Originally appeared in 1973 as part of Toward a Concept Inventory

Representation of concept networks using graph theory
Use of interactive graphic display techniques
Implications of computer augmentation of intellect
Relationship to artificial intelligence projects

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This project is concerned with the collection of entities and the indication of relationships, if any, between those entities. Expressed in these general terms, the techniques of graph theory may be used in this project. Graph theory is concerned with the "arcs" (links or relationships) between "nodes" (entities) and the various structural properties of the network so constituted.

It can be of great assistance in dealing with a broad range of combinatorial problems which occur in various economic, sociological or technological fields. It is, perhaps, that aspect of the theory of sets which can produce the most fruitful results not only for the pure mathematician, the engineer, and the organizer, but also for the biologist, the psychologist, the sociologist and many others. Graphs can be used to represent structures such as: a network of roads, an electrical circuit, communication in a group, a complex chemical molecule, circulation of documents in an organization, kinship structures, etc.(13, 14)

Its use in connection with relations between more abstract social entities such as organizations and nations is much less frequent (15-20). Its use for handling psycho-social abstractions appears to be even rarer (21).

The image of a 'network or web of ideas' to represent a complex set of interrelationships in a sphere of knowledge, and particularly culture, is a fairly familiar one. This use of 'network', however, is purely metaphorical and is very different from the notion of a network of concepts as a specific set of linkages among a defined set of concepts, with the additional property that the characteristics of these linkages as a whole may be used to interpret the semantic significance of the concepts involved.

Some features of concept networks

Points 1 to 3 below are concerned with the shape of the network, 4 to 8 with interactions within the network.

1. Centrality. A measure (in topological not quantitative terms) of the extent to which a given theoretical entity (e.g. a concept) is directly or indirectly "related" via links to other entities i.e., the extent to which it is "distant" from another entity. One can speak of a "key" concept or of a concept being "central" to the concerns of a particular discipline. It may also be considered a measure of the degree of "isolation" of the entity. A systematic analysis of the centrality of theoretical entities could indicate where new concepts are necessary to bridge conceptual gaps and link isolated domains.

2. Coherence. A measure of the degree of "interconnectedness,' or 'density" of a group of concepts. This may be considered as the degree to which a system of concepts is "complete". Differences in density would reflect the tendency for more highly coherent concept systems to appear more self-reinforcing in comparison to less organized parts of the network. In some respects this is an indication of the degree of "development" of a group of concepts.

3. Range. Some concepts are directly related to many other concepts, others to very few. The range of a concept is a measure of the number of other entities to which it is directly related. Range could be considered an indication of the "vulnerability" of a concept, to the extent that a high range concept would be less vulnerable to attack than a low range concept, since it has more bonds anchoring it to its semantic environment. High range points are therefore either key points in resistance to conceptual change or else key points in terms of which orderly change can be introduced.

4. Content. The "content" of a relationship between entities is the nature or reason for existence of that relationship. In general, different relationship contents are required for each model. Simple graphs have only one link between any two entities: multigraphs have two or more links, each of different content.

5. Directedness. A relationship between two entities may have some "direction" i.e., A to B. or B to A. There may be several types of directedness. The most important for this project is probably: A "is a subset of" B. i.e., directedness points to the more fundamental concept of a pair. In a multigraph, one link may point from A to B and the other from B to A -- where each is more significant in terms of different content.

6. Curability. A measure of the period over which a certain relationship between entities is activated and used. At one extreme, there are the links activated only on a "one-shot" basis (e.g. a "trial balloon" ideal, at the other there are links, and sets of links, which are considered stable over centuries (e.g. the concepts associated with "property").

7. Intensity. A measure of the strength of the link or bond between two entities. Two concepts may be said to be "strongly bound together". In some models, the intensity is a measure of the amount of the "flow" or "transaction" between the entities. The link from A to B may be strong, and that from B to A, weak.

8. Frequency. A link between two entities may only be established intermittently. This measure is less significant to this project (except perhaps in cyclic approaches to the history of ideas or to the activation of concepts over a 24 hour period.)

9. Rearrangeability and blocking. A connecting network is an arrangement of entities arid relationships allowing a certain set of entities to be connected together in various possible combinations. Too suggestive properties of such networks, which are extensively analyzed in telephone communications (22) are:

• rearrangeability: a network is rearrangeable, if alternative paths car, be found to link any pair of entities by rearranging the links between other entities.
• blocking: a network is in a blocking state if some pair of entities cannot be connected.

Examples of types of network patterns

Some of the above features of networks of concepts (or other entities) may be illustrated by the set of diagrams in Figure 5. Each entity is represented by a letter of the alphabet. Four simple types of entity groups are shown. Each type is further distinguished if the relationships between entities are directed.

a) In the non-directed examples of group (1), A is the central concept in (1.3), A and D in (1.4], A and F in (1.23. In (1.1l, there is no central concept.

b) In group (1), peripheral concepts are D and C in (1.2); B. C, E and F in (1.3); B. C and F in (1.4). There are no peripheral concepts in (1.1).

c) In group (1), the range of A in (1.3) is 4, in (1.4) it is 3.

d) In group t1), the reachability of A in (1.1) and (1.2) is 3, in (1.3) it is 1, and in (1.4) it is 2.

e) In all the directed examples of group (2), * is the central concept with at least B and F as direct component concepts. In all except (2.3), there are even sub-sub-components of A.

f) In all the directed examples of group t3), A is the central concept but only as a common sub-component. O is also a common sub-component in t2.13.

g) In all the directed examples of group (43, there is a chain of component/ sub-component links. In (4.1), this is continuously forming a loop. In (4.2) and t4.4), C is the major concept. in (4.33, A is the central concept but only by having F and t as sub-components and being itself a common sub-component to B and C.

The above features are all evident, almost to the point of being trivial. But most cases of interest are likely to be much more complex, with many nested levels of concepts and cross-linking relationships. These may be examined by matrix analysis techniques, particularly using computers (to which the proposed record layout is suited) (23-26). Computer programs exist to detect properties of networks.

A more complex example is illustrated by Figure 6. There is shown the manner in which two different models or conceptual structures might interlink the same concepts to form two very different patterns - which may be analyzed.

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