A dynamic network model of societal complexity and resilience inspired by Tainter's theory of collapse
Florian Schunck, Marc Wiedermann, Jobst Heitzig, Jonathan F. Donges
AA DYNAMIC NETWORK MODEL OF SOCIETAL COMPLEXITY ANDRESILIENCE INSPIRED BY T AINTER ’ S THEORY OF COLLAPSE
A P
REPRINT
Florian Schunck ∗ , Marc Wiedermann † , Jobst Heitzig ‡ , Jonathan F. Donges § February 16, 2021 A BSTRACT
In the past twenty years several events disrupted global economies and social well-being and generallyshook the confidence in the stability of western societies such as during the 2008 financial crisis andits economic aftermath. Here, we aim to identify and illustrate underlying drivers of such societalinstability or even consequential collapse. For this purpose we propose a low-dimensional and stylisedmodel of two classes of networked agents (termed “labourers” and “administrators” hereafter) thatis inspired by Joseph Tainter’s theory of collapse of complex societies. We numerically model thedynamics of societal complexity, measured as the share of “administrators”, which is assumed toaffect the productivity of energy-producing ‘labourers”. We show that collapse becomes increasinglylikely if the model society’s complexity continuously increases in response to external stresses thatemulate Tainter’s abstract notion of problems that societies need to solve. We additionally provide ananalytical approximation of the system’s dominant dynamics which matches well with the numericalexperiments, thus, allowing for a precise estimate of tipping points beyond which societal collapseis triggered. The administration’s ability for increasing productivity of labour shows to be the mostinfluential parameter for longer survival times which is additionally fostered by a minimum level ofsocial connectivity. Finally, we show that agents’ stochastic transitions between labour force andadministration, i.e. social mobility, can increase the survival time of the modeled society even further.Our work fosters the understanding of socio-ecological collapse and illustrates its potentially directlink to an ever increasing complexity in response to external shocks or stress via a self-reinforcingfeedback. K eywords societal complexity · social-ecological collapse · resilience · network model · agent-based-model Societies were always faced with challenges vastly different in nature. These include external events such as invasionsand environmental catastrophes in the case of Mesopotamia (Weiss et al., 1993) or internal challenges like corruption,rebellions and mismanagement in ancient Egypt (Butzer, 2012). Historically, both cases led to societal collapse. Otherclassic and frequently studied examples include the fall of the Western Roman Empire in the 5th century CE, the collapseof the Maya in 9th century CE or the fall of the Minoan Civilization of Crete in 14th century BCE (Tainter, 1988;Middleton, 2012). Modern societies are facing similarly severe challenges, such as global climate change, pandemics,or financial instabilities. Estimated costs of climate change (Tol, 2018) alone will put additional strain on already ∗ Helmholtz Centre for Environmental Research [email protected] † Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, FutureLab on Game Theory and Networksof Interacting Agents, P.O. Box 60 12 03, D-14412 Potsdam, Germany ‡ Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, FutureLab on Game Theory and Networksof Interacting Agents, P.O. Box 60 12 03, D-14412 Potsdam, Germany § Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, FutureLab on Earth Resilience in theAnthropocene, P.O. Box 60 12 03, D-14412 Potsdam, Germany; Stockholm Resilience Centre, Stockholm University, Kräftriket 2B,106 91 Stockholm, Sweden a r X i v : . [ phy s i c s . s o c - ph ] F e b PREPRINT - F
EBRUARY
16, 2021excessively indebted national states (World Bank, 2020). While the real estate crisis of 2008 still reverberates in severalcountries, the Covid-19 pandemic caused major impacts on health systems and potential long-term consequences foreconomies and socio-cultural systems (Trump et al., 2020).The aforementioned examples therefore raise the question whether it is possible to identify underlying principles thatdetermine a society’s ability to cope with such large-scale challenges and thus increase their resilience to collapse. Asreviewed by Cumming and Peterson (2017), several explanatory models for collapse exist. Explanations focusing onresource limitations such as the classical Malthusian trap (Malthus, 1798) and limits to exponential growth and overuseof resources have been widely discussed in the past (Meadows et al., 1972; Rockström et al., 2009; Diamond, 2011).Tainter’s theory of collapse of complex societies emphasises specifically the notion of societal complexity and itstendency to self-amplifying increase in response to stress as the main cause of collapse (Tainter, 1988, 2006). Withinthis framework, complexity emerges continuously through problem solving and manifests itself in the differentiation andspecialization of social roles, hierarchies and control of behavior, growing population, technical abilities and increasedinformation flow. These investments in complexity have an associated cost, such as calories, time, or money, all of whichcan be reduced to an abstract form of energy . Although an increase in complexity is beneficial for societies and adds towell-being, surpluses gained by complexity increments are hypothesized to decrease as the ‘low-hanging-fruits’ startgetting less. Thus diminishing marginal returns on investments in complexity (or simply return-on-complexity
ROC )ultimately drive a society into collapse. In addition, benefits of complexity may be short-natured and quickly consumedby population increases or gains in living standard. In contrast, elevated demands in energy are persistent, leading tonew problems, once previously earned energy surpluses are used up. Within the framework proposed in Tainter (1988)this process is denoted as the energy-complexity spiral (Tainter, 2006).Contemporary evidence for complexity gains in specialization and differentiation of social roles as well as technicalabilities are the decrease of labour in the agricultural sector and increase of the finance and communication sectors inthe 20th century, while simultaneously efficiency in primary sectors was increased (Feinstein, 1999). Another examplefor complexity increase is a rising bureaucracy burden in cancer research (formalities, regulations of opening studies).The added complexity diminishes ROIs of research, thus resulting in reduced innovativeness in this field (Steensma andKantarjian, 2014). Hall et al. (2009) estimates the energy return on investment (EROI) of a sustainable society to bewell above 3:1, which can be interpreted as a proxy for complexity. Despite these examples, overall empirical evidencefor societal complexity and its change over time is scarce, although it seems ubiquitous in our lives.Mathematical models and simulations are important tools to study the implications of theories of collapse and resilienceand to test competing hypotheses. In the archaeological domain, agent-based models are employed to understand theinteraction of water and land-use change (Barton et al., 2010) or to compare trajectories of biocultural evolution (Bartonand Riel-Salvatore, 2012). Dynamical systems models are frequently employed to model socio-ecological-systems(Kohler and van der Leeuw, 2007; Ostrom, 2009; Schlueter et al., 2012). Examples include a resource economicsmodel of collapse of the Rapa Nui population (Brander and Taylor, 1998) or an exploration of vulnerabilities inforager-resource systems (Freeman and Anderies, 2012). More recently, Motesharrei et al. (2014) develop a dynamicalsystems model to study trajectories to collapse in a coupled hierarchical consumer resource system. Similarly, Nitzbonet al. (2017) employed a model of coupled carbon, population and capital stocks to identify preconditions of collapseand sustainability.In the last decade, a new generation of models was conceptualized turning to larger and up to global scales andembracing socio-ecological complexity by coupling concepts from dynamic land-use global models and agent-basedmodels (Rounsevell and Arneth, 2011; Arneth et al., 2014). Building upon this framework Brown et al. (2019) identifiedsocietal breakdown as an emergent property of large-scale behavioral models of land-use change to different climate-economic scenarios. To strengthen the human side of the equation(s), much attention was payed to increase realismwhen modelling human decision making in socio-economic and socio-ecological models (Schlüter et al., 2017; Schillet al., 2019) as well as modelling collective behaviour and social tipping dynamics (Wiedermann et al., 2020). Recently,the mentioned efforts precipitated in a comprehensive modelling framework for so-called World-Earth Models (Dongeset al., 2020).In this work we explore the theory of the collapse of complex societies in terms of the mechanisms leading to increasingsocietal complexity and ultimately to collapse. For this purpose, we implement an agent-based network model consistingof productive and administrative nodes. This implementation successfully recaptures the theory’s expected behaviour ofcollapse due to diminishing marginal returns on complexity. We then go on and study its macroscopic dynamics indetail and identify two escape mechanisms from collapse: increasing the output elasticity of labour, or by allowingsocial mobility by stochastic transitions of nodes between productive and administrative states. In order to approach thetheory of societal collapse from a conceptual angle, we here propose to reduce it to three key elements:2
PREPRINT - F
EBRUARY
16, 2021
Figure 1: Sketch of the model setup. Nodes/agents are either attributed as administrators A or labourers L . Labourers that areconnected to at least one administrator are denoted as coordinated labourers C that produce higher energy outputs due to an increasedefficiency. The influence of each administrator is marked with green shading. In case the energy production falls below a criticalvalue, the coordinated labourer that is connected with most other nodes (here marked in orange) becomes an administrator as well.
1. The basic currency of any society is energy, since labour and material goods can be viewed as derived fromenergy.2. Problems need to be solved when energy availability is deficient as a consequence of stochastic events orshocks. According to the self-reinforcing process denoted as energy-complexity spiral (see above) this processalways increases societal complexity.3. Gains in complexity are resembled by an increase in administration, since it encapsulates gains in special-izations of social roles, hierarchies and control, as well as information flow. Additionally, the size of anadministrative body is a very graspable example of complexity.The remainder of this paper is organized as follows. We start with a thorough description of our proposed networkmodel in the next section. We first present the basic model and then go on to describe a model variant, which includes amechanism to counteract collapse. In the results section we first present the stochastic model by discussing exemplarytrajectories of the system followed by the derivation of an analytical approximation. Based on this approximation wepresent a comprehensive ensemble analysis of all crucial model parameters. The paper concludes with a discussion ofall relevant results and an outlook to future work.
In the following we introduce the model that is used in this work to exemplify the dynamics of collapse that are proposedin Tainter (1988). Its goal is to illustrate the increase in societal complexity as a response to the solving of problems,here the sustaining of a certain level of energy supply E . We define complexity as an increase in the diversificationof the society (the creation of a new role) and the corresponding increase in control over the behaviour of others (anincreasing administrative body). We will show that with only including these two processes, one already observes anincrease in an appropriately defined measure of complexity, and, hence, we do not need to specifically account foradditional phenomena, such as specialisation or an increase in hierarchies.In particular, we represent an artificial society by means of a complex network (Newman, 2003; Castellano et al., 2009)as complex networks have been successfully used in the past to model social systems with respect to collapse (Heckbert,2013), sustainability (Wiedermann et al., 2015; Barfuss et al., 2017; Holstein et al., 2020) and their ability to adapt tonew conditions (Auer et al., 2015; Schleussner et al., 2016). They are usually comprised of entities of two basic types,i.e., nodes and links. For our case the nodes indicate (representative) small but long-lived entities that supply labor,3 PREPRINT - F
EBRUARY
16, 2021such as family lines, in a society. Links between the nodes or entities represent some comparatively stable social tie,e.g., a professional relationship, between them. N counts the total number of nodes in the network, Fig. 1.The sole goal of the society is to produce energy E at a certain level in order to sustain its function. The correspondingper-‘capita’ (actually, per-node) energy requirement of the society is denoted as (cid:15) . In order to produce this energycertain nodes (denoted as labourers L ) harvest an individual energy source R . R is subject to external shocks thatrepresent an abstract form of a problem that needs to be solved. In order to increase the energy that is produced fromthe resource, some labourers L can also be appointed to become administrators A . The purpose of the administration isto increase the efficiency of the labourers L that are under its direct management. We assume each labourer L who isdirectly connected with at least one administrator A becomes part of a more efficient body of the labour force whichhas increasing returns to scale, expressed by an output elasticity of labour of φ > (while the rest of the labour forcehas an output elasticity of labour of ). We denote this class of more efficient labourers the coordinated labourers C , Fig. 1. While they increase the output elasticity of their immediate surrounding, the administrators A themselvesdo not produce energy any longer (thus indicating a certain cost of maintaining an administrative body). Note thatbeing additionally connected with more than one administrator A does not further increase the output elasticity of thecoordinated labourers C . Along the lines of Tainter (1988) we measure the resulting societal complexity S in terms ofthe size of the administration N A = | A | as it represents the level of behavioral control in our model society.We model the temporal dynamics within our artificial society according to the following rules. At each time step t the available resources per node, R ( t ) , are drawn from a beta distribution with parameters α = 15 and β = 1 . Thecorresponding probability density function is f ( R ) = αR ( t ) α − . (1)The stochasticity of R ( t ) here represents the occurrence of shocks or problems to be solved by society. Smaller valuesof R ( t ) represent larger shocks or problems. We have chosen f ( R ( t )) such that drawing R ( t ) mostly results in R ( t ) close to (i.e., no or small problems) with a rare chance of R ( t ) (cid:28) (i.e., large problems). Depending on the numberof labourers N L ( t ) (uncoordinated) and N C ( t ) (coordinated) the total produced energy E ( t ) is computed as E = R ( N L + N φC ) . (2)If energy per capita falls below a minimum value representing the society’s vital needs, i.e., if E ( t ) /N < (cid:15) for someparameter (cid:15) , we assume the society tries to solve this problem by appointing exactly one (additional) administrator suchthat the output elasticity of some labourers is increased. The new administrator is selected as follows:(a) If currently no administrators exist in the society (i.e., if N A = 0 ) the labourer L with the most networkconnections becomes the sole administrator.(b) If at least one administrator currently exists (i.e., if N A > ), the coordinated labourer who has the mostnetwork connections becomes an additional administrator.The model then proceeds to the next time step and the availability of R ( t ) as well as the produced energy E ( t ) is againcomputed according to Eqn. (1) and Eqn. (2).Following (Tainter, 2006) we define the potential onset of societal collapse, i.e., the vicinity to a critical state, once thesystem approaches diminishing returns on complexity ROC . We operationalize
ROC into our model by computingthe differences in energy E and complexity S after the recruitment of one additional administrator A at time t , ROC ( t ) = E ( t ) − E ( t − S ( t ) − S ( t − . (3)Note that ROC ( t ) depends on the specific structure of the underlying social network, in particular on who the newlyappointed administrator A is connected to and how many labourers turn from L to C as a consequence. As outlinedabove we measure societal complexity S simply in terms of the size of the administration N A , i.e., S ( t ) = N A ( t ) .Since the size of the administration is increased by one due to the recruitment of a single additional administrator attime t we obtain N A ( t ) = N A ( t −
1) + 1 such that S ( t ) − S ( t −
1) = 1 . Therefore the return on complexity
ROC reduces to
ROC ( t ) = E ( t ) − E ( t − . (4)Once ROC ( t ) becomes negative our artificial society is expected to begin its decline into collapse since the energyproduction E becomes smaller and approaches zero with increasing time t and a correspondingly ever increasing sizeof the administration N A . In the following section we therefore directly display results for the energy production E ( t ) PREPRINT - F
EBRUARY
16, 2021over time and interpret a positive slope as increasing returns to complexity and a negative slope as decreasing returns tocomplexity.For the initial model setup we consider an Erd˝os-Rényi random network (Erd˝os and Rényi, 1960). It consists of N ( N = 400 for our case) nodes that are all initially assumed to be labourers ( N L (0) = N , N A (0) = N C (0) = 0 ).Additionally, we place a link between each pair of nodes with a fixed probability (cid:37) (which we will vary between (cid:37) = 0 and (cid:37) = 0 . for our analyses). Thus, (cid:37) gives the expected density of links in the resulting network, while N ( N − / is the maximum number of possible links. Starting from this setup, we simulate the system for a maximumof t max = 10 , time steps following the model logic that is given above. In the case when all nodes becomesadministrators (i.e., if N A ( t ) = N ) we stop the simulation even though t max may not have been reached. In that caseno more energy is produced ( E = 0 due to the lack of labourers L ) and we consider the society to be entirely collapsed. The basic model that we introduced in the previous section represents closely the assumptions put forward in Tainter(1988). Here a society may get caught in the energy-complexity spiral as the increased or even flat demand for energyincreases its complexity (measured as the size of the administrative body N A ( t ) ) when being faced with ever newproblems.One obvious downside of the above model setup is that, once selected, an administrator A does not change its role backto become a labourer L (or C ) again. This decision is motivated by findings that social mobility is predominantly stableor upward rather than downward within (Stawarz, 2018) and across generations (e.g., Adermon et al., 2018). Similarly,it is only possible to become an administrator A if one is selected from the set of coordinated labourers C , meaning thata spontaneous jump in hierarchy is permitted within the logic of the model.However, in order to increase social mobility, we also aim to investigate a version of our model that allows for thenodes to randomly change their state with a fixed small probability. Such dynamics can be interpreted as the loss orgain in socio-economic status within or across generations. Within the logic of our model such an additional process isimplemented as follows. At the beginning of each time step t every node changes its state with probability p e eitherfrom A to L or C (depending on whether it is connected to another administrator), or from L or C to A . We expectthat this allows to regulate the complexity of the society and possibly avoid its unlimited growth (as for this settingthe complexity may also decrease as the problems/shocks become smaller again). Hence, such process should cause asustainable ratio between the size of the administration N A and the body of labourers N L + N C . Note that if we set p e = 0 , we again obtain the dynamics of the model setup that is outlined in the model description. Figure 2A shows the implementation of the original approach to Tainter’s theory of societal collapse with dynamicsdescribed above in the model description. Initially the model society benefits from its response to external shocks. Thereason is that initially each new administrator A has a much higher number of connections to labourers L , which arethus turned into coordinated labourers C producing energy output at increasing returns to scale (here φ = 1 . ). As theamount of labourers L decreases, the marginal returns on complexity ( ROC ) of each new administrator A decreasesuntil it becomes negative. Figure 2B displays the energy output as a function of administration which resembleswell the assumed underlying principle of diminishing marginal returns on complexity in Tainter (2006). In our firstapproximation to the theory of collapse, the model society can only react to shocks by choosing the best possibleadministrator each time the energy requirements are not met. We interpret the formation of an administrative bodyas an increase in societal complexity. After a period of increased energy output, marginal returns decrease whilecorrespondingly the share of administration rises, caught in an energy-complexity-spiral until the society collapses(here E = 0 ), Fig. 2A.Next, we study whether small adaptation mechanism of the model society enable the model to overcome the fastcollapse shown in Fig. 2A. For this purpose we allow for a random exploration of node states enabling the model agentsto change their state ( A −→ L/C or vice versa) with a low probability ( p e = 0.00275). For a network of N = 400 individuals this amounts to an average of 1.1 nodes changing their status at random per time step.As figure 2C shows, an exploration rate of this low magnitude can already be insufficient to overcome the collapse thatfollows from the event-triggered selection of administrators indicated by a higher survival time of the society, compareFig. 2A/C. However, energy output E is still very low at p e = 0 . , due to the high degree of administration in thenetwork over sustained periods of time. 5 PREPRINT - F
EBRUARY
16, 2021
Figure 2: Exemplary network simulations of a Tainter-like model society ( N = 400 ) according to the model description. The panelsshow only the first 5000 timesteps to focus on the initial dynamic. Blue curves show the share of administration in the network (lightblue: Administration as result of decreased resource availability, dark blue: Administration resulting from exploration with rate p e ).Orange curves show the average energy produced per node. A) shows the typical development of a network reacting to shocks bychanging one node from coordinated labourer C to administrator A . B) displays a moving average of the energy measured in A)against the size of the administration. The black curve indicates stylized parabola-shaped diminishing marginal returns as in (Tainter,2006). C) displays the network development at an intermediate exploration probability ( p e = 0 . . D) shows the developmentof a network high probability of exploration ( p e = 0 . ). We also study the case of larger rates of exploration, i.e, p e = 0 . , Fig. 2D. We find an initially sharp increase inadministration, facilitated by the high exploration probability. The trajectory then converges to a stable administratorshare slightly above N A /N = 0 . . This fraction is positively offset by additional effect based promotion of nodes,because energy production per capita is always below the threshold ( (cid:15) = 1 . ). After reaching this (meta-)stable state,the model society seems to survive for a long time (at least longer than t = 5000 ) while faced with a decreased meanenergy production at around E ≈ . .It is noteworthy that in each instance of the simulation (Fig. 2A, C, D) the society reaches an energy production abovethe energy requirements, which is then however quickly surpassed until the energy level either tends towards zero forzero or low exploration rates in Fig. 2A,C or converges to a stable regime for large exploration rates in Fig. 2D. We now derive a macroscopic approximation of the above stochastic micro-model in terms of an ordinary differentialequation describing the average time evolution of an aggregate quantity (here the total number N A of administrators).This time evolution is governed by average transition rates between the three groups. The rates are based on approxima-tions of the probabilities with which individual agents switch between the groups and the assumption that N is large sothat the law of large numbers applies.There are two processes which make the number of administrators change, exploration and targeted recruiting inresponse to energy demands Due to exploration, from the N A many administrators, on average p e N A many leave theadministration per unit time, and from the N − N A many non-administrators, on average p e ( N − N A ) many are hiredas administrators per unit time, making a balance of p e ( N − N A ) additional administrators on average per unit time.For targeted recruiting, we make the simplifying assumption that the density of links inside and between the threegroups remains approximately constant and can thus be estimated by the overall link density (cid:37) of the Erdös-Renyinetwork. Then, if N A is the number of administrators, the probability that a non-administrator is not linked to any ofthe N A administrators (and is thus un-coordinated) is approximately (1 − (cid:37) ) N A , hence the numbers of un-coordinated6 PREPRINT - F
EBRUARY
16, 2021
Figure 3: Analytic approximation of the model dynamics with the share of administration N A /N (A) and corresponding energyproduction per capita (B). Transparent solid lines show the results of an ensemble of 100 simulations of the microscopic networkmodel for different values of p e . Dashed lines display the respective analytical approximation. (C) Histograms of collapse frequenciesof the model, i.e., frequency of times at which energy production approaches zeros, E = 0 . Bars located at time t ≥ showright-censored cases of the simulation. and coordinated labourers are approximately L ≈ ( N − N A )(1 − (cid:37) ) N A and C ≈ ( N − N A )(1 − (1 − (cid:37) ) N A ) , hencewe have E = Re with e = L + C φ ≈ ( N − N A )(1 − (cid:37) ) N A + [( N − N A )(1 − (1 − (cid:37) ) N A )] φ .An additional administrator is recruited iff E/N < (cid:15) , i.e., iff
R < N (cid:15)/e . The probability that
E/N < (cid:15) is then P = F ( (cid:15)/e, β, α ) , where F ( · , β, α ) is the cumulative distribution function of the beta distribution. The expectednumber of administrators hired additionally due to shocks per time unit is then also equal to this probability P .In all, we get the approximation dN A dt ≈ p e ( N − N A ) + F (cid:18) N (cid:15) ( N − N A )(1 − (cid:37) ) N A + [( N − N A )(1 − (1 − (cid:37) ) N A )] φ , β, α (cid:19) ,C ≈ ( N − N A )(1 − (1 − (cid:37) ) N A ) ,L ≈ ( N − N A )(1 − (cid:37) ) N A as long as N A < N .Simulation results are well predicted by the approximation, Fig. 3A,B. Particularly at higher values of p e , the ap-proximation yields excellent results both for the energy production and the administration share. Only at explorationprobability of p e = 0 (dashed line and light colors in Fig. 3), the macroscopic approximation slightly overestimates thevelocity of collapse. This can be explained by one simplifying assumption of the macroscopic approximation, namelythe network degree or local link density to be homogeneous, i.e., approach (cid:37) , across the entire network. In contrast,due to the links being put at random when setting up the network, this distribution in a node’s number of connectionsbecomes heterogeneous with some nodes showing an above-average number of connections. Because those nodes arepreferred over those with few connections in the process of administration selection, initially selected administratorswill result in an above average energy return. This produces the prolonged slow-growth phase of the administration inthe numerical simulation. In contrast, the macroscopic approximation assumes that at a given point in time any newadministrator would have the same average effect.To illustrate the probability of collapse under different exploration assumptions, histograms of survival times fromthe stochastic model are computed, Fig 3C. At high rates of social mobility p e the model converges to a stable statewhere the society persists for infinite times due to its ability absorb shocks. For lower p e ≈ we also find isolatedcases of infinite survival even though the society mostly collapses comparatively early, Fig. 3C. Such observations canbe explained by a fragmentation of the network into smaller worker networks that are mutually disconnected. The7 PREPRINT - F
EBRUARY
16, 2021
Figure 4: Survival time analysis of the parameters exploration probability ( p e ), link density ( (cid:37) ) and efficiency ( φ ). (A) The effect of p e between zero and N for some select values of φ on median survival time over all computed values of (cid:37) . The vertical dashedline defines the exploration threshold of N . Lower panels ( B − ) display the relation between φ and (cid:37) for different explorationprobabilities. Grey areas in the lower panels indicate a survival time ≥ , potentially going to infinity. scattered survival times along the time axis for intermediate exploration rates illustrate stochastic survival of the modelsociety due to a random sequences of shocks. In the third step of our analysis we use the approximation proposed in the previous section to estimate model outcomesfor a broad range of parameter values (cid:37) , φ and p e . Note, that the additional parameters of resource availability ( α , β )and threshold ( (cid:15) ) have a major influence on the outcome of the model as well. Specifically, a high probability of lowresource availability leads to much shorter survival times and vice versa while a low threshold (cid:15) to appoint furtheradministrators considerably increases survival times. However, since these effects did not reveal any additionallyremarkable results (not shown), we focus our attention on the major drivers of survival time and energy output, i.e, theexploration probability p e , link density (cid:37) and output elasticity of labour φ .Panels 4B − show the survival time as a function of link density (cid:37) and output elasticity of labour φ . Figure 4B displays a parameter analysis for the case of no exploration, p e = 0 and can hence be interpreted as showing minimumrequirements of a successful administration. The most important requirement being a sufficiently large connectivityof the social network (indicated by the increasingly grey area with increasing (cid:37) ). Also, the extreme case of (cid:37) = 0 . ,i.e, a network with no connections, must be noted since in this case nodes cannot be converted at all to administratorsand thus the model remains in its initial state. As p e approaches N (Fig. 4B − ) the relevance of (cid:37) diminishes and thesurvival time becomes mainly a function of the output elasticity φ .Figure 4A demonstrates the effect of exploration for selected values of φ . We have chosen a range of values for φ thatroughly correspond to the order of magnitude of returns to scale reported in Baležentis et al. (2015); Diewert et al.(2011). For exploration probabilities p e < /N , i.e., when less than one individual per time unit switches their type,collapse is likely, which is indicated by reduced median survival times, Fig. 4A. In fact, our macroscopic approximationshows that N A = N is a stable fixed point of the approximate dynamics for p e < /N since the RHS of Eqn. 5 remainsstrictly positive for N A → N . This means that the labour force will vanish in finite time, thereby also resulting ina finite survival time. Additionally, in this “low exploration regime”, a larger output elasticity of labour φ tends toincrease the median survival time. Indeed, the term F ( · ) in Eqn. 5 is a decreasing function of φ for large N A , whichmeans a larger φ slows down the collapse towards N A = N . The effect of a p e on survival time is more difficult tounderstand because it is twofold, as can again be seen in Eqn. 5. Initially, when still N A < N/ , larger p e increases dN A /dt and speeds up the process of recruiting administrators, as seen on the left of Fig. 3A. As soon as N A > N/ ,8 PREPRINT - F
EBRUARY
16, 2021
Figure 5: Trajectories of administration share for φ = 1 . and (cid:37) = 0 . for two choices of p e below the critical value p e = N above which collapse of the system is mitigated. The trajectories show how the administration share is stabilized by exploration.This mechanism effectively slows down the total change in administration size until a critical administration size is reached at whichadministration growth enters the event triggered phase. the reverse happens and a larger p e slows down that process. For small p e below some turning point value p e < /N ,the first effect dominates, so that survival time decreases. For p e between p e and /N , however, the second effectdominates, so that survival time increases again. For some values of φ , the two effects are clearly distinguishable indifferent phases of the evolution, as can be seen in the realization depicted in Fig. 5, where the system stays a long timeclose to N A = N/ before reaching N A = A and thereby collapses. As p e approaches the critical value of /N , thiseffect becomes ever stronger and expected survival times approach infinity.Of course, since the microscopic network model is of stochastic nature, individual survival times still vary quite a bitbetween realizations for the same set of parameters. As survival time also depends on other parameters, we displayonly the central tendency of survival time in Fig. 3A. Since we terminate our simulations after t = 10000 time steps,we cannot distinguish higher survival times, which means we cannot use the arithmetic mean of survival times as themeasure of central tendency. This is why we display the median instead, which has the additional advantage of being amore robust statistic than the mean.For larger values of φ , we see that some realizations survive for very long even though p e < /N . This is becausefor large enough φ , the approximate dynamics Eqn. 5 have a second stable fixed point despite N A = N . Indeed,for φ → ∞ , the F ( · ) term vanishes and dN A /dt = 0 for some value close to N A = N/ . In that case, realizationsof the stochastic model starting with small N A are likely to stay close to N A = N/ for very long (which is thus a“metastable” state) before eventually escaping into the basin of attraction of the “collapse” fixed point N A = N due to alarge enough shock.When p e crosses the critical value /N , the collapse fixed point N A = N of the approximate dynamics becomesunstable, so that collapse in the can only occur due to a sequence of large shocks which becomes increasingly unlikely.The median survival time is then very large. For large p e , one can also see from Eqn. 5 at which levels of N A occurmost likely over the course of the simulation. Since the F ( . . . ) term is between 0 and 1, dN A /dt = 0 implies N/ ≤ N A ≤ N/ / p . In other words, when either p e or φ are large enough, one can expect that after sometransient phase, there will be only slightly more administrators than labourers on average for a long time, beforeeventually a large shock eventually causes the system collapse after all. We have developed an agent-based network model to illustrate emergent dynamics from the theory of societal collapse as postulated in Tainter (1988). In our model, problem solving increases complexity and ultimately leads to collapseof the system by diminishing marginal returns. Problems were abstractly represented by random shocks in energyproduction, which were counteracted by adding complexity to the networked social system in form of an increasingadministration. We were able to derive a well-performing macroscopic approximation of the proposed model, thusproviding a very simple mathematical description of the theory in form of an ordinary differential equation. Additionally,we carried out an analysis of the effect of random status exploration where agents can change their status from aproductive to an administrative state (and vice versa). We found that a small exploration rate of at least p e = 1 /N is already sufficient to counterbalance the progression towards a network consisting only of administrators under theinfluence of shocks and reach a stable state where survival times are very long. However, smaller exploration rates haveadverse effects on survival time, as they only facilitate the spread of administration over the network.9 PREPRINT - F
EBRUARY
16, 2021In addition, we find that a minimal link density (cid:37) in the network is essential for long survival times, but has no furthereffect beyond a threshold value. Also, large output elasticity of labour φ , i.e., the effectiveness of administratorsto increase productivity of labour, can increase the survival time. When output elasticity of labour is large enough,networks may survive for very long until a large enough shock destroys them, regardless of the exploration rate. Finally,we note that with increasing time t and correspondingly increasing share of administration in the network, the potentialto change and react to external influences was reduced because marginal returns on complexity diminished. By a similarargument Allison and Hobbs (2004) show how, for instance, the Western Australian Agricultural Region could end upin a ‘Lock-in Trap’, which is characterized by low ability to adapt to changing external circumstances.In summary, within the scope of our model two main strategies allow mitigation of collapse: (i) an increase in the outputelasticity of labour or (ii) a random transition of people between productive and administrative roles at a sufficientlylarge rate.With regard to open research topics posed by Cumming and Peterson (2017), such as moving the debate from ‘whether’to ‘why’ collapse occurred and possible strategies to “avoid, slow or hasten collapse”, we wish to contribute our findingsto the discourse of the drivers of socio-ecological resilience and collapse. Growing complexity has recently beenidentified as a source of increased societal risk due to, e.g., high degrees of socio(-ecological) interconnections thatcan lead to cascading failures (Helbing, 2013; Rocha et al., 2018). In this regard complexity is considered mainlya facilitator of collapse by shaping systems to become more vulnerable. In our research we show that complexitymay be even more directly linked to collapse since societies may reach a point where associated costs outweigh thebenefits, thereby directly causing their breakdown. This dimension is, to our understanding, an underrepresented viewof socio-ecological collapse.Similar to Motesharrei et al. (2014) we find that an unequal society in which the elite consumes more (in their case) orproduces less (in our case) than the commoners is likely to run into collapse. In addition, we found that the absenceof an elite (i.e. egalitarian society), which is comparable to the random state exploration mechanism in our model,produced more pathways to avoid collapse than a hierarchical (complex) society. In comparison, previously reportedsimulations of competing pre-modern societies showed that collapse fades once societies become complex and candevelop intensive agriculture at sufficiently large scale (Guzmán et al., 2018). This finding contrasts our result fortwo specific reasons: First, in Guzmán et al. (2018) complexity is a binary state and cannot increase further oncea society is denoted as being complex . Second and more importantly, Guzmán et al. (2018) model a warrior classcomparable to administrators in our model, which is directly coupled to available food (energy). Hence complexityin terms of hierarchies cannot exceed the level of existing energy and thereby grow to an unsustainable size. A moredetailed investigation into the coupling of complexity and economic development in future work would certainly helpin understanding the importance of complexity in collapse.Butzer (2012) points out that collapse itself should be modelled by sophisticated socio-ecological models, which donot draw on simplifying assumptions. Nevertheless, Butzer concludes that preconditions for breakdown are typicallyeconomic or fiscal decline. We agree and emphasize that the our model of the theory of collapse of complex societies offers a potential path leading to exactly such preconditions.Like in Watson and Lovelock (1983) or Granovetter (1978), the objective of our research was not to predict anyguidelines for resilient societies, but to illustrate the mechanism of collapse due to diminishing marginal returns ofan increasingly complex society. We acknowledge that our approach to modelling collapse is highly conceptual,nevertheless our results underline the relevance of understanding the emergence and consequences of complexity in theresilience discourse.In contrast to more economically spirited models, we do not include in our simple model any form of competition,economic externality, or resource depletion, which in the real world can be alternative or additional drivers of collapse.An additional important limitation of our model is that we incorporated no mechanism for reduction in bureaucracybeyond random state exploration in contrast to the automated increase in complexity due to shocks. However, one majorargument in Tainter (1988) is that endogenous reduction of complexity as problem solution strategy is not observedin historical cases, at least it is not a dominant process. Instead we assume that the model’s administration possessescharacteristics of persistent power structures (Acemoglu and Robinson, 2008), which come into existence much moreeasily than they are removed. Further we compare the administration to expanding elite bureaucracies according toParkinson (1955) and Weber (1922). As an extension, it would be interesting to consider more complex dynamics ofintra-generational and inter-generational mobility of social status and wealth and their respective change over time.Finally, we acknowledge that our proposed model only represents one feature of complexity, i.e., the emergence of oneadditional hierarchy layer, the administration. Of course, modelling efforts exist which consider much more intricatemechanisms, like human behaviour, shared resource systems, interacting societies, etc. (Arneth et al., 2014; Rounsevelland Arneth, 2011; Schill et al., 2019; Schlüter et al., 2017; Nitzbon et al., 2017; Motesharrei et al., 2014). But keeping10 PREPRINT - F
EBRUARY
16, 2021Occam’s razor in mind, we believe that this very simple model (without population increase, additional hierarchylayers, dynamic networks, etc.) is sufficient to make a case for the direct influence of complexity with respect tosocial-ecological collapse.Our model should be extended in further studies to analyze how complexity relates to collapse in potentially morerealistic scenarios that could, e.g., be represented by a further social stratification or and more complex networktopologies. In this context, complexity costs of social connectivity should be considered. Additionally, impacts ofinnovations and energy substitutes (coal, oil, etc.) on collapse could be studied to identify additional strategies to slowdown or avoid collapse. Furthermore, the interrelation between energy and complexity could also be studied in moreelaborate World-Earth and Human-Earth system models to study under which circumstances the results obtained in thiswork could be transferred to less abstract applications. For this purpose, appropriate measures of energy and complexityneed to be defined depending on the specific model and research question at hand.While increasing complexity has brought vast benefits to modern industrialized societies, our findings also raise thequestion to which degree resilient societies can increase complexity without creating large-scale risks of collapse due toan ever increasing demand for energy. Along those lines, it is of great interest to further investigate whether the rise ofcomplexity always comes with an increased risk or whether certain forms of complexity exist that do not necessarilyincrease the risk of societies for collapse. Such an assessment would allow to develop design principles for resilientinfrastructures and social structures that do not facilitate complexity in the understanding of Tainter (1988) therebyallowing modern societies to escape the energy-complexity spiral to path their the trajectory of the socio-ecologicalEarth System toward a sustainable trajectory.
Acknowledgements
The idea for this work was developed during a
Studienkolleg working group on
The Earth as a complex system fundedby the German National Academic Foundation (Studienstiftung des deutschen Volkes). The work was carried outin the scope of the COPAN collaboration at the Potsdam Institute for Climate Impact Research. M.W. J.H. andJ.F.D. were supported by the Leibniz Association (project DominoES) and the European Research Council (ERC)under the European Union’s Horizon 2020 Research and Innovation Programme (ERC grant agreement No. 743080ERA). F.S. was supported by the German National Academic Foundation. The authors gratefully acknowledge theEuropean Regional Development Fund (ERDF), the German Federal Ministry of Education and Research and theLand Brandenburg for providing resources on the high-performance computer system at PIK. For fruitful discussionswe sincerely thank Jürgen Kurths, Lea Tamberg, Adrian Lison, Sascha Haupt, Vivian Grudde, Bastian Ott and othermembers of the Studienkolleg’s working group 4. We thank Wolfgang Lucht for inspiring discussions that triggered ourinterest in Tainter’s theory.
Appendix A: Optimal number of administrators
From the analytical approximation, we know that the expected energy input E is proportional to E := ( N − N A )(1 − (cid:37) ) N A + [( N − N A )(1 − (1 − (cid:37) ) N A )] φ , (5)which has a maximum in N A where ∂ N A E = − [1 − ( N − N A ) ln(1 − (cid:37) )](1 − (cid:37) ) N A (6) + φ [( N − N A )(1 − (1 − (cid:37) ) N A )] φ − [(1 − (cid:37) ) N A [1 − ( N − N A ) ln(1 − (cid:37) )] − . (7)E.g., for N = 400 , (cid:37) = 0 . , and φ = 1 . , the optimal number of administrators is N A ≈ or ≈ , leading to an E ≈ or E/N ≈ . . Model Documentation
References
Acemoglu, D. and J. A. Robinson2008. Persistence of Power, Elites, and Institutions.
American Economic Review , 98(1):267–293.11
PREPRINT - F
EBRUARY
16, 2021Adermon, A., M. Lindahl, and D. Waldenström2018. Intergenerational Wealth Mobility and the Role of Inheritance: Evidence from Multiple Generations.
TheEconomic Journal , 128(612):F482–F513.Allison, H. E. and R. J. Hobbs2004. Resilience, Adaptive Capacity, and the “Lock-in Trap” of the Western Australian Agricultural Region.
Ecologyand Society , 9(1).Arneth, A., C. Brown, and M. D. A. Rounsevell2014. Global models of human decision-making for land-based mitigation and adaptation assessment.
NatureClimate Change , 4(7):550–557.Auer, S., J. Heitzig, U. Kornek, E. Schöll, and J. Kurths2015. The Dynamics of Coalition Formation on Complex Networks.
Scientific Reports , 5:13386.Baležentis, T., T. Li, and A. Baležentis2015. The trends in efficiency of lithuanian dairy farms: a semiparametric approach.
Management Theory andStudies for Rural Business and Infrastructure Development , 37(2):167–178.Barfuss, W., J. F. Donges, M. Wiedermann, and W. Lucht2017. Sustainable use of renewable resources in a stylized social–ecological network model under heterogeneousresource distribution.
Earth System Dynamics , 8(2):255–264.Barton, C. M. and J. Riel-Salvatore2012. Agents of Change: Modelling Biocultural Evolution in Upper Pleistocene Western Eurasia.
Advances inComplex Systems , 15(01n02):1150003.Barton, C. M., I. I. Ullah, and S. Bergin2010. Land use, water and Mediterranean landscapes: modelling long-term dynamics of complex socio-ecologicalsystems.
Philosophical transactions. Series A, Mathematical, physical, and engineering sciences , 368(1931):5275–5297.Brander, J. A. and S. M. Taylor1998. The Simple Economics of Easter Island: A Ricardo-Malthus Model of Renewable Resource Use.
The AmericanEconomic Review , 88(1).Brown, C., B. Seo, and M. Rounsevell2019. Societal breakdown as an emergent property of large-scale behavioural models of land use change.
EarthSystem Dynamics , 10(4):809–845.Butzer, K. W.2012. Collapse, environment, and society.
Proceedings of the National Academy of Sciences of the United States ofAmerica , 109(10):3632–3639.Castellano, C., S. Fortunato, and V. Loreto2009. Statistical physics of social dynamics.
Reviews of modern physics , 81(2):591.Cumming, G. S. and G. D. Peterson2017. Unifying Research on Social-Ecological Resilience and Collapse.
Trends in ecology & evolution , 32(9):695–713.Diamond, J. M.2011.
Collapse: How societies choose to fail or survive / Jared Diamond . New York: Penguin.Diewert, W. E., T. Nakajima, A. Nakamura, E. Nakamura, and M. Nakamura2011. Returns to scale: concept, estimation and analysis of japan’s turbulent 1964–88 economy.
Canadian Journalof Economics/Revue canadienne d’économique , 44(2):451–485.Donges, J. F., J. Heitzig, W. Barfuss, M. Wiedermann, J. A. Kassel, T. Kittel, J. J. Kolb, T. Kolster, F. Müller-Hansen,I. M. Otto, K. B. Zimmerer, and W. Lucht2020. Earth system modeling with endogenous and dynamic human societies: the copan:CORE open World–Earthmodeling framework.
Earth System Dynamics , 11(2):395–413.Erd˝os, P. and A. Rényi1960. On the Evolution of Random Graphs. In
Publication of the Mathematical Institute of the Hungarian Academyof Sciences , Pp. 17–61.Feinstein, C.1999. Structural change in the developed countries during the twentieth century.
Oxford Review of Economic Policy ,15(4):35–55. 12
PREPRINT - F
EBRUARY
16, 2021Freeman, J. and J. M. Anderies2012. Intensification, tipping points, and social change in a coupled forager-resource system.
Human nature(Hawthorne, N.Y.) , 23(4):419–446.Granovetter, M.1978. Threshold Models of Collective Behavior.
American Journal of Sociology , 83(6):1420–1443.Guzmán, R. A., S. Drobny, and C. Rodríguez-Sickert2018. The Ecosystems of Simple and Complex Societies: Social and Geographical Dynamics.
Journal of ArtificialSocieties and Social Simulation , 21(4).Hall, C., S. Balogh, and D. Murphy2009. What is the Minimum EROI that a Sustainable Society Must Have?
Energies , 2(1):25–47.Heckbert, S.2013. MayaSim: An Agent-Based Model of the Ancient Maya Social-Ecological System.
Journal of ArtificialSocieties and Social Simulation , 16(4):11.Helbing, D.2013. Globally networked risks and how to respond.
Nature , 497(7447):51–59.Holstein, T., M. Wiedermann, and J. Kurths2020. Optimization of coupling and global collapse in diffusively coupled socio-ecological resource exploitationnetworks.Kohler, T. A. and S. E. van der Leeuw2007.
The model-based archaeology of socionatural systems , 1st ed. edition. Santa Fe N.M.: School for AdvancedResearch Press.Malthus, T. R.1798.
An essay on the principle of population, as it affects the future improvement of society. With remarks on thespeculations of Mr. Godwin, M. Condorcet and other writers . London: J. Johnson.Meadows, D. H., D. L. Meadows, R. Jorgen, and W. W. Behrens1972.
The limits to growth: A report for the Club of Rome’s project on the predicament of mankind . New York:Universe Books.Middleton, G. D.2012. Nothing Lasts Forever: Environmental Discourses on the Collapse of Past Societies.
Journal of ArchaeologicalResearch , 20(3):257–307.Motesharrei, S., J. Rivas, and E. Kalnay2014. Human and nature dynamics (HANDY): Modeling inequality and use of resources in the collapse orsustainability of societies.
Ecological Economics , 101:90–102.Newman, M.2003. The Structure and Function of Complex Networks.
SIAM Review , 45(2):167–256.Nitzbon, J., J. Heitzig, and U. Parlitz2017. Sustainability, collapse and oscillations in a simple World-Earth model.
Environmental Research Letters ,12(7):074020.Ostrom, E.2009. A general framework for analyzing sustainability of social-ecological systems.
Science (New York, N.Y.) ,325(5939):419–422.Parkinson, C. N.1955. Parkinson’s Law.
The Economist .Rocha, J. C., G. Peterson, Ö. Bodin, and S. Levin2018. Cascading regime shifts within and across scales.
Science (New York, N.Y.) , 362(6421):1379–1383.Rockström, J., W. Steffen, K. Noone, A. Persson, F. S. Chapin, E. F. Lambin, T. M. Lenton, M. Scheffer, C. Folke, H. J.Schellnhuber, B. Nykvist, C. A. de Wit, T. Hughes, S. van der Leeuw, H. Rodhe, S. Sörlin, P. K. Snyder, R. Costanza,U. Svedin, M. Falkenmark, L. Karlberg, R. W. Corell, V. J. Fabry, J. Hansen, B. Walker, D. Liverman, K. Richardson,P. Crutzen, and J. A. Foley2009. A safe operating space for humanity.
Nature , 461(7263):472–475.Rounsevell, M. D. and A. Arneth2011. Representing human behaviour and decisional processes in land system models as an integral component ofthe earth system.
Global Environmental Change , 21(3):840–843.13
PREPRINT - F
EBRUARY
16, 2021Schill, C., J. M. Anderies, T. Lindahl, C. Folke, S. Polasky, J. C. Cárdenas, A.-S. Crépin, M. A. Janssen, J. Norberg, andM. Schlüter2019. A more dynamic understanding of human behaviour for the Anthropocene.
Nature Sustainability , 2(12):1075–1082.Schleussner, C.-F., J. F. Donges, D. A. Engemann, and A. Levermann2016. Clustered marginalization of minorities during social transitions induced by co-evolution of behaviour andnetwork structure.
Scientific Reports , 6:30790.Schlueter, M., R. R. Mcallister, R. Arlinghaus, N. Bunnefeld, K. Eisenack, F. Hoelker, E. J. MILNER-GULLAND,B. Müller, E. Nicholson, M. Quaas, et al.2012. New horizons for managing the environment: A review of coupled social-ecological systems modeling.
Natural Resource Modeling , 25(1):219–272.Schlüter, M., A. Baeza, G. Dressler, K. Frank, J. Groeneveld, W. Jager, M. A. Janssen, R. R. McAllister, B. Müller,K. Orach, N. Schwarz, and N. Wijermans2017. A framework for mapping and comparing behavioural theories in models of social-ecological systems.
Ecological Economics , 131:21–35.Stawarz, N.2018. Patterns of intragenerational social mobility: An analysis of heterogeneity of occupational careers.
Advancesin Life Course Research , 38:1–11.Steensma, D. P. and H. M. Kantarjian2014. Impact of cancer research bureaucracy on innovation, costs, and patient care.
Journal of clinical oncology :official journal of the American Society of Clinical Oncology , 32(5):376–378.Tainter, J. A.1988.
The collapse of complex societies , New studies in archaeology. Cambridge: Cambridge University Press.Tainter, J. A.2006. Social complexity and sustainability.
Ecological Complexity , 3(2):91–103.Tol, R. S. J.2018. The Economic Impacts of Climate Change.
Review of Environmental Economics and Policy , 12(1):4–25.Trump, B. D., I. Linkov, and W. Hynes2020. Combine resilience and efficiency in post-COVID societies.
Nature , 588(7837):220.Watson, A. J. and J. E. Lovelock1983. Biological homeostasis of the global environment: the parable of Daisyworld.
Tellus B: Chemical and PhysicalMeteorology , 35(4):284–289.Weber, M.1922.
Wirtschaft und Gesellschaft . Tübingen: Mohr.Weiss, H., M. A. Courty, W. Wetterstrom, F. Guichard, L. Senior, R. Meadow, and A. Curnow1993. The genesis and collapse of third millennium north mesopotamian civilization.
Science (New York, N.Y.) ,261(5124):995–1004.Wiedermann, M., J. F. Donges, J. Heitzig, W. Lucht, and J. Kurths2015. Macroscopic description of complex adaptive networks coevolving with dynamic node states.
Physical ReviewE , 91(5):052801.Wiedermann, M., E. K. Smith, J. Heitzig, and J. F. Donges2020. A network-based microfoundation of Granovetter’s threshold model for social tipping.