Controllability of the Voter Model: an information theoretic approach
CControllability of the Voter Model: an information theoretic approach
Pierre-Alain Toupance b , Laurent Lef`evre b , and Bastien Chopard a a University of Geneva, Geneva, Switzerland b Univ. Grenoble Alpes, LCIS, F-26902, Valence, France
Abstract
We address the link between the controllability or observability of a stochastic complex system and concepts of infor-mation theory. We show that the most influential degrees of freedom can be detected without acting on the system, bymeasuring the time-delayed multi-information . Numerical and analytical results support this claim, which is developedin the case of a simple stochastic model on a graph, the so-called voter model. The importance of the noise whencontrolling the system is demonstrated, leading to the concept of control length . The link with classical control theoryis given, as well as the interpretation of controllability in terms of the capacity of a communication canal.
Keywords: reachability and observability analysis, Mutual and multi-information, voter model
1. Introduction
Causality is an important concept in many areas ofscience [8]. It helps to better understand the behaviorof complex dynamical systems. In particular, it revealshow the different degrees of freedom of a system influenceeach other. In this paper we investigate how causality(considered in a pragmatic and intuitive way) can be usedto discover efficient control strategies in a complex system.The key ingredient of our approach is the concept ofthe most influential components in a complex system. Thisnotion is defined here as the impact of controlling a givenvariable on the behaviour of the other variables. For in-stance, one can measure the change in the joint probabilitydistribution (Kulback-Leibler divergence) when the valueof a selected variable is imposed. Alternatively, we canmeasure the variation of an average quantity when a per-turbation is applied. The variable for which this change isthe most important is labelled as the most influential. Fol-lowing this procedure we can rank the degrees of freedomof a system from the most to the less influential. Arguablythe notion of influence depends on the quantity used tomeasure the effect of forcing the variable. Then, to con-trol this quantity in the system, it will be more effectiveto act on the corresponding most influential nodes.The aforementioned procedure is intrusive in the sensethat it requires to act on the system to be able to deter-mine the effects of a perturbation. Here we would liketo consider a non-intrusive approach, essentially based onthe observation of the system. The non-intrusive approachwe proposed is based on a time delayed multi-informationmeasure on the free system. This procedure can be per-formed by simple sampling on the system variables, evenif the underlying dynamics is unknown, like for instance infinancial systems. For now we consider, as a benchmark system, the so-called voter model described in the next section. The mostinfluential nodes can be determined by controlling succes-sively each variables and measuring the impact on the av-erage opinion of the entire group. We will show that thesame ranking of influence can also be obtained by moni-toring the time-delayed multi-infomation.The determination of the most influential variables hasa clear connection with the well developed theory of con-trol, in which observability and controllability of a systemare defined and explored. In section 5 we make the linkbetween the standard concepts of control theory and ourpresent approach. A important element of our discussionis related to the effect of noise on the possibility to con-trol a system. The voter model shows that in presenceof noise the influential nodes cannot force the opinion ofthe far enough agents, despite the existence of a connect-ing path. This result shows the limit of some previousapproaches about the controllability of systems on a com-plex network [7].The paper is organized as follows: section 2 introducesour voter model, then section 3 demonstrates the link be-tween influence and time-delayed multi-information. Sec-tion 4 solves the 1D voter model analytically, in the mean-field regime and gives a formal link between influence anddelayed multi-information. The link between the controllength and the capacity of a communication channel is alsogiven. Section 5 proposes a formulation of the 1D votermodel in the usual framework of control theory. Loss ofcontrollability is related to the noise intensity and the costof controllability is expressed with a Gramian.
Preprint submitted to Elsevier November 18, 2019 a r X i v : . [ ee ss . S Y ] N ov . Voter Model Simple models that abstracts the process of opinionformation have been proposed by many researchers [4, 6].The version we consider here is an agent-based model de-fined on a graph of arbitrary topology, whether directedor not.A binary agent occupies each node of the network. Thedynamics is specified by assuming that each agent i looksat every other agent in its neighborhood, and counts thepercentage ρ i of those which are in the state +1 (in casean agent is linked to itself, it obviously belongs to its ownneighborhood). A function f is specified such that 0 ≤ f ( ρ i ) ≤ i to be in state+1 at the next iteration. For instance, if f would be chosenas f ( ρ ) = ρ , an agent for which all neighbors are in state+1 will turn into state +1 with certainty. The update isperformed synchronously over all n agents.Formally, the dynamics of the voter model can be ex-press as s i ( t + 1) = (cid:26) f ( ρ i ( t ))0 with probability 1 − f ( ρ i ( t )) (2.1)where s i ( t ) ∈ { , } is the state of agent i at iteration t ,and ρ i ( t ) = 1 | N i | (cid:88) j ∈ N i s j ( t ) . (2.2)The set N i is the set of agents j that are neighbors of agent i , as specified by the network topology.The global density of all n agents with opinion 1 isobviously obtained as ρ ( t ) = 1 n n (cid:88) i =1 s i ( t ) (2.3)In what follows, we will use a particular function f ,(see Fig. 1) f ( ρ ) = (1 − (cid:15) ) ρ + (cid:15) (1 − ρ ) = (1 − (cid:15) ) ρ + (cid:15) (2.4)The quantity 0 ≤ (cid:15) ≤ / G , as simple instance of a socialnetwork [2]. We use the algorithm of B´ela Bollob´as ([3])to generate this graph.Figure 2 shows the corresponding density of agentswith opinion 1, as a function of time. We can see thatthere is a lot of fluctuations due to the fact that states “all0’s” or “all 1’s” are no longer absorbing states when (cid:15) (cid:54) = 0.
3. Characterisation of the influence of an agent
In this study we would like to characterize how theopinion of one agent influences that of its neighbors and ε=0.1 p r ob t o b ec o m e Transition function, voter model
Figure 1: The probablility f ( ρ ) used in this study. The noise (cid:15) isvisible as the values f (0) and 1 − f (1).Figure 2: Graph of the time evolution of the density of opinion 1 withnoise (cid:15) = 0 ,
001 and n = 200 agents connected through a scale-freenetwork. that of the entire system. We will first propose an ap-proach based on information theory, and then measure theinfluence directly by forcing (or controlling) the opinion ofone agent. We will show that both characterizations arestrongly correlated. The information theoretic quantitiesthat will be considered are the time-delayed mutual infor-mation and the time-delayed multi-information. The pur-pose of considering a time delay is to capture the causaleffect of one element on another. Let us consider a set of random variables X i ( t ) associ-ated with each agent i , taking their values in a set A . Forinstance, X i ( t ) = s i ( t ) would be the opinion of agent i atiteration t .To measure the influence between agents i and j , we2efine the τ -delayed mutual information w i,j as w i,j ( t, τ ) = I ( X i ( t ) , X j ( t + τ )) (3.1)= (cid:88) ( x,y ) ∈ A p xy log (cid:16) p xy p x p y (cid:17) (3.2)with p xy = P ( X i ( t ) = x, X j ( t + τ ) = y ) p x = P ( X i ( t ) = x ) and p y = P ( X j ( t + τ ) = y )We also define the τ -delayed multi-information w i tomeasure the influence of one agent i on all the others w i ( t, τ ) = I ( X i ( t ) , Y i ( t + τ )) (3.3) Y i ( t + τ ) = (cid:88) k (cid:54) = i X k ( t + τ ) (3.4)These information metrics can be computed by themethod of sampling. We consider N = 10 instances ofthe system in order to perform an ensemble average. Ac-cording to the central limit theorem, we know that, withthis number of instances, we obtain a precision of 3 × − with a risk of 5% for the approximate values of the prob-abilities that we compute (see Appendix B for details). The τ -delayed multi-information can be used as a mea-sure of the influence of opinion of each node i on the vote ofthe other agents. For instance, Fig. 3 shows w i ( τ = 2) in asteady state, where the origin of time is arbitrary. We ob-serve that some agents i exhibit a more pronounced peak ofmulti-information towards the rest of the system, suggest-ing that the opinion of these agents may affect the globalopinion of all agents. Note that this results is obtainedonly by probing the systems, without modifying any of itscomponents. For this reason, we describe this approach as“non-intrusive”. The algorithms used throughout this pa-per to numerically evaluate the delayed mutual- and multi-informations in the voter model example are described inAppendix C In this section, we consider another way to measure theinfluence of an agent on the system. We call this approach“intrusive” as it implies a perturbation, and no longer justan observation.To measure the influence of agent i , its opinion is forcedto a chosen value, for instance the value 1. As a result thedensity (2.3) of opinions 1 on the system ρ ( t ) = 1 n n (cid:88) j =1 s j ( t ) (3.5)can be averaged over a large number N of independentrealizations, to give a quantity (cid:104) ρ ( t ) (cid:105) i , where the subscript Figure 3: τ -delayed multi-information w i ( τ ) ( τ = 2) as a function of i , for graph G with n = 40 agents and noise level (cid:15) = 0 . i indicate which agent has been forced to 1. If t is largeenough, (cid:104) ρ (cid:105) i no longer depends on t .The influence can be measured in a steady state, orfrom the initial state where all agents are initialized uni-formly to 0 or 1 with probability 1.The color representations of the graphs (Figures 4 and5) show that the multi-information give some informationabout the controllability and the observability of the sys-tem. In the case the multi-information is calculated fromthe initial state, these figures emphasize the link betweenthe multi-information and the influence of an agent. Wecan then identify the agents that allow the best control ofthe system when their vote is forced.The measurement obtained in the steady state for thedelayed multi-information is different from that observedin the transient regime. Low-impact agents can get a highmulti-information by being a proxi of an influential neigh-bor. In this case, the multi-information rather evaluatesthe observability than the controlability.
4. The 1D Voter model
The previous section gave an illustration of the link be-tween influence defined by intrusive forcing and the influ-ence measured by observing the time-delayed multi-information.In this section, we propose an analytical meanfield solu-tion of the voter model, in a one-dimensional topology.This solution will formally specify the proposed links. Inparticular we will introduce a characteristic control length.
We consider the case of n voters organized along a lineso that voter i looks at voter i − i = 0 has no left neighbor and will havea controlled dynamics. For instance its opinion will be3
12 34 5 678 9101112131415 16 17 1819 20 2122 23 242526 27 2829 303132 33 343536 37 3839 404142 434445 4647 4849
Intensity of influence Initial regime
Intensity of delayed multi-info (tau=4)
Figure 4: Scale free graph colored as a function of the values of the influence (left) and the τ -delay multi-information (right), for τ = 4. Inthis case, the multi-information is computed from the initial state. always 1. The other agents are initialized randomly in { , } .Since agent 1 is looking at agent 0, its next state willlikely to be 1. And so on for agent 2 , , . . . , n . Intuitively,we could expect that the entire system will become 1, dueto the control imposed by agent 1. But noise is changingthis conclusion.If p i ( t ) is the probability that agent i is 1 at time t , wecan write the equation p i ( t + 1) = p i ( t ) W → ( t ) + (1 − p i ( t )) W → ( t ) (4.1)where W a → b is the probability that the state evolves from a to b . In a meanfield approximation, we can write, W → = p i − ( t ) f (1 /
2) + (1 − p i − ( t )) f (0) W → = p i − ( t ) f (1) + (1 − p i − ( t )) f (1 /
2) (4.2)Before attempting to solve the above system analytically,we can observe its behavior numerically. We can see onFig. 6 that is the noise is small ( (cid:15) = 0), the entire systemis indeed controlled by the left-most agent whose state isalways 1. But if the noise is increased ( (cid:15) = 0 .
01) thecontrol is not effective anymore. There is a critical noise (cid:15) = (cid:15) c ( n ) below which a system of size n can be controlledby the first node, and above which the influence of thedriving node is diluted by the noise. Figure 7 shows thedensity of agents with opinion 1, as a function of time, fordifferent intensities of noise, (cid:15) . We observe in this figurethe effect of the system size. For smaller systems, the effectof controlling agent i = 1 is more effective than for larger n . We can determine the probability distribution in thecase of the linear voter model. We have p i ( t + 1) = p i ( t ) W → ( t ) + (1 − p i ( t )) W → ( t )= p i ( t )( W → ( t ) − W → ( t )) + W → ( t )With W → ( t ) = p i − ( t ) f (1) + (1 − p i − ( t )) f (1 / p i − ( t )(1 − (cid:15) ) + (1 − p i − ( t )) 12and W → ( t ) = p i − ( t ) f (1 /
2) + (1 − p i − ( t )) f (0)= p i − ( t ) 12 + (1 − p i − ( t )) (cid:15) we obtain p i ( t + 1) = (cid:18) − (cid:15) (cid:19) p i ( t ) + (cid:18) − (cid:15) (cid:19) p i − ( t ) + (cid:15) (4.3)As p ( t ) = 1, we obtain p ( t + 1) = (cid:18) − (cid:15) (cid:19) p ( t ) + 12 (4.4)Let P ( t ) be the vector of probability defined by P ( t ) = p ( t ) p ( t )... p n ( t )
12 34 5 678 9101112131415 16 17 1819 20 2122 23 242526 27 2829 303132 33 343536 37 3839 404142 434445 4647 4849
Intensity of influence Steady state regime
Intensity of delayed multi-info (tau=4)
Figure 5: Scale free graph colored as a function of the values of the influence (left) and the τ -delay multi-information (right), for τ = 4. Inthis case, the multi-information is computed when the system is in a steady state regime.Figure 6: Space-time diagram of the evolution of the states of a n = 500 agents of a voter model organized in a line. Line t of thefigure depicts the configuration of the n voter at iteration t . We cansee the first 500 iterations. Left: (cid:15) = 0. Right: (cid:15) = 0 .
01 .
With this notation, the system can be expressed in a ma-trix form P ( t + 1) = AP ( t ) + B (4.5)with A = ( 12 − (cid:15) ) . . . . . .
01 1 0 . . . ...0 1 1 . . . 0... . . . . . . 00 . . . and B = / (cid:15) ... (cid:15) This equation can be solved recursively and gives P ( t ) = A t P (0) + (cid:0) t − (cid:88) j =0 A j (cid:1) B (4.6)The explicit forms for power matrices A j are given in Ap-pendix A. Let us write Π = π π ... π n the stationary distribution. From relation (4.5), it obeys A Π+ B = Π ⇔ ( 12 + (cid:15) ) π = 12 ∀ i ∈ (cid:74) n (cid:75) , π i = 1 − (cid:15) (cid:15) π i − + 2 (cid:15) (cid:15) (4.7)It is an arithmetico-geometric sequence which can be solvedfor all agents i as π = 11 + 2 (cid:15)π i = 12 + (cid:16) − (cid:15) (cid:15) (cid:17) i − ( π −
12 ) (4.8)with π −
12 = 12 (cid:18) − (cid:15) (cid:15) (cid:19) psilon=0.0005n=500 epsilon=0epsilon=0.01 d e n s it y controlled 1D voter model epsilon=0.001n=50 epsilon=0.01 d e n s it y controlled 1D voter model Figure 7: Density of agents with value 1 as a function of time, for different noise levels, and two different system sizes, n = 500 and n = 50.The dashed lines are the predictions of the meanfield analytical approach, see eq. (4.14). Further, we can write eq.( 4.8) as π i = 12 (cid:34) (cid:18) − (cid:15) (cid:15) (cid:19) i (cid:35) = 12 + 12 exp (cid:20) − i ln (cid:18) (cid:15) − (cid:15) (cid:19)(cid:21) = 12 + 12 exp (cid:20) − i(cid:96) c (cid:21) (4.9)where (cid:96) c is defined as (cid:96) c = 1ln (cid:16) (cid:15) − (cid:15) (cid:17) (4.10)and referred to as the control length as it gives a value for i above which the exponential falls quickly to zero. It is acharacteristic distance from the controlled agent where itsinluence starts to fade.We see that, when (cid:15) approaches 1 /
2, the length of con-trol (cid:96) c converges to 0, which corresponds to a total loss ofthe controlability of the system. Figure 8 shows that (cid:96) c decreases very quickly to 0 when (cid:15) increases to 1/2. In the case of a stationnary system, we can calculatethe average density of agents with vote 1. S = 1 n n (cid:88) i =1 π i with n is the number of free agents.According to (4.9), we have S = 1 n (cid:34) n n (cid:88) i =1 (cid:18) − (cid:15) (cid:15) (cid:19) i (cid:35) (4.11) Figure 8: Control length (cid:96) c as a function of the noise (cid:15) , according toeq. (4.10). When (cid:15) (cid:54) = 0, we have n (cid:88) i =1 (cid:18) − (cid:15) (cid:15) (cid:19) i = (cid:18) − (cid:15) (cid:15) (cid:19) n − (cid:88) i =0 (cid:18) − (cid:15) (cid:15) (cid:19) i (4.12)= (cid:18) − (cid:15) (cid:15) (cid:19) − (cid:16) − (cid:15) (cid:15) (cid:17) n − (cid:16) − (cid:15) (cid:15) (cid:17) (4.13)and we obtain S = 12 + 12 n (cid:18) − (cid:15) (cid:15) (cid:19) − (cid:16) − (cid:15) (cid:15) (cid:17) n − (cid:16) − (cid:15) (cid:15) (cid:17) = 12 + 12 n (cid:18) − (cid:15) (cid:15) (cid:19) (cid:20) − (cid:18) − (cid:15) (cid:15) (cid:19) n (cid:21) (4.14)In Figure 7, we see that the simulations are in agreementwith this theoretical result.6 .5. Delayed mutual information In this section we will compute the influence of anagent based on the τ -delayed mutual information, w i,j ( τ ),between agents i and j , as defined in eq. (3.2). Thesevalues are obtained by a sampling of the simulation ofthe 1D voter model, with n = 50 agents. Measurementsare performed when the system has reached a stationnarystate, that is after t iterations such that all the probabil-ities A t P (0) are smaller than a certain threshold. In ourcase, we take the threshold at 10 − .In Fig. 9, we notice that the mutual information w i,j ( τ )is zero if j < i , has a plateau for j < i + τ , shows a peakfor j = i + τ , and decreases for j > i + τ . This observationreflects the fact that agent i can only influence agents onits right as the voting decision of an agent is based on thestate of its left neighbor. The plateau shows the influenceof the past j − i iterations. The influence of i over j ismaximum for j = i + τ as it takes τ iterations for the voteof i to travel from i to j . For j > i + τ the influence is dueto the steady state regime.In Fig. 10 we consider the behavior of w i,j ( j − i ). Itsuggests the following relationship ∀ j > i, w i,j ( j − i ) = α i exp [ − λ i ( j − i )] (4.15)where α i and λ i depend on the noise level, (cid:15) .The coefficients of correlation between ln( w i,j ( j − i ))and j , for different values of the noise are found to be be-tween − − .
99, thus confirming the relation proposedin eq. (4.15). The value of α i and λ i can be determinedwith a least squares method.Consequently, the value of the delayed mutual infor-mation w i,j ( j − i ) decreases quickly as j departs from i .This reflects the difficulty to control agent j from agent i .This interpretation is confirmed by Fig. 11 which showsthe relation between the values of λ ≡ λ and the controllength (cid:96) c defined in eq. (4.10). Each point in this figurecorresponds to a different value of the noise. The relationcan be fitted by λ ( (cid:15) ) = a (cid:96) c ( (cid:15) ) + b (4.16)with a = 0 .
973 and b = − . (cid:15) . The coefficient of correlation is 0 . λ and 1 /(cid:96) c . i = kd are forced to vote 1, where k ∈ { , , , . . . } and d is givenby the control length (cid:98) (cid:96) c (cid:99) or (cid:98) λ (cid:99) . Fig. 12 shows the simulation results for d chosen as (cid:98) λ (cid:99) . The density of agents voting 1 increases significantly.The quantity nλ is the number of controlled agents and isa good indicator to evaluate the cost to control the system. In the previous section, by evaluating the mutual infor-mation, we found that the cost of control increased greatlywhen the noise increases. This result can be related to thenotion of capacity , as defined in the standard theory of in-formation. In the linear voter model, agent i + 1 can beconsidered as a channel of communication where the inputmessage is the vote of agent i at the time t and the outputmessage is the vote of agent i + 2 at he time t + 2.The information channel capacity C is defined as (see([5]) C = sup P Xi I ( X i ( t ) , X i +2 ( t +2)) = sup β ∈ [0;1] I ( X i ( t ) , X i +2 ( t +2))(4.17)With (cid:18) P ( X i ( t ) = 1) P ( X i ( t ) = 0) (cid:19) = (cid:18) β − β (cid:19) , we obtain (cid:40) P ( X i +1 ( t + 1) = 1) = (1 − (cid:15) ) P ( X i ( t ) = 1) + (cid:15) P ( X i ( t ) = 0) P ( X i +1 ( t + 1) = 0) = (cid:15) P ( X i ( t ) = 1) + (1 − (cid:15) ) P ( X i ( t ) = 0)which we write as (cid:18) P ( X i +1 ( t + 1) = 1) P ( X i +1 ( t + 1) = 0) (cid:19) = A (cid:18) β − β (cid:19) o`u A = (cid:18) − (cid:15) (cid:15)(cid:15) − (cid:15) (cid:19) Therefore, (cid:18) P ( X i +2 ( t + 1) = 1) P ( X i +2 ( t + 1) = 0) (cid:19) = A (cid:18) β − β (cid:19) where A = (cid:18) (1 − (cid:15) ) + (cid:15) (cid:15) (1 − (cid:15) )2 (cid:15) (1 − (cid:15) ) (1 − (cid:15) ) + (cid:15) (cid:19) (4.18)= (cid:18) − (cid:15) (1 − (cid:15) ) 2 (cid:15) (1 − (cid:15) )2 (cid:15) (1 − (cid:15) ) 1 − (cid:15) (1 − (cid:15) ) (cid:19) (4.19)(4.20)This corresponds to a binary symmetric channel, with aprobability of error p e = 2 (cid:15) (1 − (cid:15) ). We know that the capacity of a binary symmetric channelwith a probabiliy of error p e is C = 1 − H ( p e ) with H ( p e ) = − p e log ( p e ) − (1 − p e ) log (1 − p e )Therefore C = 1 − H (2 (cid:15) (1 − (cid:15) )) (4.21)Now, we consider all agents from i + 1 to i + m − i et i + m . Wenote C m the capacity of this channel (it depends only on m ,7 igure 9: Delayed mutual information, w i,j ( τ ), as a function of agent j , for different values of i . The different curves correspond to i = 1 , , , ,
20 et 40, from left to right, respectively. The vote of agent i = 0 is forced to 1 and the noise is (cid:15) = 0 .
01. The delay is τ = 1(left panel), τ = 3 (middle panel) and τ = 5 (right panel).Figure 10: Delayed mutual information w i,j ( j − i ) as a function of j , for agents i = 1 , , ,
20 and 30 (curves from left to right,respectively). The vote of agent i = 0 is forced to 1 and the noise is (cid:15) = 0 ,
001 (left panel), (cid:15) = 0 ,
01 (middle panel) and (cid:15) = 0 ,
05 (right panel).Figure 11: λ as a function of 1/ (cid:96) c the length of the channel). Following the same derivationas before, we obtain (cid:18) P ( X i + m ( t + m ) = 1) P ( X i + m ( t + m ) = 0) (cid:19) = A m (cid:18) β − β (cid:19) Since A is a symmetric matrix it can be cast in a diagonalform with an orthonormal basis. The eigenvalues are λ = 1 and λ = 1 − (cid:15) . Thus, A m can be expressed as A m = P T (cid:18) − (cid:15) ) m (cid:19) P with P = 1 √ (cid:18) − (cid:19) Thus A m = 12 (cid:18) − (cid:15) ) m − (1 − (cid:15) ) m − (1 − (cid:15) ) m − (cid:15) ) m (cid:19) and we obtain a symmetric binary channel of length m with a probabily of error (cid:15) m = 12 (1 − (1 − (cid:15) ) m )Therefore, the capacity of this channel is C m = 1 + (cid:15) m log ( (cid:15) m ) + (1 − (cid:15) m ) log (1 − (cid:15) m )We know that the capacity is an upper bound of themutual information for each value of (cid:15) . In Fig. 13, thecapacity C m is shown as a function of its length m , fordifferent values of the noise. The fact that the capacity C m decreases with m and with the noise, gives anotherconfirmation of the increasing difficulty to control agent i + m by forcing the vote of agent i .
5. Linear voter model and control theory
The linear voter model analysis here above may be in-terpreted in terms of reachability or observability using8 igure 12: Evolution of the density of vote 1 during the time when only the vote of the agent 1 is forced to 1 (blues curves) and when everyvote of the agents j ∝ (cid:98) λ (cid:99) is forced to 1 (greens curves). left (cid:15) = 0 . (cid:15) = 0 .
01, right: (cid:15) = 0 .
05 . classical tools from system (control) theory (see [1], chap-ter 4 for an introduction to the control notions used here-after). Let us consider again a linear topology with n + 1voting agents. In section 4, we mostly considered the casewhere agent was forced to vote 1. Here we consider amore general case. For l, m ∈ { , . . . , n } , forcing the voteof agent l may be considered as a control action, whileobserving the vote of agent m may be considered as anoutput measurement. Since we are interested in the devi-ation from 1/2 of the probability to vote 1 (thus measuringthe influence of a forcing action, for instance), we definethese deviations as state space variables˜ p i ( t ) := p i ( t ) − ∈ (cid:20) −
12 ; 12 (cid:21) (5.1)for all t ≥ i ∈ { , . . . , n } . We will consider in thesequel, with no loss of generality, a forcing of agent 0 voteand an observation of agent n vote, since the influencein the considered linear voter is unidirectional (from leftto right). Therefore, the input variable, ˜ u , and outputvariable, ˜ y , will be defined as˜ u ( t ) := p ( t ) −
12 ; ˜ y ( t ) := p n ( t ) −
12 (5.2)Using these state space, input and output variables, thedynamical voter model (4.5) transforms into the state spacesystem˜ p ( t + 1) = A ˜ p ( t ) + b ˜ u ( t ) (5.3)˜ y ( t ) = c T ˜ p ( t )with the state vector ˜ p ( t ) := [˜ p ( t ) , . . . , ˜ p n ( t )] T ∈ R n andthe internal dynamics matrix (generator) A defined as A := (cid:18) − (cid:15) (cid:19) . . . . . .
01 1 0 . . .
00 1 1 . . . 0... . . . . . . . . . 00 . . . (5.4) The control column matrix b and observation row matrix c T are defined respectively as b := (cid:18) − (cid:15) (cid:19) and c T := (cid:0) . . . (cid:1) (5.5)For any time t ≥
0, any initial probability distribution˜ p (0) := ˜ p ∈ (cid:2) − ; (cid:3) n and any control (forcing) signalvalues ˜ u ( t ) ∈ (cid:2) − ; (cid:3) , the solution φ (˜ u ; ˜ p ; t ) of the statespace equations (5.3) may be written φ (˜ u ; ˜ p ; t ) = A t ˜ p + t − (cid:88) j =0 A ( t − − j b ˜ u ( j ) (5.6)Note that the matrix A has a unique eigenvalue λ ( A ) =( − (cid:15) ), with multiplicity n and such that | λ ( A ) | < (cid:15) satisfies 0 ≤ (cid:15) < ). Therefore the trajectory(5.6) is bounded when t → ∞ and the dynamical system(5.3) is said stable .A state ˜ p ∈ (cid:2) − ; (cid:3) n is said unobservable if the corre-sponding output can not be distinguished from the outputassociated with the zero state, that is if y ( t ) = c T φ (0; ˜ p ; t ) = 0 (5.7)for all t ≥ u is set to zero). Thewhole state space system (5.3) is said observable if the setof unobservable states reduces to { } . With the solution(5.6) and Cayley theorem, it is easy to prove that this isthe case if and only if the observability matrix O n = (cid:2) c ; A T c ; . . . ; ( A n − ) T c (cid:3) T (5.8)is full rank or when the infinite observability Gramian W o := lim n →∞ O Tn O n = ∞ (cid:88) k =0 ( A k ) T cc T A k (5.9)is strictly positive definite.9 igure 13: Capacity C m of the channel between agent i and i + m ,as a function of m , for different noise levels (cid:15) . The infinite observability Gramian gives additionnalquantitative information about how much the system or aparticular state is observable. Indeed, the largest obser-vation energy (i.e. the maximum energy for the outputsignal) is reached when t → ∞ and equals (cid:107) ˜ y (cid:107) := lim t →∞ t (cid:88) k =0 | ˜ y ( k ) | = ˜ p T W o ˜ p (5.10)for any given state space trajectory φ (0; ˜ p ; t ). Therefore,with the appropriate change of state space coordinates, thecomponents of the initial condition (or subspaces) may bere-ordered, from the less to the most observable ones. Ifsome of the infinite horizon observability Gramian eigen-value are zero, then the corresponding vector spaces areunobservable. If some of these eigenvalues are small, theninitial conditions variations in the corresponding subspaceswill cause low energy variations in the output signal.In the linear voter model example, rather than mea-suring the influence of forcing permanently the agent 0 tovote 1 (with a constant input signal ˜ u ( t ) = , ∀ t ≥
0) onthe vote of agent n , we could instead analyze to effect of considering the initial probability distribution˜ p := [1 , , . . . , T ∈ R n +1 (5.11)on agent n +1, by measuring the corresponding observationenergy. We will consider a long range time horizon k > n for which the influence of the initial state of agent 1 hasreached agent n + 1 in the line. The last row of matrix A k may be written (see Appendix A): A k ( n +1 , · ) := ( − (cid:15) ) k (cid:104)(cid:0) kn (cid:1) (cid:0) kn − (cid:1) . . . (cid:0) k (cid:1) k (cid:105) when k ≥ n ( − (cid:15) ) k (cid:104) . . . (cid:0) kk (cid:1) . . . (cid:0) k (cid:1) k (cid:105) when k < n (5.12)According to definition (5.9), since we are measuring thevote of agent n +1, we get for the components of the infiniteobservability Gramian W oi,j := ∞ (cid:88) k =0 A k ( n +1 ,i ) A k ( n +1 ,j ) (5.13)for all i, j ∈ { , . . . , n } . Measuring the influence of theinitial vote of agent 1, we start with the initial probabil-ity distribution (5.11) and get, for the agent n + 1, theobservation energy (cid:107) ˜ y (cid:107) := ∞ (cid:88) k =0 (cid:16) A k ( n +1 , (cid:17) = ∞ (cid:88) k =0 ( 12 − (cid:15) ) k (cid:16) A k ( n +1 , (cid:17) (5.14)With equation (5.12), one gets (cid:107) ˜ y (cid:107) = ∞ (cid:88) k = n ( 12 − (cid:15) ) k (cid:18) kn (cid:19) (5.15)= 1( n !) ( 12 − (cid:15) ) n ∞ (cid:88) p =0 ( 12 − (cid:15) ) p (cid:18) ( p + n )! p ! (cid:19) Using the lower bound( p + 1) n < ( p + n )! p ! (5.16)one gets 1(( n − ( 12 − (cid:15) ) n − (cid:15) )(1 + 2 (cid:15) ) < (cid:107) ˜ y (cid:107) (5.17)On the other hand, since ∞ (cid:88) p =0 ( 12 − (cid:15) ) p (cid:18) ( p + n )! p ! (cid:19) ≤ (cid:32) ∞ (cid:88) p =0 ( 12 − (cid:15) ) p ( p + n )! p ! (cid:33) = (cid:88) k ≥ n ( 12 − (cid:15) ) k − n k ( k − . . . ( k − ( n − = (cid:18) n !2 n +1 (1 + 2 (cid:15) ) n +1 (cid:19) (5.18)10e get the following upper bound for the observation en-ergy (cid:107) ˜ y (cid:107) < − (cid:15) ) n (1 + 2 (cid:15) ) n +2 (5.19)It is worthwile to notice how this upper bound behaveswith the number of agents along the line and with the noise (cid:15) . For instance, the upper bound (5.19) decreases withthe number of agents and the corresponding observationenergy is divided by two when k supplementary agents areadded in the line, with k ≥
12 1log (cid:16) (cid:15) − (cid:15) (cid:17) = (cid:96) c (cid:107) ˜ y (cid:107) = O (cid:0) (1 − (cid:15) ) n (cid:1) → (cid:15) → (cid:18) (cid:19) − (5.21)The lower bound (5.17) decreases similarly, with the sameorder, when the noise decreases. However it decreasesmuch faster with the number of agents in the voter linesince this lower bound for the observation energy is di-vided by (cid:0) − (cid:15) n (cid:1) when only one agent is added to the n previous ones.Note that we performed the observability analysis onthe linear voter model. We could as well develop the dualreachability analysis for the same example. In this anal-ysis, the initial condition is assumed to be zero and oneanalyzes the forced solution of the state space model (5.3).More specifically, one could be interested in its reachabil-ity property. A state ˜ p is said reachable when there is aan input signal ˜ u ( t ) such thatlim t →∞ φ (˜ u, , t ) = ˜ p It may be proved (see, e.g. [1]) that, among those inputsignals which can reach the state ˜ p from a zero intial con-dition, the one with minimum energy may be written as (cid:107) ˜ u ∗ (cid:107) = ˜ p T ( W c ) − ˜ p where the infinite reachability Gramian W c is defined as W c := ∞ (cid:88) k =0 A k bb T ( A T ) k (5.22)Therefore, a reachability Gramian analysis may be usedto compute the forcing of agent 1 with minimal energyrequested to reach a state ˜ p where all agents in the linevote 1, that is such that ˜ p i = 1, for all i ∈ , . . . , n + 1.However, in this case, it would be necessary to computethe sum of all the elements in ( W c ) − , which is a muchmore involved computation than the one performed forthe observability analysis. Besides, the duality betweenreachability and observability for linear systems [1] andthe particular topology of the linear voter model lead usto the conjecture that the reachability analysis would notbring any new result fundamentally different from the onesobtained through the observability analysis.
6. Conclusions
In this paper we show that time delayed mutual- andmulti-informations are promising tools to better grasp thebehavior of a dynamical system on complex networks. Inparticular it can be used to determine the most influentialdegrees of freedom and the most observable variables. Thisknowledge can be obtained without perturbing the system,by just probing its behavior.We claim that influential nodes are those that are themost interesting to control or monitor to (i) force a systemto reach a given target, or (ii) to have a proxy giving aninformation on the state of the entire system.We illustrated our approach in a simple stochastic dy-namical model on a graph, a so-called voter model, whereagents iteratively adapt their opinion to that of the ma-jority of their neighbors, with however a given noise level.We first discussed the case of a general scale-free topology,where only numerical results can be obtained. Then weconsider a 1D topology for which analytical results can beobtained. There, we rigorously showed that the influenceof an agent on the entire system can be equivalently mea-sured by actually forcing its behavior, or, in a non-intrusiveway, by measuring the time delayed multi-information ofthis agent with respect to the rest of the system. In par-ticular, we proposed the concept of a control length, whichindicates a characteristic distance above which the influ-ence of a controlled agent fades exponentially.The link with classical control theory has been pro-posed and the control length has been related to the reach-ability Gramian, thus indicating that the cost of controlbecomes intractable at large distance. The importance ofthe noise is clearly shown as being a central element in thepossibility of observing or controlling a system, as opposedto previous literature that claimed that a causality pathwas sufficient to achieve control [7].As an additional link of our approach to existing con-cepts, we showed that controllability can also be consid-ered in the framework of the capacity of communicationchannel, as defined in information theory by Shannon. Weshowed that this capacity drops as agent are separated bya distance above the control length.In a forthcoming paper we will apply our approach toother complex systems, in particular those for which theunderlying dynamics and topology of interaction are notknown. We already obtained (not shown here) that thetime delayed multi-information can be used to infer thetopology of the graph of Fig. 4. Further, we want to use theconcept of observability as a way to detect early warningsignal of possible tipping points in a complex dynamicalsystem. In simple words, we want to analyze the idea thatthe most influential degree of freedom is the best variableto observe to know in advance if a given system is likely tomove to another regime. These nodes being the most influ-ential ones, we can argue that their evolution will dictatethe evolution of the other variables.11 cknowledgment
We thank Gregor Chliamovitch and Alex Dupuis forinitiating several of the ideas developed in this paper, dur-ing the FP7 project SOPHOCLES (2012-2015), and forthe reading of the manuscript.
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Causality: Models, Reasoning and Inference . Cam-bridge University Press, 2009. ppendix A. Explicit evaluation of A p In 4.6, we need to calculate A p with A = ( 12 − (cid:15) ) . . . . . .
01 1 0 . . . ...0 1 1 . . . 0... . . . . . . 00 . . .
We have A = I n + C with C = . . . . . .
01 0 0 . . . ...0 1 0 . . . 0... . . . . . . 00 . . . C is a nilpotent matrix and ∀ k ∈ N , A k = (cid:80) kp =0 (cid:0) kp (cid:1) C p . Therefore, for p (cid:62) n − A p = ( 12 − (cid:15) ) p . . . . . . (cid:0) p (cid:1) . . . . . . ... (cid:0) p (cid:1) . . . . . . . . . ...... . . . . . . . . . 0 (cid:0) pn − (cid:1) . . . (cid:0) p (cid:1) (cid:0) p (cid:1) and for p < n − A p = ( 12 − (cid:15) ) p . . . . . . . . . . . . . . . (cid:0) p (cid:1) (cid:0) p (cid:1) (cid:0) p (cid:1) . . . . . . ...... (cid:0) p (cid:1) . . . . . . . . . ... (cid:0) pp (cid:1) . . . . . . . . . ...0 . . . . . . . . . . . . 0 ...... . . . . . . . . . . . . . . . 00 . . . (cid:0) pp (cid:1) . . . (cid:0) p (cid:1) (cid:0) p (cid:1) Appendix B. Accuracy and confidence for the numerical evaluation of probability distributions
Let us consider an attribute of the members of a population which appears with probability p . For a sample of size n drawn in this population, let F n be the random variable equal to the proportion of those elements having this attribute.According to the Moivre-Laplace theorem, the quantity F n − p √ p (1 − p ) /n converges in distribution to a Gaussian distribution P ( | F n − p | (cid:54) (cid:15) ) = 1 − α ⇔ P (cid:16) | F n − p | (cid:113) p (1 − p ) n (cid:54) (cid:15) (cid:113) p (1 − p ) n (cid:17) = 1 − α ⇔ (cid:15) √ n (cid:112) p (1 − p ) = t − α/ ⇔ (cid:15) = t − α/ (cid:114) p (1 − p ) n t − α/ is the real number defined by P ( X (cid:54) t − α/ ) = α with X ∼ N (0 , p ∈ [0 ,
1] and p (1 − p ) (cid:54) . (cid:15) (cid:54) t − α/ √ n . For N = 10 and α = 0 ,
05, we have t − α/ = 1 .
96, we obtain an approximation value of p with a precision of 0 .
03, with a risk of 5%.
Appendix C. Algorithms for the computations of mutual and multi-information
Appendix C.1. Mutual information
We consider a scale free graph G , with n agents. To compute the τ -delayed mutual information between 2 agents i and j , we generate N runs. For every run, we have a matrix S defined by: for i ∈ (cid:74) , n (cid:75) , and for j ∈ (cid:74) , t + τ (cid:75) , such that S [ i ][ j ] is the state of the agent i at the moment j .We use 4 n × n matrix (N00, N01, N10 and N11), initialized to zeros. For every run, we compare the vote of theagent i at the time t and the vote of the agent j at the time t + τ for i from 1 to n for j from 1 to n if S [ i ][ t ] = 0 and S [ j ][ t + τ ] = 0 then N i ][ j ] + + endifif S [ i ][ t ] = 0 and S [ j ][ t + τ ] = 1 then N i ][ j ] + + endifif S [ i ][ t ] = 1 and S [ j ][ t + τ ] = 0 then N i ][ j ] + + endifif S [ i ][ t ] = 1 and S [ j ][ t + τ ] = 1 then N i ][ j ] + + endifend forend forWe then compute w i,j ( t, τ ), the τ -delayed mutual information at time t between agents i and j , according to defini-tion (3.2). We obtain ∀ ( i, j ) , w i,j ( t, τ ) = N i ][ j ] N log (cid:16) N i ][ j ] × N ( N i ][ j ] + N i ][ j ]) × ( N i ][ j ] + N i ][ j ]) (cid:17) + N i ][ j ] N log (cid:16) N i ][ j ] × N ( N i ][ j ] + N i ][ j ]) × ( N i ][ j ] + N i ][ j ]) (cid:17) + N i ][ j ] N log (cid:16) N i ][ j ] × N ( N i ][ j ] + N [ i ][ j ]) × ( N i ][ j ] + N i ][ j ]) (cid:17) + N i ][ j ] N log (cid:16) N i ][ j ] × N ( N i ][ j ] + N i ][ j ]) × ( N i ][ j ] + N i ][ j ]) (cid:17) (C.1) Appendix C.2. Multi-information
To compute the delayed multi-information, as for the delayed mutual information, we execute N runs, and for everyrun, we compute the state matrix S .We use two n × n matrix, N N n is the number of agents) defined by : ∀ ( i, j ) , N i ][ j ] is equal to the numberof runs where the vote of the agent i is 0 and the number of agents who voted 1 is j − t + τ . ∀ ( i, j ) , N i ][ j ] is equal to the number of runs where the vote of the agent i is 1 and the number of agents (without theagent i ) who voted 1 is j − X i ( t ) , Y i ( t + ττ