Adversarial decision strategies in multiple network phased oscillators: the Blue-Green-Red Kuramoto-Sakaguchi model
Mathew Zuparic, Maia Angelova, Ye Zhu, Alexander Kalloniatis
AAdversarial decision strategies in multiple network phasedoscillators: the Blue-Green-Red Kuramoto-Sakaguchi model
Mathew Zuparic a , Maia Angelova b , Ye Zhu b , Alexander Kalloniatis a a Defence Science and Technology Group, Canberra, ACT 2600, Australia b School of Information Technology, Deakin University, Burwood, VIC 3125, Australia
Abstract
We consider a model of three interacting sets of decision-making agents, labeled Blue,Green and Red, represented as coupled phased oscillators subject to frustrated synchro-nisation dynamics. The agents are coupled on three networks of differing topologies,with interactions modulated by different cross-population frustrations, internal and cross-network couplings. The intent of the dynamic model is to examine the degree to whichtwo of the groups of decision-makers, Blue and Red, are able to realise a strategy of beingahead of each others’ decision-making cycle while internally seeking synchronisation ofthis process – all in the context of further interactions with the third population, Green.To enable this analysis, we perform a significant dimensional reduction approximationand stability analysis. We compare this to a numerical solution for a range of inter-nal and cross-network coupling parameters to investigate various synchronisation regimesand critical thresholds. The comparison reveals good agreement for appropriate parame-ter ranges. Performing parameter sweeps, we reveal that Blue’s pursuit of a strategy ofstaying too-far ahead of Red’s decision cycles triggers a second-order effect of the Greenpopulation being ahead of Blue’s cycles. This behaviour has implications for the dynamicsof multiple interacting social groups with both cooperative and competitive processes.
Keywords:
Kuramoto-Sakaguchi, phased oscillators, frustration, multi-network
1. Introduction
The spontaneous appearance of patterned behaviour in locally coupled dynamical sys-tems is immensely relevant to social, biological, chemical and physical systems. Notableearly examples of simple models that demonstrated such behaviours include Schelling’s [1]segregation models which display the emergence of communities based on the correlatedchoices and practices of individual decision-makers. Additionally, Watson and Lovelock’s
Daisyworld model [2] demonstrates the growth and decline of different coloured flowerswith different albedo levels that both compete for space and coordinate to stabilise global
Email addresses: [email protected] (Mathew Zuparic), [email protected] (Maia Angelova), [email protected] (Ye Zhu), [email protected] (Alexander Kalloniatis)
Preprint submitted to Commun Nonlinear Sci Numer Simulat November 20, 2020 a r X i v : . [ n li n . AO ] N ov emperatures simply by the flower’s response to variable radiation levels received from thesun. Social versions of such patterned behaviour often involve mixtures of competitiveand cooperative dynamics. Examples include the work of Abrams, Yaple and Wienerconcerning religious affiliation [3]; social opinion dynamics using the Axelrod culturalmodel by Gonz´alez-Avella et al. [4]; and examination of shifts in societal morals usingnetworked Monte Carlo simulations and mean field theory by Vicente et al. [5]. Commonacross all of these systems is the ability for seemingly unintelligent actors as representedin components of dynamical or statistical physics models to display complex patterns andbehaviours within mathematical representations of aligned or mis-aligned ‘intentions’ or‘strategies’. For a contemporary review of this topic of growing attention in the scien-tific community refer to Strogatz [6] and Chapter 2 of Ilachinski [7]. In this paper, weextend the approach of network synchronisation to modelling such complex systems witha dichotomy of cooperative and competitive processes across three sets of actors.Using the Kuramoto model [8] as the starting point of this work, we focus on the onsetof synchronisation amongst agent populations across multiple networks, where the agentsexist in cooperative and adversarial relationships according to the degree of ‘frustration’ inthe interaction. The term frustration in this work is not used in an emotive sense, rather itsapplication is similar to the term’s use when applied to condensed matter systems, whereatoms find themselves in non-trivial arrangements due to conflicting inter-atomic forces,usually referred to as geometrical frustration. Since its original inception, the Kuramotomodel has provided a paradigmatic mathematical modelling environment to explore theonset of global critical phenomena; for recent reviews refer to [9, 10, 11, 12, 13]. Therole of frustration occurs in the Kuramoto-Sakaguchi model [14, 15, 16, 17, 18], where theintroduction of phase shifts in the interaction terms changes the potential steady-statebehaviour from phase synchronisation (all phases equal) to frequency synchronisation(phases shifted by a constant amount in relation to each other) between selected oscil-lators. As this work is concerned with multiple populations, we focus on the multiplenetwork formulation of the model [19, 20, 21, 22, 23] where each sub-network has poten-tially different characteristics, such as graph topologies or natural frequency distributions.Notable examples of Kuramoto-based applications to social-organisational systems can befound in; the conformists-and-contrarians model [24, 25]; the opinion-changing-rate model[26]; network community detection using the Fourier-Kuramoto model [27]; and the mea-surement and dynamic modelling of decision cycles in military headquarters [28, 29].In this work, we extend the two-network Blue-Red model of [30] to the three-network
Blue-Green-Red (BGR) model. The model’s novelty comes from the introduction of theGreen network, which is not on equal footing with Blue or Red; we impose that Greendoes not ‘seek to be ahead of decisions’ of either Blue or Red networks through a prede-fined strategy which we characterise with the frustration parameter. This is in contrastto the Blue-Red interaction, as previously modelled in [30]. Nevertheless, as shall beshown in the following sections, Green still may stay ahead in phase as a consequenceof the nonlinear dynamics, but the mechanism for such a strategy comes from differentsources. These mechanisms include other networks pursuing a certain strategy, and/orthe structural choices Green makes with the way it interacts with Blue and Red. In each2f these networks, we distinguish ‘strategic’ (or leadership) and ‘tactical’ nodes. We alsointroduce an asymmetry into the model by imposing that the Blue and Green networksinteract entirely through their strategic nodes, whereas Red and Green interact via theirmore numerous tactical nodes. This asymmetry allows analysis of the effect of exertinginfluence on senior decision-makers via the Blue-Green interaction, versus targeting themore numerous but less influential network members via the Green-Red interaction. Ahistorical example includes the events during and after the 2001 Afghanistan war, whereNATO/Coalition forces (Blue) were engaging in military action against Taliban insur-gents (Red) whilst concurrently seeking to train wider Afghan society (Green) for theireventual assumption of responsibility for the security of their nation [31]. Our interestin applying the Kuramoto model as a window into decision-making processes is largelydue to the cyclicity of the model’s dynamic variables. While oscillations are pervasivein many physical, chemical and biological systems [32], the human cognitive process alsodisplays a fundamental cyclicity. Relevant versions of this process include the PerceptionCycle model of Neisser [33], the Observe-Orient-Decide-Act (OODA) model of Boyd [34],and the Situation Awareness model of Endsley [35]. For the majority of the paper, weanalyse the model abstracted from the specific military application context, principallybecause the results have value for other applications of such a three-network model.A key result we find through both analytic and numerical examination is that thereare regions of behaviour where Blue enjoys the advantage over Red in being advance ofthe latter decision process. However, within this, there are opportunities where Greenmay be offered initiative by Blue, which resonates with aspects of Counter-Insurgencystrategy [31].In the next section, we detail relevant parameters (networks, coupling, frequencies,frustrations) of the BGR model, and highlight how the asymmetry of the interaction ofGreen with both Blue and Red networks is manifested mathematically. We also detail asignificant dimensional reduction technique which affords us semi-analytic insight into thedynamics. Section 3 provides the specific topologies of the networks, and input parameterchoices for a use-case which runs throughout the remainder of the paper. In Section 4we provide a detailed analysis of the BGR model through the lens of specific networktopologies and parameter choices. This includes comparing the semi-analytic outputs withthe full numerical model, revealing very good agreement between both approaches, givingus the confidence to perform an extensive and computationally inexpensive parametersweep of the model revealing areas of interest from each network’s point of view. In thefinal Section we re-interpret the model behaviours back in the context of the militaryapplication, and suggest future work.
2. The Blue-Green-Red model
The three-network BGR model is given by the following ordinary differential equationsfor each of the three sets of phases: Blue, Green and Red,3 B i = ω i − σ B (cid:88) j ∈B B ij sin ( B i − B j ) − ζ BG (cid:88) j ∈G I ( BG ) ij sin ( B i − G j − φ BG ) − ζ BR (cid:88) j ∈R I ( BR ) ij sin ( B i − R j − φ BR ) , i ∈ B , (1)˙ G i = µ i − σ G (cid:88) j ∈G G ij sin ( G i − G j ) − ζ GB (cid:88) j ∈B I ( GB ) ij sin ( G i − B j ) − ζ GR (cid:88) j ∈R I ( GR ) ij sin ( G i − R j ) , i ∈ G , (2)˙ R i = ν i − σ R (cid:88) j ∈R R ij sin ( R i − R j ) − ζ RB (cid:88) j ∈B I ( RB ) ij sin ( R i − B j − φ RB ) − ζ RG (cid:88) j ∈G I ( RG ) ij sin ( R i − G j − φ RG ) , i ∈ R , (3)where each network’s adjacency matrix is denoted by B , G and R . The dynamic variables B i , G j and R k are the Blue, Green and Red phases, or decision-states, for agents ateach network’s respective node i ∈ B , j ∈ G and k ∈ R . The variables ω i , µ j and ν k are the natural frequencies, or decision-speeds of the agents in isolation, with valuestypically drawn from a particular distribution. Furthermore, the parameters σ B , σ G and σ R (all positive real valued) are referred to as the intra-network couplings, or intensityof interaction between agents. For one-network systems, the global coupling parametercontrols the phase dynamics from a totally asynchronous regime to clustered limit cycles,and finally to phase locking behaviour [36, 37, 38, 39, 40].The inter-network adjacency matrices I ( MN ) for networks M and N specify the con-nections between the nodes of network M and N . Note that throughout this work weassume that I ( MN ) = (cid:0) I ( NM ) (cid:1) T , though this assumption can be relaxed to offer moremodel generality. Furthermore, the inter-network couplings are specified by the parame-ters ζ MN ∈ R + , for networks M and N . Lastly, the strategy chosen by agents of network M to collectively stay ahead of phase, or decision-state, of agents of network N is spec-ified by the frustration parameter φ MN ∈ S . We remark that the asymmetry betweenthe Green network and Blue and Red is made clear in Eq.(1-3) by the absence of φ GB and φ GR ; this means that Green agents do not explicitly pursue a strategy to stay aheadin the phase of agents of other networks. We summarise the variables which compriseEq.(1-3), and their interpretations, in Table 1.A diagram of this scenario, with strategic and tactical sub-structures is shown inFigure 1. Strategic nodes for each network contain the highest number of connectionsin their respective graph, generally reflecting the span of control of leaders in social andorganisational settings. The Blue and Red tactical networks interact with each other,attempting to stay ahead in the phase of their adversary’s tactical nodes. In the absenceof a Green network, the adversarial dynamics between Blue and Red networks has beenexplored in [30, 41, 42, 43]. 4 able 1: Summary of the variables used in Eq.(1-3), and their physical interpretations. expression name interpretation { B, G, R } phase dynamic agent decision state {B , G , R} adjacency matrix internal network topology { ω, µ, ν } natural frequency decision-speed of agent in isolation σ M coupling of network M intensity of agent interaction in MI ( MN ) inter-adjacency matrix topology between M and N ζ MN coupling between M and N intensity of M - N agent interaction φ MN frustration M ’s strategy against N Figure 1: Diagram of
Blue , Green and
Red , their ‘strategic’ and ‘tactical’ substructures, and how theyinteract with one another.
To measure the self-synchronisation within a given population, we use local orderparameters for { B, G, R } phases, labeled as { O B , O G , O R } , respectively. The computationof the order parameters is accomplished using local versions of Kuramoto’s original global order parameter [8]: O B = 1 |B| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈B e iB j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , O G = 1 |G| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈ G e iG j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , O R = 1 |R| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ∈R e iR j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)The absolute value |M| represents the number of nodes of the generic graph M . Valuesapproaching unity represents coherence of phases in the respective networks, namely ofsynchronised decision-making of the corresponding agents. By making the assumption that each of the three networks’ phases has approximatelysynchronised, we can perform a significant dimensional reduction to Eq.(1-3). This is5chieved by assuming the following form for each network’s phase B i = b i + E B , G j = g j + E G , R k = r k + E R , { i, j, k } ∈ {B , G , R} , (5)where { b i , g j , r k } are ‘small’ fluctuations, namely b i ≈ g j ≈ r k ≈
0. The variables E B , E G and E R are the mean values of the phases for the Blue, Green and Red networksrespectively, E B = 1 |B| (cid:88) i ∈B B i , E G = 1 |G| (cid:88) j ∈G G j , E R = 1 |R| (cid:88) k ∈R R k . (6) E B , E G and E R are alternatively referred to as the corresponding network’s centroid . Thedifference between each network’s centroid value is denoted by E B − E G ≡ α BG , E G − E R ≡ α GR , E B − E R ≡ α BR = α BG − α GR . (7)The approximations specified by Eq.(5) amount to a system of |B| + |G| + |R| definingequations, with |B| + |G| + |R| + 3 variables. However, since E B , E G and E R are the meanvalue of each network’s phases, then necessarily we obtain that (cid:80) i ∈B b i = (cid:80) j ∈G g j = (cid:80) k ∈R r k = 0, thus collapsing the system dimensionality appropriately.By inserting the approximation for the phases given by Eq.(5) into Eq.(1-3), andutilising properties of the eigenvalues and eigenvectors of the resulting graph-Laplacians(details shown in Appendix B) we obtain the following expressions for the dynamics ofthe centroids˙ E B = ¯ ω − ζ BG d ( BG ) T |B| sin ( α BG − φ BG ) − ζ BR d ( BR ) T |B| sin ( α BR − φ BR ) , ˙ E G = ¯ µ + ζ GB d ( GB ) T |G| sin α BG − ζ GR d ( GR ) T |G| sin α GR , ˙ E R = ¯ ν + ζ RB d ( RB ) T |R| sin( α BR + φ RB ) + ζ RG d ( RG ) T |R| sin( α GR + φ RG ) , (8)where we have applied the notation¯ ω ≡ |B| (cid:88) i ∈B ω i , ¯ µ ≡ |G| (cid:88) i ∈G µ i , ¯ ν ≡ |R| (cid:88) i ∈R ν i , (9)for the respective means of each network’s natural frequencies. Additionally, d ( MN ) T ≡ (cid:88) i ∈M (cid:88) k ∈N I ( MN ) ik , (10)is the total number of edges shared by networks M and N . Eq.(8) approximates thedynamics of the centroids of each of the three networks completely in terms of their6ifferences. Taking the appropriate difference of each of the expressions in Eq.(8) wecollapse the dynamics of the centroids into the following two-dimensional system:˙ α BG = ¯ ω − ¯ µ − ψ BG sin( α BG − φ BG ) − ψ GB sin α BG − ψ BR sin( α BG + α GR − φ BR ) + ψ GR sin α GR , ˙ α GR = ¯ µ − ¯ ν + ψ GB sin α BG − ψ RB sin( α BG + α GR + φ RB ) − ψ GR sin α GR − ψ RG sin( α GR + φ RG ) , (11)where we have applied the notation, d ( MN ) T ζ MN |M| = ψ MN for networks M and N . (12)In Table 2 we offer a summary of the various measures which are applied in this work toanalyse and understand model outputs. Table 2: Summary of the measures used to analyse model outputs. measure name range { O B , O G , O R } local order parameter (0 , { E B , E G , E R } centroids/mean value of phases S { α BG , α GR , α BR } centroid differences S
3. Use-case
For numerical exploration of the BGR model, we construct graphs of size |B| = |G| = |R| = 21, given explicitly in Figure 2. This extends the example followed in previous Blue-vs-Red studies in [30, 42, 43]. As shown on the left side of Figure 2, the Blue populationforms a hierarchy stemming from a single root, followed by a series of four branches twolayers deep. The right side of Figure 2 shows the network for the Red population, givenby a random Erd˝os-R´enyi graph, generated by placing a link between nodes with 0.4probability. Finally, the network for the Green population, presented in the middle ofFigure 2, is given by a small-world
Watts-Strogatz graph [44] with rewiring probability0.3. These are all simplified caricatures of, respectively, military, terrorist and societalstructures for the purpose of illustrating the behaviours of the model.Focusing on the Blue network on the left of Figure 2, the particular colour, shape andnumbering of each node determines its connection to other graphs. Specifically, the nodesnumbered 1–5 are coloured green, and hence each share an edge with the correspondingnodes on the network for Green which share the same number (1–5) and shape. Thus, thetotal number of connections between the Blue and Green networks is 5. Similarly, the redcoloured triangle nodes, labeled 6–21, on both the Blue and Green networks are connected7
234 5 6 789 1011 12 131415161718 1920 21 123456 7 8 9 10 11 12 13 14 15161718192021 123456 7 8 9 10 11 12 13 14 15161718192021
Figure 2: Blue (hierarchy), Green (Watts-Strogatz) and Red (Erd˝os-R´enyi) networks used in the numer-ical simulation of Eqs.(1–3). Nodes with the same label and shape share edges with respective networks.For instance — nodes 1–5 presented as upside-down triangles on the Blue network (coloured green) arelinked with the correspondingly labeled nodes on the Green network, presented as blue upside-down tri-angles. Similarly — nodes 6–21 presented as triangles on the Blue and Green networks (coloured red)are linked with the correspondingly labeled nodes on the Red network, presented as blue/green triangles.Nodes 1–5 on the Red network are the only nodes not externally connected with other networks. to the corresponding shaped and labeled nodes on the Red network, themselves colouredblue and green. Consequently, the total number of edges shared between the Blue-Red andGreen-Red networks is 16. As indicated in Figure 1, the strategic nodes of Red, labeled1–5 and portrayed as red squares, share no edges with either Blue or Green networks.In the left panel of 3 we present the eigenspectrum of the graph-Laplacians [45],defined in Eq.(B.2), for the Blue, Green and Red networks, coloured accordingly. A keyobservation of the graph spectrum lies in the relatively lower eigenvalues of the Bluegraph, which is a direct consequence of the poor connectivity afforded by a hierarchy(total number of edges equal to 20). Contrastingly, we see that the Green and Rednetworks possess very similar Laplacian eigenvalues, much higher than Blue, reflectingtheir relatively high connectivity, with a total number of edges of 84 and 77 for Greenand Red respectively.The right panel of Figure 3 gives the values of the natural frequencies used for eachnetwork’s node. The frequency values for the Blue and Red networks were drawn from auniform distribution between zero and unity, and for the Blue vs Red model [30, 42] thedifference between the means of their respective frequencies, ¯ ω − ¯ ν , plays a critical rolein the dynamics of the oscillators. Finally, for Green, the combination of the small-worldtopology, and the replicated natural frequencies for all the nodes, µ i = 0 . ∀ i ∈ G , ischosen to emulate the Green network as a tight-knit community [44]. Numerous workshave shown that a well-connected network, with similar natural frequency values acrossthe nodes, will have very good synchronisation properties. Thus, by placing Green in themiddle of the adversarial relationship between Blue and Red, our intent is to examine theeffect a tight-knit easily-synchronisable network has on the particular strategies chosen8
10 15 20 ρ λ ρ ( B , G , R ) { ω i , μ i , ν i } Figure 3: Left panel: The eigenvalue spectrum of the graph Laplacians (defined in Eq.(B.2)) of the Blue,Green and Red networks. Right panel: The natural frequency values of the oscillators for each node onthe Blue, Green and Red networks, with the mean-average of each network being ¯ ω = 0 . µ = 0 . ν = 0 .
551 respectively. by the remaining adversarial networks.
In order to make a meaningful comparison with previously published results [30, 42],we apply the following intra-network coupling values: σ B = 8 , σ G = 0 . , σ R = 0 . , (13)which are sufficient to enable the networks to internally synchronise without inter-networkcoupling. Observe here that the high coupling for Blue compensates for the relatively poorconnectivity of the hierarchy; this reflects the real-world phenomenon that hierarchicalorganisations rely quite heavily on tight discipline and training. Contrastingly, the lowercoupling of both Red and Green reflects the less disciplined responsiveness between mem-bers of ad hoc organisations; but their lower coupling is compensated by higher, if unevenconnectivity. Additionally, we choose the inter-network coupling values: ζ BR = ζ RB = 0 . , ζ BG = ζ GB = ζ GR = ζ RG ≡ ζ ∈ R , (14)The main reason for these choices is that they are sufficiently high that synchronisationis achievable, but also interesting deviations, or disruptions to synchronisation may bedetected and examined. Furthermore, we choose the following values for the strategies ofthe adversarial networks: φ BG = φ RB = φ RG = 0 , φ BR ∈ S . (15)To compare the outputs of Eq.(11) with those of the full system given in Eq.(1-3) we set, d ( BG ) T = d ( GB ) T = 5 , d ( BR ) T = d ( RB ) T = d ( GR ) T = d ( RG ) T = 16 , (16)9hich reflects the use-case topology explained in Figure 2. Thus the variables ψ become ψ BG = ψ GB ≡ ζ, ψ BR = ψ RB = ψ RG = ψ GR ≡ ζ, ζ ∈ R (17)which allows us to understand model behaviour as we vary two key parameters: Blue’sfrustration with respect to Red φ BR , and the inter-network coupling ζ . Thus Eq.(11)becomes,˙ α BG = 0 . − ζ sin α BG + 1621 ζ [sin α GR − sin( α BG + α GR − φ BR )] , ˙ α GR = − .
051 + 521 ζ sin α BG − ζ [2 sin α GR + sin( α BG + α GR )] , (18)which is easily solved numerically.
4. Model analysis
Code was developed in
Matlab ® − π/ , π/ ζ BG = ζ GR = 0 and ζ BR = ζ RB = 0 .
4, whilst varying φ BR . Notably, dynamic behaviour (limitcycles) from steady-state was detected for φ BR > . π for these particular parametervalues.For the full system of Eq.(1-3), outputs of Eq.(4) while varying ζ ∈ (0 ,
1] revealedthat each of the networks had highly synchronised phase dynamics ( O ≥ .
95) overthis range. Although local phase synchronisation for each network is high, the centroidsdisplay dynamic limit-cycle behaviour for ζ ≤ .
1, and steady-state behaviour for ζ ≥ . . < ζ < . φ BR . Indeed, the behaviour amongst the centroids undergoes multipletransitions as the frustration parameter φ BR varies. In order to explore the observation ofmultiple behavioural changes of the system, we use the approximation given by Eq.(18)as the local synchronisation of each network is sufficiently high for the assumption givenby Eq.(5) to hold. Example outputs of Eq.(4) for the full system are given in AppendixC. Figure 4 offers plots of the difference of the centroids given in Eq.(18) for ζ = 0 .
2, wherefrustration values are increased from left to right panels. The left-most panel ( φ BR = 0 . π )shows the three centroids in a steady-state (frequency-synchronised) arrangement witheach other. Increasing φ BR to 0 . π in the middle panel, shows the system displaying limitcycle behaviour, with Green oscillating dynamically with respect to Blue and Red, whothemselves have frequency synchronised with each other. Increasing φ BR to 0 . π in theright-most panel, the system returns to a steady-state regime. The three different modesof behaviour displayed while varying the frustration parameter suggest at least two valuesof φ BR (for this particular value of ζ ) which generate a regime change. We expose themechanism of this regime change by careful examination of the steady-state solution(s)offered in Eq.(18). 10 - α
50 100 150 200t - - α
50 100 150 200t - - α α BG α GR α BR Figure 4: Example of Eq.(18), giving the difference of the centroids of the Blue, Green and Red networksfor inter-network coupling value ζ = 0 .
2. Frustration parameter values from left-most to right-mostcolumns are given by φ BR = { . π, . π, . π } . Figure 5 offers a comparison between the two methods of solution, semi-analytic andfully numerical. The black line on the top row gives the steady-state position of α at t = 2000 of the semi-analytic approach of Eq.(18) for ζ = 0 .
2, whilst varying φ BR as acontinuous variable. Overlaid on these results appearing as purple points are the corre-sponding outputs from the fully numerical system. In order to account for any degeneracyintroduced by the BGR model’s trigonometric functions, the semi-analytic and fully nu-merical outputs are both projected onto S ( − π, π ] via,2 arctan (cid:20) tan α BG ( ζ, φ BR )2 (cid:21) → α BG ( ζ, φ BR ) , (19)and similarly for α GR and α BR . The bottom row of Figure 5 presents the logarithmic plotof the modulus of the difference between the semi-analytic and fully numerical results forthe difference of the centroids of the Blue, Green and Red networks, labeled as ∆( α ).Focusing on the top row, the left-most panel of Figure 5 for α BG , displays an almostlinear increase in the angle between the Blue and Green centroids as φ BR increases inthe range (0 , . π ). The system then enters a dynamic state for the parameter values φ BR ∈ (0 . π, . π ), represented in Figure 5 as gaps where no steady-state solutioncan be found. For the interval φ BR ∈ (0 . π, π ), Eq.(18) again enters a steady-stateregime with α BR being negative in S for this range of φ BR . Focusing on the bottom-leftpanel, the fully numerical results agree with the semi-analytic results when calculating α BG , with the largest divergence appearing immediately after the steady-state has beenreestablished. The corresponding steady-state behaviour of α GR and α BR in the middleand right panels of Figure 5 similarly agrees with the semi-analytic computations. Figure 6 offers the fixed points of α BG which arise as roots of the following system, − . ζ = −
10 sin α BG + 16 [sin α GR − sin( α BG + α GR − φ BR )] , . ζ = 5 sin α BG −
16 [2 sin α GR + sin( α BG + α GR )] , (20)11 π π π π ϕ BR - - - α BG π π π π π ϕ BR - - - - - - α GR π π π π π ϕ BR α BR π π π π π ϕ BR - - - - Δ ( α BG ) π π π π π ϕ BR - - - - Δ ( α GR ) π π π π π ϕ BR - - - - Δ ( α BR ) Figure 5: Top row: plots showing the steady-state values at t = 2000 of the difference of the centroidsfor the three networks for ζ = 0 . φ BR ∈ (0 , π ). Black lines show the semi-analytic solutionresulting from Eq.(18), and purple points give the equivalent fully numerical outcome. Note that thereis no steady-state solution between φ BR ∈ (0 . π, . π ) as the system in genuinely dynamic in thatregion. All solutions have been projected in the range ( − π, π ] ∈ S . Bottom row: logarithmic plots ofthe modulus of the difference between the semi-analytic and fully numerical outputs for the difference ofthe centroids of the networks — labeled as ∆( α ). for ζ = 0 .
2, and varying φ BR continuously. Eq.(20) is obtained by inserting ˙ α BG = ˙ α GR =0 in Eq.(18). Furthermore, we project solutions for each of the roots onto S via Eq.(19).Figure 6 presents four of the six roots for α BG that stem from Eq.(20), containing both areal (solid curve) and imaginary (dashed curve) component for each root. The remainingtwo roots of α BG , and the six roots of α GR (not shown) display qualitatively similarbehaviour. The root values which coincide with the steady-state behaviour of Eq.(18),given in the top-left panel of Figure 5, are presented by the red line sections in the top-left,top-right and bottom-left panels of Figures 6.The reason for the solution jumping from one root to another is not immediatelycomprehensible from these plots. To this end, we perform stability analysis by substituting α BG = α ∗ BG + δ and α GR = α ∗ GR + δ into Eq.(18), where the constant terms α ∗ BG and α ∗ GR are the roots of the system (shown in Figure 6 for α BG ). We also assume that thetime-dependent perturbations δ and δ are small, i.e. δ ≈ δ δ ≈ δ ≈
0. Thus, Eq.(18)becomes,˙ δ = ¯ ω − ¯ µ − ψ BG sin α ∗ BG + ψ GR sin α ∗ GR − ψ BR sin ( α ∗ BR − φ BR ) + β δ + β δ , ˙ δ = ¯ µ − ¯ ν + ψ BG sin α ∗ BG − ψ GR sin α ∗ GR − ψ BR sin α ∗ BR + β δ + β δ , (21)12 π π π π ϕ BR - - - α BG - ϕ BR π π π π π ϕ BR - - - α BG π π π π π ϕ BR - - - α BG π π π π π ϕ BR - - - α BG Figure 6: Plots showing four of the six roots of α BG in the system given in Eq.(20) for ζ = 0 . φ BR ∈ [0 , π ]. Solid and dashed curves denote the real and imaginary values of each of the roots,respectively. Note that the real sections given in red in the top row correspond to the steady-statesolution given in the top-left panel in Figure 5, with the inset in the top-left panel providing a zoomed-inperspective of the highlighted steady-state solution. π π π π ϕ BR λ + π π π π π ϕ BR - - - λ - Root 1Root 2Root 3
Figure 7: Plots of the real components of the Lyapunov exponents detailed in the linearised system givenin Eq.(23) for ζ = 0 . φ BR ∈ [0 , π ]. Each of the three lines correspond to the three differentroots which have valid sections for different values of φ BR . Similar to Figure 6, the red sections of eachof the three curves detail the exponents which are valid across the φ BR region. where β = − (cid:2) ψ BG cos α ∗ BG + ψ BR cos ( α ∗ BR − φ BR ) (cid:3) ,β = ψ GR cos α ∗ GR − ψ BR cos ( α ∗ GR − φ BR ) ,β = ψ BG cos α ∗ BG − ψ BR cos α ∗ GR ,β = − (cid:0) ψ GR cos α ∗ GR + ψ BR cos α ∗ BR (cid:1) . (22)Hence the Lyapunov exponents of the linearised system are λ ± = β + β ± (cid:115) ( β − β ) β β . (23)The Lyapunov exponents which corresponds to each of the valid roots as a function of φ BR are given in Figure 7. Valid root values corresponding to the correct steady-statesolution must satisfy: • zero imaginary component of the root values, and; • negative real values of the Lyapunov exponents λ + and λ − .Given these requirements it is possible to choose the correct roots in the region φ BR ∈ (0 , . π ) ∪ (0 . π, π ), due to there being only one root which fulfils all the require-ments. It is also possible to determine that there are no valid roots in the region φ BR ∈ (0 . π, . π ), as in this region none of the roots satisfies the stability require-ments. Nevertheless, the linearised system detailed in Eq.(21) and (22) is not sensitiveenough to detect limit cycles in the region φ BR ∈ (0 . π, . π ); a small discrepancyis visible in the right-most red section of Figure 7, with its onset after the change ofsign. Indeed, all of the requirements are satisfied in this region (real-valued roots and14 - - Figure 8: Contour plots of Eq.(18) for α BG , α GR and α BR , varying both ζ ∈ [0 ,
1] and φ BR ∈ [0 , π ].Each panel is generated by calculating the t = 2000 values of α BG and α GR in Eq.(18), varying values of ζ and φ BR over an equally spaced 201 by 201 grid. Points categorised as dynamic are shown as white.Points categorised as steady state are shown in colour and projected onto S via Eq.(19). negative real components of the Lyapunov exponents), yet we know from Figure 5 thatthis region displays limit cycle behaviour. We also tested the sensitivity of the stabilityanalysis by adding additional terms to Eq.(21). The addition of quadratic terms did notincrease Eq.(21)’s ability to detect limit cycles in this region, whereas with the additionof cubic terms we were only able to additionally detect limit cycle behaviour in the region φ BR ∈ (0 . π, . π ). We forego these details for the sake of brevity. We conclude this section by presenting contour plots, given by Figure 8, of Eq.(18) for α BG , α GR and α BR , varying both ζ ∈ [0 ,
1] and φ BR ∈ [0 , π ] as continuous variables. Eachpanel in Figure 8 was generated by calculating the t = 2000 values of α BG and α GR inEq.(18), varying values of ζ and φ BR over an equally spaced 201 by 201 grid, and projectedonto S using Eq.(19). Each point on the contour plot was suitably tested whether itcould be categorised as either steady-state or dynamic. Dynamic points appear as whitein Figure 8, whereas steady-state values are coloured based on the legend appearing onthe very left of Figure 8.In the left panel of Figure 8, for values of inter-network coupling ζ ∈ [0 . , α BG is mostly positive, rarely rising above a value of unity, except for a small region where φ BR ≈ π , where the value of α BG is negative, but still small. In this region of ζ -values,one of the effects of Blue intending to be approximately π ahead of Red is for Green tosuddenly be ahead of Blue, as indicated by the appearance of colours corresponding tonegative values. Additionally, we note that in this region of ζ values, the contours for α BG vary quite smoothly. Contours start displaying more varied behaviour as the value of ζ decreases, with contour lines becoming denser. Generally, as ζ values decrease, we witnessboth greater rates of change as ζ and φ BR vary, and the appearance of more extreme valuesof α BG . In the region ζ ∈ [0 . , .
4] and φ BR ∈ [0 . π, π ], the value of α BG varies quite15rastically across all values between ( − π, π ). Of course, below a certain threshold of ζ ,which is influenced by the value of φ BR , the system becomes dynamic. Intuitively, we notethat between φ BR ∈ [0 , . π ], a higher value φ BR requires a higher value of ζ to enable asteady-state solution to Eq.(18); demonstrating that a greater frustration value generallyplaces more strain on the system, which then requires greater coupling strength to enablesteady-state solutions. For φ BR > . π , highly negative steady-state solutions appear forlower ζ values showing that the system has flipped with Blue behind Green.The middle panel of Figure 8, showing α GR , demonstrates less dramatic behaviourthan that witnessed in the left panel. Generally, as ζ decreases below 0 . α GR becomesmore negative, very rarely reaching values less than −
1. There is a small region howeverwhere α GR exhibits small positive values for ζ ∈ [0 . π, . π ] and φ BR ≈ π . Generally,however, rates of contour changes as ζ and φ BR vary in the middle panel are never asextreme as witnessed in regions of the left panel for α BG .The right panel for α BR is given by the addition of both left and middle panels.Visually, the right panel is very similar to the left panel for α BG , but lacks the moreextreme rates of change as ζ and φ BR vary. In the small region ζ ∈ [0 . , .
4] and φ BR ∈ [0 . π, π ], the value of α BR reveals Blue’s phase to be maximally ahead of that of Red.Nevertheless, in the same parameter region for α BG , we see that the exact opposite canoccur: α BG shows Blue centroid to be minimised in relation to Green. This phenomenonis an interesting second-order effect caused by the Blue network’s frustration in relationto the Red network in a model where the interactions of a third population are intrinsic.
5. Conclusions, discussion and future work
We have extended the adversarial two-network Blue-Red model of locally coupled frus-trated phase oscillators to include a third networked population of actors with vanishingfrustration. Through numerical analysis and dimensional reduction we found that as frus-trations increase or inter-population couplings decrease, the system discontinuously flips,where the steady-state phase advance of one population in relation to another would in-crease, become time-varying, and then reverse. Notably, Green’s behaviour showed thatin certain parameter ranges it may be ahead of Blue’s centroid, despite vanishing frustra-tion parameter. The ‘sweet spot’, where both Blue could be ahead in the phase of Redand Green, and Green in turn ahead of Red, was very narrow.We can interpret these results, particularly the contour plot Fig.8, through the lensthat frustration represents a strategy for decision advantage and the BGR model capturesmultiple adversarial/cooperative decision-making groups. Firstly, it shows that even anorganisational structure such as a hierarchy — which is designed for equal distributionof information loads and a single source of intent, but intrinsically poorly structuredfor synchronised decision-making against better-connected adversaries — can achieve ad-vantageous outcomes. However, to achieve this it requires tighter internal and externalinteraction. There are significant portions of parameter space in Fig.8 with high ζ whereBlue is both ahead of Green’s decision-making (thus able to exert influence on a neutralparty) and an adversary. It is intuitively plausible that as Blue seeks to be further ahead16f decision of both Green and Red (through greater φ ) then it must also increase itsintensity of interaction ζ to maintain a steady-state decision advantage.Blue may find itself behind Green, even though it maintains a decision advantagewith respect to Red, as seen in the extreme right region of the first panel of Fig.8. Weremark that this is where Blue gains its most extreme advantage over Red in the thirdpanel. Interpreted in the context of multi-party decision-making, this offers an interestingoption for Blue: that sometimes ceding initiative to the neutral group provides scope for amaximal advantage over a competitor. This phenomenon offers a qualitative (and, to thedegree that parameters in the model may eventually be measured in real-world systems,quantitative) means to examine the risks introduced by pursuing a particularly greedystrategy: by striving to be too far ahead of a competitor’s decision making, a populationmay allow non-adversarial actors to be ahead of one’s decision-making processes. Likeall risks, however, this phenomena can also be seen as an opportunity; does Blue use thefact that Green is afforded a means to stay ahead in decision-making cycles as a methodof gaining trust with the third population? An answer in the affirmative or negative is ofcourse context-sensitive, depending on the application.The model offers an intuitive conclusion: that in contexts of multiple parties withnon-consistent objectives, the sweet spot of unilateral advantage for one party over theothers may be very narrow or non-existent altogether. The implications of such decisionpositions cannot be deduced from within the model itself but how it is embedded inthe real world context — either through qualitative considerations, or by coupling thesebehaviours to an additional mathematical model of the external actions in the world.Significantly, we do not observe in the regimes of semi-stable behaviour in any regionswhere Red is ahead of Blue, even though for the two-network case such behaviours canbe found [30]. We have not performed complete parameter sweeps of the BGR model,so such regions may exist. Alternately, the introduction of the third population in theparticular asymmetric way implemented here may push such behaviours into a narrowisland inside more chaotic dynamics. This is worth further numerical investigation butmay be outside the scope for an analytical solution. However, this does imply the valueof strategic engagement with neutral parties in such a three-way contested context.Future work may consider stochastic noise in the BGR model as a means to explore theeffects of uncertainty of human decision making in an adversarial engagement. Further-more, it may be meaningful to frame the BGR model in a game-theory setting; the utilityfunctions of such a study, and their measurement, may yield novel and useful ways tothink about risk and trust between noncombatant groups caught up in inherently adver-sarial settings. Finally, the coupling of this model into a representation of the outcomes ofdecisions will yield a means of quantifying risks through the interplay between probabilityand consequences. In particular, in view of the military contextualisation we adopt withthis model there is an opportunity to couple this model with well-known mathematicalrepresentations of combat and network generalisations of them [46]. Above all, througha compact mathematical model of complexity such as this, at least partially analyticalinsights may be gained into otherwise surprising and rich behaviours.17 cknowledgements The authors would like to thank Richard Taylor, Irena Ali and Hossein Seif Zadehfor discussions during the writing of this manuscript. This research was a collaborationbetween the Commonwealth of Australia (represented by the Defence Science and Tech-nology Group) and Deakin University through a Defence Science Partnerships agreement.
Appendix A. Strategic-tactical view
Following Figure 1, namely the segregation of strategic and tactical nodes, we offer theBGR model as the following expanded set of ordinary differential equations, segregatedinto the relevant strategic (labelled by I ) and tactical (labelled by II ) components,˙ B ( I ) i = ω ( I ) i − σ B (cid:88) j ∈B ( I ) B ij sin (cid:16) B ( I ) i − B ( I ) j (cid:17) − σ B (cid:88) j ∈B ( II ) B ij sin (cid:16) B ( I ) i − B ( II ) j (cid:17) − ζ BG (cid:88) j ∈G ( I ) I ( BG ) ij sin (cid:16) B ( I ) i − G ( I ) j − φ BG (cid:17) , i ∈ B ( I ) , ˙ B ( II ) i = ω ( II ) i − σ B (cid:88) j ∈B ( I ) B ij sin (cid:16) B ( II ) i − B ( I ) j (cid:17) − σ B (cid:88) j ∈B ( II ) B ij sin (cid:16) B ( II ) i − B ( II ) j (cid:17) − ζ BR (cid:88) j ∈R ( II ) I ( BR ) ij sin (cid:16) B ( II ) i − R ( II ) j − φ BR (cid:17) , i ∈ B ( II ) , (A.1)˙ G ( I ) i = µ ( I ) i − σ G (cid:88) j ∈G ( I ) G ij sin (cid:16) G ( I ) i − G ( I ) j (cid:17) − σ G (cid:88) j ∈G ( II ) G ij sin (cid:16) G ( I ) i − G ( II ) j (cid:17) − ζ GB (cid:88) j ∈B ( I ) I ( GB ) ij sin (cid:16) G ( I ) i − B ( I ) j (cid:17) , i ∈ G ( I ) , ˙ G ( II ) i = µ ( II ) i − σ G (cid:88) j ∈G ( I ) G ij sin (cid:16) G ( II ) i − G ( I ) j (cid:17) − σ G (cid:88) j ∈G ( II ) G ij sin (cid:16) G ( II ) i − G ( II ) j (cid:17) − ζ GR (cid:88) j ∈R ( II ) I ( GR ) ij sin (cid:16) G ( II ) i − R ( II ) j (cid:17) , i ∈ G ( II ) , (A.2)˙ R ( I ) i = ν ( I ) i − σ R (cid:88) j ∈R ( I ) R ij sin (cid:16) R ( I ) i − R ( I ) j (cid:17) − σ R (cid:88) j ∈R ( II ) R ij sin (cid:16) R ( I ) i − R ( II ) j (cid:17) , i ∈ R ( I ) , ˙ R ( II ) i = ν ( II ) i − σ R (cid:88) j ∈R ( I ) R ij sin (cid:16) R ( II ) i − R ( I ) j (cid:17) − σ B (cid:88) j ∈R ( II ) R ij sin (cid:16) R ( II ) i − R ( II ) j (cid:17) − ζ RB (cid:88) j ∈B ( II ) I ( RB ) ij sin (cid:16) R ( II ) i − B ( II ) j − φ RB (cid:17) − ζ RG (cid:88) j ∈G ( II ) I ( RG ) ij sin (cid:16) R ( II ) i − G ( II ) j − φ RG (cid:17) , i ∈ R ( II ) . (A.3)18qs.(A.1-A.3) explicitly highlights the roles and interactions of the types of nodes in thenetworks by designating the strategic and tactical nodes. Appendix B. Dimensional reduction
Inserting the approximation detailed in Eq.(5) into Eq.(1-3) we obtain,˙ E B + ˙ b i = ω i − σ B (cid:88) j ∈B L ( B ) ij b j − ζ BG sin( α BG − φ BG ) d ( BG ) i − ζ BR sin( α BR − φ BR ) d ( BR ) i − ζ BG cos( α BG − φ BG ) (cid:88) j ∈B∪G L ( BG ) ij V j − ζ BR cos( α BR − φ BR ) (cid:88) j ∈B∪R L ( BR ) ij V j , ˙ E G + ˙ g i = µ i − σ G (cid:88) j ∈G L ( G ) ij g j + ζ GB sin α BG d ( GB ) i − ζ GR sin α GR d ( GR ) i + ζ GB cos α BG (cid:88) j ∈B∪G L ( GB ) ij V j − ζ GR cos α GR (cid:88) j ∈G∪R L ( GR ) ij V j , ˙ E R + ˙ r i = ν i − σ R (cid:88) j ∈R L ( R ) ij r j + ζ RB sin( α BR + φ RB ) d ( RB ) i + ζ RG sin( α GR + φ RG ) d ( RG ) i + ζ RB cos( α BR + φ RB ) (cid:88) j ∈B∪R L ( RB ) ij V j + ζ RG cos( α GR + φ RG ) (cid:88) j ∈G∪R L ( RG ) ij V j , (B.1)where { L ( B ) , L ( G ) , L ( R ) } are the graph Laplacians [45] of the Blue, Green and Red networksrespectively: L ( B ) ij = (cid:88) k ∈B B ik (cid:124) (cid:123)(cid:122) (cid:125) ≡ d ( B ) i δ ij − B ij , L ( G ) ij = (cid:88) k ∈G G ik (cid:124) (cid:123)(cid:122) (cid:125) ≡ d ( G ) i δ ij − G ij , L ( R ) ij = (cid:88) k ∈R R ik (cid:124) (cid:123)(cid:122) (cid:125) ≡ d ( R ) i δ ij − R ij . (B.2)Correspondingly, the matrices L ( BG ) , L ( GB ) etc . are the inter-network graph Laplacians,given by, L ( BG ) ij = (cid:88) k ∈B∪G I ( BG ) ik (cid:124) (cid:123)(cid:122) (cid:125) = d ( BG ) i δ ij − I ( BG ) ij , L ( GB ) ij = (cid:88) k ∈B∪G I ( GB ) ik (cid:124) (cid:123)(cid:122) (cid:125) = d ( GB ) i δ ij − I ( GB ) ij . (B.3)and similarly for ( BR ) , ( GR ) , ( RB ) and ( RG ). The integer d i , for node i , is the degree of node i (total number of edges) for the particular network or inter-network connection.Lastly, the quantity V i in Eq.(B.1) simply encodes the fluctuations for each network, V i = b i i ∈ B g i i ∈ G r i i ∈ R . (B.4)19he intra-network Laplacians present in Eq.(B.1) all come equipped with a completespanning set of orthonormal eigenvectors, which we label by e ( B,ρ ) i , ρ = 0 , , . . . , |B| − ∈ B E , (cid:88) j ∈B L ( B ) ij e ( B,ρ ) j = λ ( B ) ρ e ( B,ρ ) i ,e ( G,ρ ) j , ρ = 0 , , . . . , |G| − ∈ G E , (cid:88) j ∈G L ( G ) ij e ( G,ρ ) j = λ ( G ) ρ e ( G,ρ ) i ,e ( R,ρ ) k , ρ = 0 , , . . . , |R| − ∈ R E , (cid:88) j ∈R L ( R ) ij e ( R,ρ ) j = λ ( R ) ρ e ( R,ρ ) i , (B.5)where we distinguish between indices in the eigen-mode space {B E , G E , R E } and those inthe node space {B , G , R} . The spectrum of Laplacian eigenvalues of any given network,labeled { λ ( B ) , λ ( G ) , λ ( R ) } , is real-valued and conveniently bounded from below by zero;the degeneracy of the zero eigenvector equals the number of components of the respectivenetwork [45]. Thus, the Blue, Green and Red networks given in Figure 2 each contain asingle zero-valued eigenvalue — for the Laplacian eigenvalues of the particular networksused in this work refer to the left panel of Figure 3. The corresponding zero eigenvectors { e ( B, , e ( G, , e ( R, } , up to normalisation, consist entirely of unit valued entries.We wish to use the completeness of the Laplacians to diagonalise the system. Fora single network, namely the ordinary Kuramoto-Sakaguchi model for a single graph,the Laplacian basis elegantly separates out the collective mode, corresponding to thesynchronised system, which identifies with the Laplacian zero eigenvector. Contrastingly,the non-zero, or ‘normal’, modes turn out to be Lyapunov stable, namely exponentiallysuppressed. Thus the Laplacian neatly exposes the dynamics close to synchrony for theordinary Kuramoto model [38].In the case of multiple networks, the Laplacians do not commute and therefore do notprovide a simultaneous diagonalisation of the system. To proceed with the dimensionalreduction procedure we impose the further approximation that (cid:88) j ∈M∪N L ( MN ) ij V j ≈ , ∀ networks M and N , (B.6)i.e. all inter-network Laplacian fluctuations in Eq.(B.1) are approximately equal to zero.This approximation enables the fluctuations b i , g i and r i in Eq.(B.1) to decouple. Nev-ertheless, as mentioned in [42], this approximation is not guaranteed to completely hold,even in model regimes which enable Eq.(5) to be satisfied.We now expand the fluctuations in Eq.(B.1) via the non-zero normal modes, b i = (cid:88) ρ ∈B E / { } e ( B,ρ ) i x ρ , g i = (cid:88) ρ ∈G E / { } e ( G,ρ ) i y ρ , r i = (cid:88) ρ ∈R E / { } e ( R,ρ ) i z ρ , (B.7)and exploit the orthonormality of the spanning vectors to obtain,˙ x ρ = q ( ρ ) B ( x ρ , α BG , α GR ) ˙ y ρ = q ( ρ ) G ( y ρ , α BG , α GR ) , ˙ z ρ = q ( ρ ) R ( z ρ , α BG , α GR ) , (B.8)20here, q ( ρ ) B = (cid:88) i ∈B e ( B,ρ ) i (cid:104) ω i − ζ BG sin( α BG − φ BG ) d ( BG ) i − ζ BR sin( α BR − φ BR ) d ( BR ) i (cid:105) − σ B λ ( B ) ρ x ρ , ρ ∈ B E / { } ,q ( ρ ) G = (cid:88) i ∈G e ( G,ρ ) i (cid:104) µ i + ζ GB sin α BG d ( GB ) i − ζ GR sin α GR d ( GR ) i (cid:105) − σ G λ ( G ) ρ y ρ , ρ ∈ G E / { } ,q ( ρ ) R = (cid:88) i ∈R e ( R,ρ ) i (cid:104) ν i + ζ RB sin( α BR + φ RB ) d ( RB ) i − ζ RG sin( α GR + φ RG ) d ( RG ) i (cid:105) − σ R λ ( R ) ρ z ρ , ρ ∈ R E / { } . (B.9)We note that the time derivative of the centroids is eliminated from each expression inEq.(B.9) due to the orthogonality of the Laplacian eigenvectors; the sum of the individualentries of each non-zero Laplacian eigenvector exactly equals zero. Eqs.(B.8,B.9) give thedynamics of the normal modes of the BGR system. Eqs.(B.8,B.9) are linear in their re-spective fluctuation mode variables, but ultimately their dynamics involves the differencesof the centroids α BG and α GR . These variables themselves are completely determined bythe two-dimensional system in Eq.(11), which is a two-dimensional extension of a tiltedperiodic ratchet system [47, 48].Finally, projecting Eq.(B.1) onto the zero eigenvectors for each of Blue, Green andRed we obtain the expressions for the centroids given in Eq.(8) in the main text. Becausethe zero eigenvector projection separates out equations for E B , B G and E R in Eq.(8), wemay refer to these as the zero-mode projections of the phases B , G and R , respectively. Appendix C. Local synchronisation examples
Figure C.9 offers numerical outputs of local order parameter values of all three net-works. In the top row, for ζ = 0 . φ BR from the left-most to the right-most panel on the top row does not appearto have an appreciable influence on this behaviour other than making it slightly moreerratic. The second row of Figure C.9, for ζ = 0 .
2, presents a more interesting picture.The local order parameters switch between steady-state behaviour for the left-most panel( φ BR = 0 . π ), to periodic limit cycle behaviour on the middle panel ( φ BR = 0 . π ), andback to steady-state behaviour for the right-most panel ( φ BR = 0 . π ). The bottom rowof Figure C.9, for ζ = 0 .
3, produces steady-state model outputs, regardless of the valueof the Blue network’s strategy towards Red.
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