Consensus Problems on Small-world Networks with time delay

In this paper, we study the consensus problem on small-world network with time delay. In particular, we give the sufficient condition of the consensus problem reaching consensus base on a rigorous spectral analysis for the Laplacian matrix of small-world networks. Closed-form upper bound for the spectral radius for Watts-Strogatz type small-world network is presented. These general closed-form analytical results build the relationship between network topological characteristic and the network dynamical characteristic of time delay. Applying these results we can control the time delay given the network while we can also design the required small-world topology given the target time delay. Simulation results verify our analysis and close-form formula behave effectively in the design of the efficient small-world network with time delay robustness. Introduction Recently much attention has been focused on the on social networks theory and related dynamical behaviors. Social networks are well known to exhibit the small-world phenomenon, which comes from the observation that individuals are often linked by a short chain of acquaintances. Specifically, from Milgram's 1967 experiment [1], the average message delivery steps of United States is discover to be six, known as six degree of separation. This phenomenon in social networks has a wide application to shorten delay, improving spectral efficiency and energy efficiency. Watts and Strogatz [2] further certify that the average distance can become dramatically shorten with just few random linkages and establish the small-world models. Moveover, Newman [3] provides a comprehensive study in network structure and functions from complex networks perspective. Gu, Huang and Zhang [4] proposed a generalized small-world network model and proved the model possesses large clustering coefficient and small diameter. The consensus problem [5] [6] for a network dynamics means to reach a common aggreement of all agents in the network, where the agents act in coordination with the others through an information network. The underlying topology of network plays an important role in the analysis of convergence speed for the consensus problem [7] [8] [9]. Gu, Zhang and Zhou [10] showed that in small-world network, the consensus speed is accelerated and become ultrafast which is first discovered by Olfati-Saber [11]. However the consensus problem with delay is much more complicated. In [8], Olfati-Saber shows that the consensus condition is much more strict for the one with delay comparing with the one without delay. Moreover the consensus problem with delay on the small-world topology still remains open. In this paper, we study the consensus problem on Watts-Strogatz type small-world network with time delay. In particular, we give the sufficient condition of the consensus problem reaching consensus base on a rigorous spectral analysis for the Laplacian matrix of small-world networks. The rest of the paper is organized as follows: In next section, we review some preliminary results on the small-world networks and give topological analysis on Watts-Strogatz model. The third section illustrates the dynamical behavior of consensus problem with time delay on Watts-Strogatz network and provides with our main analytical results. In the last section, we include some numerical results and our conclusion. 5th International Conference on Information Engineering for Mechanics and Materials (ICIMM 2015) © 2015. The authors Published by Atlantis Press 121 Small-world network and The Wattz-Strogatz model It was discovered that there are two important characterizations: i.e., large clustering coefficient, and the small world phenomenon (small diameter, or short average distance) in the real complex networks. The small world effect may be dated back the famous experiments carried out by Stanley Milgram [1] in the 1960s, in which letters passed from person to person were able to reach a designated target individual in only a small number of steps. In order to describe complex networks with the two properties, Watts and Strogatz [2] [12]in 1998 proposed a famous small-world network model which possesses both small average distance and large clustering coefficient by simulation if the wired probability ] , [ 2 1 p p p∈ . Fig. 1 Watts-Strogatz network model, [a]:the regular circulant graph; [b]:WS model with p=0.1 rewiring probability The small-world network proposed by Watts and Strogatz as shown in Figure 1 can be described as follows: Let k n C , be a 2k-regular circulant graph (or network) with vertex set ) ,. , , { ) ( 2 1 , n k n v v v C V = and edge set , mod }( 1 , 1 ), , {( ) ( , n j t i k t n i v v C e t i i k n ≡ + ≤ ≤ ≤ ≤ = + if i+t>n). Then one rewires every edge of k n C , with probability p by changing one end of an edge uniformly at random. No loops and multi-edges are allowed. The importance of the Watts and Strogatz' model is due to the fact that this model, roughly speaking, possesses both large clustering coefficient and small Diameter [2] [12] when ) (logn k Θ = . In the following, we simply consider the Watts-Strogatz small-world model with n c k log = where c>0. In order to examine the graph topological property, we need the following Chernoff inequality for binomial random variables. Lemma 1: (Chernoff inequality)[4] Let ) , ( p n Bin X = and set np = μ then . ) 3 / ( 2 exp ) Pr( 2       + − a a a X μ μ Now we are ready to provide the result of maximum degree of Watts-Strogatz network which will be further applied in the network dynamic system. Lemma 2: Let n c k log = . Then the maximum degree ∆ of the Watts-Strogatz network ) , , ( p k n WS has an upper bound . log 2 9 1 3 1 log ) 2 ( n p c n c p       + + + +