Synchronization problem on networking and synthesis based on algebraic graph theory

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

Synchronization problem on networking and

synthesis based on algebraic graph theory

Richa Agarwal

1

and Shivam Agarwal

2

*

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.

1_{Applied Science, Mathematics Department, KIET Institute of Engineering, Ghaziabad, India.}

2_{IIT Kharagpur, India.}

*Corresponding author:1_{[email protected];}2_{[email protected]}

Article History: Received24December2017; Accepted21January2018 c2018 MJM.

1. Introduction

The focus of this research is to explain the utility of the graph theory to problems related to electrical engineering in the last several years. In past decannium, acceptable concen-trations have

been casted on the survey of complex dynamic networks (CDNs) because of their probable utility in many practical systems, such as social systems, neural networks,linguistic networks, and technological systems. In specific, speedly growing research regards have aimed on the synchronization problem.

2. Multicommodity Flow Problems With

Fixed Link Capacities

In traditional multicommodity network flow problems, the capacitiesclare usually assumed fixed and one is to minimize some convex function of the network flow variables subject to the set of constraints (2.1). For example, one of the most common cost functions used in the communication network literature is total delay function

Fdelay(t) =

∑

(2.1)

which is a convex function tot. In the minimum delay routing problem, the source vectorss(i.e., the load to be supported by the network) are given and one is to minimize fdelay(t) by selecting the optimal flow variablesxandtsubject to the constraints (2.1).

There is a vast literature on convex multicommodity net-work flow problems and many efficient solution methods have been developed; and referenced therein. In this chapter, how-ever, we are interested in the interplay between resources allo-cation, link capacities, and optimal routing present in wireless data networks.

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We assume that the medium access method and coding and modulation schemes of the communication system are fixed, but that we can optimize over the communication variables r. The communications variables are themselves limited by various resources constraints, such as limits on the total trans-mit power at each node or the total signal bandwidth available across the whole network.

3. outing Norms with Fault Links

Avoiding

The one of effective applications of tie-set path is the sturdy routing protocol that supports to keep connections even if failures occurred in the network. By use of the tie-set path, it is possible to construct more flexible fault tolerant network system than using two node-disjoint, detouring route around fault links-paths, since it can utilize many available links by avoiding fault links locally in each loop.

We propose to solve the SRRA problems via its dual and then recover the optimal primal solution from the optimal dual variables.

Due to rich structure of this problem, there are many ways to formulate the dual problem depending on for which constraints the Lagrange multipliers are introduced. At the first level we exploit the layered architecture of the wireless network-the network flow variablesx,s,t and the commu-nication variables r are only coupled through the capacity constraints,tl ≤/0l(rl). We will introduced Lagrange mul-tipliers to relax the capacity constraints, so that the SRRA problem is decomposed into a network routing sub problems and a communication resources allocation sub problem.

At the second level, we will notice that the two sub prob-lem themselves have structure that can be further exploited to develop distributed algorithms. In particular, the multicom-modity network routing sub problem is naturally decomposed into single commodity routing problems, and the resource allo-cation sub problem can be decomposed into smaller problems at each node using the dual decomposition method again- by

introducing Lagrange multipliers to relax the global resource constraints.

4. Basic Contributions

Abstract picture providing the whole information about the network geometry, i.e., about the interconnections between all the network elements.

Kirchhoff’s laws, when applied to such an underlying graph, provide a basis for establishing the numbers of linearly independent loop-current and cut-set-voltage equations with no relation whatsoever to the characteristics of the network elements. In addition, such a graph-theoretic presentation enabled Kirchhoff to formulate for a network with a finite number of linear resistors some shorthand methods of writing by inspection the current in any of the resistors knowing the voltage excitations located anywhere in the network. Kirch-hoff’s approach has subsequently been generalized for ac loop-current analysis of linear RLC networks without mutual couplings. In 1892, Maxwell provided a dual set of topo-logical formulas for solution of a system of node-voltage equations of RLC networks showing among others that the determinant of the node-admittance matrix of such a network is equal to the sum of all the tree admittance products. One of the most important properties of the Kirchhoff and Maxwell methods was that there was no need to write down the system equations. The required network functions, when using the so-called topological formulas can be written directly from the network by inspection. All terms appearing in these formulas have the same positive sign. The additional advantage gained by applying these methods was two-fold: reducing computa-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

Networks

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the transient analysis of general linear or nonlinear dynamic networks may, for any choice of the numerical integration algorithm, be reduced to dc analysis of a sequence of general linear resistive networks.

The prime objective of the present work is to develop a unified theory to show the network problems which occurs in our day-to-day life by using graph theory. We wish to examine the problems of networks by the point of view of graphs. Now-a-days it has become fashionable to mention that there are applications of one of field of mathematics to some other areas of mathematics. Our only aim of this present thesis is to show applications of graph theory in networks theory.

For example, we speak of a programme produced by a television company being ‘networked’ across the country. We think of the TV network as a number of transmitters liked by landlines or other means, all of which can broadcast a programme originating from any one of a number of different sources. Similarly, we speak of a railway network as a way of describing a number of stations linked by a series of railway lines which enable trains to travel along a number of different possible routes. A familiar example is the Delhi Underground Metro railway network.

6. A Special Case Water-Filling

Consider the maximum sum capacity problem over paral-lel Gaussian channels:

maximize log(1+Pi/σi) (6.1)

subject toPi=1,Pi≥0,i=1, . . . ,m.

Here the optimization variables are the powerPiandσiis the noise power for channeli. Observe that the total power constraintsPi=1 couples otherwise separate power allocation problems of the m channels. The water filling algorithm follows by exploiting this separable structure and solving this problem via its Lagrange dual.

We formulate the dual of (6.1) by introducing a Lagrange multiplierλfor the coupling constraintPi=1. The resulting Lagrangian is

L(P,λ) =log(1+Pi/σi)−λ(Pi−1)

= (log(1+Pi/σi)−λPi) +λ

and the dual function is defined as

L(P,λ)λ+sup(log(1+Pi/σi)−λPi). (6.2)

The dual problem is to minimizeV (λ). For any givenλ, the functionV (λ)can be evaluated analytically. In particular, the summands in (6.2) are decoupled small optimization problems for each channeliand can be solved separately by setting

Pi(λ) =max{0,1/λ−σi}, i=1, . . . ,m (6.3)

The optimal dual variableλ∗satisfiesPi(λ∗) =1, and the correspondingPi(λ∗)is the water filling solution to the primal problem.

An intuitive way to solve the dual problem is pricing, where the Lagrange multiplierλis interpreted as the price for the power resource. The pricing mechanism follows the law of supply and demand: ifPi(λ)<1, which means the resource is underutilized, then we decrease the priceλ; otherwise we increaseλ. This procedure will eventually approachλ∗.

Note that the CDNs are often subject to noisy environ-ment and, therefore, many researchers have recently investi-gated the synchronization problem of CDN in stochastic set-tings. For example, the pinning stability of linearly coupled stochastic neural networks was considered in [18]; the pin-ning distributed synchronization of stochastic coupled neural networks via randomly occurring control was studied in [19]; the global exponential adaptive synchronization for a class of stochastic CDNs was considered in [20]; the synchronization problem for stochastic discrete-time complex networks was investigated in [21]; the problem of pinning synchronization of nonlinear dynamical networks with multiple stochastic dis-turbances was studied in [22]; a new control strategy was pro-posed in [23] for achieving the synchronization of stochastic dynamical networks with nonlinear coupling; the synchroniza-tion control of stochastic neural networks was considered in [24]; and in particular, some important criteria for stochastic pinning control of networks of chaotic maps were derived in [25]. It should be pointed out that the linear matrix inequality (LMI) techniques or the analysis approaches of the spectral properties of salient matrices have been introduced in [1]-[38] to formulate the synchronization and controllability criteria. As a result, the topology structures or Laplacian matrices are required to be known, because they are involved in the LMIs or the inequalities related to the spectral radius. However, just as pointed out in [15] and [16], the priori knowledge of a complex topology abstracted from real-world systems is usually unavailable and unmeasurable. Therefore, a new tech-nique should be developed to relax this requirement for the stochastic CDN, which motivates the study of this paper.

7. Conclusion

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ISSN(P):2319-3786 Malaya Journal of Matematik

ISSN(O):2321-5666

Synchronization problem on networking and synthesis based on algebraic graph theory