Decentralized Systems and The Invisible State: Low Density Multiconnected Cohesion in Large-Scale Social Networks in Tlaxcala, Mexico

General Hypotheses:
(a) Biconnectivity is a source of emergent, potentially decentralized social cohesion that can occur (with observable effects) at low density in (the bicomponents of) relatively stable social networks.
(b) This is especially true for relations that have very high "currency" or life-support salience, such as relations of political influence, property transmission, or kinship and marriage connections.
(c) Hence, social class, elites, wealth-transmission, and marriage systems are especially well-suited for analysis.

• A decentralized network with biconnectivity can have more cohesive effects than a centralized network or a higher scale of cohesiveness than more locally cohesive clusters.
• The mechanism is the potential for self-amplifying or positive feedback circuits.
• Additionally, the higher the connectivity ("redundancy"), the less vulnerability to disconnection. Menger's theorem: connectivity max = cut min (minimum node removal for disconnection).
• Thus, energy, information or resources can be more effectively circulated or redistributed in higher k-components.
• Bicomponents are easy to identify in large graphs in linear time o(max(V,E)), V=vertices, E=edges.
• Bicomponents are not "closed" but radiate ties outward in tree-like connective patterns, possibly 1-connected to other bicomponents. Nodes on their perimeter may even be more central in the overall graph (node 9, in the example, connects two bicomponents).
• In a large network with sufficient (but often low) density, a giant bicomponent may be the source of strongly cohesive participation in core institutional arenas of the group (although not everyone biconnected by high-currency relations will have high cohesion). Others may be 1-connected to the giant component. This gives rise to a three-part structure:
• the giant biconnected core
• its periphery, 1-connected to the giant core
• the margins, in separate 1-components
• Connections within a bicomponent need not be homogeneous: local interactions or k-components of higher order within bicomponents may give rise to further emergent global or subgroup properties.
• k-components of higher order are also easily computable: tricomponents in linear time o(V+E) [Hopcroft and Tarjan 1973], and all the k-components in low polynomial time o(V**.5 E**2) [Even and Tarjan 1975] or near-linear time in parallel computation.

• "The Invisible State: Radial and Proximal Cohesion in Tlaxcala" (DRW with Michael Schnegg and Lilyan Brudner). 1998. International Social Networks Conference, Barcelona, Spain.
• "Kinship and Compadrazgo in Rural Tlaxcala" (DRW with Michael Schnegg, Lilyan Brudner and Hugo G. Nutini) in press, eds. Samuel Schmidt and Jorge Gil, Social Networks: Theory and Applications. U.N.A.M. Press, Mexico, D.F. Tlaxcala generic link
• "Analyzing Large Kinship and Marriage Networks with Pgraph and Pajek" (DRW, Vladimir Batagelj and Andrej Mrvar), in press, Social Science Computer Review.

tlaxcala map

The map of Tlaxcala shown below was part of the indigenous census of 1555. This was a period of during more than a century (1525-1640) when the early Franciscan Friars protected the autonomy of the indigenous kingdom. Its four principalities are marked on the map along with the proportion of nobility in each district. In Belen, in the lower left corner of the district of Tizatlan, 22% of the population at that time were nobility (pipiltin). 
The nobility quickly learned Spanish but continued to record historical events in their traditional manner, as in the Lienzo de Tlaxcala from which this map was taken.

Members of the Tlaxcalan nobility continued to live in the indigenous towns until the Tlaxcalan centralized government was abolished by the Spanish settlers after 1660, following the expulsion of the Friars. The Tlaxcalan heartlands had acquired permanent land rights, however, thanks to the Francescans and to the service of the Tlaxcalans to the Crown and to the Conquistadores in their conquest of the Aztec, the traditional enemy of Tlaxcala.

For over a century, to the 1660s, the Tlaxcalans were largely left alone (except by the Friars) by other Spanish colonialists and settlers, whose efforts to buy Indian lands were largely thwarted by the surviving Tlaxcalan state organization. This resistence to Spanish acquisition continued in the "Free Indian Lands" long after their government was abolished in the 1660s and their nobility expulsed to Saltillo, Santa Fe, and Guatemala. The Tlaxcalan population that remained constructed a decentralized social organization during their "century of isolation" from 1660 to 1750, when even the new and more secular Spanish priesthood spent little effort on the Tlaxcalan villages.
 
 
 
 
 
 
 

Network dynamics in Belen, 1500-2000: Overview of Sociocultural Adaptation and Organizational Change.
 
 
 
 

Network statics: We did our network survey in 1978. Here is a drawing of the kinship and marriage network as we found it then.  Three features of the graph are striking: (1) The 26 shortest elementary cycles are long-girth in their number of edges; none are short; (2) Families do not cluster in the center but spread in cycles of very large diameter or nodes connected to these cycles; (3) Families of all different economic ranks intermarry. The next inset is a view of this structure from the top, and shows the relative emptyness of the center given the lack of short-girth cycles.

Click for "birds-eye-view" and larger Image

The relinking of families by marriage -- as shown above right -- is almost entirely among families WITHIN the community

Note the contrast with the compadrazgo network (inset below), which is drawn with the same spring embedding algorithm as the kinship network above but has a starkly higher concentration of short cycles in the center of the graph.

The compadrazgo network also has many more links and, unlike kinship, extends 4-to-1 OUTSIDE the community.

Click for larger Birds-eye-view of the Compadrazgo Network (39K)
  Colors marking different structural positions in the compadrazgo network:
 

	People who live in Belen and are connected to the largest relinked block. ________8% (KINSHIP: 51%)
	People who live in Belen and are NOT connected to the largest relinked block. ____2% (KINSHIP: 41%)
	People from outside Belen who are integrated in the largest relinked block. ______10% (KINSHIP: 0%)
	People from outside Belen who are NOT relinked. _________________________81% (KINSHIP: 8% CURRENT)

Analysis of Cohesion: Evidence of Cohesion from Bicomponents

	Belén Ancestors‘ (generations 2-4) Core/Periphery Positions as defined by kinship and marriage			Outsiders
	Giant Relinked Bicomponent	Giant 1-component	Small Components	Not Rrelinked
26-30	3	0	0	0
7-24	15	14	0	6
1-6	9	48	8	109

Table 3A:Kinship Relinking of Ancestors Predicting Number of Descendants (r=.43, p<.005)

Table 3B:Kinship Relinking Predicting Civil/Religious councils membership (r=.44, p<.005)

Table 3C:Compadrazgo Relinking Predicting Civil/Religious town council (r=.60, p<.001)

Beyond Bicomponents: Analysis of Subvarieties of Cohesion

Average Length of Independent	Average	Level of Connectivity (generating bounded subgroups)
Paths within subgroups	Path Length overall	1-Connected (component)	2-Connected (bicomponent)	3-Connected (tricomponent)	Small World Simulation	…	k-Connected (k-component)
Very short (<< random)	very long	Proximal Tree	Proximal (2-) Cohesion	Proximal (3-) Cohesion	Local World		Proximal (k-) Cohesion
B. Short (< random)	Long	Proximal Tree	Proximal (2-) Cohesion	Proximal (3-) Cohesion	Small World		Proximal (k-) Cohesion
C. Short ( = random)	Medium	Random  Tree	Random bicomponent	Random tricomponent	Random world
D. Long (> Random )	Short	Radial Tree	Radial (2-) Cohesion	Radial (3-) Cohesion			Radial (k-) Cohesion
E. Medium (> Random )	Medium (Bounded)	Radial Tree	Radial (2-) Cohesion	Radial (3-) Cohesion			Radial (k-) Cohesion

Row B. Proximal Cohesion (example): Members of the Town Councils  Row C. Random Graph: Giant Components
(a Monte Carlo simultion within generations is also done as a baseline model for the marriage network)
 
 
 
 
 
 

Row D. Radial Cohesion (example): Bicomponent members not on the Town Councils

• a) As a graph: Emptiness of First and Second Zones

• b) In Matrix View

Deviation from the Expected Values: These numbers tend to 1.0 ("random graph") if town council participants are excluded.
 

	1. Zone	2. Zone
1. Zone	2.8	4.6
2. Zone	4.6	1.8

REVIEW - General Hypotheses

• A decentralized network with biconnectivity can have more cohesive effects than a centralized network or a higher scale of cohesiveness than more locally cohesive clusters.
• The mechanism is the potential for self-amplifying or positive feedback circuits. Additionally, the higher the connectivity ("redundancy"), the less vulnerability to disconnection.
• Energy, information or resources can be more effectively circulated or redistributed in higher k-components.
• Bicomponents are easy to identify in large graphs. They are not "closed" but radiate ties outward in tree-like connective patterns, possibly 1-connected to other bicomponents. Nodes on their perimeter may even be more central in the overall graph.
• In a large network with sufficient (but often low) density, a giant bicomponent may be the source of strongly cohesive participation in core institutional arenas of the group (although not everyone biconnected by high-currency relations will have high cohesion). Others may be 1-connected to the giant component. This gives rise to a three-part structure:
• the giant biconnected core
• its periphery, 1-connected to the giant core
• the margins, in separate 1-components
• Connections within a bicomponent need not be homogeneous: local interactions or k-components of higher order within bicomponents may give rise to further emergent global or subgroup properties.

The relations that have very high "currency" or life-support salience, studied here, are those of political and religious influence (esp. religious town council), property transmission (indigenous ownership of land, bilateral inheritance), kinship and marriage connections ("structurally" endogamous, within the community, with ca. 8% migration in and out per generation of others from Tlaxcala), and ritual kinship or compadrazgo (multiply connected in a larger "invisible community" reaching outside Belen to adjacent communities).

From the Tlaxcalan evidence reviewed below: The existence of an egalitarian social class and the absence of elite differentiation is well accounted for in this case. Wealth transmission also works on an egalitarian basis in terms of bilateral inheritance divided equally among males and females. "Structurally endogamous" relinking marriages reinforce intra-village social class solidarity. The compadrazgo relations are also of high currency in social support, and, given the greater number of ties they provide, extend the egalitarian social organization out to inter-village relations, presumably adding coordination in ritual and economic life. The core of leading participant in the ceremonial and town council positions within the village tend to be a subset of more intense local interaction WITHIN the bicomponents of kinship and compadrazgo, and are reflected in the network by proximal as opposed to radial cohesion. 

Decentralized Aspect 1: CONCENTRIC ORGANIZATION Of COHESION AND SPATIALLY RADIAL 1-CONNECTIVITY
• non-cohesive radiality (80% outside 1-connectivity) of the compadres network extends far and wide both within Tlaxcala and to urban centers of migration.
• compadres network at least radially cohesive ("invisible community" across villages)
• kinship network is radially cohesive (within community); but only 8% of parental ties are radially 1-connected: these are intra-Tlaxcalan MIGRANTS from other villages.
• within village, a proximally cohesive core, members of religious/civil town councils
• the proximally cohesive core acts as a kind of "pump," pulling in participants to religious activities, then radially extending ties to connect with those in other villages, with independent paths returning back that allow the circulation of favors via "confianza" relations of mutual trust and dyadic exchange.
• although there is some hierarchy in the town council offices, an overall "invisible state" in rural Tlaxcala operates largely through the horizontal ties of multiple connectivities in giant k-components rather than through a centralized hierarchy.
• it is out of the giant bicomponents that community participants in activities and councils are recruited; The development of "proximally redundant ties" within the compadrazgo bicomponent follows rather than precedes recruitment to office.
• The local institutional structure, local economic exchanges based on trust, and opportunity structure for residential and labor migration -- a great deal of the "economic and sociopolitical structure" -- can be seen as emergent out of the dynamics of the social networks examined here.

 
 

Decentralized Aspect 2: CROSS-CUTTING ORGANIZATION AND AUTONOMOUS AGENCY AS A FACTOR IN DISTRIBUTED INTEGRATION

• the kinship and compadrazgo networks are perfectly othogonal in cross-cutting one another, maximizing their combined bicomponent integration. No special "agency" is required for this other than locally independent choices.
• everyone in the community with the exception of very recent migrants are at least radially biconnected by the combined kinship/compadrazgo network.
• quasi-random assortment of marriages by economic rank or by occupational groupings
• integrative tendency to avoid choosing compadrazgo partners of same economic rank

 

 
 
 
 
 

Decentralized Aspect 3: COOPERATIVITY WITH RESPECT TO COMMON GOOD/COMMON RISK A KEY FACTOR IN DECENTRALIZATION

• Large-scale solidarity and consciousness of social integration into a single egalitarian social class has the key effect of preventing alienation or sale of land to outsiders
• Lindenberg's solidarity criteria fulfulled in terms of cooperative behavior with respect to the common good, sharing, responses to need, avoidance of damage to others, and explanations or repairs for failure to comply with solidarity norms
• effective social sanctions within the community exerted through solidarity "moral community"
• migration permitted between villages, migrants assimilated only after they are locally integrated by biconnectivity
• solidarity norms effective across communities for biconnected and town council members

• 1. why are decentralized systems difficult to understand? (in this case, they are)
  
  
• Controller bias? Social networks research does not suffer the bias of inferring that observed structures are centrally regulated, but its problem has been not seeing the forest for the trees. Interaction effects are strongest in dyadic relations, but predictable effects decay with path distance. Hence it has been assumed that beyond path length 2 or 3, network effects are weak or nonexistent (in physical processes, diffusive effects equipartition in 2-3 mean free paths). From this it has been inferred that for members to have interaction effects on one another, "cohesive groups" must be internally connected by short path lengths. It is further inferred that longer paths play no role in how a system operates. What has been missing in the standard social networks approach is a focus on ensemble effects, "assemblies" with rich internal feedback circuits even if the length of the feedback cycles is long (a perspective on "hierarchical systems" in physical processes).
  
  
• The economy is a good example that contradicts the idea that longer paths play no role in how a system operates. The economy depends on propagation through long paths. When oil producers raise prices, the effect reverberates through successive transactions that pass along price effects.
  
  
• Bicomponents or multiple path connectivities are also crucial to how the economy operates. The idea of competitive markets and equilibrium depends fundamentally on multiple possible buyers and sellers for each actor and multiple paths between pairs of actors. Hence the giant biconnected core and its periphery, 1-connected to the giant core, are fundamentally different structural positions in terms of the market, the former "cohesive" and competitive, the latter subject to unique relationships that are by definition noncompetitive.
  
  
• Another conceptual bias that has preventing sociology from decentralized but "large ensemble" network effects is this: Belief in the priority of "Proximal Effects" such as clique-like (proximally) cohesive groups has obscured the possibility of observing or even conceiving dispersed low density phenomena such as radial cohesion or multiple connectivities. The latter approach was present in American sociology in the 1930s but was effectively banished by the mainsteam sociological doctrines of "Middle Range Theory" (Merton 1949) and "Proximal Causes" (Homans 1950s), plus the increasing focus of networks research on small groups dynamics. Large-scale network studies of social classes or elites (e.g., those of the 1930s Temporary National Economic Committee or TNEC and its derivative studies in the National Bureau of Economic Research or Harvard Economic Studies) were eclipsed by the "pluralistic interest groups" model of weakly linked small-scale cohesive interest groups.
  
  
• The hypothesis adopted here, in contrast to most networks research, is that bicomponent assemblies based on multiple independent paths are the substratum of networked processes out of which emerge social institutions, markets, politics, social organization, logics of identity. Understanding this substratum provides the key to network phase transitions as a theory for punctuated social change within a general perspective of networked path dependencies in historical processes (cf. Brian Arthur, Douglass North, etc.).