Positive Particle Interaction Ulrich Krause FB Mathematik und Informatik, Universit¨ at Bremen, 28334 Bremen, Germany
[email protected]
Abstract. This paper treats interaction between finitely many particles where the future state of each particle is obtained from the present states of all other particles by a positive linear combination with time variant coefficients. The main result provides conditions for a common globally asymptotically stable equilibrium to exist. These conditions are, in particular, satisfied if the particles show slowly decaying interation. Since “particles” can be many things, there are many applications, for example, heat diffusion in an inhomogeneous medium, a many body problem under pseudo–gravity and consensus formation under bounded confidence.
1 Introduction: The Model “It is for positive systems, therefore, that dynamic systems theory assumes one of its most potent forms.” (D.G. Luenberger, Introduction to Dynamic Systems, page 188) Consider finitely many particles in some space with positive interaction between them in the sense that the future state of each particle is a positive linear combination of the present states of all other particles. The dynamics of such a system is quite well–understood if the coefficients of the combination do not depend on time but there are many open problems concerning the dynamics for time variant interaction. The present paper contains a stability theorem for positive linear and time variant interaction. This theorem provides conditions for a common globally asymptotically stable equilibrium that is, for any given initial states, all particles approach asymptotically the same state. Examples and applications abound because “particles” can mean many things from moving bodies in real space to inhomogeneous plates cooling by heat diffusion to human beings exchanging opinions. “Approaching the same state” then means collision of bodies or equalization of temperature or reaching a consensus, respectively.
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Ulrich Krause
Consider n interacting particles in a closed convex region D ⊂ IRd and let xi (t) ∈ D the state of particle i ∈ {1, 2, . . . , n} at time t ∈ IN = {0, 1, 2, . . .}. The dynamics of interaction for particle i is modeled as X xi (t + 1) − xi (t) = aij (t)(xj (t) − xi (t)) (1) j6=1
where the coefficients of interaction aij (t) are nonnegative P numbers. We shall assume that forces are bounded in the sense that aij (t) ≤ 1 for j6=i
all iP and P we admit the possibility of selfinteraction by defining aii (t) = 1− aij (t). Thus, we can (1) rewrite as j6=i
xi (t + 1) =
n X
aij (t)xj (t) for 1 ≤ i ≤ n
(2)
j=1
or, in matrix form, as x(t + 1) = A(t)x(t) 1
n
0
where x(t) = (x (t), . . . , x (t)) ∈ D
(3) n×n
n
and A(t) = (aij (t)) ∈ IR is the n P row–stochastic matrix of coefficients, i.e., 0 ≤ aij (t) and aij (t) = 1 j=1
for 1 ≤ i, j ≤ n and t ∈ IN. In other words, the model as given by (2) exhibits not only positive inn P teraction but positive and convex interaction because of aij (t) = 1. This j=1
happens often to be the case, as for instance in the applications mentioned above where also selfinteraction makes sense (see Section 3). For various kinds of positive systems see [4, 6, 9, 13, 14, 15], for the particular positive systems of consensus formation in one dimension see [1, 2, 3] and for opinion dynamics under bounded confidence, also in one dimension, see [5, 7, 11, 12]. In Section 2 we shall present the main result of the paper, a stability theorem for time variant interaction. In Section 3 we specialize this result to the case of slowly decaying interaction and illustrate this by some examples.
2 A stability theorem for time variant particle interaction Let k · k be an arbitrary vector space norm on IRd which is fixed in what follows. For a subset M ⊆ IRd the diameter of M is defined by ∆(M ) = sup{k m − m0 k | m, m0 ∈ M }. The convex hull of M , denoted by convM , n n P P is the set of all convex combinations αk xk , where 0 ≤ αk , αk = 1 k=1
k=1
and xk ∈ M . The following lemma extends a useful inequality known for one dimension (cf. [15, Theorem 3.1]) into higher dimensions.
Positive Particle Interaction
Lemma 1. Let x1 , . . . , xn ∈ IRd and let y i =
n P
3
aik xk for 1 ≤ i ≤ n and
k=1
A = (aij ) a row–stochastic matrix. Then the following inequality holds
∆(conv{y 1 , . . . , y n }) ≤ (1 − min
1≤i,j≤n
n X
min{aik , ajk })∆(conv{x1 , . . . xn })
k=1
(4) P
Proof. First, we show that for any two convex combinations αk uk and P d k k k βk v of points u and v , respectively, from IR for k ∈ I finite, one has that X X k αk uk − βk v k k≤ max{k ui − v j k | i, j ∈ I}. (5) P For, βk v k then P if wk= P k αk u − w k = kP αk (uk − w) k≤ max{k ui − w k | i ∈ I} and k w − ui k = k βk (v k − ui ) k≤ max{k v j − ui k | j ∈ I}. From (5) we have that ∆(conv{y 1 , . . . , y n }) ≤ max{k y i − y j k | i, j ∈ I} with I = {1, . . . , , n}. (6) P λik = Let λhk = ahk − min{aik , ajk } for h = i, j. Obviously, λhk ≥ 0 and k∈I P P min{aik , ajk }. For rij > 0 let αhk = λrhk λjk = rij with rij = 1 − and, ij k∈I k∈I P P αjk = 1. αik = hence, k∈I
Now
k∈I
P P P (aik − ajk )xk k=k λik xk − λjk xk k P P = rij k αik xk − αjk xk k for rij > 0
k yi − yj k = k
and, by (5),
k y i − y j k≤ rij max{k xk − xl k | k, l ∈ I}.
(7)
If rij = 0 then λik = λjk for all k ∈ I and, hence, aik = ajk for all k. In this case, y i = y j and (7) holds trivially. Equations (6) and (7) together prove inequality (4). For the model of positive particle interaction x(t + 1) = A(t)x(t) as introduced in the previous section we obtain the following stability result. Theorem 1. Denote for s, t ∈ IN with s < t the matrix product A(t − 1)A(t − 2) · · · A(s) by B(t, s) with entries bij (t, s). Suppose there exist a sequence 0 = t0 < t1 < t2 < . . . in IN and a sequence δ1 , δ2 , . . .in [0, 1] ∞ P with δm = ∞ such that for all 1 ≤ i, j ≤ n and all m ≥ 1 the following m=1
inequality holds
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Ulrich Krause n X
min{bik (tm , tm−1 ), bjk (tm , tm−1 )} ≥ δm
(8)
k=1
Then for arbitrary starting points x1 (0), . . . , xn (0) in D there exists an equilibrium x∗ ∈ conv{x1 (0), . . . , xn (0)} ⊆ D such that for all 1 ≤ i ≤ n lim xi (t) = x∗ .
(9)
t→∞
Furthermore, for any other starting points y 1 (0), . . . , y n (0) in D with equilibrium y ∗ it holds that k x∗ − y ∗ k≤ max{k xi (0) − y j (0) k | 1 ≤ i, j ≤ n}.
(10)
Proof. From x(t + 1) = A(t)x(t) for t ∈ IN we have for s, t ∈ IN with s < t x(t) = A(t − 1)A(t − 2) · · · A(s)x(s) = B(t, s)x(s) and, hence, x(tm ) = B(tm , tm−1 )x(tm−1 ) for m ≥ 1. n P For y i = xi (tm ) = bik (tm , tm−1 )xk (tm−1 ) and M (t) = conv{x1 (t), . . . , xn (t)} k=1
from Lemma 1 we obtain, taking assumption (8) into account, that ∆M (tm ) ≤ (1 − δm )∆M (tm−1 ). By iteration ∆M (tm ) ≤ (1 − δm )(1 − δm−1 ) · · · (1 − δ1 )∆M (0). By the mean value theorem 1 − r ≤ exp(−r) for r ≥ 0 and, hence, Ã m ! X δi ∆M (0) for all m. ∆M (tm ) ≤ exp − i=1 ∞ P
By assumption
m=1
δm = ∞ and, hence, lim ∆M (tm ) = 0. From x(t + 1) = m→∞
A(t)x(t) it follows that M (t + 1) ⊆ M (t) and, hence, ∆M (t + 1) ≤ ∆M (t). Since ∆M (tm ) converges to 0 this shows that ∆M (t), too, converges to 0. ∞ T Furthermore, since M (t), t ∈ IN, is compact we have that M (t) is non– ∗
empty. For x ∈
∞ T
t=0
M (t) it follows for every 1 ≤ i ≤ n that
t=0
k x∗ − xi (t) k≤ ∆M (t) for all t and, hence, lim xi (t) = x∗ for all i. t→∞
Obviously, x∗ ∈ M (0) = conv{x1 (0), . . . , xn (0)} and, similarly, y ∗ ∈ conv{y 1 (0), . . . , y n (0)}. Therefore, inequality (5) implies inequality (10). The above theorem and its crucial condition (8) are inspired by the treatment of consensus formation in one dimension as in [1] and [2]. (Other extensions can be found in [11] and [12].) In the next section we present more easy–to–use criteria for condition (8) to hold. Roughly speaking, these criteria require that interaction between the particles does not decay too fast.
Positive Particle Interaction
5
3 Stability for slowly decaying interaction One cannot expect a common globally asymptotically stable equilibrium if there is almost no interaction between the particles. Similarly, for time variant interaction, one cannot expect conclusion (9) of Theorem 1 to hold if interaction is vanishing too fast. More precisely, we say that for the model given as before by x(t + 1) = A(t)x(t), t ∈ IN, there holds slowly decaying interaction of degree p if there exists a base matrix A ∈ IRn×n and a + decreasing function f : IR+ → IR+ such that A(t) ≥ f (t)A for all t ≥ s for some fixed s ∈ IN and
(11)
Z∞ f (t)p dt = ∞ where p ≥ 1.
(12)
1
Thus, by this definition the interaction between the particles may become weaker in the course of time but a lower ceiling f (t) of interaction should exist which on the average is big enough. Theorem 2. Suppose that interaction is slowly decaying of degree p and that (p) for any two particles i and j there exists a third one k such that aik > 0 (p) and ajk > 0 for the entries of the p–th power of the base matrix A. Then the conclusions (9) and (10) of Theorem 1 hold. Proof. We show that the assumptions made imply condition (8) of Theorem 1. For m ∈ IN let tm = pm and suppose that m ≥ p1 + 1. From (11) we get B(tm , tm−1 ) = A(tm − 1) · · · A(tm−1 ) ≥ f (tm )p Ap taking into account that f is decreasing. For the entries bij (tm , tm−1 ) of B(tm , tm−1 ) this implies n X
min{bik (tm , tm−1 ), bjk (tm , tm−1 )} ≥ f (tm )p
k=1
n X
(p)
(p)
min{aik , ajk }.
k=1
By assumption on Ap ( δ 0 = min
n X
) (p) (p) min{aik , ajk }
| 1 ≤ i, j ≥ n
> 0.
k=1
Thus, inequality (8) is satisfied for δm = δ 0 f (tm )p and we have to show that ∞ P δm = ∞.
m=1
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Ulrich Krause
According to a general relationship between the convergence of series and ∞ P integrals (cf. [8, Theorem 3, p. 64]) we have that f (tm )p = ∞ if and only if
R∞
m=1
p
f (pt) dt = ∞. The latter follows from (12) and, hence, we arrive at
1 ∞ X m=1
δm ≥ δ 0
∞ X
f (tm )p = ∞.
s m≥ p
To conclude we mention a few examples which will be discussed in detail elsewhere. Examples 3 a) Obviously, constant interaction, i.e., A(t) = A for all t, provides an example for f identically equal to 1 and A as in Theorem 2. In particular, for d = 1, D = IR+ one obtains the Basic Limit Theorem for Markov Chains (cf. [13, page 230])which states that lim At = B for each t→∞ regular stochastic matrix, where the rows of B are all equal to a vector n P v 0 ≥ 0 with vi = 1 and v 0 A = v 0 . “Regular” means that all entries of i=1
some power Aq of A are all (strictly) positive. Obviously, a regular matrix A satisfies the assumption made on A in Theorem 2 whereas · the ¸ converse 1 0 is not true as can be seen from the simple example A = 1 1 . 2 2
b) Consider Jacobi–interaction, where the particles can be labelled in such a way that each particle interacts (strictly) positively with its direct neighbours, that is, A(t) has for every t the structure ++ + + + 0 +++ A(t) = where + indicates +++ .. 0 . ++ a (strictly) positive entry. Suppose that the smallest positive entry of A(t) 1√ is at least n−1 for t big enough. Choosing the latter as f (t) and p = n−1 t the conclusions of Theorem 2 hold. It is easy to give examples where the entries of A(t) decay too fast for the conclusions of Theorem 2 to hold. In the field of consensus formation this phenomenon is known as a fast “hardening of positions” (cf. [3]). c) To treat heat diffusion in an inhomogeneous medium consider an agglomeration of n pieces of different materials which has been heated from the outside and for which we will study the movement of heat through all the pieces. Denote by xi (t) the temperature (in Kelvin) of piece of material i at time t ∈ IN. Obviously, d = 1, and let D denote the range of relevant termperatures. By Newton’s law of cooling we have that
Positive Particle Interaction
xi (t + 1) − xi (t) =
X
aij (t)(xj (t) − xi (t))
7
(13)
j6=i
for the change in temperature of piece i, where coefficient aij (t) measures heat transfer from piece j to i and may depend on various circumstances as the materials of pieces i and j, their boundaries, the states xj (t) and time t directly via, e.g., changing room temperature. d) Consider finitely many bodies in Euclidean space which attract each other by some pseudo–gravitational force which goes inversely with a certain power of the bodies distance. If the continuous dependence on distance is replaced by an appropriate step function one obtains the nonlinear system in discrete time X xi (t + 1) = αij (x(t)xj (t) (14) j∈I(i,x(t))
where I(i, t) is the set of bodies “near” to body i at state x(t) = (x1 (t), . . . , xn (t))0 and the αij make a row–stochastic matrix by adding zeroes. Actually, equation (13) has been originally obtained within a model of consensus formation under bounded confidence ([5, 7, 11]). There the ”particles” are experts who assess a certain issue and interact by exchanging opinions.
References 1. S. Chatterjee. Reaching a consensus: Some limit theorems. Proc. Int. Statist. Inst., pages 159–164, 1975. 2. S. Chatterjee and E. Seneta. Toward consensus: some convergence theorems on repeated averaging. J. Applied Probability, 14:89–97, 1977. 3. J.E. Cohen, J. Hajnal, and C.M. Newman. Approaching consensus can be delicate when positions harden. Stochastic Proc. and Appl., 22:315–322, 1986. 4. J. Conlisk. Stability and monotonicity for interactive Markov chains. J. Math. Sociology, 17:127–143, 1992. 5. J.C. Dittmer. Consensus formation under bounded confidence. Nonlinear Analysis, 47:4615–4621, 2001. 6. L. Farina and S. Rinaldi Positive Linear Systems: Theory and Applications. Wiley & Sons, New York, 2000. 7. R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence: Models, analysis and simulation. J. Artificial Societies and Social Simulation, 5 (3), 2002. http://jasss.soc.surrey.ac.uk/5/3/2.html. 8. K. Knopp. Infinite Sequences and Series. Dover Publ., New York, 1956. 9. U. Krause. Positive nonlinear systems: Some results and applications. In V. Lakshmikantham, editor, World Congress of Nonlinear Analysts, pages 1529–1539. De Gruyter, Berlin, 1996. 10. U. Krause and T. Nesemann. Differenzengleichungen und diskrete dynamische Systeme. Teubner, Stuttgart, 1999.
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11. U. Krause. A discrete nonlinear and non–autonomous model of consensus formation. In S. Elaydi, G. Ladas, J. Popenda, and J. Rakowski, editors, Communications in Difference Equations, 227–236. Gordon and Breach Science Publ., Amsterdam, 2000. 12. N. Kruse. Semizyklen und Kontraktivit¨ at nichtlinearer positiver Differenzengleichungen mit Anwendungen in der Populationsdynamik. Ph. thesis, dissertation.de, Verlag im Internet. Bremen, 1999. 13. D.G. Luenberger. Introduction to Dynamic Systems. Theory, Models, and Applications. Wiley & Sons, New York, 1979. 14. T. Nesemann. Stability Behavior of Positive Nonlinear Systems with Applications to Economics. PhD thesis, Wissenschaftlicher Verlag, Berlin, 1999. 15. E. Seneta. Non–negative Matrices and Markov Chains, 2nd. edition. Springer, New York, 1980.
