Differential Equation of Connectedness for Massively Interconnected Systems

7/30/2019 Differential Equation of Connectedness for Massively Interconnected Systems

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Differential Equation of Connectedness forMassively Interconnected Systems

(Part of PostDoc Dissertation, Moscow 1990, Translation from Russian)Pavel Barseghyan

Abstract Considering the Massively Interconnected system (MIS) as a spatially distributedsystem of elements and interconnections among them, the differential equation of connectivity isderived. The equation is based on the local balance of densities of interconnections and elementsin the space. The capabilities of differential equation of connectedness are validated throughchecking the behavior of its solutions for different practical examples. The differential equation iscapable to reflect any connectivity behavior in the real system of coordinates if the spatialdensities of elements and interconnections among them may be considered as continuousfunctions. In addition the equation can reflect the statistical behavior of the totality of elementsand interconnections. One-, two-, and three-dimensional interconnection length distributions arethe specific solutions of this equation.

I. Introduction. Massively Interconnected Systems

Massively Interconnected systems can be characterized by a large number of elements andmassive interconnections between them [1, 2]. Such systems are to be found everywhere. Amongthe examples of MIS are the human society, Internet and World Wide Web, an organization, ahuman group, a communication system or some semiconductor chip with a millions of elementsand interconnections between them. The quantitative description of such systems becomesimperative nowadays because of the variety of potential applications of the latter. In organizationscience those practical applications are related to the structural analysis and synthesis of theorganization and development of mathematical theory of the projects, including quantitativeanalysis of the behavior of human groups.From the mathematical point of view such systems can be divided into static and dynamicMassively Interconnected Systems. While in static massively interconnected systems theconnections between elements are time independent, in dynamic massively interconnectedsystems they are time dependent.

II. Requirements to the Mathematical Models of Massively Interconnected Systems

Lets briefly discuss the requirements to the mathematical description of the MIS.1. Due to the very big number of interconnections and elements within the MIS, the

appropriate mathematical description for the connectivity must be a continuous one.2. Due to the importance of real densities and distances within the MIS, the mathematical

description of the latter must be in the real system of coordinates.3. Mathematical description of such systems must incorporate the density type of

parameters (density of interconnections, density of elements, etc.), in other words it mustdeal with the local or point characteristics of the system.

4. Due to the fact that these densities are functions of the space coordinates, the appropriatemathematical description of MIS must have the form of differential or integral equations.

III. Graphical Representations of Massively Interconnected Systems

Lets start with simple graphical representations of massively interconnected systems, having agoal to quantitatively describe the spatially distributed elements of the latter and the massive


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connections among them. For that purpose it is useful to examine the common behavior for separate elements and their connections in such systems. Each element in the system has chaoticconnections with the other elements. Elements of a massively interconnected system andconnections of one of those elements are shown in the Fig.1.

Fig.1 One element of a MIS along with its connections with other elements

Separating the element out along with its connections we will have the picture presented in Fig.2.The same may be done with the other elements and as a result we will always obtain almost thesame picture where an element emits a certain number of connections which are graduallyabsorbed by the other elements. This chaotic penetration of interconnects into the environment

Fig.2 A separate element along with its connections

with some variations is common for all massively interconnected systems. Therefore it may become a focus around which we can concentrate our modeling effort. Another characteristic for such a behavior is that the connection penetration image into the environment is very close to the


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well-known diffusion process. Thus for the same modeling purposes it is useful to make parallels between the diffusion and the process of connection penetration into the environment.

IV. Spatial Densities of Interconnections and Elements

The study of an ordinary interconnection image in MIS shows that quantitatively it can becharacterized by the spatial densities of elements and connections. If the MIS has a hierarchicalstructure with different levels of subsystems or elements, it may be described in any level of

physical hierarchy. Irrespective to the level of physical hierarchy, the MIS will represent itself asa spatial conglomeration of elements and connections. The spatial densities of elements can beconsidered as scalar quantities. But the connections cannot be characterized only by the spatialdensities, because in each point of the connectivity field they have another importantcharacteristic direction, which must be accounted in connectedness mathematical models.

V. Vector Representation of Connectedness fields as a necessity

Let s consider a simple connectedness field in the form of parallel lines (Fig.3.). In order toreveal the vector behavior of such a connectivity field lets conduct a simple geometricalexperiment. Lets consider three line segments in the field which have different angles with thedirection of the parallel lines. The goal of the experiment is to find the number of crossings of theline segments with the connectedness field that has a constant density . Quantitatively thevector is equal to the number of crossings with the field of the unit line segment, which is

perpendicular to the connections. Obviously the number of crossings with the line segment L willhave its maximum value when it is perpendicular to the connections (segment 1 in the Fig.3.).The number of crossings will have a value 0 when line segment is parallel to the connections(segment 3). In order to solve the problem in the general case when the line segment has anarbitrary angle with the connections (or X axis) it is necessary to consider the connectedness fieldas a vector field. For that it is necessary to split the vector into two components, 21 and 22 .The first one is parallel to the arbitrary line segment and the component 22 is perpendicular tothe same segment. Therefore the number of crossings of the line segment with the connectednessfield is equal to

sin222 L Lm == (1)

These calculations show that in order to account for the directions of connections in theconnectedness field the latter has to be considered as a vector field.

Fig. 3 Vector Field of connections


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VI. Generic Connectivity FieldsConsider the generic connectedness field shown in Fig.4. Each point of this field can becharacterized by a vector ),( y x that has the direction of the tangent to the connections. The

magnitude of this vector is the density of the connections in the point ),( y x .

For each point of the field the connectedness density vector can be split up into the components x and y .

),(),( y x y x x = i, + ),( y x y j, (2)where i and j are the standard basic vectors. The norm of the density vector is

= 22 y x + (3) VII. Number of Connections in the Cross-Sections of the Generic Connectedness FieldLets consider two cross-sections 1 y y = and 1 x x = in the generic connectedness field (Fig.4). Inorder to calculate the number of connections in these cross-sections we can use the same

technique of splitting the connections density vectors into x and y components. For instance,the connection density vector ),( 1 y x in the point A can be split into two vectors -

),( 1 y x y j and ),( 1 y x x i.

Fig.4 Generic connectedness field with its 1 x and 1 y cross-sections

Thus in the point A the connectedness density vector will have the form j y xi y x y x A Ay A Ax x ),(),(),( 11 += .

Similarly for the point B we can have j y xi y x y x A Ay A Ax x ),(),(),( 11 += .


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Therefore, by knowing the vector field ),( y x we can define the densities of connections at any point of the arbitrary cross-sections 1 y y = and 1 x x = , simply by splitting those connectednessdensity vectors into their components. Using those density components, we can now estimate thenumber of connections crossing any segment in the field. So with the aid of the component

),( 1 y x y we can calculate the number of connections M ym ,1 crossing the segment [ 1 y ,M].

=1

01,1 ),(

x

y M y dx y xm . (4)

Similarly we can calculate the number of connections M xm ,1 , 1,0 xm and 1,0 ym crossing thesegments [ 1 x , M], [0, 1 x ], and [0, 1 y ] respectively.

=1

1

01, ),(

y

x M x dy y xm (5)

=1

1

0,0 )0,(

x

y x dx xm (6)

=1

1

0,0 ),0(

y

x y dy ym . (7)

VIII. Functional relationship Between Spatial Density of Connections and the Number of terminals of the connectedness field

Using the above results, we can calculate the number of connections that cross the edges of therectangle [0, y1, M, x1] (Fig.4.) as a sum:

m = M xm ,1 +1

,0 xm +1

,0 ym + M ym ,1 (8)After substitution of the expressions (4), (5), (6), and (7) in (8) we have

),( 11 y xm = 1

01 ),(

y

x dy y x + 1

0

)0,( x

y dx x + +1

0

),0( y

x dy y + 1

01 ),(

x

y dx y x (9)

that represents the number of terminals of the rectangle [0,y1,M,x1]. Replacing 1 x and 1 y in (9)with the current variables x and y and finding the second order partial derivative of the totalnumber of terminals ),( y xm , we obtain

y x y x

y xm y x

+

=

),(2. (10)

This expression establishes the functional relationship between the number of terminals andspatial density of connections in a differential form.

IX. Total Length of Connections in the Generic Connectivity Fields

It is not difficult to show that for any small area dxdyds = the total interconnection length can be calculated through the formula

dL = ds = dxdy . (11)


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Integrating (11) for an arbitrary y x* connectedness field we have

= x y

dxdy y x y x L0 0

),(),( . (12)

If the connectedness field has constant density, then from (12) we can obtain a practical simple

formula for estimating the total length of connectionsL = S , (13)

Where S is the total area of the connectivity field.Also from (12) it follows that

),(),(2

y x y x

y x L =

, (14)

which establishes the functional relationship between the total length of connections and thespatial density of connections in the arbitrary connectedness field.

X. Differential Equation of Connectedness(Steady State case)

Based on the fact that the number of elements and connections between them for a certain part of the connectedness field are strongly interrelated, lets explore the balance between them in thevicinity of an arbitrary point (x, y). For doing that lets consider a small area dxdyds = (Fig.5).Connections are incoming into that area from the left and bottom sides of the area ds and areoutgoing from the top and right sides of the same area. According to the expression (5), thenumber of connections inm that enters into the small area ds from the left side can bedetermined as dy y xm xinl ),( = . (15)

Similarly, the number of outgoing connections outr m from the right side of the same small area

ds can be calculated by the formulady ydx xm xoutr ),( += . (16)

Thus the difference between the numbers of incoming and outgoing connections in the directionof x axis equals to dy y x ydx xmm x xinl outr )],(),([ += . (17)Similarly we can find the number of incoming connections from the bottom of the small area dsas

dx y xm yinb ),( = . (18)

The number of outgoing connections outt m from the top of the same small area ds can becalculated by the formula

dxdy y xm youtt ),( += . (19)Thus the difference between the numbers of incoming and outgoing connections in the directionof y axis equals to

dx y xdy y xmm y yinboutt )],(),([ += . (20)So the total difference between the numbers of incoming and outgoing connections for the smallarea ds can be calculated as the sum of the differences (17) and (20)


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inboutt inl outr inout mmmmmm += . (21)Substituting the expressions (17) and (20) into (21) we will have

=inout mm dx y xdy y x y y )],(),([ + + dy y x ydx x x x )],(),([ + . (22)

Lets analyze this expression assuming that the density of connections is a continuous function inthe small area ds and in its vicinity. Based on that we can represent ),( ydx x x + as a power series by dx

+

+=+ dx

x ydx x x x x

),( (23)

Substituting (23) into the (22) and neglecting the higher order terms we have

dxdy x

mm xinout

= . (24)

Similarly representing ),( dy y x x + as a power series by dy we have

dxdy ymmyinboutt

=

. (25)

Substituting the results obtained from (23) and (25) into the expressions (21) or (22) we will have

dxdy y x

mm y xinout )(

+

= . (26)

As it can be seen from Fig.5, the difference between the numbers of incoming and outgoingconnections is formed as a result of the termination of a portion of incoming connections in thesmall area and generation of a portion of outgoing connections in the same small area. In order toaccount for that termination and generation process of connections lets introduce the notions of densities of connection sinks ),( y x s and connection sources ),( y xn in the connectivity field.

Based on these notions we are able to calculate the same difference y b

out m dy y +

dy y

inm0 x dx dx x + a x

Figure 5: Balance between the connections and contacts of elements in the small area dxdyds =

inout mm by the following expression:

dxdy y x y xmm sninout )],(),([ = . (27)Equating the expressions (26) and (27) we will have


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),(),( y x y x y x sn

y x =

+

, (28)

which represents the above-mentioned balance between the elements and connections in the form

of a differential equation. The functions ),( y x x in the point 0= x and ),( y x y in the point 0= y can serve as a boundary condition for the equation (28), that is to say - functions

),0( y x and )0,( x y in the left and bottom edges of the connectedness field.

XI. Other Forms of the Connectedness Differential Equation

Because the elements in the connectedness field are the main sinks and sources of theconnections, it is evident that the equation (28) contains information about the elements as well,

but in an indirect (implicit) form. In order to deal with the spatial densities of elements ),( y xn lets denote the numbers of the connection sources and sinks for the elements in the vicinity of the point ),( y x by ),( y x p and ),( y xq . It is obvious that

),(),(),( y xn y x p y xn = (29)and

),(),(),( y xn y xq y x s = . (30)Substituting (29) and (30) into the main equation (28) we can have another form for the latter.

),()],(),([ y xn y xq y x p y x

y x =

+

. (31)

From mathematical point of view it is convenient to incorporate the components of the densityvector of connectedness directly into the right hand side of the equation (28). In order to do thatlets introduce two new notions: connection termination intensity ),( y x and connection

generation intensity),( y x

such thatdxdy y x ),(

is the probability of the termination of oneconnection in the area ds , and dxdy y x ),( is the probability of origination of one connectionin the area ds . Based on that we can define the connection sink and source densities by thefollowing formulas:

))(,(),( y x s y x y x += (32)and

))(,(),( y xn y x y x += . (33)Substituting (32) and (33) into the main equation (28) we can have a very useful form of thedifferential equation of connectivity.

))](,(),([ y x y x

y x y x y x

+=

+

(34)

This equation with corresponding boundary conditions is able to describe arbitrary massivelyinterconnected systems.Lets apply the equation (34) for connectedness analysis of one- and two-dimensional cases of spatially distributed elements. The main goal here is to describe the spatial connectivity of astand-alone element. Then the next step is to solve the differential equation for that element andfind its connectedness field. If a system incorporates some number of elements, then applying the


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solution for one element and the principle of superposition of the connectedness field we can findconnectedness characteristics for the arbitrary configuration or shape of connectedness fields.This approach is valid for both homogeneous and heterogeneous connectedness systems.

XII. Differential Equation for Separate Elements. One-dimensional case

Lets consider a simple homogeneous connectedness system in the form of an infinite chain of elements positioned along the x axis in the points with integer values. Each element has terminals half of which are connected with the left-hand side elements and another half of theterminals are connected with the elements on the right-hand side (Fig.6.). All the elements in sucha system have the same connectedness characteristics (Fig.7) and the system can be described by

connectedness equation (34), with 0= y

x x x x

x

)]()([ =

, (40)

where all parameters are the functions of x variable only. In this equation x represents thenumber of connections length of which exceeds x.

Fig.6. Left and right branches of the connectedness function )( x x

Lets separate the element positioned in the point 0= x along with its connections. In this case

connection origination function )( x transforms into a boundary condition2

)0( = x , and the

equation itself has the form

x x x

x

)(=

. (41)

Since by definition )( x x is the number of connections in the section x that have been

originated from element 0, the ratio )( x x / 2

will be the probability that the length of

connections of that element will exceed x.Consequently we can define the connection length distribution function as

x x F 21)( = (42)

The solution of the equation (41) has the form

dy y Exp x x

x ])([2)(

0 = , (43)


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which describes the right branch in Fig.6 and is directly related with the connection lengthdistribution function F(x), which can be described as

dy y Exp x F x

])([1)(0 = (44)

Combining the solution (43) with the principle of superposition of the connectivity field, we cancalculate connectivity characteristics for the group of N elements (Fig.7). Assume we need toestimate the number of terminals for that group of elements (each one of those has the sameconnectedness function) with the other neighboring elements.

Fig.7. Group connectedness explorationDue to the symmetric character of the connectedness functions, we can calculate the number of

terminals outgoing from the right side of the group only - )( N mr . Besides, we can representthe number of terminals as the sum (superposition) of the number of terminals for each separateelement. The number of terminals for an arbitrary element i will be the value of the function

)( i x x in the point N , i.e. )( i N m xi = .Thus the total number of terminals )( N m can be calculated as the following sum

=== N

x

N

ir i N m N m N m11

)(22)(2)( . (45)

For the large number of elements we can make a transition to the continuous analogue of thesame rule

dx x N N m N

x )(2)(0

= . (46)Differentiating relationship (46 ) with regard to the total number of elements N we can have

)(2)(

N dN

N dm x = . (47)

This expression establishes a functional relationship between connectedness characteristics of thesystem and connectedness characteristics of the element x . Taking into account the relationship

(42) between )( x x and connection length distribution function )( x F we will have

)](1[)( N F dN N dm = , (48)

which establishes a functional relationship between the number of terminals of a homogeneousone-dimensional system and connection length distribution function.

XIII. Differential Equation for Separate Elements. Two-dimensional case

Lets analyze a two-dimensional array of the identical elements as a connectedness homogeneoussystem. Similar to the one-dimensional case we can divide the analysis problem into two stages.


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First, to solve the problem with the purpose to find the connectedness field) for a separateelement that is positioned in the point (0, 0), then applying the principle of superposition of theconnectedness field to find the resulting connectivity field for many elements.For that purpose lets use equation (34). Transforming the connection origination intensity

),( y x into a boundary condition, we can describe the 2D wiring field of one element positionedin the point (0, 0) by the equation

))(,( y x y x y x

y x

+=

+

. (49)

Due to the symmetry of the connectedness field ),( y x we can easily solve this equationmaking a transition to the polar coordinates.Then we can describe the connectedness field of the element positioned in the arbitrary point

),( 11 y x as ),( 11 y y x x . Applying formulas (4)-(7) to this field we can find the number of terminals for the arbitrary element in the field. Combining solutions for any boundary segment of the field we can estimate the number of terminals for any close contour.

I. Differential Equation for the Connectedness Potential

In previous sections we discussed solutions of the connectedness equation for simplehomogeneous cases when it is possible to find the solution for one element and make a transitionfrom it to the system level connectedness characteristics. In the general case it is impossible to

solve this equation because it contains two unknown functions - x and y . Therefore there is aneed to find another general form of connectedness equation that can allow overcoming this

problem. The conventional way to do that is to find the so-called connectedness potential, a

function that has functional relationships with x and y such that GradP = , (50)where P is the connectedness potential. This simply means, that

x x P = and y y

P = . (51)

Taking into account (51) we can obtain the general form of the connectedness equation from (34)

))](,(),([22

2

2

y P

x P

y x y x y P

x P

+

=

+

. (52)

Further generalization of this equation for time dependent connectedness description leads to the followingequation

))](,(),([1

2

2

2

2

y P

x P

y x y x y P

x P

t P

D +

+

=

, (53)

where coefficient D is a constant describing the diffusion of connections into the environment being under study.

II. References

[1] P. Barseghyan, An Introduction to the Theory of Electronic Packaging and Connectedness.Moscow Aviation Institute Press. Moscow. 1989. 50 pages (in Russian).

[2] P. Barseghyan, Mathematical Basics of the Theory of Connectedness and Interconnections.Moscow Aviation Institute Press. Moscow. 1990. 114 Pages (in Russian).

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Differential Equation of Connectedness for Massively Interconnected Systems