Abstract. We investigate a simple generative model for network formation. The model is designed to describe the growth of networks of kinship, trading, corporate alliances, or autocatalytic chemical reactions, where feedback is an essential element of network growth. The underlying graphs in these situations grow via a competition between cycle formation and node addition. After choosing a given node, a search is made for another node at a suitable distance. If such a node is found, a link is added connecting this to the original node, and increasing the number of cycles in the graph; if such a node cannot be found, a new node is added, which is linked to the original node. We simulate this algorithm and find that we cannot reject the hypothesis that the empirical degree distribution is a q-exponential function, which has been used to model long-range processes in nonequilibrium statistical mechanics.

2004 Douglas R. White and Frank Harary, Collective Geodesics and Co-evolution: A Graph Theoretic Structural Model. Submitted to Advances in Complex Systems (ACS) Undergoing revision (develops the traversal algorithm used above).

Abstract. Under certain conditions, when diverse individuals (e.g., ants, individuals, agents) independently traverse a sequential decision space in reaching objectives (e.g., as modeled by a maze) they acquire synergetic properties of global problem solving (even in the absence of global knowledge about the problem space) by virtue of some form of pooling experience. The laying of pheromones on random paths taken by ants, for example, has been shown to map the set of shortest paths to a food source. This paper shows the conditions under which certain very general classes of mazes have the property of "collective advantage" to finding shortest paths by aggregating ceretain types random individual behavior (individuals have local but no global knowledge of the maze and no perception or reckoning of network distances). Of three factors considered as conditions for collective advantage, two were identified by Johnson (2000, 2001) from simulations: First is the method of traversal of the maze, and second is the method of marking trails towards a collective solution. The third - the structure of the maze - is explored here through graph theoretic concepts. These include precise definitions, theorems and observations, and simulations. They provide a language and a set of results as to the structural factors that affect collective advantage. In general, biconnected maze networks - where every node has independent paths to every other -, with many parallel paths, and many crossover paths between them, assist collective advantage. Rules developed to measure the collective advantage of a maze help to refocus the problem on the coevolution of the learning environments that endow agents with collective intelligence that is distributed across their behaviors and not condensed by selection for individual actors with better forms of global strategies or global knowledge. Results support March's (1991) findings of advantages to exploratory behaviors over selection for exploitation.

(Review Article) 2003 Douglas R. White, Ties, Weak and Strong. Encyclopedia of Community Vol. 4:1376-1379. Edited by Karen Christensen and David Levinson. Thousand Oaks, CA: Sage Reference.

Abstract. This entry reviews the relationships among the biased networks models of Rapoport, the small-world problem posed by Milgram and later addressed by Watts, and studies of community cohesion in relation to the strength of weak ties hypothesis of Granovetter

1999 "Controlled Simulation of Marriage Systems." Journal of Artificial Societies and Social Simulation. http://jasss.soc.surrey.ac.uk/2/3/5.html

1997 Lilyan A. Brudner and Douglas R. White. Class, Property and Structural Endogamy: Visualizing Networked Histories Theory and Society 25:161-208.

Abstract. This is the first use of the simulation method described above. It is also the first theoretical application of the concept of structural endogamy as identifying an empirical variable or boundary condition within social networks that is linked in causal-explanatory ways to social class formation. Using an ethnographically rich case study of an Austrian village in which oral and (ca. 100) household genealogies provide 150 years of marriage network data, while manorial archives continue the stem-line household genealogies back to the founding of the "house system" in 1517, the hypothesis is formulated that the social class boundary between farmstead owner-operators (including heirs and buyers) and secondary service occupations not linked to farmstead ownership is established and maintained through the mechanism of structural endogamy. Two principles of inheritance are in conflict in this farmstead house-system, that of passing the principal productive property intact to a principal heir (usually a son, or if not is available, a daughter), and that of the intestate rights of children to equal division of parental inheritance. The use of wills or testaments resolves his conflict through "equitable division" which maintains stem-line impartibility of farmsteads along with quitclaims to those who are not principal heirs. Structural endogamy, in this case specifically the marriage of a potential heir to a spouse who brings in divided property from another divided patrimonial stemline, is shown to be (1) a qualification for class membership via principal heirship, (2) a means of reconstituting subdivided estates, and (3) a means of social perpetuation of the two-class system which often even divides siblings within the same nuclear family. The predicted statistical relationship between class-membership, heirship and structural endogamy is confirmed empirically and implications for new approaches to studies of social class formation are discussed.

1971 Douglas R. White, George P. Murdock, Richard Scaglion, Natchez Class and Rank Reconsidered Ethnology 10:369- 388.

This 1971 reconstruction of how individuals were actually linked historically in the social networks of the Natchez people provides a classic example of how processual and network modeling can reveal and clarify inferences about social structure. The famous "Natchez Paradox" was discussed in virtually every introductory Anthropology text up to the publication of this article. See current accounts of the historical Natchez and the bibliography on the Natchez. Descendants of the Natchez today are recognized among the Southeastern American Indian groups and among mixed descendants of English colonists. Among these descendants, in turn, are today's recognized Natchez political leaders. Perhaps only the Encyclopedia Britannica is still remiss in describing Natchez social structure as a four-class system, as in Swanton's reconstruction of 1911. Swanton's reconstruction, however, implied a self-immolating social structure characterized by the Natchez Paradox as explicated in the abstract below. Only traces of this Paradox, compounded from several sources of ethnographic misunderstandings, are alive in urban legend and on the WWW today, such as Bennet's memory of discussions by Adam Przeworski (2004). The abstract that follows, simply because Ethnology publishes articles without abstracts, was written only in 2005.

Abstract Textual analysis, simulation, comparative distributional evidence, and prosopographic network methods are used here to solve the Natchez Paradox first posed by C. M. W. Hart in 1943, expressed in mathematical form by Samuel Goldberg in 1958 and summarized, in terms of analytical dilemmas, inconsistencies, and possible 'solutions,' by Jeffrey Brain in 1971. The Natchez Paradox emerged from an ethnographer's reconstruction of four Natchez social classes, three of which -- Sun rulers, Nobles, and Honoreds, as opposed to Commoners -- had been assumed by the historical ethnographer, John Swanton, to be ranked exogamous matri-descent groups. While all nobility married commoners, the children of males would be expected to belong to their mother's group. Swanton concluded from his reading of the contemporaneous historical texts of the 18th century French colonists that the children of men in the Sun, Noble, and Honored classes did not revert to commoner status but only to one level lower in the social hierarchy. The paradox shown by Hart and demonstrated even more strongly by Goldberg in his mathematical model is that given equal reproductive rates of marriages of different types over successive generations, combined with Swanton's hypothetical social rules, the Sun lineage would constitute a stable proportion of the population, the Noble lineages would increase their proportion in each generation, and the Honored lineages would increase proportionally to the proportion in the Noble lineages, thus obliterating the commoner class in relatively few generations.

What we find in our prosopographic counting of individuals mentioned by name in the historically contemporaneous French texts is that the only persons with Honored status who were mentioned in these texts were men, and consequently, without Honored women, there were no Honored lineages and no Honored class. The textual sources are clear that Honored status was a social rank for men, so that Honored matrilines (and their female members) were clearly an invention of Swanton, possibly because he did not base his analysis on mentions of individuals in the French texts, which are numerous, but only on presumed categories. The other probable mistake in inference derives from the fact that while French words in the singular indicate gender, the plural term 'les Honores' applies equally to men in the plural and to both genders in the plural. Swanton overgeneralized, in our view, in drawing the inference that there existed a social class of Honoreds that contained both men and women. Women with that status simply did not exist. Honored, we show, was a term only for male rank, not a designation for social class or for a set of distinct matrilineages.

The Natchez Paradox also arose from Swanton's erroneous rejection of a contemporaneous account given in one of the documents written by French colonists that delineated a consistent system of devolution of noble rank that depended on distance from the Royal line. By this firsthand account from someone conversant with the nobility, the children of Sun men (the royal lineage) devolved to Noble status for both men and women, but Noble status in the female line of descendants of these women devolved, after three generations, to Commoner status for women but for men to Honored rank. It was only the sons of men of Honored rank, in this account, who became commoners. Commoners, however, could also achieve Honored status by fame through their exploits in war. Part of the reason Swanton disputed this French account of Natchez nobility was because of the asymmetry of rank assigned to children of Noble men: even if such a man was a matrilineal greatgrandchild of a ruler, his sons were Honored while his daughters were Commoners. Such asymmetries, which Swanton mistakenly thought of as matters of asymmetric descent rather than of rank, seemed unlikely to Swanton. As described in one of Swanton's own publications, however, our distributional analysis of cases in the neighboring region identifies the neighboring Caddo as having asymmetric gender status of precisely the Natchez type. The Caddo and Natchez had long engaged in trade, so this asymmetric assignment of status need not be disregarded as a valid ethnographic feature of the Natchez status system. The Natchez paradox, then, was apparently the result of various compounded errors, including Swanton's assumptions about symmetries in rules of descent as concerns sons and daughters. Swanton did not differentiate clearly the different elements of rank, class, and lineage as they operated in Natchez society.

The simulation study uses difference equations, generation to generation, comparing demographic assumptions of earlier models aiming to "resolve" the Natchez paradox by differential reproductive rates, and the descent rules reconstructed from other evidence, in which case no further reproductive adjustmets are needed to "balance" the demographi composition of different social classes.

We regard these multiple sources of evidence as providing a definitive alternative description of Natchez social structure than that proposed by Swanton in this reconstruction of 1911, roughly 180 years after the dispersal of the Natchez as a distinct and integral society. This was a complex society with an hereditary aristocracy and complex rules for the devolution of status and rank. Swanton recognized only certain aspects of this complexity. When it came to Natchez principles of devolution of rank in the Royal and noble matrilines, many of which are common to royal lines, Swanton was unwilling to recognize the similarity to those found in other monarchical polities. The Natchez Paradox re-emerges, then, as an example of the use of network and mathematical models both as a check on ethnographic interpretations in the reading of historical texts and as pointing the way to better solutions in the rereading of textual data. Textual data can easily be misinterpreted. In the present case Swanton evidently overrelied on categorical inferences that fit his prior assumptions. Errors of this sort can easily overtake even the best of ethnographers, a category into which Swanton, in his extensive published works and his many ethnohistorical and ethnographic contributions, has long occupied a place. It is a credit to the Natchez historical corpus, on which Swanton relied, that some of his errors of inference can be corrected based on a reexamination of the textual evidence.

The appendix provides a network and genealogical analysis of the 20 prosopographic mentions of Sun royalty and how ranking within the royal lineage related to the center-periphery occupation of political posts in the Kingdom. It is also shown how the center-periphery structure of the ruling lineage exacerbated internal political divisions in the war with the French, and the exodus of one branch of the Natchez population to merge with Southeastern groups and towns such as those of the Creek, Chickasaw, and Cherokee.

Network Autocorrelation: A Simulation Study of a Foundational Problem in the Social Sciences. Malcolm M. Dow, Michael L. Burton, and Douglas R. White. Social Networks 4(2):169-200. (June 1982)

Abstract: It is axiomatic to the social sciences, and an essential part of the network perspective, that human performances are intricately linked with their social and enviromental contexts. Researchers in each of the disciplines have rediscovered this in the past decade with respect to a whole host of specific problem areas, under such labels as "context effects", "index utility". and "systems analysis". The earliest mention of the problem with respect to quantitative research occured, to our knowledge, in the debate between the nineteenth century cultural diffusionists and the evolutionists. The latter regarded individual societies as independent instances of uniform causation, and hoped to learn about causation from correlational studies. The former regarded their observations as embedded in an interactive network of historical relationships such as diffusion, migration, conquest, and competition, where the historical, evolutionary and ecological context of each society and the network of interconnectedness between societies plays a major role in multiple causation. In this view, events cannot be regarded as isolated or independent as if each were a context-free "independent invention" of a single society.

The same arguments, of course, apply to the interpretation of data collected in social or opinion surveys. Political science offers a recent example of the discovery of "context effects" in voting behavior (e.g. Jackson 1975). How much of voting behavior is affected by attributes of the voting unit (whether individuals or aggregates), and how much is the result of interactions between them: of the communication process, bandwagon effects, reference group behavior, or other forms of "symbolic interactionism"?

Our purpose in this paper, however, is not to attempt a review of the vast literature on context effects. Rather, we focus on the costs and benefits of either neglecting context or else incorporating it in the research design. Statistical methods such as multiple regression analysis necessarily contain mathematical axioms which either assert or deny the existence of context effects. We will explore here through simulation studies the following related questions:
(1) What are the consequences of ignoring context effects, should they be present, or ordinary least squares regression estimates, and
(2) what are some of the properties of a recently developed maximum likelihood procedure which permits context effects to be included in a regression model as network autocorrelated disturbance terms?