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A MODEL FOR CONSTRUCTING SUBGRAPHS OF HYPERCUBES

DELE OLUWADE , Member, IEEE

Abstract – Network topologies play an important role in computer communication networks. Apart from the basic topologies, some other new topologies have been proposed in the literature such as the hypercubes, meshes and trees. In this paper, a model for constructing subgraphs of hypercubes using the qualitative equivalence behavior of the first order autonomous ordinary differential equation x/ = f(x) ( where f(x) is a polynomial of degree n) is presented. The resulting topologies are based on the point-to-point primitive network topology. The topologies of particular interest are those which arise when the critical points of the above equation are complex. Index Terms – Hypercube, model, network topologies, ordinary differential equation, subgraph.*

I. INTRODUCTION Generally, a (communication interconnection) network is modelled as an undirected graph G(V,E) ( where V is the set of vertices(or nodes) and E the set of edges) in which the vertices correspond to communication ports and the edges to communication channels [1]. Communication over the network is achieved via a message passing protocol and the delay in communication is measured by the number of edges traversed. The basic network topologies (otherwise known as primitive structures) are the point-to-point topology, bus, multipoint, star and ring [2]. By linking together these primitive structures, topologies of more complexity are formed. These non-primitive topologies include the hypercubes, n-star network, meshes and trees [3]. In particular, the hypercube and the n-star are two * Manuscript received 05 February 2007. The author’s current contact address is P.O. Box 20253, University of Ibadan, OYO 200005, Nigeria Email: deleoluwade@yahoo.com

popular sophisticated topologies . A hypercube of dimension n (also known variously as n-cube, binary n-cube, cosmic cube, Boolean n-cube etc) is an undirected graph which consists of 2n vertices labeled 0 through 2n – 1 in binary such that two vertices are connected by an edge iff their labellings differ in exactly one position [4] - [5]. Some of the appealing properties of the hypercube is that it is vertex symmetric and edge symmetric (i.e the network looks the same irrespective of where it is viewed from any of its vertex), hierarchical and maximally fault tolerant, has a small degree and diameter, a high connectivity and offers simple routing algorithms. The hypercube can be recursively constructed from lower dimensional cubes. However, there is the disadvantage of high cost and complexity of building large hypercubes [1 ], [4] - [5].

An important topological property of the hypercube is that its structure allows other network topologies such as multigrids and trees to be efficiently simulated. In [5], a theoretical characterization of the hypercube as a graph is presented. It is then shown how various other topologies may be mapped into a hypercube. Generally, a graph may be constructed from a given group e.g see [6]. In [1], a group-theoretic model ( Cayley graph model ) for designing, analyzing and improving symmetric graphs (in particular, the hypercube) was developed. And in [7], a new probabilistic routing policy was presented for regular mesh-connected topologies (such as the hypercubes) in a distributed computing environment. This is an environment in which constituent subcomputers are organized in a mesh-connected topology and communication among individual computers takes place via some form of message exchange. The n-star network was proposed as an alternative to the hypercube. The properties of this network are similar to those of the hypercube. In particular, it has a smaller diameter and degree. The n-star network Sn

1-4244-0987-X/07/$25.00 ©2007 IEEE.

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is a degree n –1 , (n-1)-connected and vertex symmetric Cayley graph [8].

Local area networks (LANs) are currently being constructed with arbitrary nonstandard topology to allow a user to tailor them to his/her peculiarities with respect to such factors as the number of processors and the physical constraints. However, although most parallel machines are currently designed with a standard network topology, it is believed that future high-performance commodity interconnections would allow users to match-create custom interconnection networks like the LAN [9]. In the present paper, a novel model for constructing subgraphs of the hypercube using the qualitative theory of autonomous ordinary differential equations (herein referred to as the qualitative equivalence model) is presented (In general, it is well-known that a network topology is a graph and a subgraph of a graph is a graph ) . Essentially, the qualitative equivalence behavior of the first order autonomous ordinary differential equation (ODE) x/ = f(x) (1.1) (where f(x) is the general polynomial of degree n ∈ N (set of natural numbers) is used. This involves obtaining the critical (or equilibrium) points of the equation for a particular n ≥ 2 and drawing the resulting phase portraits using the set of phase portraits of the equation when n = 1 i.e Ψ = {Attractor(A), Repellor(R), Positive Shunt(P), Negative Shunt(N)} (1.2) as shown in Fig. 1.1, where the critical point is located at the point marked in the middle of each diagram.

(a) (Attractor)

(b) (Repellor)

(c) (Positive Shunt)

(d) (Negative Shunt)

Fig. 1.1: Possible phase portraits when n = 1

The phase portrait is the geometrical representation of the qualitative behavior of (1.1) [10]-[14].The resulting set of phase portraits for a particular n ≥ 2 is a code in which the maximum blocklength of any of its codewords is n. For some basic discussion on coded character sets and codes, see [15]. Based on the above, novel network topologies formed from the point-to-point primitive structure are developed for (1.1) when n ≤ 4. The resulting configuration is a full-duplex transmission in which information is simultaneously transferred in both directions. Except for the direction, each of the topologies may be considered as a subgraph of the hypercube.

It should be noted that substantial traffic on the Internet and on ad-hoc wireless networks consist of point-to-point communication [16]. For the purpose of this paper, a horizontal (vertical) point-to-point topology is defined to be a point-to-point topology in which all its points lie on the same horizontal (vertical) line. The more interesting topologies are those which arise when the critical points of (1.1) have complex value(s). The paper is organized as follows: In Section II, the relevance of differential equations to the design and construction of network topologies is justified. In Section III, an algorithm for the design is presented while actual constructions are carried out in Section IV. Section V concludes the paper.

II. RELEVANCE OF DIFFERENTIAL EQUATIONS

The use of differential equations in the design of network topologies is not a new phenomenon and the relevance of these equations to networks should not be surprising because of the universal language of

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the equations. For instance, mesh network geometries are formed by discretizing elliptic partial differential equations in one, two or three dimensions [5]. These topologies result in substantial savings in communication line when they are mapped into the hypercubes. Also, it should be noted that the families of trajectories and the phase portraits of some differential equations and dynamical systems are networks in the sense that they have nodes (representing their critical points) which are connected together by edges e.g the phase portraits of the autonomous dynamical system x1

/ = (2-x1-2x2)x1 (2.1) x2

/ = (2-2x1-x2)x2 (2.1) with critical points occurring at (0,0), (2,0) (0,2) and (2/3,2/3) Consider the first order autonomous ordinary differential equation (1.1) in which n f(x) = ∑ aixn-i (2.2) i=0 where a ∈ R (set of real numbers) The geometrical representation of the qualitative behavior of this equation is called the phase portrait (or phase diagram). This portrait is completely determined by the nature of the critical points. By the Fundamental Theorem of Algebra, this equation has exactly n critical points all of which may be real and equal, real and distinct or complex conjugates. Now consider the set { x/ = f(x) } (2.3) and define a relation ∼ on this set by x/ = f1(x) ∼ x/ = f2(x) iff the two distinct equations have the same phase portrait. This relation is an equivalence relation as it is reflexive, symmetric and transitive. By the Fundamental Theorem of Equivalence Relations [17], ∼ partitions (2.3) into disjoint equivalence classes known as qualitative classes. All equations of the form (1.1) which belong to the same class have the same phase portrait [11]-[14]. Thus, equations which belong to the same class are topologically or structurally the same. The phase portraits for n ≥ 2

are a combination of elements of the set of phase portraits when n = 1 i.e the set (1.2) described in Fig 1.1

III. ALGORITHM In this section, a step by step procedure for designing subgraphs of the hypercubes is presented . i. Consider the differential equation (1.1) ii. Obtain the critical points of the equation. iii. If all the critical points are real, then conclude that only horizontal point-to-point topology exists. Go to step xiv. iv. Otherwise, mark each of the critical points on the two dimensional plane as a two coordinate position (a , b) where a represents the real part and b the complex part of the critical point a + ib , i = √ -1. v. If all the critical points have complex value, then the points occur in pairs as complex conjugates. Otherwise, GO TO step ix. vi. If there exists only one pair of complex conjugates, then conclude that only vertical point-to-point network topologies exists. GO TO step xii. vii. Draw a line to connect each of the given pairs of parallel straight lines. viii. From the leftmost vertical straight line, draw a line from the top-most point of this line to join the bottom-most point of the immediate next straight line. From the latter point, draw another line to join the topmost point of the immediate next vertical line. Continue the process until all points have been connected by a line. GO TO step xii. ix If some of the critical points are real (and some complex), first join each of the pairs of complex conjugates to form parallel vertical straight lines. x. Then, starting from the left hand side, join the leftmost point to the next point or the (next) topmost point of the straight line connecting complex conjugates, as the case may be. xi. Continue joining from left to right until all the critical points are connected by lines. xii. Indicate the direction of the line joining any two successive points by considering whether they form an attractor, a repellor, a positive shunt or a

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negative shunt. The resulting structure is then a non-horizontal directed point-to-point network topology. xiii. All topologies of the same shape and structure are grouped together . xiv. STOP

IV. SOME CONSTRUCTIONS Consider (1.1). When n = 2, both critical points may be real-valued or both complex-valued. The set of qualitative classes (qc) for the real case is {AR,RA,P,N} while the set of qc for the complex case is {AR,RA}[13]. This gives a total of six topologically distinct structures as shown in Table 4.1. In the table, C represents the possible combination of critical points, Nh the number of horizontal point to point topologies, Nnh the number of non-horizontal point to point topologies while Nt represents the total number of distinct point to point topologies.

TABLE 4.1 Number of topologically distinct networks when n = 2, 3 and 4

n C Nh Nnh Nt 2 2 REAL;

OR 2 COMPLEX

4 2 6

3 3 REAL; OR 2COMPLEX, 1 REAL

10 6 16

4 4 REAL; OR 4 COMPLEX; OR 2 REAL, 2 COMPLEX

18 10 28

When n = 3, all the critical points may be real or one of them is real while the remaining two are complex. The set of qc for the real case is [11] δ = {RAR,ARA,RP,AN,NR,PA,R,A,P,N}(4.1) There are three distinct topological structures when two of the critical points are complex. The direction of each of these is in the form RAR and ARA giving six directed structures altogether. The structure of the single real critical point is a subset of δ . Thus, there is a total of 16 structures when n = 3. In the case n = 4 [14], (i) all the critical points may be real, or (ii) all are complex, existing as two pairs of complex conjugates or (iii) two of the points are real and the remaining two are complex [14]. In (i), the set of qualitative classes is {ARAR, RARA, ARPP, RANN, AR, RA, PAR, NRA, ANR, RPA, PPP, NNN, ARP, RAN, P, N, PP, NN}. In (ii), there are two distinct topological structures, the direction of each being in the form ARAR or RARA giving four distinct directed structures. And finally in (iii), there are three distinct topological structures. The direction of each is in the form ARAR and RARA thus giving six distinct directed structures altogether. Therefore, n = 4 gives a total of 28 distinct topological structures. Fig. 5.1 shows the undirected subgraphs when the critical points are complex, n ≤ 4 , where the leading coefficient is zero

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(a) (b) (c)

(d) (e)

Fig. 5.1: Undirected subgraphs resulting from (1.1) when the critical points are complex, n = 4

V. CONCLUSION

This paper has presented a novel approach to the design and construction of network topologies which are formed from the point-to-point primitive structures.. The approach exploits the algebraic and geometric properties of first order autonomous ordinary differential equation of the form x/ = f(x) where f(x) is the general polynomial of degree n ∈ N (set of natural numbers). Topologies of the equation when n = 2, 3 and 4 were then presented. An element of the set of topologies resulting from the design can be viewed as a subgraph of the hypercube. It follows that a hypercube can be formed from a superimposition of these subgraphs at their nodes and edges. This idea of superimposition can be further buttressed by considering the pictures of the 1-cube, 2-cube , 3-cube and 4-cube depicted in [5]-[6]. It has therefore been shown in the present paper how the directed 1-cube and the graphs derived from it can be generated. The qualitative equivalence model described in the paper thus presents an alternative approach to the construction of (subgraphs of) a hypercube. This is in contrast to models based on the representation of finite groups [1]. The size of a

typical topology in the paper equals the degree of the polynomial f(x).

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ODEs Of The Transcendental Type,” Proc. Int. Conf. on Appreciating Math in Contemporary World (Conf. in honor of Two Retired Academic Staff), Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria, pp. 87-93, 2004. [14] D. Oluwade, “Modelling Fractal Patterns Via The Qualitative Equivalence of A Nonlinear ODE,” Nonlinear Anal. Vol. 63, no 5-7, pp. e2409-e2414, 2005.

[15] D. Oluwade, Design and Analysis of Computer-Coded Character Sets, PhD thesis, University of Ibadan, xii + 128pp, 2004. [16] M. Adler and C. Scheideler, “Efficient Communication Strategies for Ad Hoc Wireless Networks,” Theory Comput. Systems vol. 33, pp. 337-391, 2000. [17] I. Herstein, Topics in Algebra. New York: John Wiley & Sons, 1975.

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