946

for conference blockink. One must of course define blocking load for the number of conferees; e,.g., normal blocking is two- party blocking-generally we have k-party blocking. A k-party call is blocked if at least one of the k parties cannot complete the connection to the other k -, 1 parties and their outlets are “idle” with respect to the maxiqym number of conferees al- lowed (for center-stage conferencing). Since in most applica- tions, only a very small number of conferences are in progress at any one instant, it is likely that most realistic conferencing can be accomplished with little or no increase in the number of center-stage matrices (when conferencing is allowed only in the center stage). Substantive results will await the derivation of the appropriate blocking probability equations, or more likely (considering the analytical difficulties inherent in con- ference biocking probabilities), the results of simulation runs.

REFEdNCE.5

C. Clos, “A study of nonblocking switching networks,” Bell Sys?. Tech. X , vol. 32, pp. 406-424, Mar. 1953. T. L. Bowers, “Blocking in 3-stage “folded” switching arrays,” IEEE Trans. Commun. Technol. vol. COM-13, pp. 14-37, Mar. 1965.

Message Propagation Through Random Communication Networks

M A n E I , MEMBER, IEEE; AND R. FISCHL, MEMBER, IEEE

Abstruct-This study considers the message propagation problem in a “netalert’’ situation through directed and undirected random networks having n nodes and probability of connection between the distinct pairs of nodes equal to p. The solution has been obtained by analyzing the structure of the random networks in terms of the probability dis- tribution of the number of contacted new nodes, which are separated from the originator(s) of theymessage by simple chain progressions of k links. From these probability distributions, the average message propagation, the weak connectivity, and the kth terminal reliability are obtained. To aid the computation of these quantities frst-order difference equations that approximate the average message propagation, the weak connectivity, and the kth terminal reliability are given. The application of the results is shown for random networks having several values of n and p .

I . INTRODUCTION The analysis of message propagation through a random net-

work having n nodes and aprobability p of connection between a pair of nodes is essential to such problems as the design of communication networks that are invulnerable to enemy attack, the design of reliable telephone systems, the design of reliable transportatidn systems, etc. Although partial results are available in the literature [ 11 , there is no, complete treat- ment of the problem that considers the interrelated topics of connectivity, reliability, and vulnerability of random communi- cation networks. The major difficulties most often en- countered in the study of communication networks are 1) the network models and the measures of vulnerability and reli- ability differ from application to application and from author to author and 2) the methods of solution of these combina- torial problems involve a number of combinations that be- comes prohibitively large for large systems.

A summary of the analysis and design of survivable networks

Communications Society for publication without oral presentation.

Drexel University, Philadelphia, Pa. 19104.

Paper approved by the Systems Discipline Committee of the IEEE

The authors are with the Department of Electrlcal EngineEring,

IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 1972

is given by Frank and Frisch [ 11, who show the relation be- tween the various network models and the metliods of solu- tion. This paper considers only one network model (for directed and undirected) and completely solves for all the properties of interest; i.e., the average message propagation, the weak connectivity, the strong connectivity, and the kth terminal reliability. Baran [ 21 and Wing and Demetriou [ 31 considered Monte Carlo techniques to obtain estimates of some of the properties; Prihar [4] borrowed concepts from random neural nets [ 51, [ 61 to formulate, the message propaga- tion problem; and Frank [7] extended asymptotic and re- cursion formulas to calculate a measure o € vulnerability. We present an exact mathematical method of analysis that yields all the properties of interest and then proceed to obtain approximations that aid in coriiputing these properties. We show the interrelationship among connectivity, reliability, and vulnerability of random networks by examining the structure of the networks in terms of the message propagation; specifi- cally, we obtain the probability distributions of the number of new centers that are contacted as a function of time.

In the following sections we present the detailed description of the problem, the method of solution, some examples, and the conclusions. More specifically, in Section I1 we introduce the basic definitions and the specific models investigated; in Section I11 we formulate the message propagation problem, and in Section IV we present the exact method of solution. Sections V and ‘ V I present approximations to the average message propagation and to the weak connectivity, respectively; the conclusions are found in Section VII.

11. DEFINITIONS AND NOTATION A communication network consists of a set of communica-

tion centers (stations) interconnected by wirelines and/or radio links. This system can be represented by a linear graph that contains a set of nodes N = { n j } and a set of Gnks L = { l i j } . The linear graph is interconnected so that the links intersect only at the nodes. ,

A network is said to be undirected if information flows through every link in either direction; i.e., each node has both transmission and recteption capabilities. A network is said to be directed if information flows through, evey link in only one direction. Ai undiiected (directed) network is said to be simple if it does not contain any parallel (strictly parallel) links nor any self-loops (directed loops). A communication network for which the sets N and i5 are defined uniquely is called deterministic; a network for which there may be several possible sets ofN and L , each with a probability of occurrence, is called random.

This paper considers the message? propagation problem in both directed and undirected random< networks, specifically, those.in which the set N is specified and in which each element of the set L exists, with probability p . An example of such a network can be a tactical communication system that, has undergone an attack. Before the attack, every node is con- nected to every other node (without any parallel links of self-loops); after the attack, each link has a probability p of having survived the attack and a probability q = 1 - p of being. destroyed. The value of p can vary between 0 and 1, with 0 and 1 excluded, since then the resultant network is deter- ministic. For example, if p = 0, then all the nodes are isolated.

This paper investigates the message propagation through the following network models.

Model I: Directed Random Network Consider a simple directed network; let the number of nodes

be n and consider the n(n - 1) pairs of nodes which can be formed from n. To each link Zij assign a probability p = Pr (Iii = 1 } that it connects node ni to node nj. Note that the

CONCISE PAPERS 947

probability q that lii does not connect node ni to node ni is equal to 1 - p . Furthermore, assume that each link exists independently of any other link.

Because the network under consideration is simple, there are no self-loo s, Le., Pr {Zii = 1 ) = 0, nor strictly parallel links, i.e., Pr {Zii 2 27 = 0. ' The probability Pr {Zii = 1 ) is considered to represent the result of a single experiment of a binary pro- cess having a probability of success equal to p , i.e., Pr {Zii = 1 } = p and Pr{Zii = O} = 4 . Hence the simple random network interconnection can be represented by the nodal probability matrix

1 2 3 . .- n

where the ijth element of L , is Pr {lii = 1). A sample network can be constructed by successive inde-

pendent tossing of a biased coin with probability of success (connection) p and probability of failure (no connection) 4 =. 1 - p . The number of tosses for each node is (n - 1) and since there are n nodes, the total number of tosses is n(n - 1). Because the process is binary, i.e., the outcome of each experi- ment is either a 0 or,a 1, the total number of unique sample networks is equal to 2"("-'). Clearly, the method for obtain- ing the topology and message propagation properties of a random network by examining 2"("-') sample networks of each given p is impractical for networks containing many nodes, hence another method is presented in Section 111.

Model 11: Undirected Random Network The undirected random network model is similar to the one

described for the directed case. I t consists of a simple un- directed network having n nodes interconnected by undirected links. The undirected random network can be also represented by the node matrix L p ; however, here the probability Pr {Zii = 1) = Pr {Zii = 1) and is considered to be the result of a single experiment with probability of success p . The number of tosses for the first node is (n - l), for the second node the number is (n - 2), and for the ith node the number is (n - i), where i = 1, 2 , 3, * , (n - 1). The total number of experi- ments is

n-1

i = l ( n - i ) = n ( n - 1)/2,

and represents the maximum number of undirected links that can exist for an n-node simple undirected network. Note that the node matrices of the 2"("-')/2 sample networks are symmetric and form a subset of the matrices obtained for model I.

111. FORMULATION OF THE ME~SAGE PROPAGATION PROBLEM

The message propagation, through random networks is in- directly related to the problems of reliability, vulnerability, and connectivity of random networks. Exact definitions of these terms differ from author to author [ I ] and depend on the final objective of their investigations. In the propagation of the message analysis, or the "net-alert'' problem, one is con- fronted with the questions of how the message propagates through the network and what is the final expected number of

nodes that receive the message as a function of the probability of connection. Hence, in this paper the reliability, vulner- atiiity, and connectivity of random networks are defined in tefms of the net-alert problem.

A net-alert situation is one in which one or more nodes be- come aware of the message (alert information) and then trans- mit it to the remaining nodes. It is assumed that once a node has received the message it will retransmit it to its uncontacted adjacent nodes and there is a time delay T that delays the message as it travels along each link. This time delay& called a step, for example, if the message travels via at least thFee links, it is said that it takes three steps. Note that reception and transmission of the message through the nodes are instan- taneous and that all the links contribute identical time delays, namely steps. Hence, for an n-node network, it takes at most (n - 1) steps to alert alI the nodes.

Let X ( k ) be the total number of nodes contacted up to and including step k , Y ( k ) be the total number of nodes yet to be contacted, and Z ( k ) be the number of new nodes contacted. Therefore, at each step k , where k = 0, 1, - - e , (n - 1)

and

where n is the total number of nodes in the network. Note that the random variables X@), Y(k) , and Z(k) take on values x(k), y ( k ) , and z ( k ) , respectively.

Let us consider the following four measurks of vulnerability that have been employed most often in the desimEf invulner- able networks: 1) the average message propagation X ( k ) ; 2) the kth terminal reliability r ( k ) ; 3) the weak c*$nnectivity 7 (k); and 4) the probability of having a connected rietwork' P,.

The.determination of the kth terminal &liability r(k) and of the weak connectivity y ( k ) re uires the calculations of the ex- pected valKe of Z(k1, E { Z ( k ) q , and of the expected value of X(n - l), X(n - 1) = E{X(n - l)}. These exhected values can be found by enumerating all of the unique sample networks, subdividing them into groups each containing exactly i links, finding the arithmetic averages < z ( k ) > and <x(n - 1) > for each grbup, and finally averaging across the groups. Note that the probability (needed to average across the groups) of an n-node directed network having exactly i links is ( p i qn( , - l ) - i ) and the number of networks in this group i s equal to (?I. Approximations using Monte Carlo techniques [ 21, [ 31 are usually recommended to avoid the difficult task of enumera- tion. The approach taken in this paper avoids enumeration by solving directly for the probabilities Pr{Z(k) = z ( k ) ) and the expectedvalue of Z(k). From the expected values of Z(k), the E {X(n - 1)) is found; hence the kth terminal reliability and the weak connectivity can be calculated.

The networks studied are assumed stationary throughout this analysis; that is, the number of nodes n and the probability of a link existing p do not change during the propagation of the message. However, it can be shown that different values of p can be used at each step, under the assumption that p is the same for all the links.

The average message' propagation X ( k ) is found by noting that the recursive solution to (3) is given by

-

Also called the strong connectivity of random network.

948

so that

IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 1972

. . . .

where the expected value of Z@, Z(i) is calculated from its probability distribution. Note that; since the kth terminal reliability is defined as

.;: :, ,Lr

r ( k ) = Z ( k ) / ( n - l ) , Z(0) = 1 ( 6 )

and the weak connectivity by

y = F ( n - l)/n, X ( 0 ) = 1, (7)

these two measures of vulnerability can also be calculated. The fourth measure, the strong connectivity P, can be ob- tained by using (4) and the probability distribution of Z(i). In addition, we can obtain the probability that the network is connected to one or more nodes at step k , Pr { X ( k ) = n } given Pr ( ~ ( 0 ) = x (0)).

IV. SOLUTION OF THE MESSAGE PROPAGATION PROBLEM

The solution of the message propagation problem involves the determination ofthe probability distribution of the number of new nodes contacted at step k , i.e., Pr{Z(k) = z(k)}. Before presenting the method of solution, let us examine the local structure of random networks; namely, the relation be- tween the number of links emanating from any node and the values of n and p .

The local structure of the two model networks can be described in terms of the random variables A , and Ai for model I, and A for model 11, where A , represents the number of directed links outgoing from any node, Ai the number of directed links incoming into any node, and A is the number of undirected links incident on any node. Consider networks model I and note from (1) that the probability that an out- going link exists is p , the probability that it does not exist is q = 1 - p , and the total number of trials per node is (n - 1). Therefore, the probability that A , = k (or A i = k) is given by

the average outgoing link density is

and the variance is

Note that identical results are obtained for A i when A , is re- placed by Ai in (8)-( 10). For the undirected network model (model 11) the random variable A is also described by (8)-(10), when A , is replaced by A .

The local structure of random networks describes the message propagation from a node ni to the remaining (n - 1) nodes as shown in Fig. 1. Hence, (9) simply states that if one node originates the message at the initial step, i.e., k = 0, then the average numbers of nodes that are contacted at step one is given by x, for the directed networks model I or by x for the undirected network model 11. I

The global structure of the random network can now be obtained, namely the state of the message propagation at step k . The state of the message propagation is completely described by the random variables X ( k ) and Z(k) as follows: consider a random network in which X ( k ) nodes have received the message, of which Z(k) nodes are newly contacted at step k as shown in Fig. 2. Recall from (2)-(4) that X ( k ) = X ( k - 1) + Z ( k ) and Y ( k ) = n - X ( k ) .

Fig. 1. Local structure of random networks.

Fig. 2. Random network at step k.

The basic quantity of interest is the conditional probability distribution of the number of new nodes that will be contacted at step (k + l), given that Z(k) = z(k) and X ( k - 1) = x(k - 1). First, consider each node ni in the set of Y ( k ) with respect to the newly contacted nodes Z(k) and note that ni will be in- cluded in Z(k + 1) if there is at least one link, outgoing from any of the nodes h Z(k) and incoming into ni. The probability of this event occurring p is given [ 81 by

Then, the conditional probability of Z(k + 1) = z ( k + l), given Z(k) = z ( k ) and X ( k ) = x(k) is given by [8]

where 41 = 1 - p l , and y ( k ) = n - x(k).

given by Furthermore, the expected value of Z(k + l), z ( k + 1) is

z( k + l ) = E { [ n - X ( k ) J [ l - ( 4 ) X ( k ) - X ( k - ' ) ] } . (13)

In a net-alert situation one or more nodes become aware of the information; therefore, we can let

where z(0) = x(0) is the number o f , nodes that originate the message. Given this initial condition, all the marginal prob- abilities at step k of Z(k) = z(k), = 1, 2, e . * , n - 1, can be found. For example, the marginal probabilities at step 1 of Z( 1) = z( 1) can be found as follows. First let k = 1 in (12) and multiply this equation by (14), this yields the joint probability of Z ( 1) and Z(0). Then sum the result over z(0). The marginal probabilities at step 2 are given by .

Note that the joint of Z ( 1) and Z(0) was found previously for step 1. Continuing this process for each step gives a general

CONCISE PAPERS 949

equation for the marginal probabilities of Z ( k ) = z ( k ) at step k :

Y ( 0 ) Y ( 1 ) Y ( k - 2 ) Pr(Z(k)=z(k)}= c

Z ( l ) = O ~ ( 2 ) = 0 z ( k - l ) = O

.{($]) [ I - q ~ ( o ) ] z ( l ) [ q ~ ( o ) ] Y ( ' ) }

where

A simple example may help clarify the general procedure prescribed by (15). Consider a five-node network and let one node originate the message. Fig. 3 shows the message propaga- tion, where the number in the circles indicate the number of nodes that could be connected at each step. The lines drawn from a circled number in step k to a circled number in step ( k + 1) have a probability of that particular condition occur- ring. The lines that terminate with the black dots indicate the end of the message propagation for that particular flow; the circled numbers that do not have lines drawn to the next step indicate that the message flow is complete (i.e., there are no more new nodes to be contacted). For instance, the flow be- tween points d and B in Fig. 3 indicates that the message originated at one node at step zero, propagates to two more nodes with probability (:) p 2 q2 at step one; and, at step two, the remaining two nodes will be contacted with probability (1 - 42)2. Note that the transitions occur only from step k to step ( k + 1) and that the sum of the probabilities of each flow line from any circled number equals one; i.e., all possible cases are accounted for.

The calculations of the probabilities of Z ( k ) = z ( k ) are straightforward. For example, {Pr Z(2) = ~ ( 2 ) ) is found by scanning step 2 for all the circles containing number 2; for each of these multiply the probabilities associated with each line going back to the originator of the message and add the results; for example, for p = 0.4, we have from (2)

Pr{Z(2) = 2) = [0.4096 X 0.34561 + [ 0.288 X 0.34561

= 0.241 1.

The expected values are also calculated very simply. For example, Z(2) is given by

2 (2 ) = 0.3456[0.432 + (2 ) 0.288 + (3) 0.0641

+ 0.3456[0.4608 +(2) 0.40961 + 0.1536[0.7841

= 0.9775.

The expected value of X ( k ) is found by summing z( j ) , j = 0, - - , k [see ( 5 ) ] . The kth termigal reliability, (6), is obtained using the previously calculated Z ( k ) ; for example,

r(2) =Z(2)/(n - 1) = 0.9775/4 = 0.2444.

The weak connectivity y is found from (7) by calculating 7 ( 4 ) = 3.853 and dividing by n = 5; i.e.,

y = 1(4 ) /5 = 3.853015 = 0.7706.

Another property of interest is the strong connectivity P,; i.e., the probability that there exist link progressions that con- nect every pair of nodes in the network. This corresponds, in

0 STEP 0 STEP I STEP 2 STEP 3 STEP 4

Fig. 3. Message propagation in a 5-node network.

k = n - 1, given Pr(X(0) = 1) = 1. Note that, if the network is undirected, then every node can communicate with every other node with probability P, ; however, for directed networks there is a direction associated with the flow of information and P,, represents the probability of any node ni being able to relay information to the remaining (n - 1) nodes. The exact method of solution, presented here, yields not only the strong connectivity but also the probability that a node ni can con- tact all the other nodes at any step k , where k - 1, 2, e - - , ( n - 1).

The simple example of Fig. 3 demonstrates that (15) is sufficient to calculate all the properties of random networks, Although this can be performed easily by hand for small net- works (Le., n d 7), the computational task becomes tedious for larger network_s. Examples for a 21-node network and several values of d ( o r xo) are shown' in Fig. 4. Note that most networks of interest (i.e., n < 30 and K> 3) do not require calculations for k > 6 since the expected value of the number of total contacts increases very rapidly to its final value. That is, Pr(Z(k) = 0) --f 1 for k > 6. The value of k for which calculations must be made depends on n, p , and the number of nodes originating the message, z(0) . For_example, in a IOl-node network with p = 0.1 and z ( 0 ) = 1, Z( l ) = 10 and Z(2) = 56.72. Hence, since more than half of the nodes have been contacted on the average in 2 steps, it can be assumed that, on the average, the remaining nodes may be contacted within 4 steps. This implies that the k of interest is of the order of 4.

V. APPROXIMATIONS OF THE AVERAGE MFSSAGE PROPAGATION

Since one cannot obtain simple analytical expressions for E{Z(k)), this section considers an approximate solution. This

curves have been drawn to give a better insight into the behavior of 2Note that, although the random variables are discrete, continuous

our terminology, to finding the probability that X & ) = n at random networks.

950 IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 1972

Pr [Z(k) =

o.s+ I

Fig. 4. Probability distribution of the number of new nodes contacted a_t step k. (a) n = 2 1 , 2 = 1. (b) n = 21, 2 = 2. (c) n = 2 1 , A = 3 . (d) n = 2 1 , A = 4 .

approximate solution is based on (12) and (1 3) and yields the stepwise averages, denoted by XQ(k), yu(k), and zQ(k), where the term “stepwise” implies that the message flow at each step is conditioned on the expected values of each previous step. The stepwise averages x&), yQ(k), and Z J k ) will be shown to approximate the averages X@), y ( k ) , and Z ( k ) , respectively.

In what follows, consider the probability Pr{Z(k) = z ( k ) l Z ( k - i) = Z ( k - l), . . . , Z(0) = Z(0)) and let zQ(k) be the expected number of new nodes contacted at step k with the condition that at each previous step j , where j = 0, 1, * - , ( k - l) , z( j ) = za( j ) . Furthermore, under the same conditions, let XQ(k) be the expected number of nodes con- tacted up to and including k , and let y&) be the expected number of uncontacted nodes at step k . Note that,

and

It can be shown [8] that XQ(k) and z u ( k ) have the following properties.

In directed random networks, the stepwise average of the number of nodes contacted up to and including k , zQ(k) is

given by

yQ(k) = n - [ n - ~ ( 0 ) l ( q ) z ~ ( k - ~ ) (18)

and the stepwise average of the number of new nodes at step k , Z,(k) is given by

zu(k) = FQ(0) [(q)xQ(k-2) - (q)xQ(k-l)], (19)

where FQ(0) = n - zu(0) is the initial condition. The derivation of (1 8) and ( 19) follows directly from (1 2)

by Etting z( j ) = Tu( j ) for j = 0, 1, 2, . . . , ( k - 1) and solving for Z ( k ) ; this Z ( k ) is the new Z,(k).

Note that (1 8) and (1 9) are nonlinear difference equations. Although the closed form solutions are difficult to find, once the initial condition xQ(0) (or TJO)) is given, xQ(k) or ZQ(k) can be calculated from ( 18) and (1 9).

An insight into the behavior of the stepwise average flow through random networks can be obtained by linearizing (1 8) or (19). Consider (18) and apply the binomial expansion to the term (q)xQ(k-l) = (1 - under the assumption that (p)XQ(k-’) < 1. This yields

ZQ(k> = ZQ(O) + Ta(0) pTu(k - I) ,

CONCISE PAPERS 951

where yQi,(0) = n - T,(O). Equation (20) is a linear first-order difference equation whose solution is given by

Z Q ( k ) = TQ(O) FQ(0)l k

(21) i=O

Note that since p Y , ( O ) = p [ n - zQ(0)], then if za(0)= 1, p F Q ( 0 ) = p ( n - 1) = x,, where 2, is the expected value of the diLected link density given by (9). Hence, if X,(O) = 1 and pX,(k - 1) < 1, the stepwise ave rage f the number of nodes contacted up to and including step k , XQ(k) is given by

k F a f k ) = (2,)i. (22)

$0

Note that the value of the stepwise average z , ( k ) given by (22) holds as long as

p X , ( k - l ) = p Z a ( k - 1 ) < 1. k=l

i=O

This is true for the initial values of k (e.g., k = 0, 1, 2) even though x, may be greater than 1.

The stepwise average of the number of new nodes contacted at step k , Z,(k) can be found from (17); for example,

This implies that the stepwise average message prolagates through the network as z2 at each step k. Hence, if A , < 1, then the message will stop propagating before all the nodes are contacted. The approximate values of the kth terminal re- liability rQ(k) and the approximate value of the weak con- nectivity /̂, can also be obtained from (1 8) and (1 9), that is,

zQ(O)= 1, ~ Q ( l ) = ~ Q ( l ) - ~ ~ ( o ) = ~ , , and ZJ2) = 22, etc.

and

for XJO) = 1. At this point we consider the following question: “HOW

does the stepwise average differ from the actual average?” Let us consider the double expected value of a conditional random variable R given S, i.e.,

E { E [ R I S ] } = z E [ R I S = s ] Pr{S=s}. (25)

Furthermore, let the probability of S, Pr {S = s) be given by an impulse function of the form 6 (s - S), then (25) becomes

E { E [ R I S = s ] } = E [ R I S = F ]

S

- = R . (26)

According to our notation, (26) representsthe stepwise average of R . For example, if the probability of Z ( j ) can be replaced by the impulse function 6 [z( j ) - Z ( j ) ] , j = 0, 1, . * * , ( k - I ) , %en the actual average of Z ( k ) equals the stepwise average Z,(k).

For example, a comparison of the actual and stepwise average message propagation is shown in Fig. 5(a)-(c) for an 1 I-node network with averag link densities of 1, 2, and 3, for a 21- node network with A = 1, 2, 3, and 4, and for a 3 1-node net- work with average link densities of 3, 4, and 5. Observe that

- Actual Ave.

- - - - -- Stepwise Ave .

Fig. 5. Actual and stepwise average message propagation. (b) n = 21. (c) n = 31.

(a) n = 11.

since at step zero one node originates the message with prob- ability 1 (i.e., an impulse) the actual and stepwise averages at step one are identical. Also note that the stepwise averages approach the actual averages for large values of n and p , since K = (n - 1 ) p .

952 IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 1972

VI. WEAK CONNECTIVITY The weak connectivity of random networks y is given by

y=Z(n - ~ ) / n , Pr(X(0) = 1) = 1.

Although a closed form expression for y cannot be obtained, an expression for the approximate weak connectivity ya can be obtained using the stepwise average X a ( k ) of (1 8) to approx- imate X ( n - 1). That is,

Recall from the definition of the step that when k > n - 1 no new nodes can be contacted, that is, the maximum number of steps that can separate anyJwo nodes in an n-node network is n - 1. Therefore, X a ( n ) = X a ( n - 1). Let us denote the step- wise average number of nodes contacted up to and including step k = ( n - 1) by X f g X -a ( n - 1). Then, from (18)

- Ff= n - Y,(O)(q)Xf, (28)

where Fa(0) = n - xa(0) . Hence, the approximate weak con- nectivity of (27) is given by

Ya = 1 - [Ya(O>/nl ( q P f . (29)

It should be noted that ya 2 y, since %f 2 x ( n - 1). The ex- pression for the weak connectivity given by (29) can be com- pared to that used by several authors [ 5 1, [ 71 , namely,

7 = 1 - exp [-yA,l (30)

TO compare ”/a given by (29) with y given by (30) expand the term (q)’f of (29) as follows3 :

($.f = 1 - p x f t- [XdX,- l)p21/(2!)

- [Xf(Zf- 2)p3] / (3!) +. . . . If (xf)ipi >> (yf)i-l pi, then

- ( q ) X f % exp [ - p X f ] = exp [-Z,Tf/(n - 111 % exp [-yaAo].

Furthermore, if Xa(0) = 1 and n are large, then ?,(O)/n e 1, so that (29) can be approximated by

- ^la z 1 - exp I- y a A o l .

Hence, for large networks ya = y. The comparison of y a and 7 for several values of n is shown in Fig. 6. Note that for large n the approximate value of the weak connectivity of random networks ya is identical to that previously reported.

VII. CONCLUSIONS

The message propagation problem for both directed and undirected random networks is . formulated in terms of a simple model. This model represents an n-node simple ran- dom network whose structure is described by an n X n node matrix L p whose ijth entry represents the probability p of a link connectlng node ni to node nj. Furthermore, this model considers the undirected random networks to be a subset of directed random networks. Hence, the results obtained for directed random networks are also applicable to undirected random networks. The solution of the message propagation is found in terms of the probability distribution, at time step k ,

’Note that q = 1 - p and p z < 1.

0 I t f : A.

A W E LINK DRiSITY

Fig. 6. Weak connectivity of random networks.

of the number of new nodes that are separated from the originator of the message by simple chain progressions of time length k. Note that, when exactly one node originates the message, the probability distribution at the first time step (i.e., k = 1) describes the local structure of the random network in terms of the number of links emanating from the node. As a result, the average link tensity is found to be directly de- pendent on n and p (i.e., A = ( n - l ) ~ ) . This result shows that the previously reported results [ 41, [ 91 , which assume models having constant values of A and p independently, are oversimplified.

Based on the above results, a step-by-step computational method was,developed to calculate the probability distribution of the number of new nodes at each time step, the average message propagation, the weak connectivity, and the kth terminal reliability. Note that this method allows one to cal- culate the probability distribution of the total number of nodes contacted up to and including step k from the prob- ability distribution of the number of new nodes at step k , k = O , I ; - . , n .

An explicit approximate expression has been derived for the average number of nodes contacted up to and including time step k for all k = 0, . . * , ( n - 1). This approximate value is almost equal to the actual value calculated from the exact computational method.

From the above approximation, explicit expressions have been obtained that give approximate values of the average number of new nodes contacted at step k , Z ( k ) , approximate values of the kth terminal reliability ra(k) , and approximate values of the weak connectivity ya .

REFERENCES

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