The data collected together in the sections of this publication has been deliberately organized in a manner which stresses the interrelationships between the entities within a section and between those in different sections. (Each section is characterized by entities of a different type, and several types of relationship may exist between the same two entities). In effect, therefore, the entities and relationships in each section constitute a network, possibly composed of many subnetworks. Similarly, since entities in each section may be linked to those in other sections, the whole is constituted by a system of interlinked networks in which the relationships have a limited number of distinct meanings. The entities and relationships are currently held in computer files in a form which should facilitate analysis of these networks. It is hoped that the availability of data in this form will encourage the development of new types of analysis more appropriate to the structural complexity portrayed, especially since both the quantitative data and the mathematical functions representing the nature of particular relationships under different conditions (which are a precondition for the application of current methods of quantitative analysis of social systems), are absent and in most cases unavailable.

As François Lorrain notes the abstract notion of a network is undoubtedly called to play a role in the social sciences comparable to the role played in physics by the concept of euclidean space and its generalizations. But the poverty of concepts and methods which can currently be applied to the study of networks stands in dramatic contrast to the immense conceptual and methodological richness available for the study of physical spaces. A whole reticular imagery remains to be developed. At this time a network is understood to contain simply nodes and links and little else. An attempt to define anything like a reticular variable results in very little. This is not surprising, since to succeed would require the establishment of a general mathematical theory of networks which as yet has been little developed. In contrast to this situation, consider the multitude of spatial variables which are available: coordinates, length, surface, volume, curves, classes of curves, classes of surfaces, parameters of curves, parameters of surfaces, and so on, and all these in a space of any number of dimensions and manifesting any type of curvature.

**1. Social networks**

The types of network which occur in the social sciences are of such a diverse nature that only a purely formal definition of this notion is of sufficient generality.

A network is constituted by a certain set of points. In the social sciences these points may represent any or all of the following: individuals, groups, organizations, beliefs, roles, *etc*. In this exercise they represent: international organizations, multilateral treaties, world problems, strategies, concepts (human development, integrative, patterns), metaphors, symbols, modes of awareness, values. Such points may represent the existence of entities at the present time, or they may represent the existence of entities at some past or future time (or such points may also be used to represent intervals of time).

The points in a data set may be linked by one or more kinds of relationship. In this exercise three basic types of relationship are distinguished: (i) **Simple relationship, **namely A is related to B which implies that B is related to A; (ii) **Hierarchical relationship, **namely A is a part of B which implies that B is in contextual relationship to A; (iii) **Functional relationship, **namely A acts on B which implies that B is acted upon by A.

In the first case above a relationship is further defined by the types of entity between which it occurs, namely whether they are of the same type, or whether they are of different types. In the second and third case, a relationship is further defined by distinguishing the direction of the relationship, which is further developed in the third case by distinguishing several ways in which A can act upon B.

**2. Analysis of networks**

Classical mathematics, summarizing François Lorrain's remarks, is not able to handle complex structural features characteristic of social systems. Organization is best depicted as a network. The mathematical theory of networks derives largely from certain branches of topology and abstract algebra rather than from analysis, which underlies classical mathematics. The theory of graphs is often presented as a kind of general theory of networks with numerous possible applications in the social sciences. However, other than in the area of operations research, the theory of graphs has not proved itself to be very useful in sociology. The reason is probably that the theory has mainly been developed in the context of relatively limited problems in such a way that the results collected under the graph theory label, although numerous and of great interest, have little unity. In addition, the theory rarely handles networks with several distinct types of relationships each with its own configuration of links. It is precisely such networks which are of most interest in sociology. The theory also tends to exclude networks in which some of the points have links back to themselves when it is often just such networks which are important in representing social structures.

A final disadvantage of the theory of graphs is that it only offers a fairly limited number of means of global analysis of networks. It seriously neglects an important aspect of the study of any type of mathematical structure, namely the level of transformation relations between graphs. Because of its composition, a category possesses a richer structure than a simple graph, and it is therefore possible to define more rigorous and fruitful criteria of transformation (namely the concepts of function and functional reduction). In addition a set of points and a set of relations can be treated in their totality and simultaneously, in contrast to the methods of graph theory which considers individual paths between particular points in the graph. In the universe of categories (the universe of objects and relationships), transformations between categories may also be considered as relationships within a category whose objects are themselves categories, and so on. All this emerges from consideration of the global structure resulting from the manner of composition which relates the relationships themselves, thus providing a dialectic of levels of structure and a new imagery of networks. At all levels of this universe, the functional relationships between categories play a central role. They are the fundamental instruments which may be used in the exploration of structural complexity and the tools for extraction of information in global studies.

**3. Use of graph theory methods**

Despite the limitations noted above, graph theory methods have been applied to the analysis of social structures although such applications are not very common (see references below).

The image of a "network or web" of problems (or organizations, *etc*) to represent a complex set of interrelationships 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.

**4. Some features of graphs **

Using graph theory, a number of characteristics of networks can be determined. Points 1 to 3 below are concerned with the shape of the network, 4 to 8 with interactions within the network.

*(a) Centrality:* A measure (in topological not quantitative terms) of the extent to which a given entity (*eg* a problem) is directly or indirectly "related" via links to other entities (*ie*, the extent to which it is "distant" from another entity). One can speak of a "key" problem or of an organization being "central" to the concerns of a particular complex. It may also be considered a measure of the degree of "isolation" of the entity. A systematic analysis of thecentrality of entities in a network could indicate where new entities are necessary to bridge gaps and link isolated domains.

*(b) Coherence:* A measure of the degree of "interconnectedness" or "density" of a group of entities. This may be considered as the degree to which a system of problems is "complete". Differences in density would reflect the tendency for more highly coherent problem 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 problem system.

*(c) Range:* Some entities are directly related to many other entities, others to very few. The range of an entity 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 problem to the extent that a high range problem would be less vulnerable to attack than a low range problem, since it has more relationships anchoring it to its problem environment and preserving it in existence. High range points are therefore either key points in resistance to problem change or else key points in terms of which orderly change can be introduced.

*(d) Content:* The "content" of a relationship between entities is the nature or reason for existence of that relationship. Simple graphs have only one link between any two entities; multigraphs have two or more links, each of different content.

*(e) Directedness:* A relationship between two entities may have some "direction" (*ie*, A to B, or B to A). There may be several types of directedness. Two types are important for this project: A is a sub-element of B; A acts on B. In a multigraph, one link may point from A to B and the other from B to A.

*(f) Durability:* 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 (*eg* a single crisis), at the other there are links, and sets of links, which are considered stable over centuries (*eg* between the more permanent problems).

*(g) Intensity:* A measure of the strength of the link or bond between two entities. Two problems may be said to be "strongly bound together". In some cases, 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.

*(h) Frequency:* A link between two entities may only be established intermittently.

*(i) Rearrangeability and blocking:* A connecting network is an arrangement of entities and relationships allowing a certain set of entities to be connected together in various possible combinations. Two suggestive properties of such networks, which are extensively analyzed in telephone communications, are: (a) rearrangeability (a network is rearrangeable, if alternative paths can be found to link any pair of entities by rearranging the links between other entities); (b) blocking (a network is in a blocking state if some pair of entities cannot be connected).

**5. Graph analysis software packages**

An increasing number of applications of graph theory have emerged in the social sciences. GRADAP is an especially powerful package for the definition and analysis of large networks of social entities. Many other packages exist but few are able to handle more than a few hundred nodes or relationships between them. Like all of them however GRADAP offers no means of actually mapping the networks in a visually comprehensible form. The results are presented as indicators or tables.

**6. Implications of artificial intelligence research**

In considering the possibility of analyzing networks of problems (organizations, concepts, *etc*), it is important to benefit as much as possible from related work on artificial intelligence, and possibly pattern recognition. Artificial intelligence projects to simulate human personality or belief systems have had to develop mathematical techniques and computer programmes which can handle and interrelate entities such as concepts and propositions, some of which may be positively or negatively loaded to represent positive values and perceived problems (the credibility and importance of a belief in a network, and the intensity with whichit is held, may also be indicated). Clearly the objective of such projects is not achieved once a simple inventory of entities can be examined, even if it is highly structured in the form of a thesaurus. Of particular interest is the work on "dialogues" with such belief systems, some of which are established over a period by extensive interviews with individuals and others which are specially constructed to simulate paranoia, for example (see references). Presumably it would be possible to conduct somewhat similar dialogues with the collective beliefs constituted by problem/value networks such as might be developed during the course of this project.

**7. Comment**

Despite the available techniques noted above, and others which have been applied to non-social networks, much would seem to remain to be accomplished, as François Lorrain's (1) remarks indicated, in order to grasp networks in their totality.

The question is what it would be useful to know about networks at this time. What indicators would it be useful to attach to individual problems (organizations, *etc*) to indicate the characteristics of their relationship to the network(s) in which they are embedded? What similar indicators would be useful in describing the relationships between relatively dense networks and the larger network in which they themselves are embedded? What sort of concept about networks need to be embodied in a **network vocabulary** so that such matters can be discussed intelligently and unambiguously in public debate? In other words, what are the elements of an adequate vocabulary of structure and in what disciplines has the basis for such a vocabulary already been established: chemistry, crystallography, architecture, design in general, *etc*? What can be learnt from biologists about the growth and development of the many reticular structures they encounter (*eg* radiolaria)? More interesting perhaps, in which occupations do some individuals develop a special (instinctive or intuitive) sensitivity to the structural and dynamic characteristics of the networks with which, or within which, they work: airline pilots, urban bus drivers, electricity grid controllers, counter- espionage directors, factory process controllers, computer-based data network designer/controllers, telephone exchange designer/controllers, institutional fund controllers, *etc*? What do such people say, or want to say, about their networks? Why has the term "networking" suddenly sprung into common use and consequently what could "to network" mean? It is questionable whether any adequate organizational response (a **network strategy**) to the world problem complex can be elaborated until such rich experience is collected together and matched to an elaborated, mathematically-based concept structure, and an associated vocabulary. A conceptual quantum jump is required to grasp problem (and other organized) structures in their totality and be able to communicate such insights.

It is hoped that the availability of the data in this publication will help to stimulate such fresh thinking on the conceptual containment of societal networks. The challenge is to relate the results of such analysis to the needs of users of a database on a network of entities. What for example might be its **significance from a policy perspective** -- especially where redundant links or absent links have budgetary implications?