A classification scheme for vulnerability and reliability parameters of graphs

M&Z. Cornput. Modelling Vol. 17, No. 11, pp. 13-16, 1993 Printed in Great Britain. All rights reserved

0895-7177193 $6.00 + 0.00 Copyright@1993 Pergamon Press Ltd

A CLASSIFICATION SCHEME FOR VULNERABILITY AND RELIABILITY PARAMETERS OF GRAPHS* K. S. BAGGA Ball State University, Muncie, IN 47306, U.S.A. L. W.

BEINEKE,

R.

Indiana University-Purdue

E.

PIPPERT

University Fort Wayne

Fort Wayne, IN 46805, U.S.A. M.

J. LIPMAN

Office of Naval Research, Arlington,

VA 22217, U.S.A.

Abstract-The purpose of this paper is to introduce a classification scheme for measures of vulnerability and reliability in networks. We use these terms in a general sense. Since many network properties are actually properties of the underlying graph, we restrict this discussion to undirected graphs. 1. INTRODUCTION

There are two distinct types of questions that arise relative to a set of parameters of graphs: analysis and synthesis. Analysis refers to determining the values of graphical parameters and establishing relationships among them, while synthesis refers to constructing graphs that are in some sense optimal with respect to given parameters. The classification scheme we are offering here provides one approach to the analysis problem. The scheme itself, in its infancy, is presented in the next section. Some comments are offered and some questions raised in the last section. The appearance of gaps in some of the patterns in this classification scheme suggests possible new measures. These patterns emphasize the similarities as well as the differences among many of the interesting measures. The scheme we offer here demonstrates that the topics of network vulnerability and reliability are rich and ripe for further investigation. Finally, we hope to help establish conventions anticipating further suggestions from readers.

for terminology

in the area.

We will offer some,

2. CLASSIFICATION The

necessity

for a classification

scheme

becomes

apparent

as soon as we have a partial

list

of parameters which measure vulnerability and reliability or which are used in constructing such measures. We begin with a list in Table 1; it is undoubtedly incomplete. Due to the preliminary nature of this survey, we shall not attempt to provide definitions for the many parameters, nor a bibliography. We would especially appreciate information regarding the earliest uses of any of them specifically as a measure of vulnerability or reliability. In this list we include both vertex and edge versions of a parameter if we are aware of their use in the present context. Some parameters have additional versions, which we denote by use of a “wild-card” *. For example, *-connectivity can represent cycle-connectivity. Our preliminary classification scheme involves three major categories: cutting, covering, and closeness. Within each category, parameters are further classified into a two-dimensional array determined by whether they are vertex or edge parameters and whether they are deterministic (related to vulnerability) or probabilistic (related to reliability). *Research supported

in part by an Office of Naval Research Grant No. N00014-86-K-0412

13

K.S. BAGGA et al.

Table 1. Parameters used in the measurement of vulnerability or reliability. Center

Connectivity Edge-connectivity

Median

*-connectivity

Toughness

Reliability

Disjointedness Edge-connectivity

*-reliability Complexity

(number of spanning trees)

vector

Separation vector

Edge density

Independence number

Diameter

Resilience

Radius

Cutting center

Eccentricity

Bondage number

Persistence

Bisection width

Edge-persistence

Cohesiveness

Irredundance

Integrity

Domination number

Edge-integrity

Edge-domination

Vertex-covering

number

number

Domatic number

Edge-independence

Edge-covering number

*-domination

number

number

Ratio of disruption

3. COMMENTS We make no claim

that

our classification

AND

QUESTIONS

is optimal;

some parameters

straddle

a line, while

the placement of others is questionable. Nevertheless, it clearly shows the value inherent in classifying the various parameters. For example, we observe that just as all-terminal reliability is the probabilistic version of edge-connectivity, so the classification scheme suggests investigating a probabilistic

version

of the domination

number

of a graph.

Some parameters may prove to be more meaningful in the pseudographs, or hypergraphs than just graphs. This may be arising from gaps in the classification scheme. Of course, many extensions to networks in which edges are directed or weighted.

broader context of multigraphs, especially true of some of those of the parameters have natural Additional measures of vulnera-

bility or reliability may be obtained from parameters which standing alone are not adequate but which may prove useful in combination forms. A simple example is the combining of the order of a separating set with the number of components it yields to obtain toughness. At this point we raise the important question of terminology. Widely adopted conventions are to everyone’s advantage. In spite of a couple of attempts to make “vulnerability” a specific measure, efforts to keep it generic have been fairly successful. The question seems to be whether or not it should be the “umbrella” term for all work in this area. We have used “vulnerability” to denote the class of deterministic parameters and “reliability” to denote the corresponding class of probabilistic parameters, in accordance with the consensus of a collection of network researchers, including some who have used “reliability” in a much more specific sense as illustrated in Table 2. This has some appeal, but also some drawbacks. One is that it may take some time for “reliability” to move from the specific to the general. Another drawback is that “vulnerability” is by nature a negative term in that a graph of high vulnerability is structurally weak while a graph of high reliability is structurally strong, that is, “reliability” is a positive term. Although some researchers have directed their attention to such objects as the unreliability polynomial, that doesn’t resolve the difference between the two terms. Furthermore, if “vulnerability” is to be used in this restricted sense, a new term will probably be needed to refer to the entire area. We suggest “strength.” It has the virtue of simplicity, which we find desirable for so broad a concept. There may be superior alternatives, however, and we remain open to suggestions. Within the classification scheme itself, the distinction between cutting measures and covering measures seems natural. Other measures related to parameters such as diameter, radius, and centrality have been collected under the term “closeness.” It is not yet clear how useful or appropriate such measures may be, nor is it obvious that “closeness” is the most appropriate term. Perhaps additional classes are needed as well.

15

Parameters of graphs

Table 2. Cutting measures. vertex

Vulnerability

Edge Measures

Measures

Connectivity

Edge-connectivity

*-connectivity

*-edge-connectivity

Toughness

Edge-connectivity

Integrity

Bisection width

Ratio of disruption

Edge-integrity

Cutting number

Separation vector

vector

Cohesiveness Mixed connectivity

All-terminal

Reliability

reliability

*-reliability Resilience Table 3. Covering measures

Vulnerability

Vertex Measures

Edge Measures

Domination

Edge-domination

*-domination

number

*-edge-domination

number

Irredundance

Bondage

Domatic number

Edge density

Independence

Complexity

number

number

(# of spanning trees)

Edge-covering

Binding number

number

number

Edge-independence

number

Reliability

Table 4. Closeness measures. Ve&x

Vulnerability

Edge Measures

Measures

Edge-persistence

Persistence Center Median Cutting center

Diameter Radius Eccentricity

Reliability llcn 17:11-c

K.S. BAGGA et al.

16

Within

a given class, it is natural

to distinguish

deterministic

measures

from probabilistic

ones

as well as vertex measures from edge ones, although versions using a combination of vertices and edges (as in mixed connectivity) certainly deserve consideration. This provides the classifications in Tables 2, 3, and 4. Perhaps additional criteria would be useful. For example, some parameters seem to be determined by local properties of graphs, while others are more global in nature. However, these distinctions are not sufficiently precise to allow classification along these lines at this stage. On the other hand, it may be helpful to identify measures which incorporate the size or order of the graph, that is, measures which have been “normalized” in some sense. For example, an edge-connectivity of 5 seems to have different implications if the graph has 10 vertices or if it has 1000. The parameter “toughness” is normalized in a different sense; it could be considered a normalized version of connectivity, in that the order of a separating set is divided by the number of components resulting from its removal. Another possibility is to distinguish positive measures from negative ones. Consistent with our earlier comments about reliability being positive, we can define a measure to be positive if it is nondecreasing under the process of adding edges to a graph. This leads us to the general question of what characteristics are desirable in a measure of vulnerability or reliability. Some form of monotonicity seems to be on almost everyone’s list. Are normalized measures better than those which take no account of the size or order of a graph? Furthermore, some type of symmetry is usually desirable in computer networks; is it important in other applications? If so, are there or ought there be measures which incorporate symmetry? We would like to identify characteristics which make measures useful in certain applications. We conclude with a brief mention of algorithms. In the literature we observe that most

pa-

rameters of interest are difficult to compute over the class of all graphs. In fact, relatively few parameters have known polynomial time algorithms over even interesting restricted classes of graphs. Granted that network vulnerability and reliability are important ideas in part for practical reasons, just how important is computational complexity in determining measure? Are there useful measures which have polynomial complexity?

the usefulness

of a