Defining the complexity of a project management network

Marquette University

With the development of the PERT system for the Polaris missile project in 1954, many managers have become users of project management control techniques. They have become users not necessarily out of their own disposition. In many cases the government contracting agencies have mandated the use of these techniques. The Atomic Energy Commission in the early 1960’s required the use of sequential event control management systems for all of their projects that were contracted through the private sector. The Department of Housing and Urban Development has similarly mandated as recently as November, 1970. Frequently, private contractors are also requiring a demonstration of this capability for their contracted construction activities. In any event, few will question the arrival on the scene of a great variety of these techniques and their enforced use.

Many practitioners have discovered that the technique itself is not a panacea for solving management’s control problems with projects. Instead they have found that they have been forced to staff the project management function with young college graduate mathematicians, engineers, operations researchers, and computer systems analysts. These personnel frequently are not familiar with what is involved in the project itself or the management team is not sufficiently familiar with the technique of project management or PERT or precedence networking. Thus, all too often, the original network is drawn, a few runs are made through the computer, and finally in disgust, desperation, and frustration, the project control scheme is shelved.

One must immediately ask the question why a sophisticated and powerful tool becomes so bogged down that it serves no useful purpose? The key to the answer frequently lies in a poorly prepared network at the onset. This is not to say that the managers are not doing their best or that the analysts are not doing their best. To the contrary, it simply means that the neophyte or first time user of this technique frequently does not know what to include and what not to include in the final network. There appears to be some potential value for one to develop a measure to suggest that the network is adequate for the task at hand. The purpose of this paper is to describe some preliminary work in the development of the Coefficient of Network Complexity. This coefficient, or CNC, suggests an initial philosophy for the classification of these networks and attempts to explain in simple terms how it might be described for managerial use. To do so, the remainder of this paper is organized to 1) briefly review required terminology, 2) define the Coefficient of Network Complexity, 3) give some examples in terms of the procedures and 4) describe its potential usefulness.

Review of Terms

The Coefficient of Network Complexity is equally applicable to both PERT type networks and to Precedence type networks. Figure 1 displays a simple PERT network consisting of Events and Activities. Figure 2 shows a simple precedence network consisting of Work Items and Preceding Work Item relationships. These networks are both described in Tables 1 and 2.

Defining the Coefficient of Network Complexity

For PERT Networks


For Precedence Networks


Thus, the Coefficient of Network Complexity is a measure of the degree of inter-relationship (or complexity) of the network itself. In both of the simple networks, CNC = 1 as there are six activities and six events in one case and there are six preceding work items and six work items in the other. This is evident by examination of Tables 1 and 2.

Demonstration of Calculation

Figure 3 takes the precedence network relationship that was originally shown in Figure 2 and makes it more complex.

Table 3 gives a tabular description of this complex precedence network. Typically one would feel that the network shown in Figure 3 is certainly more descriptive of what is actually taking place in a project. The variety of precedence relationships demonstrated also serve to enhance the value of the network itself.

Now it is possible to calculate a coefficient of this network complexity described earlier:


For the simple network CNC = 6/6 = 1.0. For the more complex precedence network, CNC = 12/6 = 2.0.


Event Preceding Event Succeeding Event
0 - 1
1 0 2, 3
2 1 4
3 1 4
4 2, 3 5
5 4 -








Work Item WI Preceding Work Item PWI
01 -
12 01
13 01
24 12
34 13
45 24, 34


This very simple measure has given us some feeling for the degree of entanglement between various aspects of a given PERT network. It is perhaps appropriate now to discuss the potential uses for this measure, as unrefined as it might be.

First, the coefficient of network complexity may serve as an indicator of how carefully the project management team has been in preparing the PERT network itself. A low CNC will suggest a large number of end to start relationships which do not explore all the potential difficulties and lagging of starts and finishes that might take place. Therefore, a high CNC may just suggest a better and more finely tuned network.

Secondly, the length of time required by a computer to process a network is largely determined on its complexity. If the network is very straight-forward and consists of a large number of parallel paths, then the computer processing time is going to be minimal. If on the other hand, the network has a great deal of entanglement, then the computer processing time will be higher; growing perhaps at a geometric rate. It would not be unexpected that one could use the coefficient of network complexity in conjunction with the overall size of the network as a predictor for computer processing time.


Work Item WI Preceding Work Items PWI's Precedence Relation
12 01 END - START
12 01 END - END
13 01 END - START
13 01 END - END
24 12 END - START
24 13 END - END
34 13 END - START
45 24 END - START
45 24 END - START
45 34 END - START
45 34 END - END

Third, the coefficient of network complexity might be used as a strong indicator of the required frequency of review for that network. A high CNC suggests that more frequent updates should be run. One would be dealing with a much more complex network and hence would have the need to have a greater frequency review. A low CNC suggests that the project is more simple and might be monitored on a less frequent basis.


This paper has defined the term Coefficient of Network Complexity, CNC, as the quotient of the networks activities divided by its events or its preceding work items divided by its work items. The reason for the derivation of such terms is to offer the infrequent user of precedence networks for project management an early indication of whether or not the planning has been carried out at a sufficient level. Several side benefits are also available. The first being that the CNC may be used as a predictor of computer processing time and as an indicator of the frequency of review requirements. It must be pointed out that the CNC must yet stand the test of actual application beyond the simple networks shown here.

This material has been reproduced with the permission of the copyright owner. Unauthorized reproduction of this material is strictly prohibited. For permission to reproduce this material, please contact PMI.



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