Advances in Scientific Knowledge : Group Dynamics, Social Networks and Cultural Capital
(c) Douglas R. White 1998
site under construction: see also new abstracts, Longitudinal Studies
1. The empirical network of kinship and marriage ties constitutes a new object of study for social class, ethnicity, social organization and stratification, with network implications for material, social and cultural capital.
2a. The hypothesis of significant kinship links betwen family units in contemporary society as open low-density interconnected ego networks rather than as small close knit groups receives independent confirmation in a study by Eric Widmer, This entails the existence of significant low-density kinship links as large-scale connected networks. We can thus study the existence of giant components in kinship networks and the possibility of giant multiply connected components within them.
2b. One of our structural variables, membership in a multiply connected component of a marriage and kinship network, is discussed below. Briefly, this variable predicts as emergent phenomena, in different ethnographic and historical studies:
3. The prior-ties distance d between marriage partners (which may vary to infinity), which may create a multiply connected (2-) component subgraph whose diameter is d+1, is a dynamic variable under study. We (DRW and Art DeVany) hypothesize a Bose-Einstein evolution of distances approaching in the limit a Pareto-Levy stable distribution with infinite variance. Distances in such distributions are usually short, very occasionally long, but their variance is infinite! Such structures (see simulation) are associated with self-organizing systems.
4. Structural dynamics:- Membership in a multiply connected component is a dynamic structural outcome of the marriage-distance variable as a temporal process; it accounts for the merger of two components (e.g. social classes), and for the fissioning of components due to lack of relinking (terms defined below) after a number of generations.
5. Measurement:- To test our hypothesis of a power law for marriage distance, we need to analyze the 150 or so empirical datasets in our possession to measure distribution characteristics.
2. For large populations this graph is of low density since people usually have two parents, but number of children and of spouses will affect the average number of edges of each node.
3. The theory of Graphical evolution (Ed Palmer 1985) shows that as the number of edges added to large graphs surpasses the number N of nodes (at density 1/N, which is extremely low), the size of the largest component begins to increase exponentially. If the giant component evolves by the addition of random edges, it soon consumes almost all of the nodes in the network, still at low density.
4. Extrapolating from graphical evolution theory, as the giant component evolves by the addition of random edges, small worlds begin to form: subsets of nodes with low diameter, diameter being the maximal shortest-path distance between any pair of nodes.
5. New edges added to a graph do one of two things. Either they connect components into larger components (or the giant component), or they reconnect nodes within a component, shrinking its diameter. The sets of reconnected nodes of a graph are its cycles. Cycles create multiple independent paths or redundant connections between nodes.
6. Edges within a cycle are redundant in that no one edge is any longer necessary for the nodes in the cycle to be connected. Similarly, the nodes within a cycle are redundant in that the removal of any one edge no longer disconnects the other nodes in the cycle. The minimum number of independent paths between any two pairs of nodes is a measure of the redundancy or connectivity of a graph or subgraph.
7. The minimum number of nodes needed to disconnect a component is its cut-number. It is a measure of the cohesion of a graph or subgraph.
8. One of the fundamental theorems of graph theory is that the measure of the redundancy of a graph or subgraph is identical to the measure of its cohesion. If we define the cut-set of a graph as the number of its nodes whose removal increases the number of its components (i.e., results in disconnection), then: The smallest cut-set of a graph is the minimum number of independent paths between any pair of its nodes. A k-component of the graph is a maximal graph with a cut-set of size k and with k or more independent paths between any pair of its nodes. A k-component has cohesion k and redundancy (or connectivity) k.
9. The analysis of cohesion and redundancy in large graphs can proceed by the detection of nested k-components: the time needed to find the 1- and 2-components of a graph is simply a function of the number of edges, and no more difficult to compute for graphs of 100 million than for graphs of 100, except for a linear increase in time. For computing all the higher orders of k-connectivity, however, the computational problem is NP-complete, solved only by ever-increasing combinatorials.
1. The good news, computationally, is that due to the limitations on the number of parents, the connectivity of the parental relation in kinship and marriage systems is never greater than 2. This fact means that ALL of the k-connected components in a large kinship and marriage graph can be easily computed, and it is a fact that is easy to demonstrate. In any kinship network with connectivity 2 via the parent edges, there will always be a younger generation with only 2 edges, those to their parents. These "weakest links" always insure that the network is at most 2-cohesive and 2-redundant (multiple connectivity 2).
2. The good news, scientifically, is that even at the low densities exemplified by kinship and marriage networks in large complex societies, "small worlds" are likely to form with low diameters, considerable average redundancies, and overlapping cycles.
3. In a multiply connected 2-component with N nodes and K edges, there are necessarily K-N+1 independent cycles that use one or more distinct edges in the graph. The independent cycles also overlap in that in pairs they use at least K-(N-1)/2 edges in common: such redundancy begins to increase exponentially when K>3(N-1)/2). Each time we choose a pair of independent cycles that have an edge in common we can eliminate the common edges to constitute a new non-independent cycle formed by the remaining edges. Hence the "redundancy of cycles" of low density graphs may be considerable.
4. Cycles and redundant cycles are of fundamental importance to transmission and feedback processes -- amplification and dampening -- in behavioral systems. At low densities, interactive processes in large networks tend to take on the properties of self-organization: they lead to dynamic instabilities with considerable pattern coherence, but subject to further change due to local oscillations in reactive behavior.
5. In such networks -- large, low density, with large components and small worlds having high redundancies -- we have the greatest likelihood of finding patterns of emergent behavior.
Marital relinking and structural endogamy are two concepts developed by anthropologists (see Brudner and White 1997 for references) to characterize the existence of cyclic patterns in kinship and marriage networks. Relinking refers to marriage within a family, or multiple marriages between two families or among a larger set of families, connected in a cycle. White (1997) defines a structurally endogamous group as a set of persons in which each pair is connected by multiple independent paths of parent/child links.
1. Since kinship and marriage networks have just those properties expected to lead to coherent and self-sustaining emergent-divergent patterns of behavior, we may expect as an empirical hypotheses to find a great variety of social forms whose differences are connected to different institutional configurations in human society.
2. Some hypotheses about the boundaries of structurally endogamous groups are that they may correspond to:
FORMS OF POLITICAL SUCCESSION:- STRUCTURAL SUCCESSION
1. Ordinary forms of succession to office are classified as to hereditary versus nonhereditary (Murdock), or ascribed vs. achieved. Hybrid succession is the case of a mix of the two principles. "Clan-based succession" is succession within a bounded subgroup that recognizes common descent from either a single ancestor or a set of common ancestors whose descendants have intermarried to form an endogamous clan or "deme" (Murdock 1949, 1967). terms. Again, clan-based succession is to be distinguished from the ordinary forms of hereditary succession that follow a definite rule, e.g., patrilineal, matrilineal, bilateral.
2. In many cases succession to office is ascribed by social class, but achieved within the class. Where office remains within a structurally endogamous segment of the population (White 1997), we may speak of "structural succession." In such cases offices may pass to close or distant relatives (including affines, affines of affines), not following a definite rule of hereditary succession, but constrained within a set of intermarrying families.
FORMS OF WEALTH CONSOLIDATION:- STRUCTURAL WEALTH-HOLDING GROUPS
1. What do we call a set of wealth-holidng families (e.g., who have a patrimonial type of wealth), where wealth-holders for the families succeed one another either by inheritance or by wealth transfers of marriage, and where these wealth transmissions are within a bounded structurally endogamous group? In such a case there is clearly a larger "structural" unit of wealth management, which is not itself necessarily a corporation or corporate groups, but which may be termed a structural wealth-holding group.
Structural endogamy, emergent from a network of marriage ties, is a very
different concept that ordinary endogamy, which is usually defined or measured
within a certain territory of category of people. It implies self-organizing
boundaries of a social group, precise delimitation of (2-connected) core members,
and others connected to the core who are ranked by distance. While the core
boundary is precise, the core accommodates marriages to non-core members
since relinking can be through parents and children, and is not necessarily
dependent on spouses. It also accommodates offspring of core members who leave
Maintenance of core membership in a structurally endogamous group may, as we have seen, be linked to political succession (control of political offices) and to the consolidation of wealth and estates in a network of families. It may also be linked in crucial ways to processes that consolidate or maintain cultural capital in a variety of ways.
CULTURAL CAPITAL:- STRUCTURAL IMPLICATIONS
1. ...under development
(c) Douglas R. White 1998 P-graph home page