Boltzmann type opinion consensus

1. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Boltzmann type control of opinion consensus Mattia Zanella Department of Mathematics and Computer Science, University of Ferrara, Italy Joint research with: G. Albi (Munich, Germany) L. Pareschi (Ferrara, Italy) XXXIX Summer School on Mathematical Physics Ravello, September 15-27, 2014 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 1 / 26

2. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Sketch of the presentation 1 The Boltzmann Equation Complexity reduction SHBEMM 2 Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control 3 Fokker-Planck Modeling 4 Numerical results Test 1 Test 2a Test 2b 5 Conclusions 6 Bibliography Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 2 / 26

3. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography The Boltzmann Equation Let us consider D R3 open, limited and regular, and we consider x 2 D and v 2 R3 Then for t 2 [0; T] the Boltzmann Equation is @f(x; v; t) @t + v rxf(x; v; t) = Q(f; f) f(x; v; 0) = f0(x; v); where we interpret f : D R3 [0;+1) ! R+ as a probability density function and where we de

4. ned the collision operator Q(f; f)(x; v; t) = Z R3S2 [f(x; v; t)f(x;w; t) f(x; v; t)f(x;w; t)] B(jv wj; v w jv wj )ddw; with v;w post collisional velocities de

5. ned as v = v + [(w v) ] w = w + [(w v) ] : Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 3 / 26

6. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Complexity reduction Hyp.1: Spatial homogeneity of f0(x; v) f0(x; v) = f0(v) 1D(x) jDj ) f(x; v; t) = f(v; t) 1D(x) jDj ; Hyp.2: Maxwellian molecules B jv wj; v w jv wj = b v w jv wj ; Hyp. 3: Grad cut{o Z 1 1 b(x)dx +1: Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 4 / 26

7. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography SHBEMM Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules 1 2 8 : @f @t (v; t) = Z R3S2 [f(v; t)f(w; t) f(v; t)f(w; t)] b v w jv wj u(d)dw f(v; 0) = f0(v); v 2 R3; t 0 For a suitable test function ' 2 Cb(R3) SHBEMM has the following weak formulation d dt Z R3 f(v; t)'(v)dv = Z R3 Q(f; f)(v)'(v)dv := (Q(f; f); ') 1C.Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, in Handbook of Mathematical Fluid Dynamic Vol.1, 01 2C. Cercignani. The Boltzmann Equation and Its Applications, Springer 88 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 5 / 26

8. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Constrained self-organized systems We consider problems of collective behavior related to the process of alignment, like in opinion consensus dynamics. 3 Dierently to the classical approach we are interested in such problems in a constrained setting. 4 Mean{

9. eld control theory has raised lot of interest in recent years. 5 The general setting consists in a control problem involving a very large number of agents where both the evolution of the state and the objective functional of each agent are in uenced by the collective behavior of all other agents. Classical examples in socio-economy and biology are given by persuading voters to vote for a speci

10. c candidate, by in uencing buyers towards a given food or asset or by forcing animals to follow a speci

11. c path. 3G. Toscani 06, S. Motsch - E. Tadmor 13 4G. Albi - L. Pareschi - M. Herty 14, M. Caponigro - M. Fornasier - B. Piccoli - E. Trelat 13 5A. Bensoussan - J. Frehse - P. Yam 13, P. Degond - J.-G. Liu - C. Ringhofer 14, M. Burger - M. Di Francesco - P.A. Markowich - M.-T. Wolfram 14 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 6 / 26

12. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Public opinion De

13. nition Aggregate of the individual views, attitudes, and beliefs about a particular topic, expressed by a signi

14. cant proportion of a community. a aEncyplopdia Britannica Online Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 7 / 26

15. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Opinion control through leaders Let us consider a population of NF followers and NL leaders with opinions wi; ~ wk 2 I for i = 1; : : : ;NF and k = 1; : : : ;NL w_ i = 1 NF XNF j=1 P (wi;wj) (wj wi) + 1 NL XNL h=1 S (wi; ~ wh) ( ~ wh wi) ; _~ wk = 1 NL XNL h=1 R( ~ wk; ~ wh) ( ~ wh ~ wk) + u; wi (0) = wi;0; ~ wk (0) = ~ wk;0: where the control u(t) characterizes the leaders strategy u = argmin ( 1 2 Z T 0 NL XNL ( ~ wh wd)2 + h=1 NL XNL ( ~ wh mF )2 h=1 # ds Z T 0 2 u2ds mF is the followers' mean opinion, S;R are compromise functions and ; 0 are s.t. + = 1, represents the importance of the control on the overall dynamic. Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 8 / 26

16. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Instantaneous binary control Split the time interval [0; T] in M intervals of length t, let tn = t n and solve the optimal control problem in each time interval 8 : wn+1 i = wn l )( ~ wn l wn i + P(wn i ;wn j )(wn j wn i ) + S(wn i ; ~ wn i ) wn+1 j = wn j + P(wn j ;wn i )(wn i wn j ) + S(wn j ; ~ wn l wn l )( ~ wn j ) 8 : ~ wn+1 k = ~ wn k + R( ~ wn k ; ~ wn h)( ~ wn h ~ wn k ) + 2un ~ wn+1 h = ~ wn h + R( ~ wn h; ~ wn k )( ~ wn k ~ wn h) + 2un where = t=2; i; j are the indexes of the interacting followers, l the index of an arbitrary leader, h; k the indexes of two interacting leaders and the control un is given by the solution of n un = argmin 2 X p=fh;kg ( ~ wn p wd)2 + 2 X p=fh;kg ( ~ wn p mn F )2 + (un)2 o : Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 9 / 26

17. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography The feedback control In order to solve the minimization problem, we can adopt a standard Lagrange multipliers approach to compute explicitly un.6 2un = X p=fk;hg 22 ( ~ wn+1 p wd) + ( ~ wn+1 p mn+1 F ) : which can be written explicitly as 2un = X p=fk;hg

18. 2 [ ( ~ wn p wd) + ( ~ wn p mn F )]

19. 2 (R( ~ wn k ; ~ wn h) R( ~ wn h; ~ wn k ))( ~ wn h ~ wn k ); if we further approximate mn+1 F with mn F and denote

20. = 42 + 42 : 6G. Albi - M. Herty - L. Pareschi, Kinetic description of optimal control problems in consensus modeling, 2014 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 10 / 26

21. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography The Boltzmann dynamic Let us introduce the followers and leaders' distributions fF (w; t) and fL( ~ w; t) such that R I fF (w:t)dw = 1 and R I fL( ~ w; t)d ~ w = 1. Then post collisional opinions are given by Leader-leader ( ~ w = ~ w + R( ~ w; ~v)(~v ~ w) + 2u + ~1 ~D ( ~ w) ~v = ~v + R(~v; ~ w)( ~ w ~v) + 2u + ~2 ~D (~v); where ~i are random variables with zero mean and

22. nite variance ~2, 0 ~D () 1 and the feedback control is de

23. ned as 2u =

24. 2 [ (( ~ w wd) + (~v wd)) + (( ~ w mF ) +(~v mF ))]

25. 2 (R( ~ w; ~v) R(~v; ~ w))(~v ~ w); Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 11 / 26

26. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography The Boltzmann dynamic Follower-follower ( w = w + P(w; v)(v w) + 1D(w); v = v + P(v;w)(w v) + 2D(v); Follower-leader ( w = w + S(w; ~v)(~v w) + ^ ^D (w) ~v = ~v Where we introduced additional noise components 1;2; ^ with zero mean and

27. nite variances 2; ^2 respectively, and functions ~D ;D; ^D : [1; 1] ! [0; 1] represent the local relevance of diusion. Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 12 / 26

28. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography System of Boltzmann equations For a suitable choice of test functions ' 2 Cb(I) we describe the evolution of fF (w; t) and fL( ~ w; t) thanks to the system of Boltzmann equation in weak form d dt Z I '(w)fF (w; t) = (QF (fF ; fF ); ') + (QFL(fL; fF ); ') d dt Z I '(~v)fL(~v; t)d~v = (QCL (fL; fL); ') If we assume P(v:w) = P(w; v) and S 1 we obtain the system for the evolution of mean opinions mL and mF 8 : d dt mL(t) = L

29. f ~(wd mL(t)) + (mF (t) mL(t))g d mF (t) = FL~(mL(t) mF (t)); dt where ~L = L; ~FL = FL Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 13 / 26

30. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography System of Boltzmann equations We introduced the Boltzmann collisional operators 7 (QF (fF ; fF ); ') = F Z I2 ('(w) '(w))fF (w)fF (v)dwdv (QFL(fF ; fL); ') = FL Z I2 ('(w) '(w))fF (w)fL(~v)dwd~v (QL(fL; fL); ') = L Z I2 : ('( ~ w) '( ~ w))fL( ~ w)fL(~v)d ~ wd~v 7L. Pareschi and G. Toscani, Interacting Multiagent Systems:Kinetic equations and Monte Carlo methods, Oxford University Press Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 14 / 26

31. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Long time behavior For the second order moments EF (t) = R I w2fF (w; t)dw and EL(t) = 1 R I ~ w2fL( ~ w; t)d ~ w in absence of diusion we have d dt EF (t) =2F( 1)(EF (t) m2 F (t)) + ~FL2(EL(t) + EF (t) 2mL(t)mF (t)) + 2~FL(mF (t)mL(t) EF (t)) and assuming R 1 we get d EL(t) =~L dt 2( 1)(EL(t) m2 L(t))

32. 2 (2

33. )(EL(t) + m2 L(t)) +2

34. (1

35. )( wd + mF (t))mL(t) +

36. 2( wd + mF (t))2 and since mF (t);mL(t) ! wd as t ! +1 it follows that EF (t);EL(t) ! w2 d. Then the quantities Z I ff (w; t)(w wd)2dw ! 0; Z I fL( ~ w; t)( ~ w wd)2d ~ w ! 0 i.e. the steady state solutions are Dirac delta centered in wd. Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 15 / 26

37. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Quasi invariant opinion limit Let consider the scaling parameter 0, the quantities involved in our constrained model can be scaled as follows 8 = ; = ; 2 = 2; ^2 = ^2; ~2 = ~2; F = 1 cF ; FL = 1 cFL ; L = 1 cL ;

38. = 4 + 4 : The introduced scaling corresponds to the situation where the interaction operator concentrates on binary interactions which produce a very small change in the opinion of the agents. From a modeling viewpoint, we require that the scaling in the limit ! 0 preserves the main macroscopic properties of the kinetic system. 8G. Toscani 06 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 16 / 26

39. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Fokker-Planck equation (followers) In the limit ! 0, integrating back by parts the scaled system of Boltzmann equations we obtain the Fokker-Planck equation for the followers' opinion distribution 9 @fF @t + @ @w 1 cF KF [fF ](w) + 1 cFL KFL[fL](w) fF (w) = 1 2 @2 @ ~ w2 2 cF D2(w) + ^2 cFL ^D fF (w); 2(w) where KF [fF ](w) = Z I P(w; v)(v w)fF (v; t)dv; KFL[fL](w) = Z I S(w; ~ w)( ~ w w)fL( ~ w)d ~ w: 9B. During - P. Markowich - J.F. Pietschmann - M.T. Wolfram 09 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 17 / 26

40. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Fokker-Planck equation (leaders) Similarly we obtain for the leaders' opinion distribution @fL @t + @ @ ~ w cL H[fL]( ~ w) + 1 cL KL[fL]( ~ w) fL( ~ w) = 1 2 @2 @w~2 ~2 cL ~D 2( ~ w)fL( ~ w) where K[fL]( ~ w) = Z I R( ~ w; ~v)(~v ~ w)fL(~v; t)d~v H[fL]( ~ w) = 2 ( ~ w + mL(t) 2wd) + 2 ( ~ w + mL(t) 2mF (t)) : Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 18 / 26

41. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Test1: leaders driving followers Kinetic densities evolution for a single population of leaders with constant interaction functions P;R and S. Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 19 / 26

42. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Test 1: leaders driving followers Kinetic density followers 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 12 10 8 6 4 2 0 Kinetic density leaders 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 25 20 15 10 5 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 20 / 26

43. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Test 2a: multiple leaders populations Let M 0 be the number of families of leaders, each of them described by the density fLp ; p = 1; :::;M such that Z I fLp ( ~ w)d ~ w = p: d dt Z I '(w)fF (w; t)dw = (QF (fF ; fF ); ') + MX p=1 QFL(fLp ; fF ); ' d dt Z I '( ~ w)fLp ( ~ w; t)d ~ w = (QL(fLp ; fLp ); '); p = 1; : : : ;M: Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 21 / 26

44. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Test 2a: multiple leaders populations Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 22 / 26

45. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Multiple leaders populations Kinetic density followers 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 9 8 7 6 5 4 3 2 1 0 Kinetic density leaders 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 20 18 16 14 12 10 8 6 4 2 0 Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 23 / 26

46. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Test 2b: Two leaders' populations with time-dependent strategies Kinetic densities with time-dependent strategies p(t) = 1 2 Z wdp+ wdp fF (w)dw + 1 2 Z mLp+ mLp fF (w)dw; p = 1; 2: Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 24 / 26

47. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Conclusions and future developments We introduced a general way to construct kinetic descriptions of optimal control problems for large systems of interacting agents. The main feature of the method is that the control is explicitly embedded in the resulting interacting dynamic. Dierent generalizations of the approach are possible: like the application of this control methodology on social networks. A fundamental problem in understanding the physics of this complex systems is the impact of errors, or uncertainty in data as parameters values or initial values. The development of ecient numerical methods for such stochastic models is essential because of our incomplete knowledge of the underlying physics. Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 25 / 26

48. Boltzmann type control of opinion consensus Mattia Zanella The Boltzmann Equation Complexity reduction SHBEMM Constrained self-organized systems Opinion control through leaders The Boltzmann-type optimal control Fokker-Planck Modeling Numerical results Test 1 Test 2a Test 2b Conclusions Bibliography Bibliography G. Albi, L. Pareschi, M. Zanella. Boltzmann type control of opinion consensus through leaders. To appear in Philosophical Transactions of the Royal Society A. Pre-print. arxiv.org/abs/1405.0736 G. Albi, L.Pareschi, M. Zanella. Social network based mean-

49. eld control models. In preparation. L. Pareschi, M. Zanella. Uncertainty quanti

50. cation in kinetic models of collective behavior. In preparation. Mattia Zanella (University of Ferrara) Boltzmann type control of opinion consensus Ravello September 2014 26 / 26

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