American Institute of Mathematical Sciences

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March  2021, 16(1): 139-153. doi: 10.3934/nhm.2021002

Periodic consensus in network systems with general distributed processing delays

Yicheng Liu 1,, , Yipeng Chen 1, , Jun Wu 2, and  Xiao Wang 1,

1. 

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, P. R. China
2. 

College of Mathematics and Statistics, Changsha University of Science Technology, Changsha, 410076, P. R. China

* Corresponding author: Yicheng Liu

Received  September 2020 Revised  November 2020 Published  March 2021 Early access  December 2020

Fund Project: This work was partially supported by National Natural Science Foundation of China (11671011 and 11428101)

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.

Keywords: Weak consensus, periodic consensus, differential system, processing delay, Gamma distribution delay.
Mathematics Subject Classification: Primary: 93C23, 93D09, 93C95.
Citation: Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks and Heterogeneous Media, 2021, 16 (1) : 139-153. doi: 10.3934/nhm.2021002
References:
[1]

M. H. DeGroot, Reaching a consensus, J. Am. Stat. Assoc., 69 (1974), 118-121. 

[2]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control, 49 (2004), 1465-1476.  doi: 10.1109/TAC.2004.834433.

[3]

Q. Feng, S. K. Nguang and A. Seuret, Stability analysis of linear coupled differential–difference systems with general distributed delays, IEEE Transactions on Automatic Control, 65 (2020), 1356-1363.  doi: 10.1109/TAC.2019.2928145.

[4]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[5]

J. Jost, F. M. Atay and W. Lu, Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Networks and Heterogeneous Media, 6 (2011), 329-349.  doi: 10.3934/nhm.2011.6.329.

[6]

F. A. Khasawneh and B. P. Mann, A spectral element approach for the stability of delay systems, Int. J. Numer. Meth. Engng., 87 (2011), 566-592. 

[7]

N. A. Lynch, Distributed Algorithms, San Francisco, CA: Morgan Kaufmann, 1996. doi: 10.1108/IMDS-01-2014-0013.

[8]

F. Mazenc, M. Malisoff and H. $\ddot{O}$zbay, Stability and robustness analysis for switched systems with time-varying delays, SIAM J. Control Optim., 56 (2018), 158-182.  doi: 10.1137/16M1104895.

[9]

M. Michiels, I. C. Morărescu and S.-I. Niculescu, Consensus problems with distributed delays, with application to traffic flow models, SIAM J. Control Optim., 48 (2009), 77-101.  doi: 10.1137/060671425.

[10]

I. C. Morarescu, W. Michiels and M. Jungers, Synchronization of coupled nonlinear oscillatiors with gamma-distributed delays, in American Control Conference, ACC 2013, (2013), Washington, United States.

[11]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113.

[12]

A. V. Proskurnikov, Average consensus in networks with nonlinearly delayed couplings and switching topology, Automatica, 49 (2013) 2928–2932. doi: 10.1016/j.automatica.2013.06.007.

[13]

D. Serre, Matrices, Graduate Texts in Mathematics, 216 (2010), Springer. doi: 10.1007/978-1-4419-7683-3.

[14]

J. W. H. So, X. Tang and X. Zou, Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33 (2002), 1297-1304.  doi: 10.1137/S0036141001389263.

[15]

O. Solomon and E. Fridman, New stability conditions for systems with distributed delays, Automatica, 49 (2013), 3467-3475.  doi: 10.1016/j.automatica.2013.08.025.

[16]

T. Vicsek, A Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

M. H. DeGroot, Reaching a consensus, J. Am. Stat. Assoc., 69 (1974), 118-121. 

[2]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control, 49 (2004), 1465-1476.  doi: 10.1109/TAC.2004.834433.

[3]

Q. Feng, S. K. Nguang and A. Seuret, Stability analysis of linear coupled differential–difference systems with general distributed delays, IEEE Transactions on Automatic Control, 65 (2020), 1356-1363.  doi: 10.1109/TAC.2019.2928145.

[4]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[5]

J. Jost, F. M. Atay and W. Lu, Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Networks and Heterogeneous Media, 6 (2011), 329-349.  doi: 10.3934/nhm.2011.6.329.

[6]

F. A. Khasawneh and B. P. Mann, A spectral element approach for the stability of delay systems, Int. J. Numer. Meth. Engng., 87 (2011), 566-592. 

[7]

N. A. Lynch, Distributed Algorithms, San Francisco, CA: Morgan Kaufmann, 1996. doi: 10.1108/IMDS-01-2014-0013.

[8]

F. Mazenc, M. Malisoff and H. $\ddot{O}$zbay, Stability and robustness analysis for switched systems with time-varying delays, SIAM J. Control Optim., 56 (2018), 158-182.  doi: 10.1137/16M1104895.

[9]

M. Michiels, I. C. Morărescu and S.-I. Niculescu, Consensus problems with distributed delays, with application to traffic flow models, SIAM J. Control Optim., 48 (2009), 77-101.  doi: 10.1137/060671425.

[10]

I. C. Morarescu, W. Michiels and M. Jungers, Synchronization of coupled nonlinear oscillatiors with gamma-distributed delays, in American Control Conference, ACC 2013, (2013), Washington, United States.

[11]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113.

[12]

A. V. Proskurnikov, Average consensus in networks with nonlinearly delayed couplings and switching topology, Automatica, 49 (2013) 2928–2932. doi: 10.1016/j.automatica.2013.06.007.

[13]

D. Serre, Matrices, Graduate Texts in Mathematics, 216 (2010), Springer. doi: 10.1007/978-1-4419-7683-3.

[14]

J. W. H. So, X. Tang and X. Zou, Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33 (2002), 1297-1304.  doi: 10.1137/S0036141001389263.

[15]

O. Solomon and E. Fridman, New stability conditions for systems with distributed delays, Automatica, 49 (2013), 3467-3475.  doi: 10.1016/j.automatica.2013.08.025.

[16]

T. Vicsek, A Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

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 $)
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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 $)
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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 $)
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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 $
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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 $
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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 $)
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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
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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)
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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)
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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)
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Table 1.  The values of $ k^* $ and $ y_{im} $ for some special cases
Cases $ 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
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Cases $ 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
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).
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$ 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).
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
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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
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
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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
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