114 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 1. 1996 MARCH

A Unified Domination Approach for Reliability Analysis of Networks with Arbitrary Logic in Vertices

Arkadi A. Chernyak

Zhanna A. Chernyak State Economic University, Minsk

University of Informatics & Radioelectronics, Minsk

Key Words - Network reliability, Topological formula, Domination theory, Directed acyclic graphs.

Summary & Conclusions - The concept of a probabilistic graph G has been used as a universal model for network reliabili- ty. Most of the literature on this subject concentrates on computing various reliability measures (such as k-terminal reliability or all- terminal reliability) which are the probability that vertices of G can communicate with other specified vertices. primarily, this paper provides a common theoretical framework for addressing k- terminal reliability problems. To this end, it extends the model ad- mitting vertex functions of G to be arbitrary monotone Boolean functions: In this case G is called a monotone (&&graph. In such graphs the signal passability across a node is carried out in accor- dance with collections of signals delivered on the node inputs from other ones, the collections are subjected to some logic principle realized by a Boolean function. Monotone (S,t)-graphs include all known classes of directed multi-terminal network reliability models. The main result of this paper is the reliability expression for com- puting the probability of an acyclic monotone ($&graph G being operational. The expression uses the local domination parameters introduced here. That reduces the system level of consideration to the element level, providing a unifying understanding of the com- binatorial nature of some results based on the domination theory and developed earlier for ordinary networks.

1. INTRODUCTION

The concept of a probabilistic graph G has been used as a universal model for network reliability. The literature on this subject is vast. Most of it concentrates on computing various reliability measures (such as k-terminal reliability and all- terminal reliability) which are the probabilities Pr (G) that ver- tices of G can communicate with other specified vertices. A few efficient attempts to systematize such problems were made in [6-71. In particular, the source-to-k-terminal reliability was shown to be the most general one and a unified formula was derived for solving it [6]. Ref [6] used the ideas developed in [8 - 101 which have become classical [2].

Primarily, this paper provides a common theoretical framework for addressing k-terminal reliability problems. To this end, it extends the model admitting vertex functions of G to be arbitrary monotone Boolean functions: in this case a graph G is called a monotone (S,t)-graph. In such graphs the signal passability across a node is carried out in accordance with col- lections of signals delivered on the node inputs from other nodes,

the collections are subjected to some logic principle realized by a Boolean function. For instance, it might be a majority prin- ciple defined by a threshold number of operational nodes deliver- ing signals on inputs of the node under consideration (in or- dinary networks any node can be interpreted as having the threshold number 1 ; see section 2). So monotone ( S , t ) -graphs include all known classes of directed multi-terminal network reliability models (see section 2).

The main result of this paper is the reliability expression for computing the probability of an acyclic monotone ( S , t ) - graph being operational (Theorem 1). Theorem 1 uses the local domination parameters introduced in this paper. That reduces the system level of consideration to the element level, providing a unifying understanding of the combinatorial nature of some results based on domination theory and developed for ordinary networks. In particular, all the results from [5, 7 - 101 concer- ning acyclic graphs follow straight-forwardly from those of cor- ollaries 1 - 3 . The simple formula for all-terminal reliability of acyclic graphs due to [1] results from corollary 4.

All graphs considered in this paper are directed, acyclic graphs without multiple arcs and without isolated vertices.

Abbreviations

thres-kit threshold kit thres-num threshold number,

Notation

r general name of a graph m, Dl? set of [vertices, arcs] of I’ D + ( v , r ) , D - ( v , r ) subset of D r directed [into, out of]

vertex v r-DF graph obtained from l? by removing the arcs DF and

all vertices VF\W F is a graph having some common vertices & arcs with

R(F) set of regular subgraphs of r d ( r ) domination of I’ d ( v , r ) local domination of I’ with respect to vertex v g (v,l?) vertex function of I’ S set of source nodes S a member of S t terminal (sink) node Adj (v , r ) set of vertices adjacent to v in r th(v) 9( e )

r

set of thres-kits of vertex v indicator function: S(True) = 1,5(False) =O.

Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

0018-9629196/$5.00 01996 IEEE

815 CHERNYAWCHERNYAK: A UNIFIED DOMINATION APPROACH FOR RELIABILITY ANALYSIS OF NETWORKS

Nomenclature & Dejnitions

(The dejned concept is in italics.)

then g(v,H) = gv(xl ,..., x,,O ,..., 0).

That is, the vertex function g(v,H) is induced by g(v,I’). A e

a

a

a

e

e

a

a

a

e

a

e

e

e

e

e

e

e

e

e

Arcs & vertices of An item (eg, element, graph) is operational iff it is able to perform all of its functions. An item (eg, element, graph) is failed iff it is not operational. A monotone (S,t)-graph I’ is minimal iff is failed when any element of r is failed. If a minimal graph A is contained in r ( A C_ r) , then A is a minimal subgraph of l?. A monotone (S,t)-graph r is regular if every element of r is in at least one minimal subgraph. The formation of a regular graph r is a set of its minimal subgraphs whose union is r [ 101. The minimal subgraphs are components of the formation. A formation is [odd, even] iff the number of its minimal subgraphs is [odd, even]. The ‘domination of l?’ = d(l?) = ‘number of odd forma- tions of r’ - ‘number of even formations of l?’. A local formution of a ‘vertex v SE S C W’ is a subset of thres-kits from th(v) whose union is Adj(v,l?). A local formation is [odd, even] iff the number of thres-kits in it is [odd, even]. d ( v,r) = ‘number of odd local formations of v’ - ‘number of even local formations of v’. d( v , r ) = 1 for every v E S. A vertex w is adjacent to a vertex u iff there is an arc (w,u) directed into U from w. A thres-kit of a vertex v is a subset of vertices adjacent to v. Thresh-kits are elements of set th(v). Consider a special symmetric case: a) all thresh-kits from th(v) have the same cardinality k; and b) every k-element subset of Adj (vJ) is contained in th(v). Then k is the thres-num of the vertex v. The conjunction x L l . . .xi, corresponds to thres-kit {wil,. . . , wLk} s Adj(v,r). The vertex function g (vJ) of v is the disjunction of 1 th(v) 1 elementary conjunctions corresponding to thres-kits of th(v) . Let g ( v X ) = gv(xl,. . . J,) be expressed as the disjunction of ( g ) distinct elementary conjunctions of the same car- dinality k; then g(v , r ) is a symmetricfunction of degree k. The degree of g(v,I’) is the same as thres-num of v. ( S,t) -graphs admitting arbitrary monotone Boolean vertex functions are monotone (S,t) -graphs.

are the elements of r’. symmetric function induces a symmetric function of the same

The traditional network reliability models are limited by degree. 4

the form of vertex functions.

2. MODEL

2.1 General Models

Assumptions

1. A directed acyclic (S,t)-graph G has unreliable vertices & arcs. The vertices from the set S are sources and the vertex t is the sink (terminal).

2a. Each vertex and each arc is either operational or failed. 2b. All vertices & arcs of G are subject to mutually

2c. A graph is either operational or failed. 3. Given vertex v E VG\S, it is assigned to v, the set th(v)

of thres-kits. The set th(v) associated with v consists, in turn, of subsets of vertices adjacent to v.

s-independent random failure.

4. th(v) = 0 for any v E S. 5. The union of all thres-kits of v is Adj(v,G) . [Otherwise

4 G would have an irrelevant component.]

Vertex w is attainable from vertex v iff -

1. v is adjacent to w; and 2. vertex w and arc (v,w) are operational; and 3. either ‘v E S and v is operational’ or ‘v is attainable

4

A vertex v that is attainable from each vertex of a thres- kit of th(v) is simply called attainable. Now the criterion for operational is: G is operational iff the sink vertex t is attainable.

At last, the above model is described more formally with the vertex function: Every ‘vertex v e S’ having arcs el,. . . ,e,, directed into v is associated with the vertex function g (v, G) = g (x~ , . . . ,x,,) ‘disjuntion of (th(v) 1 elementary conjunc- tions corresponding to thresh-kits of th(v) ’ ; g (v, G ) 3 1 for every v E S.

Let the Boolean variable zv characterize the state of the vertex v: zv= 1 iff v is operational. Recurrently define the out- put signal of v as:

from each vertex of a thres-kit of th(v) ’ .

The variable xi corresponds to the arc ei = (w,v) directed into v ; thus xi = 1 iff ei is operational and the ‘output signal yw of vertex w’ = 1. The xi does not characterize simply the ‘state of arc ei’; rather it indicates both the ‘state of ei’ and the ‘out- put signal of w’.

Finally, the criterion of being operational is: G is opera- tional iff the ‘output signal yt of the sink vertex t’ = 1.

116 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 1, 1996 MARCH

2.2 2-Terminal Reliability Model 3. MAIN RESULTS

This is a special case of section 2.1. Let: Pzeorem 1. Let G be an acyclic monotone (S,t)-graph. Then,

s= Is}

thres-num = 1 for all vertices of 6, except for s whose thres- Pr{G) =

num = 0. The attainability of the vertex t is equivalent to d(H).Pr{H}; (1)

H E R ( G )

the existence of an operational minimal path from s to t. Indeed, since t is attainable there is an attainable vertex v1 adjacent to t , in this case the vertex t and the arc ( v l , t ) are operational. Likewise, the attainability of v1 implies the existence of an attainable vertex v2, etc. Eventually, because the graph is acyclic, the result is the sequence of operational vertices & arcs,

((2)

Corollary 1 [5]. Let G be an acyclic monotone (S , t ) -graph Then

d ( ~ ) = IT d ( v , ~ ) . v f H

and let every vertex function g ( v* (1) holds, and

Of be

vn=s, v,-1, ..., v1, v,=t;

( vi,vi- i = n,. . . ,1; Notation

ID+ ( v ,W I , that is, an operational minimal path of G. rv k v degree of g ( v , H ) . Let all vertex functions of G be elementary disjunctions:

Let vertex v # s have n arcs el, . . . , e, directed into v then, d

Corollary 2 [IO]. Let G be an acyclic probabilistic ( s , t ) - graph. Then (1) holds, and

g(v,G)= X,V ... vx,, v # s;

g(s,G) = 1.

Thus y t= 1 iff there is an operational minimal path from s to t .

2.3 Source-to-k-Terminal Reliability

This is a special case of section 2.1 with K= (u l , ...,

Add a perfect vertex t, and k perfect arcs (ui, t ) ; the resulting graph is denoted by G. Assign the thres-num k to vertex t , and thres-num 0 to vertex s. All other vertices have thres-num = 1. The attainability of sink vertex t is equivalent to the attainability of vertices u l , ..., uk. Indeed, the same arguments as in section 2.2 show the existence of k opera- tional minimal paths from s to u l , ..., uk, respectively. In other words these paths form the tree of G rooted at s, all pendant vertices of which belong to the set K. (Such trees were called K-trees in [9]). Vertex t is attainable in G iff there is an operational K-tree in G.

Let vertex functions of vertices ‘v # s,t C e’ be elemen- tary disjunctions with appropriate variables depending on arcs directed into them. Let,

U k ) .

R ( G) E set of p-subgraphs of G defined as unions of simple paths from s to t ,

Corollary 3 [7 - 91. Let G be an acyclic (s, K) -graph, and let Pr { G} be the source-to-K-terminal reliability. Then (1) & (3) hold.

R (G) = set of K-subgraphs of G defined as unions of directed trees rooted at the source vertex s and having K as the set of their pendant vertices,

Corollary 4 [l] . Let G be an acyclic graph with perfect source vertex s, other perfect vertices U,, and unreliable arcs e,,,, i = l , ..., n ; j = l , ..., d,,

d

4 = I D f ( ~ , , G ) I ,

D+(u i ,G) = { q j : j=1 , ..., di }

Pr { G} E source-to-all-terminal reliability

Pr{G} =

g ( t ,G) = XIX;?. . .x,, Then,

g(s,G) = 1. d, 1

Then yt= 1 iff there is an operational K-tree of G.

CHERNYNCHERNYAK: A UNIFIED DOMINATION APPROACH FOR RELIABILITY ANALYSIS OF NETWORKS

For the set R ( G) in theorem 1, an algorithm [4] generates all graphs from R( G ) without duplicates. Thus (l), if kept in symbolic form, involves only non-cancelling terms which cor- respond one-to-one with the subgraphs from R ( G ) .

We conjecture that the techniques presented here can be extended to cyclic structures.

ACKNOWLEDGMENT

This work was partially supported by the International Soros Science Education Program (ISSEP). We are very grateful to Professor M.O. Ball and anonymous referees for valuable comments that improved the content of our paper.

APPENDIX

Notation

S P

[S,e a coherent system.

A formation of S is a set of minimal paths whose union is S; it is [odd, even] if the number of minimal paths is [odd, even]. The domination, d[S,PI, of system [ S , q = 'number of odd formations of S' - 'number of even formations of S'.

{ 1,. . . ,n} : a set of elements {PI,. . . ,P,} : a family of minimal paths such that Pi C S, P, C Pj for i # j

A.l Lemma I

Let P consist of minimal paths:

P1= {1}, 1=1, ..., t;

P i J = {i , j} , i=t+ 1 , . . . , in, and j = m + 1 ,..., k, fork > m > t.

(1-element paths Pl may be absent.)

Then d[S,P] = (-l)k. 4

Proof. By induction on k. The lemma is proved if k = m + 1. Suppose k > m+ 1. Let

P(') = {Pl,Pi,j: l= 1 ,..., t; i=t+ 1 ,..., m; j=m+ 1 ,..., k- l}.

The P(2) consists of the minimal paths of P not containing k; whereas P(') is obtained by deleting k from all minimal paths of P and then discarding any supersets which are now present.

According to the Pivoting Domination Theorem [3] :

d[S,P] = d[S,P]+k - d[S,Pl-k.

But [S,P] + contains the irrelevant component m + 1. Hence,

d[S,Pl+k = 0 PI ;

d[S,P]-k = (-1) k - l by inductive supposition.

It follows that d[S,P] =(-l)k.

A.2 Proof of Theorem 1

Q. E. D

Let {Al,. . . ,A,} be the set of minimal subgraphs of G. Use the Inclusion-Exclusion formula, and write the reliability of G:

Pr{G} = Pr{Ai} - Pr{Ai U Aj} +... i i<j

+ (-1 .Pr{Al U .. . UA,} . (4)

By definition, a subgraph H belongs to R( G) iff,

H= Ai,U ... UAi,, for some 1 5 il <... < ik 5 n. Hence (4) & (1) are equivalent.

We prove (2) by induction on the number of elements of H. If VH=SU { t } then

d ( H ) = d( t ,H) and the theorem follows.

Suppose that VH\S # 0. Among vertices from VH\S, choose vertex v of the least rank. H2 = 'the graph such that DH2 = D+(v,H) , and VH2 = Adj(v,H) U {v}'.

Hi H - DH2.

Because

then for any U from VHl fl VH2, the d(u,H,) = 1; and d ( H 2 ) = d(v,H2) = d(v,H). In view of this and the inductive sup- position, it suffices to prove the equality:

d ( H ) = d(H,)*d(Hz).

Consider a formation R, of graph H. Let C E R,. If v e C then the graph C1 = C is a minimal subgraph of HI. If v E C then C can be partitioned in two subgraphs C1 & C2 such that,

DCi = DC n DHi, for i = 1,2.

Thus Ci is a minimal subgraph of Hi, i= 1,2. In both cases, C induces C1 & C,. Let Ra,i be a formation of Hi, i = 1.2. The pair (R,,1,Ra,2) is induced by the formation R, of H if every minimal graph from Ra,i is induced by some minimal graph from R,, i = 1,2.

Introduce the mapping f: R, 3 (R,,l,Ra,2) where (Ra,l,Ra,2) is induced by R,; f is a surjective mapping.

118

For any set T of some subsets,

m ( T ) = 'number of odd cardinality subsets of T - 'number of even cardinality subsets of T.

La f - ' (~a , l iRa ,2 )

It follows from the surjectivity off that

d ( H ) = m(La). (5) (Ro IA 2 )

We now show that m ( L a ) = (-1) I R a I I + i R n z l . Let,

Ra, 1 E {B,,. . . ,Bt,Bt+ 1,. . . , B m } ,

Ra,z E {B~+I>...,B~};

v E B,, i = l , ..., t ; v E B,, i= t+ 1 ,..., m; k > m > t.

The following assertion is true:

Assertion 1

R, E La iff -

a. for any C from R,, either C= B,, 1 5 i I t or C= B, U BJ,

b. every B, ( i = 1,. . . , k ) is mentioned in #a at least once, 4

We interpret R, in terms of a formation of the coherent

t + l 5 i 5 m, m + l I j I k;

provided C runs through the elements of R,.

system [S,P] with,

P {Pl, P i j : Z ~ [ l , t ] , i ~ [ t + l , m J , j ~ [ m + l , k l } ,

Pl E {Ill}, PiJ = (B,,Bj}.

Assertion 1 provides the natural one-to-one correspondence be- tween elements 'Rx of La' and formations of [S,P]; the parity is invariant under this correspondence. Thus m ( La) = d[S,P]. But by lemma 1, d[S,P]= It follows from (5 ) that,

A.3 Proof of Corollary 1

It was shown [3] that:

d ( v , H ) = ( - l ) r u - k b , . (2:;).

IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. I, 1996 MARCH

This corollary follows from theorem 1. Q. E. D.

A.4 Proof of Corollary 2

All vertex functions of G, except g ( s , G) are taken to be of the degree 1. Let rv & k, be the same as in corollary 1. Then,

q = r, = ( r v - l ) + n-1. vcms v E m s

It follows that,

,Ems

because k,= 1 for any vertex v. This corollary follows from corollary 1. Q. E. D.

A.5 Proof of Corollary 3

Add a 'vertex t' and ' k perfect arcs (u,, t) , i = 1, ..., k' to the (s,K)-graph G where K = {ul,. . . ,uk } . The resulting graph is 6. Let g ( t , G ) xl,. . . , xk . AH other vertex functions of G excepting g (s, e) are set to degree 1. G is a monotone ( s , t ) - graph and Pr(G} = Pr{G). Because the set { ( u l , t ) : i = 1,. . . ,k} is contained in every minimal subgraph of 6, then K-subgraphs of G are in one-to-one correspondence with minimal graphs of e. Thus d ( H ) = d(&. Besides, r,=k,. It follows from corollary 1 that,

d ( H ) = d ( f i ) . IT ( - l ) 'v-kv = (-1)qPn+';

vEms

the last equality is in the proof of corollary 2. Q. E. D.

A.6 Proof of Corollary 4

Let,

6 be the same as in section A.5.

Since the degree of g ( t , G) is n, then every regular graph H from R ( 6) uniquely determines the regular graph from R ( G) . Hence H is uniquely determined by the characteristic vector t = ( tl, 1 >. . . 7 tn,d, 9

t i j = 4 ( e i j E U ) ,

$Ii E 4 ti,j > 0.

j = 1

Thus,

n d;

CHERNYAWCHERNYAK A UNIFIED DOMINATION APPROACH FOR RELIABILITY ANALYSIS OF NETWORKS 119

n

d ( H ) = r-I (-1p-1. i = l

According to (1) ,

/I

[8] A. Satyanarayana, J. Hagstrom, “Combinatorial properties of directed graphs useful in computing network reliability”, Networks, vol 11, 1981,

[9] A. Satyanarayana, J. Hagstrom, “A new algorithm for the reliability analysis of multi-terminal networks”, IEEE Trans. Reliability, vol R-30, 1981 Oct, pp 325-334.

[lo] A. Satyanarayana, A. Prabhakar, “New topological formula and rapid

pp 357-366.

-I

algorithm for reliability analysis of complex networks”, IEEE Trans. Reliability, vol R-27, 1978 Jun, pp 82-100. Pr{G} = Pr{G} = n (-1)61-1- H p t i ” ZJ

t i = l j = 1

AUTHORS

Dr. Arkadi A. Chernyak; ul. Russiyanova, 10, 299; Minsk, RB 220141 Rep. BELARUS.

Arkadi Chernyak was born (1955) in the Soviet Union. He earned his MMath (1977, honors) at the State Univ. of the (former) Soviet Union, Minsk,

d,

= Ifi: [’ - n (‘-PZ,J)]’ r = l J = 1

The last equality is immediately checked by removing the Q‘E‘D’ parentheses.

, REFERENCES

M.O. Ball, J.S. Provan, “Calculating bounds on reachability and con- nectedness in stochastic networks”, Networks, vol 13, 1983, pp 253-278. R.E. Barlow, “Mathematical theory of reliability: Historical perspec- tives”, IEEE Trans. Reliability, vol R-33, 1984 Apr, pp 16-20. R.E. Barlow, S. Iyer, “Computational complexity of coherent systems and the reliability polynomial”, Prob. Engin. Inform. Sci., vol2, 1988,

A.A. Chernyak, “Combinatorial-graph method for reliability analysis of complex systems with positive Boolean functions”, Remote and Con- trol, N 4, 1991, pp 165-174 (in Russian). A.A. Chernyak, “A new graph-combinatorial method for reliability analysis of monotone graphs”, Proc. 8Ih Int’l Symp. Reliability in Elec- tronics, 1991, pp 135-140; Budapest. C.J. Colbourn, The Combinatorics of Network Reliability, 1987; Oxford Univ. Press. A. Satyanarayana, “A unified formula for analysis of some network reliability problems”, IEEE Trans. Reliability, vol R-31, 1982 Apr, pp

pp 461-469.

23-32.

and PhD (1986) in Discrete Mathematics at the Mathematical Inst. of Belarus Academy of Sciences. From 1977-1993 he was with the Inst. of Reliability Machines in Minsk. Since 1993 he has been with the State Economic Univ. in Minsk where he is a Senior Lecturer. Dr. Chernyak is the author of over 50 refereed articles. His research interests include combinatorial algorithms, graph theory, complexity theory, and network reliability.

Dr. Zhanna A. Chernyak; Dept. of Higher Mathematics; State Univ. of Infor- matics & Radioelectronics; Minsk, RB 220027 Rep. BELARUS.

Zhanna Chernyak was bom (1955) in the Soviet Union. She earned her MMath (1977, honors) at the State Univ. of the (former) Soviet Union, Minsk, and PhD (1984) in Discrete Mathematics at the Mathematical Inst. of Belarus Academy of Sciences. From 1977-1980 she was with the Dept. of Mathematics at the State Belarus Univ. Since 1984 she has been with the Dept. of Higher Mathematics at the Univ. of Informatics & Radioelectronics in Minsk, where she is Associate Professor. Her research interests include graph theory, matrix theory, and mathematical education problems.

Manuscript received 1994 December 5

Publisher Item Identifier S 0018-9529(96)-02247-6 4 T R b

A Little Learning Is A Dangerous Thing ... “A little learning is a dangerous thing; Drink deep, or taste not the Pierian spring; There shallow draughts intoxicate the brain, And drinking largely sobers us again.”

- Alexander Pope, ca 1710

“The theory of probabilities is at bottom nothing but common sense reduced to calculus.’q

- Pierre Simon de Laplace, ca I820

The Editorial Staff suggests that you learn more than “a little” before writing papers about probability and statistical-inference concerning reliability. You will be lucky if Laplace didn’t figure it out before you did. Check out the papers & books written ca 1958 - 1977; most of basic reliability/probability math was developed then (several times). If you take a short course in reliability, or even a semester or two, you still have only “a little learning”. The odds are very good that several people have published your great idea at least 20 years ago. 4TRb