Proof: We follow thearoach develoed in [4]. We adot a useful but non-intuitive notion of time; a bin with z balls at time t receives its next ball at

Transcription

1 A Scaling Result for Exlosive Processes M. Mitzenmacher Λ J. Sencer We consider the following balls and bins model, as described in [, 4]. Balls are sequentially thrown into bins so that the robability that a bin with x balls obtains the next ball is roortional to x for some constant >. Secifically, we consider the case of two bins, in which case the state (x; y) denotes that bin has x balls and x bin has y balls. In this case, the robability that the next ball lands in bin is. x +y This model is motivated by the henomenon of ositive feedback. In economics, ositive feedback refers to the situation where a small number of comanies comete in a market until one obtains a non-negligible advantage in the market share, at which oint its share raidly grows to a monooly or near-monooly. One loose exlanation for this rincile, commonly referred to as Metcalfe's Law, is that the inherent otential value of a system grows suer-linearly in the number of existing users. Positive feedback also occurs in chemical and biological rocesses; for more information, see e.g. []. Here we consider ositive feedback between two cometitors, with the strength of the feedback modeled by the aramter. Our methods can also be alied to similar roblems with more cometitors. It is known that for the model above thatwhen>eventually one bin obtains a monooly in the following sense: as the total number of balls n goes to infinity, one bin obtains n o(n) balls with robability [, 4]. Indeed, with robability there exists a time after which all subsequent balls fall into just one of the bins [4]. Given this limiting behavior, we now ask what is the robability thatbinwilleventually obtain the monooly starting from state (x; y). We rovide an asymtotic analysis, based an examining the aroriate scaling of the system. This aroach is reminiscent of techniques used to study hase transitions in random grahs, as well as other similar henomena. Let a =(x + y)=. We show that when x = a + a that in the limit as a grows large, the robability that x obtains the monooly converges to Φ( 4 ), where Φ is the cumulative distribution function for the normal distribution with mean 0 and variance. Theorem For the balls and bins theorem described above, from the state (x; y), the robability that bin obtains the eventual monooly is Φ( 4 )+O(= a). Λ Suorted in art by an Alfred P. Sloan Research Fellowshi and NSF grants CCR , CCR-0870, and CCR-054.

2 Proof: We follow thearoach develoed in [4]. We adot a useful but non-intuitive notion of time; a bin with z balls at time t receives its next ball at a time t + T z, where T z is a random variable exonentially distributed with mean z. From the roerties of the exonential distribution, we can deduce that this maintains the roerty that the robability that the next ball lands in a bin with x balls is roortional to x. Secifically, the robability that the minimum of two exonentially distributed random variables T x with mean x and T y with mean y x is T x with robability. x +y Moreover, from the memorylessness of the exonential distribution, when a ball arrives at bin (resectively, bin ) the time T x (T y ) until the next ball arrives at bin (bin ) is still exonentially distributed with the same mean. The exlosion time for a bin is the time under this framework when a bin receives an infinite number of balls. If we begin at time 0, the exlosion time F x for bin satisfies F x = T j = j=a+ a and similarly for bin. Note that E[F x ] and E[F y ] are finite; indeed, the exlosion time for each bin is finite with robability. It is therefore evident that the bin with the smaller exlosion time at some oint obtains all balls thrown ast some oint, as roven formally in [4]. We first demonstrate that for sufficiently large a, F x and F y are aroximately normally distributed. This would follow immediately from the Central Limit Theorem (secifically, thevariation where random variables are indeendent but not necessarily identically distributed) if the sum of the variances of the random variables T j grew to infinity. Unfortunately, Var[T j ]= T j j < ; and hence standard forms of the Central Limit Theorem do not aly. Fortunately, wemay aly Esseen's inequality, a generalization of the Central Limit Theorem, which can be found in, for examle, [3][Theorem 5.4]. be indeendent random vari- Lemma (Esseen's inequality) Let X ;X ;:::;X n ables P with E[X j ] = 0, Var[X j ] = ff, and E[jX j jj 3 ] < for j = ;:::;n. P n B n = i=0 ff, F (x) n P =Pr(B = X j n j= j <x), and L = B 3= n E[jX n j= jj 3 ]: Then for some universal constant c. su jf (x) Φ(x)j»cL x Let In our setting, let X j = T x+j (x + j ). We note that there are no roblems

3 alying Esseen's theorem to the infinite summations of our roblem. Consider F x (z) =Pr P (T j j ) qp <za : j That is, F x (z) is the robabilitythatf x, aroriately normalized to match a standard normal of mean 0 and variance, is less than or equal to z. Then we have su jf x (z) Φ(z)j»O x : z Hence F x (z) aroaches a normal distribution as x grows large. We also have E[F x ]= E[T j ]= Z j = j dj + O = x x + ); O(x and Var[F x ]= Var[T j ]= Z j = j dj + O = x x + ): O(x We wish to determine the robability thatf x F y < 0. Now F x F y is (aroximately) normally distributed with mean μ where μ = E[F x ] E[F y ]= (a + a) (a a) + O(x )+O(y ) = (a a) (a + a) + O(a ) (a a) and variance ff where = a = + O(a ) ff =Var[F x ]+Var[F y ]= (a + a) + (a a) + O(x )+O(y ) = (a a) +(a + a) + O(a ) (a a) = + O(a a ) Hence the robability that F x F y < 0isΦ( 4 +O(= a)) + O(= a), which is just Φ( 4 )+O(= a). 3

4 We also resent a heuristic argument, which yields the same result. While this argument does not (yet) rovide a formal roof, we believe that variations of this heuristic aroach may beuseful for studying similar questions. Let G( ; a) be the robability that bin achieves monooly when x = a + a and y = a a. From state (x; y), after the next is ball thrown, the new average a 0 equals a 0 += and we have a new values x 0 and 0 so that x 0 = a a 0. With (a+ a) robability (a+ a) +(a a), x0 = x +,and hence a a 0 = a ++ a 0 = =+ a a += : Similarly, with robability (a+ a) a), we have +(a a) (a Hence, G( ; a) = 0 = =+ a : a += + G( + 3;a+=) + G( 3;a+=) (a+ a) for = (a+ a) +(a a), == a +=, and 3 = a= a +=. Suose that there exists a continuous and twice differentiable function F ( ) such that F ( ) =lim a! G( ; a). So F ( ) = + F ( + 3)+ F ( 3): Now in the limit as ; ; and 3 go to 0asa grows large, with 3 = o( ) 0= F ( + 3) F ( ) F ( ) F ( 3) + (F ( + 3) F ( 3)) = F 0 ( )( 3) F 0 ( 3)( + 3) + (F ( + 3) F ( 3)+F( 3) F ( 3)) = ( + 3) F 00 ( 3) 3F 0 ( )+ (F 0 ( 3)+F 0 ( 3)): Hence we have the limiting equation F 00 ( )+ 4 3 F 0 ( ) =0: 4

5 As a grows large, 4 = converges to 4 and 3= converges to. So we require F 00 ( )+(4 ) F 0 ( ) =0: Substituting H( ) =F 0 ( ), it is easy to solve to find F ( ) =Φ( 4 ); which is indeed the correct limiting result. We rovide an examle demonstrating the accuracy of Theorem in Table. We consider initial states with 00 balls in the system, with the first bin containing between 0 and 0 balls (so ranges from 0. to ). We calculate the exact distribution when there are 60,000 balls in the system for the case =, using methods described in []. With this data, we make the very accurate aroximation bin eventually achieves monooly if it has 53% of the balls at this oint. We also aly symmetry; if at this oint bin has k balls with robability and bin has k balls with robability <, then bin reaches monooly at least out of this + fraction of the time. This aroach is sufficent to accurately determine the robability that the first bin eventually reaches monooly to four decimal laces. Comaring these results demonstrates the accuracy of the normal estimate. Table shows similar results for the case of =:5; here we calculate the distribution with 640,000 balls in the system and use a 5% cutoff to estimate the robability of monooly, and the numbers are correct to four decimal laces. Again, the normal estimate rovides a great deal of accuracy. References [] B. Arthur. Increasing Returns and Path Deendence in the Economy. The University of Michigan Press, 994. [] E. Drinea, A. Frieze, and M. Mitzenmacher. Balls and Bins Models with Feedback. In Proceedings of SODA 00. [3] V. Petrov. Limit Theorems of Probability Theory. Oxford University Press, 995. [4] J. Sencer and N. Wormald. Exlosive rocesses. Draft manuscrit. 5

6 x Calc Φ( 4 ) x Calc Φ( 4 ) Table : A calculation vs. the asymtotic estimate of our theorem when a = 00 and =. x Calc Φ( 4 ) x Calc Φ( 4 ) Table : A calculation vs. the asymtotic estimate of our theorem when a = 00 and =:5. 6