Applied Mathematics and Computation 219 (2012) 1569–1575

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Winner-take-all based on discrete-time dynamic feedback

Shuai Li a,⇑, Jiguo Yu b, Mingming Pan c, Sanfeng Chen d

a Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USAb School of Computer Science, Qufu Normal University, Rizhao, Shandong 276826, Chinac China Electric Power Research Institute, Beijing, Chinad Key Lab of Visual Media Processing and Transmission, ShenZhen Institute of Information Technology, Shenzhen, Guangdong, China

a r t i c l e i n f o a b s t r a c t

Keywords:Winner-take-allDynamic feedbackDiscrete-time systemNonlinear

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.07.057

⇑ Corresponding author.E-mail addresses: lshuai@stevens.edu (S. Li), jigu

In this paper, we investigates a simple discrete-time model, which produces the winner-take-all competition. The local and global stability of the model are both proven theoreti-cally. Simulations are conducted for both the static competition and the dynamic compe-tition scenarios. The numerical results validate the theoretical results and demonstrate theeffectiveness of the model in generating winner-take-all competition.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Winner-take-all refers to the phenomena that agents in a group compete with each others for activation and only the onewith the highest input stays active while all the others deactivated. The winner-take-all models many competition phenom-ena existing in nature [1–3] and finds applications in many engineering fields [4–6]. It is remarkably that the winner-take-allcompetition is computationally powerful and can generate some useful functions required in computational intelligenceapplications [7,8]. Due to the importance of winner-take-all competition in engineering applications, there have been manyattempts to design circuits for its implementation [9–11].

There have been various models proposed by researchers to explain or generate the winner-take-all behavior. In [12,13],the N species Lotka–Volterra model is used for the explanation. Inspired by the great success of recurrent neural networks[14–16], recurrent neural networks are utilized to investigate the winner-take-all competition [17,18]. In [19–21], theFitzHugh–Nagumo model, which is able to demonstrate the interactive spiking, is used to study the winner-take-allbehavior. In [22,23], the winner-take-all problem is regarded as the solution of an optimization problem and the result isgenerated by solving such a problem [22,23]. Although many mathematic models have been presented to explain or generatethe winner-take-all competition, it is still an open problem to find a simple model describing such a nonlinear phenomenawith rigorous analysis on its performance. In this paper, we present a simple model described by nonlinear differenceequations to generate the winner-take-all competition. The solution of the equilibrium points, the local stability and theglobal stability are resolved in theory. Due to the simplicity of this model, it has strong potentials to be implemented inhardware with less hardware complexity compared with some existing models.

The remainder of this paper is organized as follows: in Section 3, the analytical model is presented and the underlyingcompetition mechanism is explained from a selective positive–negative feedback perspective. In Section 4, the competitionbehavior and the stability results are proven rigourously by means of nonlinear stability tools. In Section 5, illustrative exam-ples are given to show the effectiveness of the proposed model. The paper is concluded in Section 6.

. All rights reserved.

oyu@sina.com (J. Yu), panmingming@epri.sgcc.com.cn (M. Pan), chensanf@sziit.com.cn (S. Chen).

1570 S. Li et al. / Applied Mathematics and Computation 219 (2012) 1569–1575

2. Problem definition

In this paper, we define the winner-take-all competition in the following way: the agent with the largest input finallywins the competition and keeps activated while all the other agents withe smaller inputs are deactivated to zero eventually.

This definition is a mathematical abstraction of many competition phenomena found in nature and society, such as thegrowth competition in plants [1], competitive decision making in the cortex [24,2], foraging and mating in animal societies[3], the competition between companies [25], neural activity competition in visual systems [26].

3. The model

The proposed model has the following dynamic for the ith agent in a group of totally n agents,

x1iðt þ 1Þ ¼ uix2iðtÞ;

x2iðt þ 1Þ ¼ x1iðt þ 1Þkx1ðt þ 1Þk ;

yiðt þ 1Þ ¼ x2iðt þ 1Þ; ð1Þ

where t represent time instant, ui 2 R is the input and ui P 0; ui – uj for i – j; x1ðtÞ ¼ ½x11ðtÞ; x12ðtÞ; . . . ; x1nðtÞ�T 2 Rn;

x1iðtÞ 2 R and x2iðtÞ 2 R for i ¼ 1;2; . . . ;n denote the first and the second state value of the ith agent at time t respectively,

kx1ðtÞk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

11ðtÞ þ x212ðtÞ þ � � � þ x2

1nðtÞq

denotes the Euclidean norm of x1ðtÞ; yiðtÞ 2 R represents the output of the ith agent at

time t.The dynamic Eq. (1) can be written into the following compact form by stacking up the state for all agents,

x1ðt þ 1Þ ¼ u � x2ðtÞ;

x2ðt þ 1Þ ¼ x1ðt þ 1Þkx1ðt þ 1Þk ;

yðt þ 1Þ ¼ x2ðt þ 1Þ; ð2Þ

where u ¼ ½u1;u2; . . . ;un�T 2 Rn; x1ðtÞ ¼ ½x11ðtÞ; x12ðtÞ; . . . ; x1nðtÞ�T 2 Rn; x2ðtÞ ¼ ½x21ðtÞ; x22ðtÞ; . . . ; x2nðtÞ�T 2 Rn; yðtÞ ¼ ½y1ðtÞ;y2ðtÞ; . . . ; ynðtÞ�

T 2 Rn, the operator ‘�’ represents the multiplication in component-wise, i.e., u � x ¼ ½u1x1;u2x1; . . . ;unxn�T .

4. Theoretical results

In this section, theoretical results on the dynamic system (1) are presented. We first examine the equilibrium point of thedynamic system and then investigate its local stability around the equilibria. After that, we turn to the proof of the globalstability of the system.

On the equilibrium points of the dynamic system (1), we have the following theorem:

Theorem 1. The discrete-time dynamic system (1) has equilibrium points at ðx�1; x�2; y�Þ ¼ �ðuiei; ei; eiÞ for i ¼ 1;2; . . . ;n, whereui P 0 is the ith input, ei 2 Rn is a n dimensional vector with the ith element equal 1 and all the others equal zero.

Proof. In Eq. (2), letting x1ðt þ 1Þ ¼ x�1; x2ðt þ 1Þ ¼ x2ðtÞ ¼ x�2; yðt þ 1Þ ¼ y�, we get the following equations for the equilib-rium points,

x�1 ¼ u � x�2; ð3aÞ

x�2 ¼x�1kx�1k

; ð3bÞ

y� ¼ x�2: ð3cÞ

According to the definition of the operator ‘�’. Eq. (3a) can be written as

x�1 ¼ diagðuÞx�2; ð4Þ

where diagðuÞ is defined as the matrix with u as the diagonal elements and all the other elements zero. Together with Eq.(3b), we have,

x�1 ¼ diagðuÞ x�1kx�1k

ð5Þ

i.e.,

kx�1kx�1 ¼ diagðuÞx�1: ð6Þ

S. Li et al. / Applied Mathematics and Computation 219 (2012) 1569–1575 1571

Clearly, this is an eigen-equation for the matrix diagðuÞ. The existence of the solution requires x�1 being the eigenvector ofdiagðuÞ and kx�1k being the corresponding eigenvalue. As the matrix diagðuÞ is diagonal, its eigenvalue and normalized eigen-vector pairs can be easily got as u1 with e1, or u2 with e2, or, u3 with e3; . . . ; or un with en, respectively. Comparing the eigen-value and eigenvector pairs of diagðuÞ with (6), we get the solution of x�1 as: x�1 ¼ �u1e1; x�1 ¼ �u2e2; . . . ; or x�1 ¼ �unen. From(3b) and (3c), it can be observed that both x�2 and y� equal the normalized vector of x1, i.e., the corresponding solutions of x�2are e1; e2; . . . ; or en and y� takes the same solution. To summarize, the equilibrium points are ðx�1; x�2; y�Þ ¼ �ðuiei; ei; eiÞ fori ¼ 1;2; . . . ;n. This completes the proof. h

We have the following results on the local stability of the system 1:

Theorem 2. ðx�1; x�2; y�Þ ¼ �ðujej; ej; ejÞ is an unstable equilibrium point of the discrete-time dynamic system (1) for j ¼ 1;2; . . . ;nj – k� where k� ¼ argmaxi¼1;2;...nui; uj P 0 is the jth input, ej 2 Rn is a n dimensional vector with the jth element equal 1 and allthe others equal zero.

Proof. Without losing generality, we only consider the equilibrium points ðx�1; x�2; y�Þ ¼ ðujej; ej; ejÞ. For the rest equilibriumpoints ðx�1; x�2; y�Þ ¼ �ðujej; ej; ejÞ, the local stability can be analyzed in the same way.

The system (2) is a nonlinear difference equation due to the presence of the normalization operation. We use theliberalization technique to analyze the local stability. According to Theorem 2, there are totally n equilibrium points for thedynamic system (1) and the jth one is ðx�1; x�2; y�Þ ¼ ðujej; ej; ejÞ. From (2), we get the dynamics of x2 as follows,

x2ðt þ 1Þ ¼ diagðuÞx2ðtÞkdiagðuÞx2ðtÞk

: ð7Þ

At the equilibrium point ðx�1; x�2; y�Þ ¼ ðujej; ej; ejÞ, we have kdiagðuÞx�2k ¼ kdiagðuÞejk ¼ kujejk ¼ jujj ¼ uj. Accordingly, we havethe following approximate dynamics around this equilibrium point,

x2ðt þ 1Þ ¼ diagðuÞx2ðtÞuj

: ð8Þ

This is a linear system with 1uj

diagðuÞ as the system matrix. 1uj

diagðuÞ is a diagonal matrix and thus its eigenvalues areu1uj; u2

uj; . . . ; un

uj. For j – k�, we have, uj < uk� according to the definition of k�. The k�th eigenvalue of (8), which is uk�

uj> 1. The lin-

ear system (8) is unstable since its system matrix has an eigenvalue outside the unit circle. Therefore the nonlinear systemwith (8) as its linear approximation is also unstable. Thus, we conclude that the system (1) is unstable atðx�1; x�2; y�Þ ¼ ðujej; ej; ejÞ for j ¼ 1;2; . . . ;n; j – k�. Following the same procedure, we can also conclude that the system (1)is unstable at ðx�1; x�2; y�Þ ¼ �ðujej; ej; ejÞ for j ¼ 1;2; . . . ;n; j – k�. This completes the proof. h

The global stability results on the system (1) are stated as follows,

Theorem 3. For any random initializations, the output yi for i ¼ 1;2; . . . ;n of the discrete-time dynamic system (1) converges to 1for i ¼ k� when x2ið0Þ > 0, converges to �1 for i ¼ k� when x2ið0Þ < 0 and converges to 0 for other is with x2ið0Þ < x2k� , where k�

defines the label of the winner, i.e., k� ¼ argmaxi¼1;2;...;nðuiÞ.

Proof. From (2), we get the dynamics of x2 as follows,

x2ðt þ 1Þ ¼ diagðuÞx2ðtÞkdiagðuÞx2ðtÞk

: ð9Þ

By iteration, we have,

x2ðt þ 1Þ ¼ diagðuÞx2ðtÞkdiagðuÞx2ðtÞk

¼ diag2ðuÞx2ðt � 1Þkdiag2ðuÞx2ðt � 1Þk

� � � ¼ diagtþ1ðuÞx2ð0Þkdiagtþ1ðuÞx2ð0Þk

¼1

uk�diagðuÞ

� �tþ1x2ð0Þ

k 1uk�

diagðuÞ� �tþ1

x2ð0Þk

¼diag u

uk�

� �tþ1x2ð0Þ

kdiag uuk�

� �tþ1x2ð0Þk

: ð10Þ

Note that diag uuk�

� �is a diagonal matrix with the ith diagonal element being ui

uk�with j ui

uk�j < 1 for i – k� and j ui

uk�j ¼ 1 for i ¼ k�.

Accordingly, we can compute that diag uuk�

� �tþ1is a diagonal matrix with the ith diagonal element equal to ð ui

uk�Þtþ1 and we

obtain that limt!1ð uiuk�Þtþ1 ¼ 0 for i – k� and limt!1

uiuk�

� �tþ1¼ 1 for i ¼ k�. Therefore, we get limt!1diag u

uk�

� �tþ1¼ diagðek� Þ

with ei 2 Rn defined as a n dimensional vector with the ith element equal 1 and all the others equal zero. Together with(10), we further get,

Fig

1572 S. Li et al. / Applied Mathematics and Computation 219 (2012) 1569–1575

limt!1x2ðt þ 1Þ ¼ limt!1

diag uuk�

� �tþ1x2ð0Þ

kdiag uuk�

� �tþ1x2ð0Þk

¼limt!1

diag uuk�

� �tþ1x2ð0Þ

klimt!1

diag uuk�

� �tþ1x2ð0Þk

¼ diagðek� Þx2ð0Þkdiagðek� Þx2ð0Þk

¼ x2k� ð0Þek�

kx2k� ð0Þek� k¼ x2k� ð0Þjx2k� ð0Þj

ek� ¼ek� when x2k� ð0Þ > 0;�ek� when x2k� ð0Þ < 0;

�ð11Þ

which means that x2iðtÞ converges to 0 for the losers with x2ið0Þ < x2k� and converges to 1 for the winner i ¼ k�. Recalling thatyðtÞ ¼ x2ðtÞ, the proof is completed. h

5. Numerical examples

In this section, numerical examples are used to further explore the winner-take-all competition phenomena generated bythe discrete-time dynamic (1). Like in [27], we consider two sceneries: one is static competition, where the input u is con-stant and one is dynamic competition, where the input u is time-varying.

5.1. Discrete-time static competition

For the static competition problem, we consider a problem with n = 10 agents under time invariant input. The input u is ran-domly generated between 0 and 1, which is u=[0.1982,0.1951,0.3268,0.8803,0.4711,0.4040,0.1792,0.9689,0.4075,0.8445],

0 10 20 30 40 50 60

−1

−0.5

0

0.5

1

iteration

y(t)

system output v.s. iterations

. 2. The output of y(t) in all dimensions in the static competition scenario with 10 agents under a different random initialization from Fig. 1.

0 10 20 30 40 50 60

−1

−0.5

0

0.5

1

iteration

y(t)

system output v.s. iterations

Fig. 1. The output of y(t) in all dimensions in the static competition scenario under a random initialization with 10 agents.

0 10 20 30 40 50

−1

−0.5

0

0.5

1

iteration

y(t)

Fig. 4. Time history of the three-agent system with small perturbations on y3ð0Þ around ½�1;0;0� or ½0;�1;0�.

−1 −0.50 0.5 1

−1−0.50

0.51

−1

−0.5

0

0.5

1

y1y2

y3

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

y1

y2−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

y2

y3

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

y1

y3

Fig. 3. Phase plot of the three-agent system starting from f�1;�0:5;0;0:5;1g3.

S. Li et al. / Applied Mathematics and Computation 219 (2012) 1569–1575 1573

and the state is randomly initialized between�1 and 1, Fig. 1 shows the evolution of output values of all agents with time (in thefigure, the value of outputs is marked as ‘+’ in each time step). From the figure, it can be observed that only a single output (cor-responds to the 7th agent, which has the largest value in u) reaches 1 eventually and all the other output values are suppressedto zero. Fig. 2 shows the evolution of output values of all agents with time with the same u but different initialization of statesfrom that used in Fig. 1. In Fig. 2, all output values converges to zero except the output of the 7th agent, which has the largestvalue in u. It is noteworthy that the ultimate output value of the winner is 1 in Fig. 1 but is�1 in Fig. 2. This observation is con-sistent with the theoretical conclusion drawn in Theorem 3 since the 7th agent, which is the winner in this set of simulations, isinitialized in x2 with a positive value in Fig. 1 but a negative value in Fig. 2 as can be observed in the two figures.

Like in [27], we next consider a three agent competition problem for the convenience of visualization. A three agent sys-tem with u = [0.5598,0.3008,0.9394] is simulated (in this case, the third agent has the largest input and thus is the winner).Fig. 3 shows the phase plot of the output in three-dimensional space and its projections in two-dimensional space. In thefigure, the red spots highlight the ultimate value of the output. Clearly, we can see that y1; y2 and y3 converge to 0, 0 and1 for the cases with y3ð0Þ > 0 (note that yð0Þ ¼ x2ð0Þ according to Eq. (2)) and they converges to 0, 0 and �1 for the caseswith y3ð0Þ < 0. This observation also validates the conclusion drawn in Theorem 3. It is noteworthy that for the cases with

0 500 1000 1500 2000 2500 3000−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

iteration

y(t)

u(t)

agent 1agent 2agent 3agent 4agent 5

Fig. 5. Inputs and outputs of the dynamic system in the dynamic competition scenario.

1574 S. Li et al. / Applied Mathematics and Computation 219 (2012) 1569–1575

y3ð0Þ ¼ 0; y converge neither to ½0;0;1� nor to ½0;0;�1�. Actually, this is due to the fact concluded from Theorem 2 that theequilibrium points ½�1;0;0�; ½0;�1; 0� (which appear as the ultimate values highlighted in red in Fig. 3) are locally unstable.To further show that these points are indeed unstable, we perturb the initial value of the system to one very close to theseequilibriums, and check whether the ultimate value is still attracted to the equilibrium points. Fig. 4 shows the time historyof the outputs for the three agents with the initial values of the third agent perturbed by a zero mean magnitude 0.01 ran-dom variable from the equilibrium points ½�1;0;0�; ½0;�1;0�. From this figure, we can see that no matter how close of y3ð0Þto zero, y3ð0Þ either goes to 1 or �1 instead of 0 under the initialization of ½y1ð0Þ; y2ð0Þ� ¼ ½�1;0� or ½y1ð0Þ; y2ð0Þ� ¼ ½0;�1� onlyif y3ð0Þ– 0.

5.2. Discrete-time dynamic competition

In this part, we consider the scenario with time-varying inputs. Note that the winner-take-all system should run in agreater sampling rate in order to successfully track time-varying signals. In the simulation, we consider n ¼ 5 agents withinput uiðtÞ ¼ 2:5þ sin 2p

1000 t þ 2p5 i

� �for i ¼ 1;2;3;4;5 and for t ¼ 0;1;2; . . . The initial value of y in all dimensions are ran-

domly generated as a positive number between 0 and 1 to guarantee the winner converges to 1 instead of �1. To avoidthe output is stuck at the unstable equilibrium points due the computation error, we add an extra zero mean 0.005 magni-tude random disturbance in the update of x2 (the second equation in (1)). The five input signals and output yðtÞ are plotted inFig. 5. The figure implies the system can successfully find the winner in real time.

6. Conclusions

In this paper, a simple dynamic system described by a difference equation are proposed to reach the winner-take-all com-petition among agents. This model is in contrast to the model proposed in [27], which is described by a differential equationwith continuous time dynamics instead of a difference equation with completely different discrete-time dynamics. Also, it isdifferent from the existing interactively spiking FitzHugh–Nagumo model [19], optimization based model [23], neural net-work model [18], etc. in modeling the problem. The equilibrium points are solved analytically and their local stability isinvestigated theoretically. In addition, global stability of the model is also rigorously studied in theory. Numerical simula-tions are performed and the results validate the effectiveness of the dynamic equation in describing the winner-take-allcompetition.

Acknowledgments

The authors would like to acknowledge the support by the National Natural Science Foundation of China under GrantNo. 61172165 and Guangdong Science Foundation of China under Grant Nos. S2011010006116 and 10151802904000013.Shuai Li would like to acknowledge the persistent motivation by the lyric ‘‘If pride has not been patted down callously bythe billows of reality, how can we profoundly realize how pains-taking it is to reach the target afar; if dream has not suf-fered the extreme of falling off cliff, how can we know who is with grim determination is actually endowed with invisiblewings.’’

S. Li et al. / Applied Mathematics and Computation 219 (2012) 1569–1575 1575

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