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This paper addresses the development of a semigroup model of social networks. Data matrices which represent the perceived relationships between members of a social network are used to construct a (possibly infinite) data semigroup of derived relations defined by (real) matrix multiplication. This complex structure is analyzed by forming interaction semigroups. These semigroups are homomorphic images of the data semigroup. The corresponding congruences are generated by identifying products of finite order which are highly positively correlated. Several methods of generating the interaction semigroups are examined and are shown to generate nonhomomorphic semigroups. For each congruence, an associated triple of numbers can be defined which may serve as an indicator of the validity and/or a measure of the stability of the semigroup model. A series of hypothetical examples is developed to study how the algebraic properties of interaction semigroups reflect and uncover properties of associated networks. Specifically, relationships between homogeneity of a network and the algebraic structure of the corresponding interaction semigroup are addressed. The applicability of the above techniques to blockmodels is demonstrated. [ABSTRACT FROM AUTHOR]

In policy networks actors use access relations to influence preferences of other actors. Establishment and shifts of access relations and their consequences for outcomes of decisions are the main focal points In this paper. Unlike most policy network studies, we therefore do not take the network and its relations as given and constant. Instead we device computer simulation models to account for the dynamics in policy networks. We compare different models and investigate the resulting network structures and predicted outcomes of decisions. The choice among the alternative models is made by their correspondence with empirical network structures and actual outcomes of decisions. In our models, we assume that all relevant actors aim at policy outcomes as close as possible to their own preferences. Policy outcomes are determined by the preferences of the final decision makers at the moment of the vote. In general, only a small fraction of the actors takes part in the final vote. Most actors have therefore to rely on access relations for directly or indirectly shaping the preferences of the final decision makers. For this purpose actors make access requests to other actors. An access relation is assumed to be established if such a request is accepted by the other actor. Access relations require time and resources. Actors are therefore assumed to be restricted in the number of access requests they can make and the number of requests they can accept. Moreover, due to incomplete information and simultaneous actions by other actors, actors have to make simplifying assumptions in the selection of their "best" requests and learn by experience. We device two base models that correspond to two basic views on the nature of political processes. In the first view politics is seen as power driven. Corresponding to this view, actors aim at access relations with the most powerful actors in the field. They estimate their likelihood of success by comparing their own resources with those of the target actors. Power also determines the order in which actors accept requests. In the second view, policy matters and actors roughly estimate the effects access relations might have on the outcome of decisions. Actors select requests to "bolster" their own preference as much as possible. We will show that these base models and some intermediate ones result in fundamentally different network structures and predicted outcomes. Moreover, we will show that the policy driven models do fundamentally better than the power driven models. [ABSTRACT FROM AUTHOR]

In the paper four semirings for solving social networks problems are constructed. The closures of the matrix of a given signed graph over balance and cluster semirings can be used to decide whether the graph is balanced or clusterable. The closure of relational matrix over geodetic semirings contains for every pair of vertices u and v the length and the number of u - v geodesics; and for geosetic semiring the length and the set of vertices on u - v geodesics. The algorithms for computing the geodetic and the geosetic closure matrix are also given. [ABSTRACT FROM AUTHOR]

People often make choices or form opinions depending on the social relations they have, but they also choose their relations depending on their preferred behavior and their opinions. Most of the existing models of coevolution of networks and individual behavior assume that actors are homogeneous. In this article, we relax this assumption in a context in which actors try to coordinate their behavior with their partners. We investigate with a game-theoretic model whether social cohesion and coordination change when interests of actors are not perfectly aligned as compared to the homogeneous case. Using analytical and simulation methods we characterize the set of stable networks and examine the consequences of heterogeneity for social optimality and segregation in emerging networks. [ABSTRACT FROM AUTHOR]

In general this comment tackles the problems and difficulties potentially combined with an application of formal economic models and constructs, such as the Walras equilibrium and microeconmic demand theory, to pure sociological contexts. In particular, this is done by analyzing a further attempt, as recently suggested by Braun (1993 and 1994), to extend the well-known Coleman Model by incorporating the embeddedness of social transactions in incomplete social network structures. "Pars pro toto" it is proved that Braun's conceptualization contains some weaknesses which imply that fundamental conclusions drawn in his article have to be revised. [ABSTRACT FROM AUTHOR]

The idea of partitioning a network in terms of a specific conceptualization of equivalence has taken a powerful hold on the imagination of network analysts. Frequently, an empirically established blockmodel is assessed in terms of its consistency with a particular visualization of a network. We demonstrate that, while a visual representation of a network can be helpful, this also constrains powerfully our image of the structure of that network. This implies that a particular picture of a network is not sufficient for establishing the adequacy of a blockmodel. We argue that once committed to a specific form of equivalence, a network analyst must be committed also to an explicit method of assessing the extent to which a blockmodel is consistent with the selected form of equivalence. We provide a method for doing this. Additionally, and perhaps more importantly, efforts to measure the fit of a blockmodel in terms of a sin- Sic form of equivalence reveal a serious weakness in the idea of using only a single form of equivalence to partition a network. It follows that this idea must be reconsidered. An appropriate generalization of the equivalence idea is one where each block, of a particular image in a blockmodel, is free to conform to a different form of equivalence. We provide a general criterion function, together with a local optimization procedure, for establishing such a generalized blockmodel. This criterion function also provides an appropriate measure of fit. Finally, we propose partitioning a network into a generalized blockmodel where each block, again in an image, can also have a particular pattern within which each equivalence type is a special case. Again, we provide a method for establishing such a model and assessing its fit. [ABSTRACT FROM AUTHOR]

The article presents a discussion on the relationship between mathematics and social networks analysis. The latter is recognized as one of the fastest growing specialties in social science. Social networks analysis is based on traditions of different branches of subjects including anthropology, communications science, human geography, and information science. Mathematics has proved to be successful in research work in all the fields particularly physics. The subject focuses on the abstract structure or form of an argument. There are number of mathematicians who are well acquainted with the social networks area. There are several social scientists who are working in the networks area who know mathematics well.