Organisational Social Influence on Directed Hierarchical Graphs, from Tyranny to Anarchy
TTyranny to Anarchy:Regimes of Organisational Influence on Directed Hierarchical Graphs
Charlie Pilgrim
The University of Warwick ∗ Weisi Guo
The University of Warwick, Alan Turing Institute
Samuel Johnson
The University of Birmingham, Alan Turing Institute (Dated: October 24, 2019)Social organisation is critical to coordinated human behaviour. There are a diverse range oforganisational structures, which can be thought of as power structures with “managers” and “sub-ordinates”. Often a change in one part can cause cascades throughout the organisation, which canbe desirable or can lead to inefficiencies. As organisations change in size, complexity and structure,we analyse how their resilience to disturbances is affected. Here, we consider majority rule dynamicson organisations modelled as hierarchical directed graphs, where the directed edges indicate influ-ence. We utilise a topological measure called the trophic incoherence parameter, q, which effectivelygauges the stratification of power structure in an organisation. We show that this measure boundsregimes of behaviour. There is fast consensus at low q (e.g. tyranny), slow consensus at mid q (e.g.democracy), and no consensus at high q (e.g. anarchy). These regimes are investigated analyticallyand empirically with diverse case studies in the Roman Army, US Government, and a healthcareorganisation. Our work has widespread application in the design and analysis of organisations.
I. INTRODUCTION
Social and political systems display different typesof order and structure, with very different outcomes.Small-scale informal organisations might be skill or powerbased, whereas large-scale social systems involve com-plex politics. In almost all social systems, some formof formal or latent hierarchy is present, with an organi-sational power structure of managers and subordinates.Traditional military organisations are perhaps the pro-totypical example of rigid hierarchy, with a very orderedstructure allowing instructions to be quickly passed fromtop to bottom [1]. While these kinds of singular hier-archies abound historically [2], in the 18th Century theinfluential treatise “The Spirit of the Laws” laid downa political theory that rejected hierarchical structure ofgovernment and called for a separation of powers [3], i.e.balancing power between multiple hierarchies. Influencedby this treatise and other enlightenment thinking, theUS Constitution, which among other things dictates thestructure of the US Government, prescribes a series ofchecks and balances with the explicit intention of pre-venting a singular tyrannical exploitation of power [4].On the other end of the scale, political anarchy has beendescribed as a rejection of any form of hierarchy [5]. An-archy should not be dismissed as disorganised chaos -there are movements in management [2] and organisa-tional [6] science that encourage self-empowered individ-uals working autonomously or in dynamically forming ∗ [email protected] teams, with an organisational structure resembling thatof political anarchy. One can regard anarchy as a non-equilibrium form of hierarchy, whereby at any particularquasi-static state, a power hierarchy exists. In betweenthese extremes, modern democracies (on average) can beconsidered more distributed than rigid tyrannical hier-archies and more ordered than political anarchy, some-where between the two. In fact, we can find a myriad oforganisational structures [2], ranging from the structured(military, churches, schools), to those that are consumerdriven, to those that have an ad-hoc agile decision pro-cess. An important question in all of these social struc-tures is how effective leadership is at influencing change,or how fast a change in policy spreads through the net-work, if at all. The same question sheds light on how fasta disruption can spread through an organisation, withresilient ones able to dampen its propagation quickly,whereas non-resilient organisations can suffer long-termcascading confusion. A. A Network Based Socio-Physical Model
Social systems display rich and complex behaviour, farbeyond our ability to capture in mathematical models.In order to investigate the problem we necessarily makesimplifying assumptions. Our approach is informed bysociophysics, the application of models and methods fromthe physical sciences to social systems [7, 8]. The mostcommon and valid criticism of sociophysics is that thenecessary assumptions made in order to reduce the so-cial system to a tractable model can often overlook thesubtleties and richness of the social dynamics [7]. A re- a r X i v : . [ n li n . AO ] O c t sponse to this criticism, and the stance that we aim for,is to avoid the positivist pitfall of thinking that our mod-els completely describe or predict social behaviour, andinstead to look for simple models that reveal salient fea-tures of social systems, that can then be used to informfurther research and more qualitative approaches. Withthis in mind, we consider networks with nodes as peo-ple and political influence moving along directed edges.We are interested in the dynamics on these networks, inparticular the speed to consensus following a change inpolicy from the leadership. Our hope is to link the topo-logical structure of social systems with the dynamics ofinfluence within. B. Novelty & Contribution
There has been much work in the past investigating thestructure and dynamics of complex networks, with manyexcellent overviews including [9–11]. Dynamics such asstability [12, 13], consensus [14], disease spread [15] andpercolation [16] have been examined in relation to topo-logical measures such as spectral analysis [9, 12], con-nectivity [14], clustering [9], core-periphery [8], degreedistributions [9] and trophic coherence [17], among manyother studies. Joint entity dynamics and and topologicalanalysis have been conducted in recent studies, especiallyfor one-dimensional population dynamics [18, 19].Our approach, which will be fully explained in the fol-lowing section, is along the lines of previous work on mod-elling opinion dynamics in social systems by consideringthe dynamics of the Ising model in complex networks (see[7] for an overview). Our unique contribution will be toexamine time to consensus on these networks in relationto the topological measure of trophic coherence, whichprovides a measure of power stratification in social andorganisational systems.
II. MODELS
We will model social systems as directed graphs, withnodes as people (or small groups/entities within organisa-tions) and with influence spreading along directed edges.Here, influence is an asymmetric property (can be two-ways) that can be the delegation of a task, the transmis-sion of a command, or the sharing of information. Wedefine a directed graph as G = ( V, E ), where V is the setof vertices or nodes in the graph, and E is the set of di-rected edges between nodes. This can be fully describedby an adjacency matrix A , such that the elements of A are a ij = (cid:40) i to j S or n to denote the size, or number ofnodes, of a network. L will be used for the number of edges. B will be used for the number of basal nodes,which are nodes with no in-edges. From a social systemsperspective, basal nodes are considered “leaders” who arenot influenced by any other nodes. A. Opinion Dynamics
The majority rule model [7] allows each node to bein one of two states, which we will denote as state − x i ∈ {− , } . We define the state vector asthe state of all S nodes, x = [ x , x , ..., x n ]. We considernodes being influenced along the directed edges, with thestate of a node changing based on some form of influencedynamics. Considering discrete timesteps, the state ofnodes in a timestep are determined by some function onthe graph topology and the state of the nodes in theprevious timestep [7]. x t = f ( G , x t − ) (1)We will use a simple majority rule algorithm to up-date the states of the nodes at each timestep. At eachtimestep, t , a node is uniformly randomly selected and itsstate is updated based on the state of its parent nodes,where a directed link goes from a parent to a child node.The algorithm will update a node’s state to match thestate of the majority of its parent nodes in the previoustimestep [7]. If there is a tie then the node’s state willbe determined by a coin flip, with a 50% probability ofstate − B. Power Stratification Measured by TopologicalTrophic Coherence
In different social organisation structures, the clarityor coherence of power stratification depends on the num-ber of feedback loops subordinates have to managers.In our abstract model, if a subordinate has equal feed-back power as the manager has delegate power, we regardthem as equals. When the network is large, the feedbackcan arch across different levels and it becomes difficultto quantify: (1) the number of power levels, and (2) thecoherence of the power levels. To capture the degree ofpower stratification we will consider trophic coherence, atopological property of directed networks which capturesthe extent to which edges are aligned, as in a top-downhierarchy, or more disorganised, as usually seen in a ran-dom graph [13, 20]. To describe trophic coherence, wefirst need to describe trophic levels.
1. Inferring Power Levels using Trophic Levels
The concept of trophic levels was originally developedin relation to food web networks, which classify speciesin accordance to their predation relationships [13]. In afood web, the nodes are species and directed edges gofrom prey to predator. Each species can be assigned atrophic level, which signifies how far “up” the food webthe species is [21]. Basal species, with trophic level 1, arethose that generate energy directly from the environmentsuch as plants and algae [21]. Species that feed only onbasal species, such as sheep, are assigned trophic level2. A wolf that predates only on sheep would be giventrophic level 3. Some species do not fit neatly into aninteger trophic level, for example a scavenger like a ratmay feed on plants as well as the dead bodies of sheep andwolves [13]. The concept of describing directed networkswith energy flow has been extended beyond food webs, forinstance to inferring multi-scale stability in both trans-portation networks [17] and water distribution systems[22].Consider the food web network adjacency matrix, A ,such that a ij is the amount of biomass that species j predates from species i . The trophic level of a species, s j , is defined as the weighted mean of the trophic levelsof the species that j preys upon, plus one [13]. s j = (cid:80) i a ij s i (cid:80) i a ij + 1 (2)This describes a linear system that can be solved with aunique solution with the sufficient condition that at leastone species is basal, ∃ j : ( a ij = 0 ∀ i ). A full derivationis in the Appendix.
2. Power Incoherence via Trophic Incoherence Parameter, q We define the trophic incoherence parameter, q , alongthe lines of Johnson et al (2014) [13]. The trophic differ-ence of an edge in the graph is the difference in trophiclevels between predator and prey. d ij = s j − s i (3)The mean of the trophic difference of edges in any di-rected graph will be equal to 1 (see [23] for a proof).In fact, in a perfectly ordered graph, the trophic differ-ence of every edge will be 1 [13]. In less ordered graphs,the mean of the trophic differences of edges will remainequal to 1, but with some variance. The trophic incoher-ence parameter, q , is defined as the standard deviationof the distribution of trophic differences over all edges inthe graph [13]. Conceptually, directed graphs that havehigh trophic coherence (i.e. low trophic incoherence pa-rameter) are tree like, and can be drawn with all edgespointing in the same direction. Directed graphs with lowtrophic coherence (i.e. high trophic incoherence param-eter) do not have edges pointing in one clear direction,and appear more random. Figure 1. Trophic incoherence’s relation to topological struc-tures. ( a ) and ( b ) show counts of cycles of length 5 and theproportion of back edges in graphs generated using the gen-eral Preferential Preying Model with 100 nodes, 750 edgesand 5 basal nodes, over a range of trophic incoherence. Atlow q ( c ), graphs are tree-like, analogous to a tyrannical socialsystem. At high q ( e ), graphs are random with little coherentstructure, analogous to anarchy. At mid q ( d ), there is somecoherence, analogous to democracy. The concept of trophic coherence was initially intro-duced as a way to potentially solve May’s paradox, along standing problem in ecology [13]. As a randomgraph grows in size and connectivity, it reaches a thresh-old beyond which it is almost certainly unstable (consid-ering the Lyapunov stability of the community matrix)[12]. The experience of ecologists suggests that natureseems to behave the opposite way - ecosystems do notbecome less stable with size and complexity and can be-come more stable [13, 24]. Light was shed on this appar-ent paradox in 2014 by Johnson et al, who showed thattrophic coherence determines the stability of food-webs[13]. Since then, trophic coherence has been associatedwith the presence of cycles in networks [20], and has beenlinked to the distribution of motifs [25].
3. Trophic Incoherence and Topology
Trophic coherence has been called a measure of howsimilar a graph is to a hierarchy [17], with a hierarchy be-ing maximally coherent with trophic incoherence, q = 0(Figure 1(c)). In a hierarchy, all nodes are in a cleartrophic level with edges all pointing from top to bottom.Conversely, graphs with a very high trophic incoherencewill appear random, with no clear structure of nodes andedges going in all directions (Figure 1(e)). Between thesetwo extremes we have graphs which have a discernible or-der but are not fully hierarchical, with some edges goingagainst the flow (Figure 1(d)).Figure 1 shows relationships between trophic incoher-ence and topological structures. Johnson et al showeda theoretical relationship between cycles and trophic co-herence [20], which suggested that graphs in the low qregime will have very few cycles, this is demonstratedempirically in 1(a). A discussion of how the number ofcycles in a graph was calculated is included in the Ap-pendix.A “back edge” is a link from a node of higher trophiclevel to a node of lower trophic level i.e. the trophicdifference of this edge will be below zero. The trophic in-coherence parameter, q , is the standard deviation of thedistribution of the trophic differences of all edges in thegraph [13]. Recalling that the mean of the trophic differ-ence of all edges in any graph will always be equal to 1,the probability of an edge having trophic difference belowzero should therefore increase with the trophic incoher-ence parameter, q . Figure 1(b) shows that, empirically,the expected number of back edges appears to be an in-creasing function of trophic incoherence, q , as expected.The empirical number of cycles and back edges supportour conceptual view of trophic coherence. This gives us 3regimes of structure, each of which has a social analogus:1. Low q regime. Hierarchical, tree-like graphs withfew cycles or back edges. The social analogue isTyranny.2. Medium q regime. Broadly hierarchical, but withincreasing numbers of cycles and back edges. Thesocial analogue is Democracy.3. High q regime. Little hierarchical structure, lotsof cycles and back edges. The social analogue isAnarchy. C. Graph Generation Models
1. Erd˝os-R´enyi, Niche and Cascade Models
One of the earliest and most influential random graphgeneration models is the Erd˝os-R´enyi model [26]. TheErd˝os-R´enyi graph generating algorithm with node count S and edge count L uniformly randomly places L di-rected edges across all possible edges between nodes[27]. The ensemble includes all possible configurations of G = ( V, E ) with those node and edge counts [27]. How-ever, Erd˝os-R´enyi graphs have a very low probability ofhaving high trophic coherence. While it is theoretically possible to investigate the entire state space with Erd˝os-R´enyi graphs, we seek a more reliable way of tuning ran-dom graphs with specific trophic incoherence parameters, q . The niche and cascade models arose from the literatureon ecology, as attempts to generate artificial food webssimilar to empirical food webs using simple rules [13].These models generate directed graphs with trophic in-coherence parameters in the range 0 . (cid:47) q (cid:47)
2. However,we need to generate graphs with a wider range of trophicincoherence.
2. Null Model: General Preferential Preying Model
Johnson et al’s 2014 paper that introduced the trophicincoherence parameter, q , also introduced the Preferen-tial Preying Model [13]. This was updated in 2016 asthe general Preferential Preying Model [28], which wewill use as our main graph generating model. The Pref-erential Preying Model was inspired by considering howecological food webs form in nature [23], and it has theadvantage of generating graphs over a controllable rangeof trophic incoherence.The general Preferential Preying Model algorithm re-quires an input of the number of nodes, S , the numberof basal nodes, B , the number of links, L and a Tem-perature parameter, T , which is used to tune the trophicincoherence of the resulting graph [28]. The algorithmproceeds in 2 phases.The algorithm begins with B basal nodes, which aregiven a temporary trophic level, ˆ s i = 1. Each of theremaining S − B nodes are added one at a time, with anedge created to the new node, j , from an existing node, i , which is chosen uniformly randomly. The temporarytrophic level of the new node is set to ˆ s j = ˆ s i + 1. Onceall nodes are added we have a graph with S nodes and( S − B ) edges [28]. In phase two, L − ( S − B ) more linksare added, the relative probability of a link being addedfrom node i to node j is given by: p ij ∝ exp (cid:18) − (ˆ s j − ˆ s i − T (cid:19) (4)The general Preferential Preying Model has several ad-vantages over other graph generating models. It is able togenerate graphs over a wide range of trophic coherence,with the ability to tune the trophic incoherence param-eter, q . Also, a graph generated with this model willalways have a basal node, which means that a unique so-lution can be found for the node trophic levels, so that thetrophic incoherence parameter can be calculated. Withthese advantages in mind, the general Preferential Prey-ing Model is the model that we will use during our inves-tigation. D. Social Structures
For our purposes, trophic coherence is a useful measureto describe different types of social systems, and in par-ticular how coherent a social structure is. The trophic in-coherence parameter, q , gives us a one-dimensional mea-sure of the slightly abstract concept of coherence, whichwill allow us to investigate the structure of social systemsin a novel and quantitative way.The general Preferential Preying Model generatesgraphs from a particular ensemble [20]. Real social sys-tems may exist outside of this ensemble, and may havedynamic structures that change over time. It may bethat there is some hidden feature of graphs generatedusing the general Preferential Preying Model which is re-sponsible for any results or conclusions we find. We willconsider graphs generated using the general PreferentialPreying Model as a null model, and be wary of over gen-eralising our results to all social systems. III. ANALYSIS
We will analytically investigate how we expect major-ity rule dynamics to propagate on a series of simplifiedmodels, which we can then compare to empirical results.We will look at 4 types of simple graphs, as shown inFigure 2.1. Strings.2. Trees.3. Cycles.4. Cliques.For each type of simple graph, we will consider initial-ising the system so that the basal node is in state -1 andall non-basal nodes are in state 1. We will find expres-sions for v ( t ), the number of nodes in state 1 (not yetinfluenced by the basal node) as a function of timesteps, t . A. Strings
The simplest form of a graph with high trophic co-herence (and low trophic incoherence parameter, q ) is astring, a line of nodes with edges down the line in onedirection. We can initialise this string so that all nodesare in state 1 except the basal node, in state -1 (Fig-ure 2). We let v ( t ) be the number of nodes in state 1as a function of timesteps. Each timestep can be consid-ered a Bernoulli trial, with probability of success equal tothe probability of the algorithm selecting the next nodein the string that will spread the influence of the basalnode, p = n . Success in this Bernoulli trial will reduce v ( t ) by 1, with v (0) = ( n − Figure 2. Selected types of simple graphs for analysis. Orangenodes are basal nodes, which will begin in a different state tothe non-basal nodes. trails gives a Binomial distribution [29], so that we canwrite( n − − v ( t ) ∼ Binomial ( t, n ) v ∈ { , , .., n − } (5)and the expected value of v ( t ) can be easily found andwritten as < v ( t ) > = (cid:40) n − − tn if t ≤ n ( n − t > n ( n −
1) (6)Figure 3 shows this expected value against simulations,with close agreement.
B. Trees
We define a tree as a directed graph with no cycles, andwith directed edges going in one direction only from basalnodes to leaf nodes, where a leaf node is a node with onlyin-edges. Again, we initialise the tree with basal nodesin state -1 and non-basal nodes in state 1 (Figure 2).We are interested in v ( t ), the number of nodes in state1 as a function of timesteps under simple majority ruledynamics.In a tree, the number of nodes influenced by the basalnode can only increase, as all edges go from the basal Figure 3. Progress of influence dynamics on a directed stringof length 100, with one basal node beginning in state -1 andall non-basal nodes beginning in state 1. The orange lineshows the analytic expected proportion of nodes in state 1 asa function of timesteps. The blue points show the empiricalmean of 50 simulated time series at 1000 timestep intervals.The light grey lines are 5 of those simulated time series. node outwards. As the basal node’s influence spreads,there is a “fringe” of nodes that have yet to be influencedby the basal node, but who have a chance of switchingin the next timestep if they are chosen. This fringe, andso the probability of spreading influence per timestep,changes over the progress of the simulation. The spreadof influence can be modelled as a series of Bernoulli trialswith a varying probability of success, p tree . If we assumethat p tree is Beta distributed, then the number of nodesinfluenced by the basal node as a function of time willbe Beta Binomial distributed [30], with the form of p tree determined by the Beta distribution parameters α and β ,which will depend on the topology of the tree in questionand the progress of the simulation.( n − − v ( t ) ∼ Binomial ( t, p tree ) v ∈ { , , .., n − } (7)where p tree is a Beta distributed random variable p tree ∼ Beta ( α, β ) (8)The number of timesteps to consensus can be modelledby considering the number of failed Bernoulli trials be-fore 0 . n − B successes (consensus requires 90% of nodesto agree with the basal nodes). In a network of 100 nodesand 5 basal nodes, consensus is reached when 90 nodesagree with the basal nodes, which requires 85 success-ful Bernoulli trials, or 85 successful algorithm timestepswhere a node is switched state to agree with the basalnodes. This kind of process with a fixed p will have aNegative Binomial distribution [30]. As discussed, p isnot fixed for trees. If we approximate p tree as being a Beta distributed random variable, then the time to con-sensus can be approximated as a Beta Negative Bino-mial distribution [30], which can be written in terms ofGamma functions as f ( k | α, β, r ) = Γ( r + k )Γ( α + r )Γ( β + k )Γ( α + β ) k !Γ( r )Γ( α + r + β + k )Γ( α )Γ( β ) (9)We expect this distribution for timesteps to consensusfor tree-like graphs, this will be investigated empiricallyin the Results section. C. Cycles
Cycles are loops of nodes with directed links pointingaround the loop in one direction. We will consider a cycleof nodes with one basal node outside the cycle, influenc-ing one node in the cycle (Figure 2). The basal node isalways in state -1. Intuitively, it can be seen that therate at which the basal node influences the cycle is pro-portional to the number of nodes in the cycle in state 1,yet to be influenced. This leads to an exponential rela-tionship for the expected value of v ( t ), a full derivationis given in the Appendix. < v ( t ) > = ( n − e − ln n ( n − t (10)Figure 4 shows this analytic prediction against sim-ulated results for simple majority rule dynamics. Thesimulated results appear to be randomly oscillating butwith an expected value over many simulations close tothe analytic prediction. In the simulations, the randomoscillations of v ( t ) mean that the system can reach theabsorbing state of v = 0. The analytic expected valueis in the limit of large n, where the probability of thesystem reaching the absorbing state is small. D. Cliques
We define a clique in a directed graph as a group ofnodes such that edges link each node to all other nodesin the group, in both directions. We consider one basalnode in state -1 influencing one node within the clique,all of which are in state 1 (Figure 2). In a clique abovesize 2 each node reinforces the state of the other nodes,and a single basal node is unable to influence any of thenodes in the clique, which all therefore stay in their initialstate of 1. There is no possibility of consensus with thebasal node. These kinds of structures within graphs canblock any influence spreading from the basal nodes. v ( t )will stay at its initial value of n − v ( t ) = n − Figure 4. Progress of influence dynamics of a directed cycleof length 100, with one additional basal node beginning instate -1 and all non-basal nodes beginning in state 1. Theorange line shows the analytic expected proportion of nodesin state 1 as a function of timesteps. The blue points show theempirical mean of 50 simulated time series at 2500 timestepintervals. The light grey lines show 5 of those simulated timeseries.
IV. RESULTSA. Time to Consensus
We are interested in how quickly a network will reachconsensus following a change in policy from the leader-ship. We consider here basal nodes, with no parent nodes(no in-edges), as “leaders”. We initialise the state of thebasal nodes to be x basal = − x non − basal = 1. We then run a simulation based onmajority rule dynamics, and record how many timestepsit takes for 90% of the nodes to agree with the state ofthe basal nodes, which we consider to be consensus. Ifconsensus is not reached before a set number of timestepsthen the simulation is stopped. We repeat this for graphsgenerated using the general Preferential Preying Model,with a range of trophic coherence.Figure 5 shows the time to consensus for graphs gener-ated with a range of trophic coherence and a fixed numberof nodes, S , edges, L , and basal nodes, B . This kind ofpattern is consistent across a range of values for S, B andL.From a visual inspection of Figure 5, it can be seenthat at low trophic incoherence, q (cid:47)
1, there is verylittle change in time to consensus with changing q . As q increases above 1, we increasingly see graphs with longertimes to consensus. At around q = 1 .
5, we start seeinggraphs take much longer to reach consensus, and somethat do not reach consensus at all. In the results here,simulations that ran for 2000 timesteps were consideredas not reaching consensus. When simulations were ranfor 5000 timesteps instead, there were some that reached consensus after 2000 timesteps in the 1 . (cid:47) q (cid:47) . . ≤ q ≤
1. Fast Consensus Regime, “Tyranny” - Low q
At low q , we see graphs with fast consensus. Thegraphs here should all be tree-like, and conceptually it isexpected that they should reach consensus quickly. Fig-ure 5(a) shows a reasonably good fit of the Beta Nega-tive Binomial distribution to the empirical distributionof time steps to consensus of an ensemble of graphs withtrophic incoherence parameter 0 . < q < .
3. Thismatches what we predicted in the Analysis section. TheBeta Negative Binomial distribution here was fit with thenumber of successful trials, r = 0 . S − B = 85. The Betadistribution shape parameters of α =60 and β = 276 wereadjusted manually. These values give quite a narrow Betadistributed p tree value, with < p tree > = 0 .
2. Slow Consensus Regime, “Democracy” - Medium q
In the medium q range the dynamics of consensuschange. Consensus is reached, but it takes longer. FromFigure 1 we can see that cycles begin to appear in largenumbers in networks with trophic incoherence abovearound q = 1, and back edges appear from about q = 0 . v ( t )is not necessarily a strictly decreasing function. Dur-ing the simulation, nodes can change states to disagreewith the basal nodes, so that v ( t ) can go up, stay thesame or go down. Each timestep is no longer a Bernoullitrial, as there are 3 possible outcomes. Conceptually, itmakes sense that consensus takes longer in these kinds Figure 5. The grey highlighted sections in the left figure correspond to the figures on the right, marked ( a ) and ( b ). The leftfigure shows the result of simulations of majority rule dynamics on graphs generated using the general Preferential PreyingModel, with each simulation representing one point. Simulations that did not reach consensus after 2000 timesteps are markedat 2000 timesteps in red. ( a ) shows the distribution of time to consensus for graphs generated in the range between 0 . < q < . r = 85 and α and β adjusted manually, with the constraint that the empirical mean and distribution mean remain equal. ( b ) shows thedistribution of time to consensus in the range 1 . < q < . q regime, and the r , α and β parameters were all adjusted manually. of graphs. Despite the fact that the underlying assump-tions of Bernoulli trials no longer hold, the Beta NegativeBinomial distribution is still a reasonable fit to the em-pirical data, as shown in Figure 5(b). Consensus takeslonger than in the low q regime, with a wider distribu-tion, and not as good a fit. The number of successful tri-als, r = 130, to consensus was not predetermined duringthe distribution fit, because if some nodes switch againstthe basal nodes state then it will take more than 85 suc-cesses to reach consensus. α = 21 and β = 108 give awider Beta distribution for the value of p tree than for thelow q regime, with < p tree > = 0 .
3. No Consensus Regime, “Anarchy” - High q
As the trophic incoherence rises above around q = 1 . q = 2 .
5, all graphs are in the no consensus regime.The basal nodes, or leaders, are unable to influence therest of the nodes. At these high q values, we expect tosee a high number of cycles and back edges (Figure 1).We hypothesise that the dynamics are similar to thosefound in the cliques in the Analysis section. This may becaused by nodes with a high number of links from nodesof a higher trophic level, which act to block influencespreading from the basal nodes, preventing consensus.
B. Network Size and Average Degree
We have shown that influence dynamics, and time toconsensus, are linked to the trophic coherence of a graph.In our analysis so far, we have concentrated on networkswith 100 nodes, 600 edges and 5 basal nodes. It makessense to begin the investigation by keeping everythingconstant except for the trophic incoherence parameter,however the question arises of whether similar results arefound if we vary the size of the networks, S , and theaverage degree, L/S .
1. Average Degree
Figure 6 shows the regimes of consensus and no consen-sus for simulations on graphs with fixed node count andbasal node count, with varying average degree,
L/S andtrophic incoherence parameter, q . There are clear areasof consensus and no consensus, with some overlap. Withhigher average degree, the no consensus regime beginsat lower trophic incoherence. A hypothesis is that moreedges mean that it is more probable to find situationswhere nodes self-reinforce each other and block influencespreading, as seen in cliques in the Analysis section. Thegeneral Preferential Preying Model is unable to generategraphs with high trophic incoherence parameters and lowedge counts, so the region in the bottom-right of the fig- Figure 6. Each point represents a simulation of majority ruledynamics on graphs generated using the general PreferentialPreying Model, either reaching consensus or not. All networkshave 100 nodes, 5 basal nodes and varying average degree,
L/S , and varying trophic incoherence, q . ure is not well populated.
2. Network Size
An obvious question to ask is whether the size of a net-work has any effect on whether consensus is reached ornot. Figure 7 shows simulations of majority rule dynam-ics on graphs with fixed average degree,
L/S = 6, fixedbasal nodes, B = 5, varying network size, S and vary-ing trophic incoherence, q . With a fixed average degree L/S = 6, the number of nodes doesn’t seem to have astrong effect on whether consensus is reached or not, andthe start of the area of no consensus is around q = 2 forall network sizes, although it does appear to be slightlyless at lower network sizes. V. REAL SOCIAL STRUCTURES
We mapped some real social structures onto graphsand measured their trophic coherence.
A. The Roman Army
An army is the prototypical hierarchical structure.Figure 8 shows the organisational structure of a Cohortin the Roman Army following the Marian reforms in theLate Roman Republic [1]. A legion was made up of 10cohorts, a total of around 4500 men [1]. The RomanArmy was incredibly effective, and employed a fightingstyle that required a high degree of coherent action be-tween individuals and groups [1]. The highly coherent
Figure 7. Each of 46,000 simulations is marked as a point,either reaching consensus or not. All networks have 5 basalnodes, an average out degree of 6, and varying number ofnodes and trophic coherence.Figure 8. A cohort of the Roman Army from the 1st CenturyBC. Each approximately 480 man Cohort is made of 6 Cen-turia, each of which composed of 10 Conterbernia of 8 men[1]. The structure has perfect trophic coherence, with trophicincoherence parameter q = 0. hierarchical structure with a large branching ratio waswell suited to the demands and culture of the RomanArmy, allowing orders from above to quickly reach allindividuals. B. The US Government
The structure of the US Government is prescribed bythe US Constitution, with an intention of creating a sys-tem of checks and balances that would prevent a dema-gogue from capturing too much power and becoming atyrant [4]. Figure 9 shows the high-level structure of the0
Figure 9. The general high-level structure of the US Govern-ment [4]. The Enfranchised People occupy the basal, leader-ship, node. The President’s position does not stand out asparticularly special in the network structure.
US Government, with a trophic incoherence of q = 0 . C. One 2 One Midwives
The Buurtzorg healthcare model aims to improve pa-tient care by embedding teams of around 12 healthcareworkers within communities [6]. The individual health-care professionals are encouraged to help and supporteach other and are empowered to make autonomous de-cisions based on their professional training and patientknowledge [6]. For a case study we look at One 2 OneMidwives, a healthcare organisation inspired in part bythe Buurtzorg healthcare model. One of the midwiveswas interviewed to discuss the organisational structure.
Figure 10. The structure of One 2 One Midwives, a Buurt-zorg inspired healthcare organisation based in Liverpool, UK.The manager, top basal node, influences all midwives. Allmidwives influence each other.
Each of 13 midwives operate autonomously to care fora caseload of pregnant women, giving advice and sup-port to each other. In addition a manager coordinatesthe midwives and can influence each of them. Figure10 shows the organisational structure of One 2 One mid-wives, with trophic incoherence, q = 3 .
46. The real struc-ture was reported by the interviewed midwife as morecomplex and dynamic, with different types of influencebetween manager to midwife and midwife to midwife.This kind of structure allows each midwife to operateautonomously, responding quickly and effectively to pa-tient needs, while accessing support if needed. There isnot a strong need for rapid consensus to influence fromthe manager (based on interview with H. Davey, August6, 2019).
VI. DISCUSSION
The trophic incoherence parameter, q , was shown to beuseful in capturing the general topological coherence ofdirected graphs in a one-dimensional parameter. In thecontext of influence spreading in social networks, threeregimes of behaviour and topology were described:1. High trophic coherence, low q. Tree-like graphswith fast consensus. “Tyranny”.2. Medium trophic coherence, medium q. Some cyclesand back edges with slow consensus. “Democracy”.3. Low trophic coherence, high q. Many cycles andback edges with no consensus. “Anarchy”.For the low q regime, the dynamics of influence weresuccessfully modelled as a series of Bernoulli trials withchanging p , with a close fit between the Beta Negative1Binomial distribution and the empirical distribution oftime to consensus.In the medium q regime, the assumptions underpin-ning Bernoulli trials were stretched. The Beta NegativeBinomial distribution was still a reasonable fit, but notas closely as in the low q regime. The presence of cy-cles and back edges was hypothesised as a reason for thispoorer fit, and the slower consensus.In the high q regime, it was hypothesised that clique-like groups of nodes were blocking influence from spread-ing, with some nodes having more in-links from nodes ofa higher trophic level than those of a lower trophic level.The location of the boundary between consensus andno consensus, in terms of the trophic incoherence param-eter, q , was found to be largely independent of networksize, given a fixed average degree. While the number ofedges was found to have an effect, with more connectedgraphs reaching the point of no consensus at lower valuesof the trophic incoherence parameter.Several real social structures were investigated with in-teresting insights, demonstrating that trophic coherencecan be a useful tool in the description of real social sys-tems. It would be interesting to expand this analysis andto see how time to consensus in real social systems relatesto trophic coherence.There are myriad forms of social structures, each withtheir strengths and weaknesses. A very coherent struc-ture that is good for the objectives of an army may be apoor choice for healthcare workers, and vice versa. Con-sidering trophic coherence, it is striking that the currentpolitical status quo, in the West at least, is not an ex-treme but instead can be thought of as a balance betweenthe efficiency of tyrannical hierarchy and the freedom ofdistributed anarchy. Appendix A: Trophic Levels
Considering the food web network adjacency matrix, A , such that a ij is the amount of biomass that species j predates from species i . The trophic level of a species, s j , is defined as the weighted mean of the trophic levelsof the species that j preys upon, plus one [13]. s j = (cid:80) i a ij s i (cid:80) i a ij + 1 (A1)This can be simplified by considering the weightedbiomass transfer along edges, p ij , found by normalisingthe biomass transfer along edges to each predator species, p ij = a ij (cid:80) i a ij (A2)So that equation A1 becomes s j − (cid:88) i p ij s i = 1 (A3) Equation A3 describes a linear system. − p − p . . . − p n − p − p . . . − p n . . . . . . . . . . . . . . . − p n − p n − p n . . . s s . . .s n = . . . (A4)This can be written in matrix notation, where P is thematrix with entries p ij and is a vector of 1s. s = (( I − P ) − ) T (A5)This linear system can be solved with a unique solutionwith the sufficient condition that there is at least onebasal species that does not prey on any other species.Equation A5 can be arrived at through considerationof the linear system. Alternatively, the trophic level canbe defined in terms of how many levels biomass travelsthrough to get to each species [21], by considering thebiomass weighted adjacency matrix, P . s j = 1 + (cid:88) i p i,j + (cid:88) i p (2) i,j + .. + (cid:88) i p ( N ) i,j (A6)Where N is the maximum number of steps in a foodweb chain that we wish to consider. This can be writtenas s = ( I + P + P + ..P N ) T (A7)Comparing this to equation A5, it is clear that equa-tion A7 is equivalent if( I + P + P + ..P N ) = ( I − P ) − (A8)Multiplying both sides by ( I − P ), we see that thisis equivalent if P N → N → ∞ . The condition ofhaving basal species means that at least one column in P must be zeroes. The matrix elements, p ij , were defined asbeing normalised along j so that the non-zero columns of P sum to 1. These conditions are sufficient for P N → N → ∞ , and so equations A5 and A7 are equivalent. Appendix B: Calculation of Cycles in a Graph
If we consider the squared adjacency matrix of a di-rected graph, A . The elements of this adjacency matrixcan be written as: a (2) ij = S (cid:88) k =1 a ik a kj (B1)That is, a (2) ij gives the number of paths of length 2 inthe graph from node i to j . Through induction, it can2be seen that a ( n ) ij gives the number of paths of length nin the graph from node i to j . The diagonal entries of A n , a nii , will therefore give the number of paths of length n that begin and end at node i , which is the definition ofa cycle of length n . In order to find the total number ofcycles of length n cycle in a given graph we can sum thediagonals of the graph’s adjacency matrix raised to thatpower, given by the Trace.cycle count = T r ( A n cycle ) (B2)Equation B2 was used to calculate the cycle counts inFigure 1(a). Appendix C: Derivation of < v ( t ) > for Cycles1. Pure cycle We will begin by considering influence dynamics in apure cycle, with no basal node. In a pure cycle, all nodeshave a single parent node, and so when the majority rulealgorithm randomly selects a node it takes on the state ofits sole parent. A node will switch state if and only if itis in the opposite state of its parent node. Through sym-metry considerations, there must be the same number ofnodes ready to switch from state 1 to -1 if selected as viceversa. (The number of “fringes” between blocks of 1 and-1 nodes must be the same on a circle). At each timestep,the transition probabilities of v ( t ) are therefore: P pure ( v → v + 1) = p f P pure ( v → v −
1) = p f P pure ( v → v ) = 1 − p f (C3)Where p f is the probability of selecting a child nodeat a “fringe” where the parent has a different state tothe child. There are absorbing states at v ( t ) = 0 and v ( t ) = n , where all nodes are in the same state and thereare no fringes. There is some path dependency involvedin p f ( v ), as certain configurations can only be reached byother configurations. The form of p f ( v ) is not necessaryfor our analysis. From the symmetry between equationsC1 and C2 we can see that < v ( t ) > = v (0) (C4)
2. Cycle with a basal node
Now, we add one basal node that influences one of thenodes within the cycle, which we will call the “hot” node.
Figure 11. The basal node is always in state -1. The hotand pocket nodes can be in one of 4 configurations of states.Orange nodes here are in state -1, the same as the basal node.Blue nodes are in state 1, the initial state of nodes in the cycle.
The other important node to consider is the parent of thehot node within the cycle, which we will call the “pocket”node (See Figure 11). We will consider how adding thebasal node changes the dynamics from the pure cycle. Ateach timestep, the only node effected by the basal nodeis the hot node, and so we can consider the dynamics inthe rest of the cycle as proceeding as before.The basal node is always in state x basal = −
1. Thisgives 4 possible configurations for states of the hot andpocket nodes, as shown in Figure 11. For each of theseconfigurations we will write down how the transitionprobabilities of v ( t ) differ compared to the pure cycle. a. Configuration a. x hot = 1 , x pocket = 1 . Giventhat the hot node is selected by the algorithm, and thatthe system begins in configuration a, there is a 50%chance that the basal node will influence the hot node. P basal ( v → v | hot, a ) = 12 (C5) P basal ( v → v − | hot, a ) = 12 (C6)Comparing this to the pure cycle in configuration a(without a basal node), given that the hot node is selectedby the algorithm, there is no chance of any change in v ( t ) P pure ( v → v | hot, a ) = 1 (C7) b. Configuration b. x hot = − , x pocket = 1 . Fol-lowing similar considerations as in Configuration a, wecan see that P basal ( v → v | hot, b ) = 12 (C8)3 P basal ( v → v + 1 | hot, b ) = 12 (C9) P pure ( v → v + 1 | hot, b ) = 1 (C10) c. Configuration c. x hot = 1 , x pocket = − P pure ( v → v + 1 | hot, c ) = P basal ( v → v + 1 | hot, c ) = 1(C11) d. Configuration d. x hot = − , x pocket = − P pure ( v → v | hot, d ) = P basal ( v → v | hot, d ) = 1 (C12)Considering equations C5-C12, we can adapt the purecycle transition probability equations C1, C2 and C3 tothe basal cycle, in terms of the probability of selectingthe “hot” node, P ( hot ) and the probability of being inconfiguration m , P ( m ): P basal ( v → v + 1) = p f − P ( hot ) P ( b ) (C13) P basal ( v → v −
1) = p f P ( hot ) P ( a ) (C14) P basal ( v → v ) = 1 − p f + 12 P ( hot ) P ( b ) − P ( hot ) P ( a )(C15)Assuming large n, we can write the change in the ex-pected value of v as: d < v >dt ∝ P ( v → v + 1) − P ( v → v −
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