relational pattern translated into relations between actors. This means that all abstract models of social structure are quantitatively linked to observed relations. Models abstracting role-sets among statuses from observed relations among actors are at all times subject to empirical test as descriptions of observed relational patterns. 2. Analysis of network and covariance structures merges in a natural and' useful manner. Actors jointly occupying a status provide inter­ changeable representations of the role-set defining the status and accord­ ingly provide interchangeable indicators of distance to the status. The actor-status epistemic linkage is thus logically isomorphic with the in­ dicator-concept epistemic correlation. As a result of this isomorphism, the well developed statistics of multiple indicator structural equation models (e.g. Bielby & Hauser 1 977) can be used to test hypotheses regarding the status /role-set duality. Used in equations 1 5 and 16 to test structural equivalence as a statistical hypothesis, this isomorphism is also 29In contrast to Lewin's ( 1936) discussion of a psychological topology as interrelated sets of perceptions, a social topology refers to relational patterns in a system. An argument can be made for synthesizing the two via symbolic interaction and ethnomethodology since perceptions are intimately connected to relational patterns (e.g. Burt 1 980b); however, it is difficult to match psychological with social topologies since the latter are stated with considerably more rigor than the former.

Models of Network Structure

Positions in Networks

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

A stochastic model for change in group structure

This paper develops and tests a continuous time discrete state Markov Chain as a model for change in group structure. Movement of triads among different states is focused upon. It is shown formally that a triad cannot move to a state where all the pair relationships are mutual without incurring intransitivity. An empirical test on the change in sociometric structures provides support for the model.

A Stochastic Model for Change in Group Structure

The conditions are found under which the removal of an intransitivity in one triad of a sociometric or cognitive structure may be accomplished without in the process creating new intransitivities elsewhere in the structure. The results are applied to modify and elaborate the theories holding intransitivity to be unstable so that sociometric or cognitive structures become more nearly transitive over time, and also to shed some light on the question of the extent to which observed intransitivities may properly be attributed to measurement error. Properties of tetrads play a central role in this theory, which thus challenges the widespread notion due to Simmel and others, that once group size exceeds three there are no significant new properties to consider. Empirical data document the relevance of this theory: “correction” of “errors” in the data or change in truly intransitive triads turns out not to significantly reduce the overall amount of observed intransitivity.

David D. McFarland

Papers

Higher order stability conditions in mathematical models of sociometric or cognitive structure