signals) was exhibited by any ofthe conflict participants for two minutes from the last such event. Fights involve multiple individuals,ranging in size from two to twenty-eight individuals. Fights can be conceived of as small networks thatgrow and shrink as pair-wise and triadic interactions become active or terminate, until there are no moreindividuals fighting under the above-described two minute criterion. As described in the Methods (Sec.7.1.1, only data on time of fight onset and the individuals involved in the fight are used in these analyses.No time data are available within fights; although the order of an individuals entry was noted duringdata collection, this information was not used in our analyses. Fight onset and termination time (usingabove-described criterion) were noted in hours, minutes, and seconds (see Sec. 7.1.1 for further detail).Our interest is in whether timing influences the decision of an individual to join fights. However,because the average number of fights per individual is low, it is hard to detect a signal using the Lomb-Scargle periodogram at the individual level. Hence for most of our analyses, we aggregate individuals intodemographic classes, according to biological and social characteristics, and ask whether, taken collectively,individuals of a given class exhibit a timescale on which they join fights.The biological classes we consider include age-class (socially-mature individuals, and two subclassesof the socially-mature set: subadults and adults), sex, and matriline (female and all daughters one year ofage or older). The criteria we use to define social classes – social power [12] and performance of policingrole – have been shown in previous work to be important factors in structuring social interactions in thestudy group [9, 11]). Demographic class sample sizes are provided with each analysis. For further detailson these demographic classes and for definitions of power and policing, see Materials and Methods.We calculate P ( ω ) for each of the demographic classes described above. For each demographic class,the Lomb-Scargle Periodogram takes as input a discrete series of measurements from the conflict timeseries. The timing of an event, t i , is set to be the onset of a fight in the observations; the h i is adiscrete variable: the presence (1) or absence (0) of an individual, or, in the case of classes, the numberof individuals involved in a fight at t i from that demographic class. In effect, a conflict is considered to“sample” the dispositions of demographic class in question. Detection of a signal using the Lomb-Scargleperiodogram indicates a timescale on which members of that class join or avoid fights.
Conflicts are short, with a median duration of only 15 seconds. The scales we recover span nearlysix orders of magnitude in timescales – between tens and and tens of millions of seconds. Of that range,the range of scales between 10 seconds (tens of minutes) and 10 seconds (days) is most accessible. Onmuch longer scales, measurements of nearby periodicities are strongly correlated, meaning there are fewindependent measurements to be made. Meanwhile, on the very shortest scales, the finite duration ofconflicts tends to wash out signals.In previous work [8], we found evidence that the decision to join fights made by individuals in thisstudy group depended on the properties of the preceding conflict event. The median time between conflictsis 255 seconds and so decisions to join conflicts correspond to the shortest timescales accessible to ouranalysis. Ref. [8] tested a set of alternative causal strategies, or behavioral production rules – formallydenoted as C ( n, m )+ AND/OR – that could be giving rise to these time scales, and found C (2 , AND to bea dominant strategy. This indicates that an individual decides to join the the current conflict becausea specific pair of individuals appeared in the previous conflict. This rule applies to all individuals thegroup. As many, though not all, adjacent fights are separated by only a few minutes, this finding suggeststhat second-to-minute reasoning scales can be of great importance to conflict dynamics.Can timescales longer than this be found directly in this ruleset? Because any particular rule isinvoked so infrequently, detecting shifts becomes difficult, if not impossible. The analysis we present inthis paper does not need to measure whether individual decision-making rules change, and so does not3uffer from the same signal-to-noise issues.
As described in Sec. 3, the output of the Lomb-Scargle method is a plot – a periodogram – of thefluctuation power as a function of frequency (or inverse period). Power at a particular frequency or rangeof frequencies indicates the presence of fluctuations with those characteristic timescales. For example, ifan individual or demographic class is characterized by a tendency to shift behaviors (from, for example,less conflict prone to more conflict prone and back) on timescales of roughly one hour, one would see anabove the null model bump in the periodogram in that range (see Fig. 1, for an example of a periodogramwith significant detections in a number of logarithmically-spaced bins.)As in previous work, the highly correlated nature of the system means the choice of adequate nullmodels is crucial. In presenting our results, we consider two null models: the mixed-strategy null, and astronger, daily-forcing null. The former, discussed in Sec. 5.1, looks for signatures of changing behavioraldispositions; the latter, discussed in Sec. 5.2, tries to explain these changes by a model of context-insensitive daily shifts in conflict behavior. Features unexplained by either null – and associated withdecision-making that is sensitive to fluctuations about mean behavior – are of particular interest. Wefocus on them in Sec. 5.3.In analyses such as these, where many bins are searched for signal, a distinction arises between thestatistical significance of a single-bin detection, and the statistical significance of the detection overall. Asan extreme example, if one searches one hundred bins, each considered equally likely to harbor a signal,a p value of 10 − in a single bin does not imply an overall significant detection. In some cases, such asthe subadult male demographic class (see Fig. 2) a single bin shows a strong above-null detection, butcombining that p value with many non-detections in other bins reduces the significance. Without strong priors on timescales of variability – or an intuition for the expected signal strength inthe periodogram – we first compare the observed variability against a null model that retains only thetime-independent properties of the data.We produce a set of null periodograms by shuffling the time series. We keep the timing of fights thesame, but shuffle their internal compositions, so a fight at time t in the data will correspond to a fightat time t in all null sets, but will have the composition of a different fight, drawn (without replacement)from a different time t ′ . The normalization of the Lomb-Scargle periodogram is such that the mean valueof the null is unity; further statistical issues associated with null model estimation are discussed in theMaterials and Methods.In a game theoretic context, this null model corresponds to assigning individuals a constant mixedstrategy (in the game-theoretic sense of mixed): time-independent probabilistic play of one of two strate-gies, “join conflict” or “avoid.” Note that Lomb-Scargle analyses the data in terms of periodic functions;failure to reject the null suggests that the animals are playing probabilistic strategies without strongperiodicities.In the case of a demographic class of size n , the equivalent mixed strategy is for the group as awhole, and amounts to a probabilistic choice of n + 1 options – “none of us join,” “one of us joins,”and so forth to “all n of us join.” This is a distinct process from averaging, over a demographic class,the periodograms obtained for individuals. It is sensitive to the timescales for collective behaviors of ademographic class, which may be different from the timescales of its individuals. For example, in thecase that two individuals in a demographic class alternate their participation – perhaps because eitheris sufficient to play a particular functional role – the observed timescale for the group will be faster, andmore coherent, than that of either of the two individuals taken independently.4 igure 1. Timescales of the decision to join fights for the socially-mature individuals considered as ademographic class ( n = 47). Top panel: Lomb-Scargle Periodogram for the socially-mature demographicclass. The data are shown as the solid red line. The (darker) blue band shows the p = 0 .
05 confidencefor the mixed-strategy null of Sec. 5.1; the (lighter) green band shows the p = 0 .
05 confidence for thedaily forcing null of Sec. 5.2. Bottom panel: one-sided p -value significance levels for the mixed (solidline) and daily (dashed line) null models, showing evidence for α and γ oscillations, between 10 seconds and 2 . p ∼ − ; the fluctuations are consistent with daily forcing.Evidence for non-null behavior indicates failure of the assumption of stationary and memorylessplay; it is, among other things, prima facie evidence against the convergence to a stationary solutionconcept [16] – unless, of course, the “game” is assumed to take place on timescales longer than thosedetected in the data.In behavioral terms, this choice of null allows us assess whether there are non-stationary features ofbehavior over and above static properties that tell us about an individual’s or demographic class’ overallwillingness to engage in conflict.Additionally, as discussed in the Materials and Methods, the null allows us to bound the influence ofsystematic effects, due to the sampling strategy or the correlations induced by noise, that might affectnaive estimates of statistical significance.Of the 47 socially mature adults we consider in these analyses (see Methods), 6 show significant ( p < .
01) deviations from the mixed-strategy null when their individual patterns of behavior are examined. Byanalyzing at the demographic class level, we increase our signal-to-noise and are able to detect significantpatterns in the timescale spectra.Fig. 1 shows the periodogram for the aggregated data of the 47 socially-mature individuals; the toppanel shows the (smoothed) power at each timescale, whereas the bottom panel shows the significance ofany above-null power. The p -values are computed for the two conceptually distinct null models. Strongsignals at two well-separated scales are the first evidence for timescales of behavior. The faster, α , scale,is at one hour; whereas there are broad γ -scale oscillations between eleven and twenty-four hours.As an example of the mixed-strategy null, Fig. 2 shows the periodograms for two biologically-defineddemographic classes – the subadult females (36 to less than 48 months old), and the subadult males (48 toless than 60 months old). The blue bands show the p = 0 .
05 confidence levels for the mixed-strategy null.Although these two demographic classes appear in a similar numbers of conflicts, in similar frequencies,the periodogram uncovers striking differences in their timescales.Whereas the subadult males show some evidence for an α -scale oscillation ( p ∼ − in a single bin),5 igure 2. Demographic classes defined by sex and age show different timescales. Top left pair:periodogram, and p -values, for the subadult females ( n = 11.) Bottom left pair: adult females ( n = 22.)Top right pair: the subadult males ( n = 6.) Bottom right pair: the adult males ( n = 8.) As before, theperiodogram data are shown in red, and the blue band shows the p = 0 .
05 confidence for themixed-strategy null of Sec. 5.1; the green band shows the p = 0 .
05 confidence for the daily forcing nullof Sec. 5.2. The p -values are one-sided, and for the mixed (solid line) and daily forcing (dashed line)nulls discussed in the text. 6heir overall behavior is consistent with the mixed-strategy null. The subadult females show strong γ oscillations on scales between eight and twenty-four hours, with a number of bins with p ≪ − . Thesubadult females, in particular, have an overall p -value, against the mixed-strategy null, of p . − .The adult females (48 months and older) and adult males (60 months and older) show similarly distincttimescales. The adult females show a strong α -scale oscillation that overlaps with the subadult males;they also show evidence for γ oscillations. The adult males show γ oscillations, as well as ( p ∼ . α and γ oscillations; their α waves are similar to the subadult males; their γ waves are slightly weakerthan the subadult females (but still detectable.)In addition to the sex/age-defined demographic classes, the demographic classes defined in terms ofsocial power show important differences. Grouping individuals by power [11] reveals additional com-plexities in the timescale structure of conflict decision-making. We also find important structure in thefunctionally-defined policing class (four high power individuals that perform the majority of effectivepolicing interventions [9].In particular, whereas the policing class shows similar α and γ oscillations to all 47 socially-matureindividuals considered collectively, the α band signal is absent in the remaining eight individuals thatmake up the top power quartile. The second quartile in power shows no evidence for either of these scales.Instead, this second tier shows evidence for an intermediate β -scale oscillation around three hours.Interestingly, there is far less evidence for timescales inherent to particular matrilines. Of the elevenmatrilines present in the study group, only two show evidence for strong intrinsic timescales; these areshown in the Materials and Methods, Figs. 5 through 7. Since members of these matrilines are naturallyincluded in other demographic classes that do have strong timescales, lack of evidence for matriline-leveltimescales suggests that the timescales on which individuals within any particular matriline decide tojoin or avoid conflicts differ enough that, when taken collectively, the signals are washed out. We presentthe full results in the Materials and Methods. In the previous section, we found evidence for multiple behavioral timescales in our data. These timescalesshow evidence for systematic modulation of the behavior of different individuals and demographic classes.What is the nature of such modulation?As the name suggests, the daily-forcing null is intended to capture shifts in behavior due to externalor systematic internal cues that act, over the course of the day, identically from day to day. Such forcingmight be generated by daily shifts in ambient temperature, by a regular feeding time, or by internalprocesses such as fatigue that naturally accrue over the course of a day. Hence, the daily-forcing null ismuch more demanding – i.e. , conservative – on the time series than the mixed-strategy null, as it allowsfor temporal inhomogeneity.Observationally, the daily-forcing null is equivalent to a time-varying mixed strategy in which thevariation is constrained to be the same from day to day. The variation of the mixed-strategy is measured,for the demographic class in question, from the data itself (see Materials and Methods). In our analysis,we allow the shifts to occur on timescales as fast as (but no faster than) 15 minutes – sufficient to model,if possible, even the fastest, α scales. The null does not, of course, specify what particular processes leadsto these daily shifts. It can be a combination of external, internal, and social factors (the behaviors ofothers that shift due to their own external and internal factors).Deviations from the daily forcing null can be accounted for in two ways. On the one hand, sincethe null has no day-to-day variations, deviations might indicate forcing on longer timescales such asœstrus, or that learning is causing accumulated shifts in behavior. These effects would be visible, in theperiodograms themselves, as strong signals beyond 10 seconds. There is some evidence, in the subadultfemale demographic class (Fig. 2), for these longer scales; there, fluctuations at and above 24 hours rejectthe daily forcing null at p . − , and the overall significance has p ∼ . strategic , decision-making. Mathematically, rejection of this null wouldmean that shifts in the average behavior are not driving the system on these scales. Instead it is thesystem’s correlated responses to fluctuations about that average. Purely random fluctuations about theaverage will generate spurious timescales, but these are accounted for by the sampling-with-replacement,and so violations the daily-forcing null indicate correlations in those responses.To demonstrate the importance of the daily forcing null, consider the top panel of Fig. 1, which showsthe periodogram of the 47 socially-mature individuals considered collectively. There are strong signals oftimescales in the α band (around 1 hour) and in the γ bands (at 11 hours and 24 hours) – sufficientlystrong that the mixed-strategy null is ruled out at high confidence ( p < ∼ . p < .
01) deviations from the dailyforcing null when patterns of behavior are independently examined.The strongest deviations from daily forcing are found in some of the most important demographicclasses in the study group. The top left pair of panels in Fig. 3 show the timescales associated with thepolicing class. In addition to the longer γ scales, there is strong evidence for timescales, over and abovethat of daily forcing, that between 1 hour and 2 hours. We return to these strategic time signatures inSec. 5.3.These results suggest that context-free models are adequate explanations for the collective decision-making pattern exhibited by the 47 socially-mature individuals, but fail as descriptions for the morespecific decision-making patterns exhibited by other demographic classes.Whereas violations of the daily-forcing null indicate strategic timing, it is also worth noting thatfailure to reject the daily forcing null does not necessarily mean the absence of strategy. Individuals anddemographic classes structuring their conflict behavior in response to others whose behavior is drivenby daily forcing would show patterns consistent with this rather stringent null and would result in afalse negative (Type II error.) Even behavior consistent with the mixed null may have strategic aspectsinvisible to this analysis, if the relevant contexts are uncorrelated with the size of involved demographicclass. Hence, whereas rejection of the daily forcing null indicates a strategic timescale, failure to rejectthe daily-forcing null does not necessarily mean the absence of a strategic timescale.
The possibility of strategic decision-making over and above daily forcing is implied in the short timescalesof the policing class, which is shown in the top left panels of Fig. 3. These scales are much shorter thancan be accounted for by multi-day shifts in behavior. This policing timescale appears in the fastest, α band. As shown in Table 1 and Fig. 2, the adult female demographic class also shows strong evidencefor violations of the daily-forcing null in the α band.The timescales of conflict-related activity can thus be mapped onto demographic classes playingfunctional – in our case, conflict management – roles as well as biological and other socially defineddemographic classes. The rest of Fig. 3 shows schematically how individuals with differing social power(in a system in which the power distribution is heavy-tailed [12]) show context-sensitive timescales ofvariation. Lower-power classes show both long and short timescales; the second power quartile shows a β -scale oscillation at ∼ β -scales. 8 igure 3. Uncovering strategic timescales for power and policers. Top left pair: periodogram, and p -values, for the policing class – four top-quartile power individuals who effectively interveneimpartially, and break up, conflicts. Top right pair: similarly, for the top quartile in power minus policers ( n = 8.) Bottom left pair, similarly for the second power quartile ( n = 12). Bottom right pair:for the third power quartile ( n = 12). In all cases, the demographic classes show significant deviationsfrom the mixed-strategy null. In the policing class, and in the top power quartile, there is alsosignificant deviation from daily forcing, suggesting that the timing of the decision to join or avoid afight is strategic. There is also evidence of strategic timing, in particular, at the α -scale oscillations seenfor the 47 socially-mature individuals, when treated collectively (Fig. 1).9emographic Class Timescales Timescales Overall SignificanceMixed-Strategy Daily Forcing (10 sec to 2 . p < .
01) ( p < . α , γ – p mix = 0 . p force > . Age & Sex
Subadult Females γ γ p mix = 0 . p force > . α - p mix > . p force > . α , γ α p mix = 0 . p force > . α , γ γ p mix = 0 . p force = 0 . Social Power & Role
Policers α , γ α , γ p mix = 0 . p force = 0 . γ - p mix = 0 . p force > . β – p mix = 0 . p force > . α , γ α p mix = 0 . p force > . p mix > . p force > . α , β – p mix (min) = 0 . p force > . Table 1.
Summary of conflict decision-making timescales detected in the study group. Detections inthe various bands ( α , β , and γ ) are shown for the two null models. In addition, we show the overallsignificance of the detections. The primary timescales found are α (between 30 minutes and 2 hours)and γ (above 6 hours). One demographic class the second power quartile, shows a significantintermediate, β , scale between 2 hours and 6 hours. Taken collectively, the 47 socially-matureindividuals, as well as a number of smaller demographic classes, show significant evidence overall fornon-stationary behavior (above the mixed-strategy null); overall violations of the daily-forcing null,indicating fluctuation-sensitive behavior, are rarer and found in only in the adult males and in thepolicing class. In the eleven matrilines, there are two detections of α and β scales; only one matrilinehas an overall significance against the mixed-strategy null ( p = 0 . .4 Summary Table Table 1 summarizes our results for the different demographic classes.
Periodic behavior at multiple time scales [15] is a fundamental feature of biological systems. Biologicalperiodicities range from cellular activity measured at the scale to milliseconds, through ultraridian cyclesmeasured at time scales of months or years. Many of these cycles derive from fundamental physiologicaland biochemical processes, and are observed across distantly related taxa [3]. This study presents, to thebest of our knowledge, the first evidence for multiple, periodic timescales associated with social decision-making and behavioral patterns in an animal society, and the first empirical study of social systems todemonstrate the existence of periodicities that are not directly coupled to environmental cycles or knownultraridian mechanisms. Rather we find that for particular sets of individuals playing important conflictmanagement roles, the timescale on which they decide to join or avoid fights is strategic . By “strategic”we mean that, collectively, these individuals time their decision to join fights in response to correlatedfluctuations around the mean pattern of conflict decision-making shown by the rest of the group.We find three main results. First, whereas some demographic classes have no timescale structureat all, a number of classes show well-separated fluctuations at either short, α , or long, γ , timescales;intermediate β scales are seen only rarely in the demographic classes we consider. Secondly, differentdemographic classes have different timescale signatures, with, for example, subadult females showingstrong γ -band fluctuations whereas the subadult males show fluctuations on in the α band. Finally, andmost strikingly, we find a strategic timescale associated with a functional role: a subset of individuals whoperform effective policing show a conflict decision-making timescale that is tuned to the mean pattern ofconflict in the group and is on the order of one to two hours.The longest time scales, the γ band, are largely but not completely driven by external or internalsystematic periodicities, like day-night cycles, feeding cycles, œstrus, and context-independent fatiguethat accrues over the course of a day. These scales are not observed for all group members. The adultfemales show the clearest day-scale, or γ , periods in their behavior.The γ activity in the subadult females that is absent in the subadult males suggests some intriguingsex-related differences in conflict decision-making. The males appear to behave more randomly (in sofar as they, as a group, are indistinguishable from the homogeneous mixed-strategy) than the females.Females manifest systematic variation in their willingness to join fights over the course of days. Maleson average appear to be more opportunistic in their decisions to joint conflicts in that their decisions aretime invariant. These results are consistent with the data on opportunistic coalition formation in malesin several primate species [7, 32].The β activity on multi-hour scales seen in some of the intermediate- and low-social power groups couldrelate to simple daily variation in mood, variable sensitivity to hidden triggers in the environment, orgroup-level variability in temperament that manifests in a variety of behaviors, including conflict-relatedbehavior.Of particular interest are the α -scale behaviors that can not be explained by daily forcing. On theseshorter timescales, we find demographic class periodicities associated with the management of conflict bypowerful individuals. These policers appear regularly in the time series at timescale of an hour to twohours.This result is consistent with previous results showing policers preemptively forestall the escalationof aggression by checking conflicts through impartial interventions [13, 22]. It appears that the policersdampen conflict not only by intervening in the regular cycles of fighting, but also by dampening fluctua-tions about these cycles by making regular appearances in fights. That policing has a signature timescale11aises the interesting question of whether policing is predictable by individuals in the society. If so, indi-viduals might be able to tune their conflict decision-making strategies to avoid or facilitate interventionby policers. A body of work in neuroscience [2,20], and preliminary results from the study of social niche constructionin animal societies [1, 14], suggest that multiple timescales within what is typically considered a “level”(e.g. the “neural level” or the “behavioral level”) play an important role in the collective construction ofaggregate patterns as well as in inter-individual coordination during communication [17]. However, be-yond the neural level and the study of biological rhythms, where multiple timescales are well documented,little is quantitatively known about the number, distribution, or significance of timescales.This is particularly true at the behavioral level, where there has been little explicit considerationof the role of time in structuring social interactions or in constraining or facilitating the emergence ofcoordinated aggregates. An important exception is the study of spatial patterning in groups, such asschooling in fish or flocking in birds [5]. Our analysis differs from these studies in that it stresses periodicvariability in a strategic state space rather than non-periodic variability in an explicitly spatial domain.There are many important timescale-related questions in the study of social evolution, and manyof these concern whether such scales are non-functional emergent properties of collective dynamics, orfunctional features that serve to better coordinate complex societies. If timescales are functional, howdo individuals influence the timescales of behavior of a large group? This study provides provisionalevidence that policers, for example, function to modify aggression in the group by performing policinginterventions and appearing in fights at regular, predictable intervals. This explanation is consistentwith the results of an experiment showing that even though the proportion of fights that receive policinginterventions is relatively small, aggression increases when the policing function is disabled by “knockout”of the policers [9].The question of why social and other systems display a range of timescales, as opposed to simplercases where a single strongly coherent oscillation – such as the circadian rhythm – dominates a system,is also of a great interest. It can indicate, among other things, the presence of spin-glass behavior [4].Near-critical spin-glass properties have been found in the dynamics of neural networks [34], and behaviorsguided by social interactions may have similar properties. Given the hypothesized role of fluctuation-correlated behavior that violates the daily forcing null, it is also of interest that such glassy systems shownon-trivial responses to their own internal fluctuations [6].Timescale separations due to the emergence of slow variables at the aggregate level – e.g , the emergenceof a slowly changing power structure from a network of status signaling interactions – are thought to be ameans for reducing social uncertainty generated by fluctuations at a lower-level in fight outcomes [1,10,14].In many cases, timescale variability appears to emerge from combinations of connectivity and constraintsamong populations of components – in the case of power, for example, this corresponds to a network ofindividuals signaling about their dominance status. We remain ignorant, however, of mechanisms thatmight channel variation in timescales at the aggregate level back to influence the timescales on whichindividuals make decisions. To answer these kinds of questions, we need a means of combining inductive,game-theoretic models of the kind presented in Ref. [8] with the spectral properties of the highly-resolvedbehavioral time series as we have presented them here.
Here we cover the methods of data collection protocol, as well as details on the statistical analysisassociated with the Lomb-Scargle periodogram and the two null models.12 .1 Data Collection
The data collection protocol was approved by the Emory University Institutional Animal Care and UseCommittee and all data were collected in accordance with its guidelines for the ethical treatment ofnonhuman study subjects.
Fight: includes any interaction in which one individual threatens or aggresses a second individual. Aconflict was considered terminated if no aggression or withdrawal responses (fleeing, crouching, screaming,running away, submission signals) was exhibited by any of the conflict participants for two minutes fromthe last such event. A fight can involve multiple individuals. Third parties can become involved inpair-wise conflict through intervention or redirection, or when a family member of a conflict participantattacks a fourth-party (See Methods). Fights in this data set ranged in size from two to 28 individualsand can be represented as small networks that grow and shrink as pair-wise and triadic interactionsbecome active or terminate, until there are no more individuals fighting under the above described twominute criterion. In addition to aggressors, a conflict can include individuals who show no aggression ( e.g. recipients or third-parties who either only approach the conflict or show affiliative / submissive behaviorupon approaching). Because conflicts involve multiple actors, two or more individuals can participate inthe same conflict but not interact directly.In this study only information about fight composition (which individuals were involved) and time offight onset are used. Our analyses focus only on the decision to fight. We do not in this paper considerwhether this decision is made with respect to starting a fight or to joining an ongoing fight. We also donot consider any internal aspects of the fight, such as who does what to whom. No time data are availablewithin fights; although the order of an individual’s entry was noted, the information was not used in thisanalysis. The median duration of fights is 15 seconds. The minimum timescale we consider is on theorder of 1000 seconds. Given the median duration and this minimum time criterion, deviations betweenthe fight start time and the time of entry of any individual into the fight should not be problematic.Fight onset and termination time were noted in hours, minutes, and seconds. Timing accuracy – isat worst, on the order of seconds for fight onset time, and so accuracy to this level is more than sufficientfor the range of timescales we investigate here.Demographic Classes: The demographic classes we consider include age-sex classes (see below), ma-trilines (see below), power quartiles (see below) and policers (see below).Age-sex Classes: With the exception of the matriline analyses, all animals in our analyses are “socially-mature”. Socially-mature males were at least 48 months and socially-mature females were at least 36months by study start. Subadult males were males between 48 and 60 months; adult males were atleast greater than or equal to 60 months. Subadult females were at least 36 months but less than 48months; adult females were at least greater than or equal to 48 months. These thresholds correspond toapproximate onset of social maturity in pigtailed macaques.Matriline: an adult female and her daughters. In the study group, all females in a matriline wererelated through the maternal line. Only females one year or older were included in the matriline analyses.Power: the degree of consensus among individuals in the group about whether an individual is capableof using force successfully. Consensus is quantified by taking into account the total number of subor-dination signals an individual receives and multiplying this quantity by a measure of the diversity ofsignals received from its population of signalers (quantified using Shannon Information) [12]. In the pig-tailed macaque, the subordination signal is the silent bared teeth display [10]. The distribution of power13n our study group is heavy tailed. The first power quartile corresponds to the top 12 individuals of the 48.Policers: four individuals (one female, three males) who preform the majority of effective policing in-terventions (sit towards the tail in a log normal distribution of the frequency of effective policing inter-ventions). A policing intervention is an impartial intervention performed by a third party into an ongoingconflict. [9]. These individuals occupy the top four spots in the power structure and sit toward the tailof the distribution.General note about demographic classes: Our results showing that demographic classes that have signa-ture timescales suggests that there are empirical grounds for treating them as coherent units with setsof actions. This is similar to the concept of coalitions in cooperative form games [26], and the finding isconsistent with results of a previous study in which we showed that the triad, not the individual or thedyad, is a the fundamental unit of conflict dynamics in this group [8].
During observations all individuals were confined to the outdoor portion of the compound and were visibleto the observer, JF. The ≈
150 hours of observations occurred for up to eight hours daily between 1,100and 2,000 hours over a twenty-week period, comprising roughly 122 days, from June through October1998 and were evenly distributed over the day. This span allows us to study a wide range of scales onwhich behavior can change. The sampling is sparse relative to the total number of hours (150 of 2928)in the data collection period; it is also irregular, in that observational periods are not separated by thesame number of days and have different lengths and gaps. Fight and status signaling data were collectedusing all-occurrence sampling.Provisioning occurred before observations, and once during observations at the same time each day.The group was stable during the data collection period (defined as no reversals in status signaling inter-actions resulting in a change to an individual’s power score, see [12]). One animal, Ud, was removed fromthe group for health reasons towards the end of the study; as this sudden removal (and thus zeroing outof all behavior data) is likely to produce strong, but spurious signals of behavioral variation, we excludedher from the analysis. Q Values and Timescale Coherence
In common with many spectral analysis methods, Lomb-Scargle takes as basis functions the sine andcosine, phase shifting them to find the optimal fit. A pure sine-wave signal, for example, would amountto an extremely sharp spike at the relevant frequency in the periodogram.Although precise oscillations are unlikely to be found in the noisy and non-ergodic environmentswe consider here, more realistic behaviors are also mapped to the relevant portions of the plot; forexample, repetitive excitation and subsequent exponential decay would map to a peak centered aroundthe excitation period. If one also allowed there to be jitter in the exponential decay – random variationboth in the time-constant and in the precise timing of the excitations following decay – the peak wouldbroaden further.One sometimes defines a Q -factor, a measure of the width of a peak in a periodogram, as f / ∆ f ;here f is the peak center and ∆ f the width of the peak at half-maximum. Very “pure” oscillations –close to a sinusoidal variation – have high Q -factors; conversely, purely damped systems that dissipateoscillations – such as a suspension system in a car or building – have Q less than unity.Man-made systems such as optical cavities for lasers can have Q factors in the millions and billions;mechanical vibrating systems such as a tuning fork have Q of of order 10 . Meanwhile, natural phenomenatend to have much lower Q -factors, indicating the presence of noise and blended signals at different scales.For example the Q -factors of brain oscillations measured from an EEG of a sleeping human of can be of14rder 10 or 100; the quasi-periodic phenomena found in neutron star systems can have Q of order 10 orlower [25, 35].In the system we study here, we find Q factors of larger than, but of order, unity – i.e. , slightly lesscoherent than the quasi-periodic oscillations of the human brain. These are similar to the Q -factors onecan estimate for the bacterial motors of E. coli measured in Ref. [21], and higher than the Q -factors seenfor the signaling networks that control them. The Lomb-Scargle periodogram, introduced in Sec. 2, forms the center of the data analysis in this paper.Traditionally it has been applied to the detection of high- Q signals such as the detection of orbital periodsof stellar systems. We discuss here some of the statistical tools employed to uncover much the broaderfeatures of the spectrum observed in our study system.It turns out that despite the discreteness of the measurement values – and the correlations betweenconflict behaviors of different individuals at the same time – the distribution of P ( ω ) under the mixed-strategy null of Sec. 5.1 is also approximately both exponential and of mean unity. The daily forcingnull is far more structured, and induces strong correlations between bins that can be seen visually in theplots.If one is then seeking a signal at a precise frequency – i.e. , a sinusoidal oscillation with an extremelyhigh Q -value – then it becomes simple to determine a threshold power above which a detection is con-sidered significant with a certain p -value. The approximate value is P ( ω ) ≈ − ln p/M , (4)where M is the number of independent frequencies. The dependence on M comes from the fact thatrare events become more likely the more one samples – in the words of Ref. [28], “look long enough, findanything!”However, for signatures that are more broad-band – that are expected to cover a range of frequencies– this simple method is too conservative. The presence of correlated noise makes the analytic estimationeven harder, since, in contrast to the Gaussian, there is no simple version of the multivariate exponentialdistribution that allows for correlations between arbitrary numbers of different frequencies. Insteadof analytic approximations, then, a Monte Carlo estimate of bin-by-bin significance is made: manyinstantiations of the null model are produced, and their distribution compared to measured value, bin-by-bin, to produce an estimate of the p -value.Given a set of such p -values for all bins, we then wish to estimate, in Sec. 5.1, the overall p -value for adetection of non-null timescales. The Bonferroni correction for combining p -values is generally consideredto be too conservative. In a search for periodicity in gene expression data, Ref. [18] used an order statisticon p -values. Here, following Ref. [24], we use the χ (2) test of Fisher, which works well where one isseeking evidence for strong signals in one or two bins. All these methods require that the p -values beindependent (in the null).Failures of null independence – and, in general, failures of null p -values to fall in a U (0 ,
1) distribution– can lead to both Type I and Type II errors for any method of combination. We check the validityof the χ (2) test by running our same analyses on null data alone. We find that the Fisher methodworks reasonably well, though not perfectly, and can underestimate p -values when p ≤ − . We silentlyinsert these corrections to reported p -values in the main paper, so that our stated p -values for overallsignificance are our best estimates, and do not rely on the strong assumptions of combination methodssuch as the χ (2) test.Our bins are logarithmically spaced; we search the range between 10 seconds and 2 . In Figs. 4 through 7, we show the periodograms for all the remaining demographic classes discussed inthe paper. These results are also summarized in Table 1.16 igure 4.
Top pair: the top quartile in power, including police class ( n = 12; in the main text thisquartile is split to show the influence of the policing functional class.) Bottom pair: the lowest quartilein power ( n = 11.) 17 igure 5. Matrilines 1 through 4. n = { , , , } igure 6. Matrilines 5 through 8. n = { , , , } igure 7. Matrilines 9 through 11. n = { , , } Acknowledgements
We thank Frans de Waal for support during data collection, and the staff of the Yerkes National PrimateResearch Center, for help with the data collection.
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