Synchronization Problem on Networking and Synthesis Based on Algebraic Graph Theory

Malaya Journal of Matematik, Vol. S, No. 1, 50-54, 2018

https://doi.org/10.26637/MJM0S01/10

Synchronization problem on networking and synthesis based on algebraic graph theory

Richa Agarwal1 and Shivam Agarwal2*

Abstract This paper is concerned with the adaptive pinning synchronization problem of stochastic complex dynamical networks (CDNs). Based on algebraic graph theory and Lyapunov theory, pinning controller design conditions are derived, and the rigorous convergence analysis of synchronization errors in the probability sense is also reﬂected. This paper describes uses of graph theory to ‘some of the problems of network synthesis and analysis, beginning with the ancient period of network theory. The initial portion is devoted to the analytic results related to the topological analysis of linear, submissive, and converter less arrangements. Then it goes over to handout dealing with moderation of these methods to group with correlative services and animated components. Keywords Graph theory, Lyapunov theory and Pinning controller design.

1Applied Science, Mathematics Department, KIET Institute of Engineering, Ghaziabad, India. 2IIT Kharagpur, India. *Corresponding author: [email protected]; [email protected] Article History: Received 24 December 2017 ; Accepted 21 January 2018 c 2018 MJM.

Contents 2. Multicommodity Flow Problems With Fixed Link Capacities 1 Introduction...... 50 In traditional multicommodity network flow problems, the 2 Multicommodity Flow Problems With Fixed Link Ca- c pacities...... 50 capacities l are usually assumed fixed and one is to minimize some convex function of the network flow variables subject 3 outing Norms with Fault Links Avoiding...... 51 to the set of constraints (2.1). For example, one of the most 4 Basic Contributions...... 51 common cost functions used in the communication network 5 Topological Synthesis of Resistive Networks...... 51 literature is total delay function

6 A Special Case Water-Filling...... 52 Fdelay(t) = ∑ (2.1) 7 Conclusion...... 52 which is a convex function to t. In the minimum delay routing References...... 52 problem, the source vectors s (i.e., the load to be supported by the network) are given and one is to minimize fdelay(t) 1. Introduction by selecting the optimal flow variables x and t subject to the constraints (2.1). The focus of this research is to explain the utility of the There is a vast literature on convex multicommodity net- graph theory to problems related to electrical engineering in work flow problems and many efficient solution methods have the last several years. In past decannium, acceptable concen- been developed; and referenced therein. In this chapter, how- trations have ever, we are interested in the interplay between resources allo- been casted on the survey of complex dynamic networks cation, link capacities, and optimal routing present in wireless (CDNs) because of their probable utility in many practical data networks. systems, such as social systems, neural networks,linguistic In a wireless system, the capacities of individual links de- networks, and technological systems. In specific, speedly pend on the media access scheme and the selection of certain growing research regards have aimed on the synchronization critical parameters, such as transmit powers, bandwidths or problem. time slot fractions, allocated to individual link or group of link. Synchronization problem on networking and synthesis based on algebraic graph theory — 51/54

We assume that the medium access method and coding and introducing Lagrange multipliers to relax the global resource modulation schemes of the communication system are fixed, constraints. but that we can optimize over the communication variables r. The communications variables are themselves limited by 4. Basic Contributions various resources constraints, such as limits on the total transmit power at each node or the total signal bandwidth available Abstract picture providing the whole information about across the whole network. the network geometry, i.e., about the interconnections between all the network elements. Kirchhoff’s laws, when applied to such an underlying 3. outing Norms with Fault Links graph, provide a basis for establishing the numbers of linearly Avoiding independent loop-current and cut-set-voltage equations with no relation whatsoever to the characteristics of the network The one of effective applications of tie-set path is the elements. In addition, such a graph-theoretic presentation sturdy routing protocol that supports to keep connections even enabled Kirchhoff to formulate for a network with a finite if failures occurred in the network. By use of the tie-set path, number of linear resistors some shorthand methods of writing it is possible to construct more flexible fault tolerant network by inspection the current in any of the resistors knowing the system than using two node-disjoint, detouring route around voltage excitations located anywhere in the network. Kirch- fault links-paths, since it can utilize many available links by hoff’s approach has subsequently been generalized for ac avoiding fault links locally in each loop. loop-current analysis of linear RLC networks without mutual We propose to solve the SRRA problems via its dual and couplings. In 1892, Maxwell provided a dual set of topo- then recover the optimal primal solution from the optimal dual logical formulas for solution of a system of node-voltage variables. equations of RLC networks showing among others that the Due to rich structure of this problem, there are many determinant of the node-admittance matrix of such a network ways to formulate the dual problem depending on for which is equal to the sum of all the tree admittance products. One of constraints the Lagrange multipliers are introduced. At the the most important properties of the Kirchhoff and Maxwell first level we exploit the layered architecture of the wireless methods was that there was no need to write down the system network-the network flow variables x, s, t and the commu- equations. The required network functions, when using the nication variables r are only coupled through the capacity so-called topological formulas can be written directly from the constraints, t ≤ /0 (r ). We will introduced Lagrange mul- l l l network by inspection. All terms appearing in these formulas tipliers to relax the capacity constraints, so that the SRRA have the same positive sign. The additional advantage gained problem is decomposed into a network routing sub problems by applying these methods was two-fold: reducing computa- and a communication resources allocation sub problem. tional effort by not computing unnecessary terms which have to be cancelled, and eliminating the errors that might occur in the cancellation process. The largest capacity region of the Gaussian broadcast channel is achieved by CDMA, with superposition coding and interference cancellation. However most CDMA system in practice are design without interference cancellation, e.g, direct sequence CDMA systems. Here we concentrate on the practical model of interference limited channels, while leaving the discussion of the information theoretical model with interference cancellation to Appendix A. The kind of interrelation between the network to be studied and the auxiliary graph may vary significantly. Historically, it started in 1847 by Kirchhoff with the direct relation, in which the graph presented a simplified,

5. Topological Synthesis of Resistive At the second level, we will notice that the two sub prob- Networks lem themselves have structure that can be further exploited to Research in the theory of resistive networks has attracted develop distributed algorithms. In particular, the multicom- considerable attention and achieved some interesting results in modity network routing sub problem is naturally decomposed the late ﬁfties and during the sixties. One of the motivations into single commodity routing problems, and the resource allo- for looking for efﬁcient algorithms in analysis of resistive cation sub problem can be decomposed into smaller problems networks may be found in the help they provide in the study at each node using the dual decomposition method again- by of transient phenomena in electric networks. As is well known,

51 Synchronization problem on networking and synthesis based on algebraic graph theory — 52/54 the transient analysis of general linear or nonlinear dynamic An intuitive way to solve the dual problem is pricing, networks may, for any choice of the numerical integration where the Lagrange multiplier λ is interpreted as the price for algorithm, be reduced to dc analysis of a sequence of general the power resource. The pricing mechanism follows the law of linear resistive networks. supply and demand: if Pi (λ) < 1, which means the resource The prime objective of the present work is to develop a is underutilized, then we decrease the price λ; otherwise we unified theory to show the network problems which occurs increase λ. This procedure will eventually approach λ ∗. in our day-to-day life by using graph theory. We wish to Note that the CDNs are often subject to noisy environ- examine the problems of networks by the point of view of ment and, therefore, many researchers have recently investi- graphs. Now-a-days it has become fashionable to mention gated the synchronization problem of CDN in stochastic set- that there are applications of one of field of mathematics to tings. For example, the pinning stability of linearly coupled some other areas of mathematics. Our only aim of this present stochastic neural networks was considered in [18]; the pin- thesis is to show applications of graph theory in networks ning distributed synchronization of stochastic coupled neural theory. networks via randomly occurring control was studied in [19]; For example, we speak of a programme produced by a the global exponential adaptive synchronization for a class of television company being ‘networked’ across the country. We stochastic CDNs was considered in [20]; the synchronization think of the TV network as a number of transmitters liked problem for stochastic discrete-time complex networks was by landlines or other means, all of which can broadcast a investigated in [21]; the problem of pinning synchronization programme originating from any one of a number of different of nonlinear dynamical networks with multiple stochastic dis- sources. Similarly, we speak of a railway network as a way of turbances was studied in [22]; a new control strategy was pro- describing a number of stations linked by a series of railway posed in [23] for achieving the synchronization of stochastic lines which enable trains to travel along a number of different dynamical networks with nonlinear coupling; the synchroniza- possible routes. A familiar example is the Delhi Underground tion control of stochastic neural networks was considered in Metro railway network. [24]; and in particular, some important criteria for stochastic pinning control of networks of chaotic maps were derived in 6. A Special Case Water-Filling [25]. It should be pointed out that the linear matrix inequality (LMI) techniques or the analysis approaches of the spectral Consider the maximum sum capacity problem over paral- properties of salient matrices have been introduced in [1]-[38] lel Gaussian channels: to formulate the synchronization and controllability criteria. As a result, the topology structures or Laplacian matrices are maximize log(1 + Pi/σi) (6.1) required to be known, because they are involved in the LMIs subject to Pi = 1, Pi ≥ 0, i = 1,...,m. or the inequalities related to the spectral radius. However, Here the optimization variables are the power Pi and σi is just as pointed out in [15] and [16], the priori knowledge of the noise power for channel i. Observe that the total power a complex topology abstracted from real-world systems is constraints Pi = 1 couples otherwise separate power allocation usually unavailable and unmeasurable. Therefore, a new tech- problems of the m channels. The water filling algorithm nique should be developed to relax this requirement for the follows by exploiting this separable structure and solving this stochastic CDN, which motivates the study of this paper. problem via its Lagrange dual. We formulate the dual of (6.1) by introducing a Lagrange 7. Conclusion multiplier λ for the coupling constraint Pi = 1. The resulting Lagrangian is In this paper, the problem of adaptive pinning synchronization control of stochastic CDN with unknown topology L(P,λ) = log(1 + Pi/σi) − λ(Pi − 1) structure has been considered. By combining graph theory = (log(1 + Pi/σi) − λPi) + λ and Lyapunov theory, pinning controller design conditions have been derived and the rigorous convergence analysis of and the dual function is defined as synchronization errors in the probability sense have also been

L(P,λ)λ + sup(log(1 + Pi/σi) − λPi). (6.2) conducted. It is shown that the number of pinning nodes and the selection of nodes for pinning depend on the unknown The dual problem is to minimize V (λ). For any given λ, the lower bounds of coupling strengths. In particular, by further function V (λ) can be evaluated analytically. In particular, the constructing an appropriate Lyapunov functional, the above summands in (6.2) are decoupled small optimization problems controller design conditions have been proven to be still valid for each channel i and can be solved separately by setting for the stochastic CDN with coupling delay. Finally, the simu- lation results on a Chua’s circuit network have been given to Pi(λ) = max{0,1/λ − σi}, i = 1,...,m (6.3) illustrate the effectiveness of the obtained theoretical results. ∗ ∗ The optimal dual variable λ satisﬁes Pi (λ ) = 1, and the Further study on application of the theory to practical net- ∗ corresponding Pi (λ ) is the water ﬁlling solution to the primal work management problems and development of distributed problem. algorithms are left to the future.

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