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Periodic consensus in network systems with general distributed processing delays
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Abstract
How to understand the dynamical consensus patterns in network systems is of particular significance in both theories and applications. In this paper, we are interested in investigating the influences of distributed processing delay on the consensus patterns in a network model. As new observations, we show that the desired network model undergoes both weak consensus and periodic consensus behaviors when the parameters reach a threshold value and the connectedness of the network system may be absent. In results, some criterions of weak consensus and periodic consensus with exponential convergent rate are established by the standard functional differential equations analysis. An analytic formula is given to calculate the asymptotic periodic consensus in terms of model parameters and the initial time interval. Also, we post the threshold values for some typical distributions included uniform distribution and Gamma distribution. Finally, we give the numerical simulation and analyse the influences of different delays on the consensus.
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Keywords:
- Weak consensus,
- periodic consensus,
- differential system,
- processing delay,
- Gamma distribution delay.
Mathematics Subject Classification: Primary: 93C23, 93D09, 93C95.Citation: 
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Figure 1. Consensus and periodic consensus with a uniform distribution delay.
$ \varphi(s) = \frac{1}{\tau} $ ,$ k^* = \frac{\pi^2}{2} $ (Tab. 1). According to Theorem 2.1, if$ \tilde{\lambda}\tau(1-\frac{8}{9})<\frac{\pi^2}{2} $ , the system achieves a consensus(left:$ \tilde{\lambda} = 9\pi^2 $ and$ \tau = 0.3 $ ). When$ \tilde{\lambda}\tau(1-\frac{8}{9}) = \frac{\pi^2}{2} $ , the system achieves a periodic consensus(right:$ \tilde{\lambda} = 9\pi^2 $ and$ \tau = 0.5 $ )
Figure 2. Consensus and periodic consensus with an exponential distribution delay.
$ \varphi(s) = \frac{\alpha e^{\alpha\tau}}{e^{\alpha\tau}-1}e^{\alpha s}(\alpha = 2,\tau = 1) $ ,$ k^* = 116.7278 $ . The critical condition is that$ \tilde{\lambda}\tau(1-\frac{8}{9})<116.7278 $ . Thus, the left one is a consensus($ \tilde{\lambda} = 270 $ and$ \tau = 1 $ ) and the right one is a periodic consensus($ \tilde{\lambda} = 1050.5502 $ and$ \tau = 1 $ )
Figure 3. Consensus and periodic consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^2e^{\alpha\tau}}{e^{\alpha\tau}-\alpha\tau-1}|s|e^{\alpha s} (\alpha = 2,\tau = 1) $ ,$ k^* = 3.8152 $ . Similarly, the left one is a consensus($ \tilde{\lambda} = 13.5 $ and$ \tau = 1 $ ) and the right one is a periodic consensus($ \tilde{\lambda} = 34.3368 $ and$ \tau = 1 $ )
Figure 4. Consensus and periodic consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^3e^{\alpha\tau}}{2e^{\alpha\tau}-(\alpha\tau+1)^2-1}s^2e^{\alpha s} (\alpha = 2,\tau = 1) $ ,$ k^* = 2.7019 $ . The left one is the case$ \tilde{\lambda} = 9 $ and$ \tau = 1 $ . The right one is the case$ \tilde{\lambda} = 24.3171 $ and$ \tau = 1 $ 
Figure 5. Consensus and periodic consensus with a Bernoulli distribution delay.
$ \varphi(s) = 0 $ for$ s\in (-\tau,0] $ and$ \varphi(s) = 1 $ for$ s = -\tau $ ,$ k^* = \frac{\pi}{2} $ . The left one is the case$ \tilde{\lambda} = 9\pi $ and$ \tau = 0.3 $ . The right one is the case$ \tilde{\lambda} = 9\pi $ and$ \tau = 0.5 $ 
Figure 6. Clustering consensus with a uniform distribution delay.
$ \varphi(s) = \frac{1}{\tau} $ ,$ k^* = \frac{\pi^2}{2} $ . According to Theorem 2.1, if$ \tilde{\lambda}\tau(1-\frac{5}{6}) = \frac{\pi^2}{2} $ , the nodes in Group 1(blue line) achieve a consensus and the ones in Group 2(red line) achieve a periodic consensus(left:$ \tilde{\lambda} = 6\pi^2 $ and$ \tau = 0.5 $ ). When$ \tilde{\lambda}\tau(1-\frac{13}{15}) = \frac{\pi^2}{2} $ , the nodes in Group 1(blue line)achieve a periodic consensus and the others in Group 2(red line) are divergence(right:$ \tilde{\lambda} = \frac{15\pi^2}{2} $ and$ \tau = 0.5 $ )
Figure 7. Clustering consensus with an exponential distribution delay.
$ \varphi(s) = \frac{\alpha e^{\alpha\tau}}{e^{\alpha\tau}-1}e^{\alpha s}(\alpha = 2,\tau = 1) $ ,$ k^* = 116.7278 $ . Similarly, the left one is the case$ \tilde{\lambda} = 700.3668 $ and$ \tau = 1 $ , which is that Group 1(blue line) achieves a consensus and Group 2(red line) achieves a periodic consensus. The right one is the case$ \tilde{\lambda} = 875.4585 $ and$ \tau = 1 $ , which is that Group 1(blue line)achieves a periodic consensus and Group 2(red line) is divergence
Figure 8. Clustering consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^2e^{\alpha\tau}}{e^{\alpha\tau}-\alpha\tau-1}|s|e^{\alpha s} (\alpha = 2,\tau = 1) $ ,$ k^* = 3.8152 $ . In the case of$ \tilde{\lambda} = 22.8912 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case$ \tilde{\lambda} = 28.614 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right)
Figure 9. Clustering consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^3e^{\alpha\tau}}{2e^{\alpha\tau}-(\alpha\tau+1)^2-1}s^2e^{\alpha s} $ $ (\alpha = 2,\tau = 1) $ ,$ k^* = 2.7019 $ . In the case of$ \tilde{\lambda} = 16.2114 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case$ \tilde{\lambda} = 20.2643 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right)
Figure 10. Clustering consensus with a Bernoulli distribution delay.
$ \varphi(s) = 0 $ for$ s\in (-\tau,0] $ and$ \varphi(s) = 1 $ for$ s = -\tau $ ,$ k^* = \frac{\pi}{2} $ . In the case of$ \tilde{\lambda} = 6\pi $ and$ \tau = 0.5 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case$ \tilde{\lambda} = \frac{15\pi}{2} $ and$ \tau = 0.5 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right)Table 1. The values of
$ k^* $ and$ y_{im} $ for some special casesCases $ k^* $ $ y_{im} $ Descriptions $ \varphi(s)=\frac{1}{\tau} $ $ \frac{\pi^2}{2} $ $ \frac{\pi}{\tau} $ Uniform distribution $ \varphi(s)=\frac{2e^{2}}{e^{2}-1}e^{2s} $ 116.7278 16.8680 Exponential distribution $ \varphi(s)=\frac{4e^{2}}{e^{2}-3}|s|e^{2s} $ 3.8152 2.8801 Special $ \gamma $ -distribution$ \varphi(s)=\frac{4e^{2}}{e^{2}-5}s^2e^{2s} $ 2.7019 2.3530 Special $ \gamma $ -distribution$ \varphi(s)=\left\{\begin{array}{ll} 0, & s \in(-\tau, 0] \\ 1, & s=-\tau \end{array}\right. $ $ \frac{\pi}{2} $ $ \frac{\pi}{2\tau} $ Bernoulli distribution | Show Table
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Table 2. Initial values
$ x_i(\theta)(i = 1,2,...,N) $ ,$ \theta\in[-\tau,0] $ $ x_1(\theta) $ $ x_2(\theta) $ $ x_3(\theta) $ $ x_4(\theta) $ $ x_5(\theta) $ 7.0605 0.3183 2.7692 0.4617 0.9713 $ x_6(\theta) $ $ x_7(\theta) $ $ x_8(\theta) $ $ x_9(\theta) $ $ x_{10}(\theta) $ 8.2346 6.9483 3.1710 9.5022 0.3445 where the numbers are randomly selected in interval (0, 10). | Show Table
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Table 3. The numerical simulations for Case Ⅰ
Cases $ \tilde{\lambda} $ $ \tau $ Results Uniform distribution (Fig. 1) $ 9\pi^2 $ $ 0.3 $ consensus $ 9\pi^2 $ $ 0.5 $ periodic consensus Exponential distribution(Fig. 2) $ 270 $ $ 1 $ consensus $ 1050.5502 $ $ 1 $ periodic consensus Special $ \gamma $ -distribution 1(Fig. 3)$ 13.5 $ $ 1 $ consensus $ 34.3368 $ $ 1 $ periodic consensus Special $ \gamma $ -distribution 2(Fig. 4)$ 9 $ $ 1 $ consensus $ 24.3171 $ $ 1 $ periodic consensus Bernoulli distribution(Fig. 5) $ 9\pi $ $ 0.3 $ consensus $ 9\pi $ $ 0.5 $ periodic consensus | Show Table
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Table 4. The numerical simulations for Case Ⅱ
Distribution cases $ \tilde{\lambda} $ $ \tau $ Group 1(blue) Group 2(red) Uniform (Fig. 6) $ 6\pi^2 $ $ 0.5 $ consensus periodic consensus $ \frac{15\pi^2}{2} $ $ 0.5 $ periodic consensus divergence Exponential (Fig. 7) $ 700.3668 $ $ 1 $ consensus periodic consensus $ 875.4585 $ $ 1 $ periodic consensus divergence Gamma 1(Fig. 8) $ 22.8912 $ $ 1 $ consensus periodic consensus $ 28.614 $ $ 1 $ periodic consensus divergence Gamma 2(Fig. 9) $ 16.2114 $ $ 1 $ consensus periodic consensus $ 20.2643 $ $ 1 $ periodic consensus divergence Bernoulli(Fig. 10) $ 6\pi $ $ 0.5 $ consensus periodic consensus $ \frac{15\pi}{2} $ $ 0.5 $ periodic consensus divergence | Show Table
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References
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Figure 1. Consensus and periodic consensus with a uniform distribution delay.
$ \varphi(s) = \frac{1}{\tau} $ ,$ k^* = \frac{\pi^2}{2} $ (Tab. 1). According to Theorem 2.1, if$ \tilde{\lambda}\tau(1-\frac{8}{9})<\frac{\pi^2}{2} $ , the system achieves a consensus(left:$ \tilde{\lambda} = 9\pi^2 $ and$ \tau = 0.3 $ ). When$ \tilde{\lambda}\tau(1-\frac{8}{9}) = \frac{\pi^2}{2} $ , the system achieves a periodic consensus(right:$ \tilde{\lambda} = 9\pi^2 $ and$ \tau = 0.5 $ ) -
Figure 2. Consensus and periodic consensus with an exponential distribution delay.
$ \varphi(s) = \frac{\alpha e^{\alpha\tau}}{e^{\alpha\tau}-1}e^{\alpha s}(\alpha = 2,\tau = 1) $ ,$ k^* = 116.7278 $ . The critical condition is that$ \tilde{\lambda}\tau(1-\frac{8}{9})<116.7278 $ . Thus, the left one is a consensus($ \tilde{\lambda} = 270 $ and$ \tau = 1 $ ) and the right one is a periodic consensus($ \tilde{\lambda} = 1050.5502 $ and$ \tau = 1 $ ) -
Figure 3. Consensus and periodic consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^2e^{\alpha\tau}}{e^{\alpha\tau}-\alpha\tau-1}|s|e^{\alpha s} (\alpha = 2,\tau = 1) $ ,$ k^* = 3.8152 $ . Similarly, the left one is a consensus($ \tilde{\lambda} = 13.5 $ and$ \tau = 1 $ ) and the right one is a periodic consensus($ \tilde{\lambda} = 34.3368 $ and$ \tau = 1 $ ) -
Figure 4. Consensus and periodic consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^3e^{\alpha\tau}}{2e^{\alpha\tau}-(\alpha\tau+1)^2-1}s^2e^{\alpha s} (\alpha = 2,\tau = 1) $ ,$ k^* = 2.7019 $ . The left one is the case$ \tilde{\lambda} = 9 $ and$ \tau = 1 $ . The right one is the case$ \tilde{\lambda} = 24.3171 $ and$ \tau = 1 $ -
Figure 5. Consensus and periodic consensus with a Bernoulli distribution delay.
$ \varphi(s) = 0 $ for$ s\in (-\tau,0] $ and$ \varphi(s) = 1 $ for$ s = -\tau $ ,$ k^* = \frac{\pi}{2} $ . The left one is the case$ \tilde{\lambda} = 9\pi $ and$ \tau = 0.3 $ . The right one is the case$ \tilde{\lambda} = 9\pi $ and$ \tau = 0.5 $ -
Figure 6. Clustering consensus with a uniform distribution delay.
$ \varphi(s) = \frac{1}{\tau} $ ,$ k^* = \frac{\pi^2}{2} $ . According to Theorem 2.1, if$ \tilde{\lambda}\tau(1-\frac{5}{6}) = \frac{\pi^2}{2} $ , the nodes in Group 1(blue line) achieve a consensus and the ones in Group 2(red line) achieve a periodic consensus(left:$ \tilde{\lambda} = 6\pi^2 $ and$ \tau = 0.5 $ ). When$ \tilde{\lambda}\tau(1-\frac{13}{15}) = \frac{\pi^2}{2} $ , the nodes in Group 1(blue line)achieve a periodic consensus and the others in Group 2(red line) are divergence(right:$ \tilde{\lambda} = \frac{15\pi^2}{2} $ and$ \tau = 0.5 $ ) -
Figure 7. Clustering consensus with an exponential distribution delay.
$ \varphi(s) = \frac{\alpha e^{\alpha\tau}}{e^{\alpha\tau}-1}e^{\alpha s}(\alpha = 2,\tau = 1) $ ,$ k^* = 116.7278 $ . Similarly, the left one is the case$ \tilde{\lambda} = 700.3668 $ and$ \tau = 1 $ , which is that Group 1(blue line) achieves a consensus and Group 2(red line) achieves a periodic consensus. The right one is the case$ \tilde{\lambda} = 875.4585 $ and$ \tau = 1 $ , which is that Group 1(blue line)achieves a periodic consensus and Group 2(red line) is divergence -
Figure 8. Clustering consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^2e^{\alpha\tau}}{e^{\alpha\tau}-\alpha\tau-1}|s|e^{\alpha s} (\alpha = 2,\tau = 1) $ ,$ k^* = 3.8152 $ . In the case of$ \tilde{\lambda} = 22.8912 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case$ \tilde{\lambda} = 28.614 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right) -
Figure 9. Clustering consensus with a Gamma distribution delay.
$ \varphi(s) = \frac{\alpha^3e^{\alpha\tau}}{2e^{\alpha\tau}-(\alpha\tau+1)^2-1}s^2e^{\alpha s} $ $ (\alpha = 2,\tau = 1) $ ,$ k^* = 2.7019 $ . In the case of$ \tilde{\lambda} = 16.2114 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case$ \tilde{\lambda} = 20.2643 $ and$ \tau = 1 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right) -
Figure 10. Clustering consensus with a Bernoulli distribution delay.
$ \varphi(s) = 0 $ for$ s\in (-\tau,0] $ and$ \varphi(s) = 1 $ for$ s = -\tau $ ,$ k^* = \frac{\pi}{2} $ . In the case of$ \tilde{\lambda} = 6\pi $ and$ \tau = 0.5 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case$ \tilde{\lambda} = \frac{15\pi}{2} $ and$ \tau = 0.5 $ , the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right)