American sociologist Harrison White created network theory using mathematical models. The focus of his study is the real interconnectedness of actors who are at a level above the simple group but below the level of the whole society. In the book An Anatomy of Kinship (1963), White uses algebraic modeling of social relations, where the social structure consists of "structures of cumulated roles" of individuals. He formulates eight axioms that connect the structure of the Australian Aboriginal clan with their rules of marriage. Later, he extended this model to apply to the institutions of the modern West and thus came to the ideal-type model of all possible societies that correspond to these axioms. Finally, he compares his model with available anthropological data.

In the book Chains of Opportunity (1970), he creates an "algebra" of semi-groups, to make his earlier scheme applicable to organizations and informal groups in a modern, Western context. He defines role structures as the positions of social actors through multiple networks of social relations (such as friendship, hostility, and providing assistance). White uses the concept of duality to denote the invariant in models of social structures and processes that relate to cases where individuals change their positions within the workplace or change jobs. Free positions, not individual persons, are what, indeed, have the freedom to move between categories toward fixed transitional probability. Therefore, conventional mobility models should be applied to the analysis of vacancies, not directly to the mobility of individuals.

White applied this scheme of analysis to the study of the mobility of priests between Episcopal, Methodist, and Presbyterian churches. Career vacancies are not only dual but have a causal priority over the careers of individuals. White introduces the term "structural equivalence", which refers to groups of individuals who are in similar positions in relation to other such groups of individuals, so it is possible to aggregate their relationships and mutual changes through multiple networks of relationships. This equivalence allows complex networks, which are obtained by aggregation of equivalent actors, to be presented in the form of "reduced-form images".

In the article "Social Structures from Multiple Networks" (1976), White further simplifies mathematical modeling and applies it to many cases of analysis of social networks. The goal of this modeling, which is called "blockmodeling", is to create an image of a specific social structure through the analysis of the relationship between social positions and social roles within multiple networks. In Identity and Control (1992), White focuses on action, actors, cultural meanings, and institutional practices. The search for and struggle for identity and control are taken as occasions for overall social action. He observes the connections in the networks in their narrative aspect, which leads to the social network being viewed as a network of meaning. In the book Markets from Networks (2002), White states that classical economic models were too abstract. Contrary to such models, he strives to present an economic model of market functioning that would be more based on empirical data, but which would also theoretically connect any particular market with broader, global, economic relations. White is also known for creating his theory of artistic production, and for his theory of language use which emphasizes switching between cultural, social, and idiomatic areas within networks of discourse.

American sociologist Mark Granovetter is best known for introducing the concept of "strength of weak ties" into network theory. In two papers, „The Strength of Weak Ties“ (1973) and „The Strength of Weak Ties, A Network Theory Revisited“ (1983), Granovetter studied the impact that "weak ties" have on individuals in society. Weak relationships are those in which there is little emotional closeness and little intensity of contact between individuals. The standard sociological view was that the connections that individuals have with those closest to them, that is, "strong ties", have a decisive influence on success in life, because such persons will first and foremost provide support and help.

However, Granovetter claims that strong connections have one flaw, which he explained through network theory. If all people with strong connections make one social network, then they will have the same resources and share the same information. Contacts with people who are not extremely close to us are what will allow us to use some other resources and information. This will make it easier for highly educated professionals to find out about a new job through distant relatives or acquaintances. Individuals who come from poor families, lower classes, or isolated communities can progress more easily through weak relationships because those relationships give them access to resources and information that they can never get within strong relationships. In addition to individuals, weak ties play a significant role at the societal level. Weak ties help strengthen communities, spread ideas, and information, and often serve as the basis for broader collective action.

Granovetter is also known for introducing the concept of "embeddedness" into the network and economic theory. He emphasizes that in society and the economy, in addition to the formal hierarchy and structure of relationships, personal interpersonal relationships, and networks are important, as well as the position that individuals have in them. Granovetter views these informal relationships as ingrained. Empirical and theoretical study of the structure of informal relations is important for understanding the behavior of actors and wider economic and social consequences.

Authors: Granovetter Mark, White C Harrison. Berkowitz S. D., Burt Roland, Faust Katherine, Mizruchi Mark, Wasserman Stanley, Wellman Barry.

Literature: 

Granovetter, Mark.  „The Strength of Weak Ties”, in American Journal of Sociology (1973);

     -     „Threshold Models of Collective Behavior”, in American Journal of Sociology (1978);

     -     „The Strength of Weak Ties: A Network Theory Revisited”, in Sociological Theory (1983);

     -     „Economic Action and Social Structure: The Problem of Embeddedness”, in American Journal of Sociology (1985);

     -     „Economic Institutions as Social Constructions: a Framework for Analysis“, in Acta sociologica (1992);

Harrison, White. An Anatomy of Kinship: Mathematical Models for Structures of Cumulated Roles (1963);

    -     Chains of Opportunity: System Models of Mobility in Organizations (1970);

     -     “Social Structure from Multiple Networks”, in American Journal of Sociology (1976);