Proceedings o the Internationa Conerence o Contro, Dynamic Systems, and Robotics Ottawa, Ontario, Canada, May 7-8, 2015 Paper o. 156 Leader-Foower Consensus Modeing Representative Democracy Subhradeep Roy, icoe Abaid Department o Biomedica Engineering and Mechanics (MC 0219 495 Od Turner Street, Virginia Tech, Backsburg, VA 24061 sdroy@vt.edu; nabaid@vt.edu Abstract- In this work, we study a eader-oower consensus protoco where both eaders and oowers negotiate their states over a stochasticay-switching network. The mode incorporates the phenomenon o numerosity, which imits the perception o exact numbers. We derive a cosed orm expression or the asymptotic convergence actor that provides a necessary and suicient condition or convergence and is used to study the expected decay rate o disagreement among agents. These resuts are vaidated with Monte Caro simuations and we expore the dependence o the asymptotic convergence actor on mode parameters using numerica simuations. This system can be used to mode decision making in a representative democracy, where representatives negotiate among themseves and drive the opinion o the popuation. Keywords: Communication networks, Consensus, Leader oower, Stochastic systems 1 Introduction Coective decision making can be modeed as a consensus protoco, deined as an agorithm or a mutiagent system with an equiibrium when a agents hod a common state. The wide range o engineering and science appications or consensus protocos, such as unmanned aeria vehices (Beard et a. 2002 and autonomous underwater vehices (AUVs (Maczka et a. 2009, is supported by a arge theoretica iterature exporing these probems (Ren & Beard 2007, Abaid & Poriri 2011, Pereira 2010, Abaid & Poriri 2012. Within this iterature, conditions to reach consensus have been studied varying the underying network o agent communication. etworks may be static (Sipahi & Acar 2008 or switching, and the atter case may be urther divided into those updating rom a deterministic sequence (Ren & Beard 2007 or rom reaizations o a random variabe (Abaid & Poriri 2011. Within the consensus iterature, cosed orm resuts or consensus conditions and convergence speed are imited to a sma set o known topoogies, or exampe, Erdos-Renyi random networks in (Pereira 2010 and numerosity-constrained (C networks in (Abaid & Poriri 2011, 2012. In socia systems, a variety o constraints restricts the communication between individuas rom a-to-a. Among these constraints, the perception o numbers impacts many socia species, incuding ish (Tegeder & Krause 1995 and humans (Piazza & Izard 2009. The so-caed numerosity constraint, deined in (Piazza & Izard 2009, imits the perception o exact numbers across species. Another striking eature o socia groups is eadership by an individua or subset o the group, studied or exampe in ish schoos (Couzin et a. 2011. This group behavior may be modeed as eader-oower consensus, which partitions the agents into two types: eaders and oowers (Abaid & Poriri 2012, Xiaohong & Qinghe 2013. In genera, the eaders have access to more inormation and attempt to drive the entire system to a desired common state through their updating protoco. We comment that the eaders whose states are time-variant and update dynamicay 156-1
according to a consensus protoco are rarey considered in the iterature (Song et a. 2013 and cosed orm expressions reating the consensus among eaders to that o the whoe system are particuary absent. In this paper, we study a eader-oower consensus protoco over a stochasticay-switching network to mode the scenario, where individuas in socia groups share inormation with a dynamic and stochastic set o peers, inspired by the numerosity constraint in bioogica systems. In (Abaid & Poriri 2012, the authors studied a simiar mode, where the eaders have a common state which is constant with time. In this work, the eaders states are time-variant and update as they interact with other eaders, whie the oowers interact with any other agent irrespective o eader or oower. We deine the dynamic eaders to be more inuentia than oowers as they ony negotiate with a subset o the tota group, whie their states propagate through the popuation via the directed interactions they may have with the oowers. This probem modes a representative democracy, where poicies are decided upon by a subset o representatives whose decisions are disseminated to the entire popuace (Mezey 2008. This ideaized system incorporates the utimate goa o democracy, in that consensus among representatives is required or making pubic poicy and consensus among a agents represents a popuace whose opinions aign with its representatives. In addition, we incorporate the numerosity constraint, which is known to impact decision making across socia species. Here, we study such a system with dynamic eaders and estabish necessary and suicient conditions or consensus in terms o the mean square stabiity o the disagreement among agents. A cosed orm expression or the asymptotic convergence actor, which measures the rate o convergence to consensus, is estabished. It is important to note that, in (Abaid & Poriri 2012, the state o consensus o the entire system was ixed by the set eaders common state, whereas in this work the eaders states start rom random initia conditions and converge over time. 2 Probem Statement We consider a system having agents, with agents serving as eaders and agents as oowers, where, 3 and + =. The sets F = {1,2,, }, L = { +1, +2,,}, and = F L are used to denote the indices o oowers, eaders, and tota number o agents, respectivey. The agents communicate over a stochasticay-switching directed network which is numerosity constrained (Abaid & Poriri 2011 during discrete time steps. At each time step, agents communicate with n randomy seected neighbors where n {1,...,min{ 1, 1}} is constant over a time steps and agents in the system. The oowers are assumed to communicate with n neighbors which are seected rom both eaders and oowers, whereas the eaders communicate with n other eaders. The communication network, at each time step k Z +, is deined through the graph Lapacian L k R. The irst rows o L k i.e. row i F, deine the oowers interaction graph, where each row represents how a oower is connected to the rest o the ( 1 agents in the system. The ast rows o L k represent the eaders interaction graph, i.e. when i L, and show the connections are restricted to the eaders. Each row o L k has a diagona entry equa to n and o-diagona entries comprising n 1 s and n 1 0 s, by deinition. Speciicay, when i F, 1 can appear aong the 1 o-diagona positions with equa probabiity, whereas when i L, the irst coumns are 0 s and the appearance o 1 s aong the remaining 1 o-diagona positions is equay ikey. In other words, the irst coumns in the eaders sub-system has a the eements equa to zero, since eaders do not receive inormation rom the oowers. Due to unidirectiona communication among the agents, L k is not necessariy symmetric, but it has zero row sum. Thus, L k 1 = 0, where the vector 1 R 1 have a entries equa to 1 and the vector 0 R 1 have a entries equa to zero. At time step k, the agents states are given by the vector x k = [x T k x T k ] T R 1, where x k R 1 represents the state vector or the oowers and xk R 1 represents the state vector or the eaders. The 156-2
state vector o the entire system is updated according to the discrete-time consensus protoco x k+1 = (I E L k x k, (1 where I is an identity matrix o size and x 0 is a random initia condition. The diagona matrix E R consists o the constant diagona entries ε > 0. The parameter ε, aso caed persuasibiity, acts as a weighting parameter and determines how the agents update their states using the inormation received rom the neighbors at each time step. We say that the system reaches consensus when agents attain common state variabe, x = s1, where s R. 3 Anaysis In this section, we deine and derive the cosed orm expression or the asymptotic convergence actor or the consensus protoco in (1. This quantity provides a necessary and suicient condition or consensus and captures the rate o convergence to consensus. 3.1 Preiminary resuts Considering a discrete-time inear system, we write x k+1 = W k x k, (2 where W k R are independent, identicay distributed random matrices. For (2 to be a consensus protoco, W k must have the property W k 1 = 1, that is, eements in span(1 are equiibria o the system. Foowing (Abaid & Poriri 2011, we project the consensus probem (1 on the disagreement space, in terms o a disagreement variabe ξ k deined as ξ k = Q T x k R 1. The matrix Q R ( 1 has the properties Q T 1 = 0, Q T Q = I 1, and QQ T = R where R = I 1 1 T. Thus, the disagreement dynamics is given by the reation, ξ k+1 = W k ξ k, where W k = Q T W k Q R ( 1 ( 1. Foowing (Zhou & Wang 2009, the asymptotic convergence actor can be written in terms o the disagreement dynamics as ( [ E ξk 2] 1/k r a = sup im k ξ 0 2 (3 ξ 0 0 where E[ ] is the expected vaue. In (Abaid & Poriri 2011, it is shown that the asymptotic convergence actor is ess than one i and ony i the disagreement system is mean square stabe, that is, i the system in (2 is mean square consentabe. We deine, using the same notation o (Abaid & Poriri 2011, G = (R R(I 2 ε(e[l] E[L] + ε 2 E[L L] (4 where and denote Kronecker product and Kronecker sum, respectivey. From (Abaid & Poriri 2011, we know that the asymptotic convergence actor is equa to the spectra radius o G. Thus we have the oowing proposition or assessing the consentabiity o the system: Proposition 1. The system (1 is consentabe in the mean square sense i and ony i Proo. The proo o the proposition can be ound in (Abaid & Poriri 2011. r a = ρ(g < 1. (5 In the next subsection, we compute G by a counting technique. Then, we write a cosed orm expression or the asymptotic convergence actor or the consensus protoco in (1 by cacuating the eigenvaues o G and associated eigenvectors and appying Proposition 1. 156-3
3.2 Main resuts The set o a possibe distinct Lapacian matrices is denoted as ˆL = {L (1,L (2,,L (p }, where p corresponds to the tota number o unique reaizations o the Lapacian matrices or a given set o parameters,, and n. Foowing simiar steps in (Abaid & Poriri 2011, 2012, Poriri 2011, 2012, we cacuate E[L] and E[L L] using a counting technique assuming that the appearance o each o these p matrices is equay ikey. The matrix E[L] has diagona components equa to n and and the o-diagona i j th components, where i j, are given by n/( 1 when i F and j ; n/( 1 when i L and j L ; and 0 when i L and j F. Thereore we can write where R = q=1 e q e T q 1 1 1 T and R = E[L] = q= +1 n 1 R + n 1 R, (6 e q e T q 1 1 1 T, the vector e q R 1 has 1 in the qth row and zeros in the remaining rows, and 1 = q=1 e q, and 1 = q= +1 e q. To compute E[L L], we note that L i j {0, 1,n}, and L L has terms o the orm L i j L km rom the deinition o Kronecker product. Thereore, L L can have vaues {0,1, n,n 2 }. To cacuate the distinct vaues or eements o E[L L], we consider the six possibe cases or the indices i, j,k, and m, that is, 1 i = j, k = m; 2 i = j,k m; 3 i j,k = m; 4 i j,k m,i = k, j = m; 5 i j,k m,i = k, j m; and 6 i j,k m,i k. In genera, the diagona bocks o E[L L] have the orm E[L L] ii = n2 1 R + n2 1 R when i, (7 and the o-diagona bocks can have three dierent orms, depending on the vaues o i and j, E[L L] i j = n2 ( 1 2 R n 2 ( 1( 1 R + 1 e i e T j 1 1 e i(1 T e T i, when i F, and j ; (8a E[L L] i j = 0, when i L, and j F ; E[L L] i j = n 2 ( 1( 1 R n2 ( 1 2 R + 2 e i e T j 2 1 e i(1 T e T i, when i L, and j L ; n( n 1 where 1 = ( 1( 2 and n( n 1 2 = ( 1( 2. Substituting (7 and (8a to (8c in (4, we derive the matrix G in bock orm. We notice that the bocks have six cases: diagona bocks G ii can have i F or i L and o-diagona bocks can have i and j beonging to either F or L. The bocks o G when both i and j F are given as oows (8b (8c G ii =θ 1 I + θ 2 1 1 T + θ 3 Iˆ + θ 4 1 1 T + θ 5 1 1 T + θ 6 Î + θ 7 1 1 T + θ 8 1 e T i + θ 9 1 e T i + θ 10 e i e T i + (i F θ 11 e i 1 T + θ 12 e i 1 T + θ 13 1 e T i + θ 14 1 1 T ; (9a G i j =θ 29 I + θ 30 1 1 T + θ 31 Iˆ + θ 32 1 1 T + θ 33 1 1 T + θ 34 Î + θ 35 1 1 T + θ 36 e i e T j + θ 37 1 e T j + θ 38 e i 1 T (i, j F, i j + θ 39 e i e T i + θ 40 1 e T i + θ 41 1 e T j + θ 42 e j e T j + θ 43 e j 1 T + θ 44 e j 1 T + θ 45 1 e T j + θ 46 1 1 T + θ 47 e i 1 T ; (9b 156-4
where Iˆ,Î R are diagona matrices and Iˆ has irst diagona entries equa to one and the remaining zeros, and Î has ast diagona entries equa to one and the remaining zeros. For G ii, when i L, the orm o (9a is retained and the coeicients θ 1,...,θ 14 are repaced with θ 15,...,θ 28. Simiary, or the o-diagona bocks o G when i F and j L ; i L and j F ; and i L and j L the orm remains the same as that o (9b and the coeicients θ 29,...,θ 47 become respectivey θ 48,...,θ 66 ; θ 67,...,θ 85 ; and θ 86,...,θ 104. The coeicients, θ i or i = 1,...,104 can be ound in the Appendix. It can be veriied that G has at most tweve distinct eigenvaues and associated eigenspaces. The eigenvectors beonging to the eigenspaces have the orm v = [v T 1 v 2T v T ] T and satisy the eigenvaue equation Gv = λv. The tweve distinct eigenvaues are given as oows λ 1 = ε 2 κ 2 n + ε 4 κ 3 n 2 ε 3 κ 5 n 2 + ε 2 κ 4 n 2 εκ 1 n + 1, (10a λ 2 = ε 2 κ 7 n εκ 6 n + 1, (10b λ 3,12 = γ 1 + (r 3,12 γ 5, (10c λ 4 = θ 1 + θ 6 θ 29 θ 34, (10d λ 5 = θ 15 + θ 20 θ 86 θ 91, (10e λ 6 = ε 2 κ 2 n ε 4 κ 3 n 2 ε 3 κ 5 n 2 + ε 2 κ 4 n 2 εκ 1 n + 1, λ 7,8 = τ 9 + (r 7,8 τ 10, (10 (10g λ 9 = θ 15 + θ 17 (θ 48 + θ 50 + ( 1(θ 86 + θ 88, (10h λ 10 = θ 1 + θ 3 θ 29 θ 31, λ 11 = 0. (10i (10j where the parameters κ 1...κ 7,γ 1,γ 5,τ 9,τ 10 are provided in the Appendix, r 3,12 are the two soutions o r o the quadratic equation: (γ 2 + γ 3 + γ 7 + r(γ 4 + γ 6 + γ 8 + γ 5 γ 1 r 2 (γ 5 = 0 and r 7,8 are the two soutions o r o the quadratic equation: τ 1 + τ 3 + r(τ 2 + τ 5 + τ 6 + τ 8 τ 9 r 2 (τ 10 = 0. The expicit deinitions o the eigenspaces associated to these eigenvaues are provided in the Appendix. Here we comment that r 1, r 6, r (1 2, and r (2 2 have highy intractabe orms and hence we do provide the expicit expressions or these ratios. The eigenvectors corresponding to the eigenspaces Γ (1, Γ (2, and Γ (6 can be shown to satisy the eigenvaue equation Gv = λ v without the expicit expressions or these ratios. It can be veriied that the eigenspaces Γ (1, Γ (2, and Γ (6 are mutuay ineary independent, as are the tripet Γ (7, Γ (8, and Γ (9 and the pair Γ (3 and Γ (12. The remaining eigenspaces are mutuay orthogona to each other. ext, we ind the eigenspace dimensions by counting the number o degrees o reedom or eigenvectors in each eigenspace, which are as oows: 1 or Γ (1, Γ (2, and Γ (6 ; 1 or Γ (3 ; 2( 1( 1 or Γ (4 ; ( 1( 2 1 or Γ (5 ; 1 or Γ (7, Γ (8, and Γ (9 ; ( 1( 2 1 or Γ (10 ; 2 1 or Γ (11 ; and 2 2 or Γ (12. Since the eigenspaces are a pairwise ineary independent, their direct sum has dimension 2. Hence, G has 2 ineary independent eigenvectors and has a spectrum comprised o {λ i } 12 i=1. The main resut oows rom Proposition 1. Theorem 1. For the C eader-oower consensus protoco in (1, with, 3, n {1,...,min{ 1, 1}}, and the associated matrix G with eigenvaues in (10a to (10j, the asymptotic convergence actor r a is given as r a = max { λ i }. (11 i=1,...,12 4 Simuations and Discussion Figure 1(a presents Monte Caro simuations or a network with = 8, = 4, n = 3,ε = 0.1 and ixed initia conditions. We observe as the disagreement system converges to zero, the magnitude o the disagreement vector decreases ineary on a ogarithmic scae. We compute a best it ine in ogarithmic scae over time 156-5
10 5 0.8 10 0 0.6 0.4 ξk 10 5 og[ra] 0.2 0 10 10 0.2 10 15 0 20 40 60 80 100 k (a 0.4 0.6 0 0.2 0.4 ǫ 0.6 0.8 (b Fig. 1: (a Magnitude o the disagreement vector or Monte Caro simuations with constant initia conditions and = 8, = 4, n = 3, and ε = 0.1. Individua markers show two hundred dierent reaizations and the back soid ine depicts the average disagreement. (b Base-ten ogarithm o r a varying with ε and = 8, = 4, n = 3. steps [10,20] and ind that the square o the disagreement norm decreases as (0.789 k, thus conirming the anaytica prediction in (11 which gives r a = 0.794 or the same set o system parameters. To study the dependence o the asymptotic convergence actor on mode parameters, we choose a system with parameters = 8, = 4, and n = 3 and pot r a varying with ε in Figure 1(b. We observe that the curve or r a as we vary ε has a characteristic shape or a admissibe vaues o,,, and n. In particuar, og[r a ] equas zero when ε = 0, decreases up to a certain negative vaue o ε, and then increases unbounded as ε. By deinition, convergence speed increases as r a decreases and the systems with r a > 1 do not converge to consensus. Thereore, we denote the maximum convergence speed as r a, where r a is minimum, and its corresponding persuasibiity as ε. To urther expore the behavior o the asymptotic convergence actor on the system parameters, we consider three speciic cases or a numerica study: (a oowers are twice the number o eaders = 2, (b eaders are twice the number o oowers = 2, and (c number o eaders is same as that o the oowers =. In Figure 2(a, Figure 2(b, and Figure 2(c, we ix and n and pot the asymptotic convergence actor in base-ten ogarithmic scae or the three aorementioned cases. In each o these pots, we observe that og[r a ] decreases as the proportion o eaders increases in each system, which means the maximum convergence speed increases as we increase the reative number o eaders with a other system parameters hed constant. Moreover, increasing the proportion o eaders resuts in the decrease o ε, which indicates that in the presence o higher proportions o eaders, the agents must be ess persuasibe or in other words more stubborn to attain maximum convergence speed. However, increasing the reative number o eaders not aways resuts in achieving aster convergence speed or system perormance at a set persuasibiity (or vaue o ε. Figure 2(a demonstrates that increasing the proportion o eaders resuts in a sight decrease o convergence speed when ε = 0.31. Aso, in Figure 2(b, and Figure 2(c, there exists ranges o ε ([0.036, and [0.293,, respectivey where og[r a ] is not susceptibe to dierent proportion o eaders. ext, to investigate the eect o group size on these trends o the asymptotic convergence actor, we compare every case o Figure 2(a with the corresponding case in Figure 2(b. We observe that when numerosity is increased proportionay with group size, both r a and ε decrease. In other words, by increasing group size, we observe maximum convergence speed increases i numerosity is kept proportionay constant, but the agents must be more stubborn to achieve it. Further, in Figure 2(a and Figure 2(c, we ix the nu- 156-6
1 1 1 0.5 0.5 0.5 og[ra] 0 og[ra] 0 0.5 0.5 1 1 1.5 0 0.2 0.4 0.6 0.8 0 0.02 0.04 0.06 0.08 0.1 ǫ ǫ 0 0.5 (a (b (c og[ra] = 2 = = 2 1 0 0.2 0.4 ǫ 0.6 0.8 Fig. 2: Comparative study o the asymptotic convergence actor or three dierent proportions o eaders and oowers and (a = 18 and n = 4, (b = 180 and n = 40, (c = 180 and n = 4. merosity and increase the group size, which resuts in a sight decrease o maximum convergence speed and corresponding persuasibiity. This is in support with the previous argument that as each agent interacts in a arger group, maximum convergence speed decreases i numerosity or each agent is not increased. Finay, we study the dependence o asymptotic convergence actor on the numerosity o individuas keeping the other system parameters constant. Comparing each case o Figure 2(b with that o Figure 2(c, we observe ra and ε decrease between the igures at the constant group size. This demonstrates that, as we ix the group size and increase numerosity or each agent, the system achieves a aster maximum convergence speed at a signiicanty ower vaue o persuasibiity. In other words, as a consequence o increasing numerosity, the agents are enabed with increasing inormation exchange among themseves, and this resuts in aster maximum convergence speed, whie simutaneousy requiring that the agents to be more stubborn. 5 Concusion In concusion, we deine a discrete-time eader-oower consensus protoco, where both eaders and oowers negotiate their states over a stochasticay-switching network. We determine a cosed orm expression o the asymptotic convergence actor, which measures the rate o convergence to consensus. Finay, we expore the dependence o the asymptotic convergence actor on mode parameters, which are group size, proportion o eaders, numerosity, and persuasibiity using numerica simuations. We ind that the system achieves consensus aster with a higher proportion o representatives but this convergence rate is achieved ony when a individuas are more stubborn during the decision making process. In addition, increasing the popuation size necessitates increasing the connectedness o each individua to maintain the system s abiity to reach consensus, when the proportion o representatives is kept constant. Acknowedgments This work was supported by the ationa Science Foundation under grant EEC-1342176 and by the Institute or Critica Technoogy and Appied Science at Virginia Tech. Reerences Abaid,. & Poriri, M. (2011, Consensus over numerosity-constrained random networks, IEEE Transactions on Automatic Contro 56(3, 649 654. 156-7
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6 APPEDIX The eigenspaces o G in (9 are Γ (1,(6 = {v R 2 : v i = (µ 1 µ 2 e i + µ 2 1 + µ 3 1 : (µ 1 µ 2 + µ 2 + µ 3 = 0, µ 3 µ 2 = r 1,6 when i F and v i = µ 3 1 + µ 4 1 : µ 3 + µ 4 = 0 when i L }; Γ (2 = {v R 2 : v i = (µ 1 µ 2 e i + µ 2 1 + µ 3 1 : (µ 1 µ 2 + µ 2 + µ 3 = 0, µ 2 µ 3 = r 2 (1 when i F and v i = µ 3 1 + µ 4 1 + (µ 5 µ 4 e i : µ 3 + µ 4 + (µ 5 µ 4 = 0, µ 3 µ 4 = r 2 (2 when i L }; (12b Γ (3 = {v R 2 : v i = k= +1 µ k e k : j= +1 ( β +n + +m (m n = r3 (µ +n + µ +m 2 Γ (4 = {v R 2 : v i = µ i je j : j= +1 j= +1 µ j = 0 wheni F and v i = µ i 1 + k= +1 β i k e k : β +m +m = (r 3 µ +m, (12a where m,n L when i L }; (12c µ i j = 0, i=1 µ i j = 0 when i F and v i = µ i je j : µ i j = 0, j=1 j=1 µ i j = 0 when i L }; (12d i= +1 Γ (5 = {v R 2 : v i = 0 when i F and v i = Γ (7,(8 = {v R 2 : v i = (µ i 1 + where m,n F and j=1 Γ (9 = {v R 2 : v i = (µ i 1 + and j=1 k= +1 µ i k e k : k= +1 β i k e k : β k k = (r 7,8 µ k, β n m(m n = µ j = 0 when i F and v i = µ j = 0 when i F and v i = Γ (10 = {v R 2 : v i = β i k e k : β k k = 0,β n m(m n = µ i k e k : µ i i = 0, µ k i = 0, µ k i = 0, µi i = 0 when i L }; (12e i= +1 ( + r7,8 2 (µ n + µ m µ k e k when i L }; (12 ( (µ n µ m where m,n F µ k e k when i L }; (12g µ i k = 0, Γ (11 = {v R 2 : v = ω 1 and v = 1 ω,ω R }; Γ (12 = {v R 2 : v i = k= +1 µ k e k : j= +1 i=1 µ i k = 0 when i F and v i = 0 when i L }; (12h µ j = 0 when i F and v i = ν i 1 + k= +1 β i k e k : (12i ( β +m (µ +n = +n + ν +m and ν j = 0 where m,n L when i L }; j= +1 (12j The parameters used in (9 and in (10 are ( 1 α 1 = ε 2 n 2 ( 1 2 + 1 εn; α 2 = α 1 + ε2 n 2 ( 1 2 ; α 3 = α 5 = ε2 n 2 ( ( 1 2 + εn ( 1 ; α 6 = ε 2 n 2 ε 2 n 2 ( 1( 1 + θ 6; α 4 = ε2 n 2 ( 2 ( 1 2 + 2 ( 1( 1 1 1 1 1 ( 1 2 + εn 1 ; + εn 1 ; α 7 = 156-9
ε 2 n 2 ( 1 1 1 + 2 + εn ( 1 ( 1 ; θ ( 1(1 εn εn( 1 1 = ( 1 ; θ 2 = + ( 1θ 11 θ 1 ; ( ( α1 θ 3 = α 1 + θ 11 ; θ 4 = + θ 11 ; θ 5 = θ ε 2 n 2 1 3 ; θ 1 + 1 εn( 1 6 = ( 1 ( 1 ; θ 7 = θ 6 ; θ 8 = 1 1 1ε 2 ; θ 9 = θ 8 θ 10 ; θ 10 = 1ε 2 1 ; θ 11 = 1ε 2 ( 1 ; θ 15 = θ 1 εn 1 ; θ 16 = α 2 2 + θ 11 θ 15 ; ( α2 θ 17 = α 2 + θ 11 ; θ 18 = + θ 11 ; θ 19 = θ ( 17 ; θ α3 20 = α 3 + θ 26 ; θ 21 = + θ 26 ; θ 23 = θ 8 θ 27 θ 24 ; θ 24 = 2ε 2 ( 1 ; θ 26 = θ 24 ; θ 27 = 2ε 2 ; θ 28 = ( 2θ 26 ; θ 29 = 2εn ε n 1 1 + 1 ; θ 30 = 1 ( α4 ( +( 1θ 11 + θ 10 θ 29 ; θ 31 = α 4 + θ 11 ; θ 32 = α 4 θ 11; θ 33 = α 4 + θ ε 2 n 2 11 2 ; θ 34 = ( 1( 1 + εn ( 1 ; θ 35 = θ 34 ; θ 36 = 1 ε 2 ; θ 37 = θ 8 θ 10 θ 36 ; θ 48 = 1 ( α εn( 5 1 ; θ 49 = + θ 11 + θ 10 θ 48 ; ( 1 θ 50 = α 5 + θ 11 ; θ 51 = α 5 θ 11; θ 52 = α ( 5 + θ 11 θ72 ; θ 53 = θ 26 + θ 72 ; θ 54 = + θ 26 ; θ 73 = θ 72 ; θ 56 = θ 8 θ 27 θ 24 θ 36 ; θ 68 = + ( 1θ 11 θ 48 θ 86 = εn ( ( 1 + 1 ( 1 + 2 ( 1 + 1 1 ; θ 87 = ( α6 θ 89 = + θ 11 ; θ 90 = α 6 + θ ( 11 α7 ; θ 92 = + θ 26 α 5 α 6 α 1 ; θ 72 = εn ( 1 ε2 n 2 ( ( 1( 1 ; θ 96 = 2ε 2 1 ; + θ 11 θ 86 ; θ 88 = α 6 + θ 11 ; θ 91 = α 7 + θ 26 ; ; θ 93 = 2 ε 2 ; θ 94 = θ 8 θ 27 θ 24 θ 93 ; θ 103 = ( 2θ 26 + θ 96 ; θ 12 = θ 40 = θ 43 = θ 44 = θ 59 = θ 81 = θ 82 = θ 11 ; θ 13 = θ 14 = θ 25 = θ 45 = θ 46 = θ 47 = θ 62 = θ 66 = θ 74 = θ 76 = θ 77 = θ 78 = θ 83 = θ 84 = θ 85 = θ 95 = θ 100 = 0; θ 22 = θ 41 = θ 60 = θ 79 = θ 98 = θ 8 ; θ 39 = θ 42 = θ 58 = θ 80 = θ 38 = θ 57 = θ 10 ; θ 55 = θ 36 ; θ 61 = θ 99 = θ 24 ; θ 63 = θ 97 = θ 101 = θ 26 ; θ 65 = θ 28 ; θ 67 = θ 48 ; θ 69 = θ 50 ; θ 70 = θ 51 ; θ 71 = θ 52 ; θ 75 = θ 9 ; θ 102 = θ 64 = θ 27 ; θ 104 = θ 96 ; K 1 = 2n( 1( + 1 ( 2 ( + 2 + 3 3 2 + 3 2 + n 2 ( 4 ( 2 2 3 2 ( 2 3 6 2 + 5 + 2 + 2 ( 3 2 + 2 + ( 2 ( 2 + 1 2 + ( 2( 1 2 ( + 1 2 ; K 2 = n ( ( ( 2 ( 2 3 + 3 2 3 + 2 2 + ( 1( + 2( + 1; K 3 = ( 2 3 + 2 ( 2 n + (n + 1 + n ( 2 + 1 + ( 1 2 ; κ 1 = 2 + (3 2 + 2( 2 ; ( 2( 1 K 3 κ 2 = 2( 2( 1 3 ; κ 3 = κ 7 = K 1 4( 2( 1 4 ; κ 4 = 2 ( 1 2 ; κ 5 = K 2 ( 2( 1 3 ; κ 6 = 2 1 ; n ( 1 2 + n + 1; κ 8 = κ 1 κ 6 ; κ 9 = κ 7 κ 2 ; κ 10 = κ 2 9 κ 3 ; κ 11 = κ 2 8 κ 4 ; κ 12 = κ 5 + 2κ 8 κ 9 ; γ 1 = (θ 103 + θ 19 θ 26 θ 28 θ 90 + θ 15 + θ 17 θ 86 θ 88 ; γ 2 = (θ 103 + θ 19 θ 21 θ 26 θ 28 θ 90 + θ 92 ; γ 3 = ( θ 26 + θ 48 + θ 72 θ 86 θ 91 ; γ 4 = θ 15 + θ 20 θ 86 θ 91 ; γ 5 = γ 6 = θ 23 θ 26 θ 94 ; γ 7 = 2 θ 96 ; γ 8 = θ 93 θ 96 ; τ 1 = ( θ 29 θ 31 + θ 48 + θ 50 ; τ 2 = θ 1 θ 29 θ 31 + θ 3 ; τ 3 = θ 33 θ 5 ; τ 5 = θ 10 θ 36 ; τ 6 = θ 11 θ 37 + θ 9 ; τ 8 = θ 10 ; τ 9 = θ 15 + θ 17 (θ 48 + θ 50 + ( 1(θ 86 + θ 88 ; τ 10 = θ 10 ; 156-10