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Polyhedral Empowerment of Networks through SymmetryPsycho-social implications for organization and global governance- / - Associated with Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors (2008), Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization (2008), Configuring Global Governance Groups: experimental visualization of possible integrative relationships (2008) and Configuring Global Governance Groups: experimental animations and video sequences (2008). Context Challenge of network connectivity and networking efficiencies Polyhedral approaches to social network analysis Polyhedral networks: designing for robustness and survivability Epistemic networks, simplicial complexes and polyhedra Polyhedral dynamics and Q-analysis Polyhedral theory Polyhedral computing: optimizing responses to complexity Polyhedral methods of conjoint analysis Polyhedral design of computer memory utilization processes Polyhedral databases: operational significance Polyhedral patterns: representation of complex numerical abstractions Polyhedral networks: strategic significance Polyhedral relationship networks? Polyhedra-based sense of identity? Missing link: self-reflexive closure? Symmetry: competition vs complementarity? Polyhedral empowerment: "eliciting the sparkle from networks"? Recognition of curvature as fundamental to a polyhedral psycho-social universe? References ContextIn the further exploration of the possibilities highlighted in the above-mentioned papers, reference is made here to the current applications of "polyhedral approaches" to networks, their operation, and to their significance for information organization in situations calling for higher orders of efficiency and robustness -- and to the insights offered for new approaches to psycho-social organization. The main emphasis below has been to indicate fields of study and application which may not necessarily be well-connected, however relevant they may be with respect to any potential psycho-social implications. Given that the concept of "network" has proven over past decades to be as significant as a metaphor for social organization as it has as an analytical framework for the development of such organization, it is possible that the potential of "polyhedra" should be similarly understood. The basic argument is that in psycho-social usage "network" is relatively unstructured and has not achieved much of what was hoped initially in contrasting it with hierarchical modes of organization. Whilst seemingly quite unrelated, the faces and edges linking vertices of any 3-dimensional polyhedron can be mapped in 2-dimensions as a network (a polyhedral net) -- by "unwrapping" the polyhedron. Studies of social networks show that desirable properties such as robustness and information transfer efficiencies can be achieved with networks that take the form of polyhedra having properties such as symmetry. Symmetrical polyhedra in 3-dimensions are those of greatest aesthetic appeal and are intuitively comprehensible as a whole, despite their possible complexity. There is therefore a case for considering how, in the light of the various polyhedral approaches (considered below), networks might be "polyhedrally empowered" by using polyhedra as structural (or dynamic) templates. Of particular related interest is the degree to which complex multicriteria decision-making is now dependent on such approaches -- suggesting that the comprehensibility and communicability of a solution to any strategic dilemma might be associated with a polyhedral form reflecting its "goodness of fit" as a pattern in a design sense. This would then have important implications for governance. Challenge of network connectivity and networking efficiencies"Tensing networks": In response to early optimism regarding the merits of social "networking", in contrast to the problematic aspects of hierarchical social organization, attention was focused on the inefficiencies of untensed networks and associtated "networking diseases" (Tensing Associative Networks to contain the Fragmentation and Erosion of Collective Memory, 1980; Implementing Principles by Balancing Configurations of Functions: a tensegrity organization approach, 1979; Tensed Networks: balancing and focusing network dynamics in response to networking diseases, 1978). These concerns resulted in a continuing preoccupation with tensegrity organization, namely ensuring a degree of tensional integrity within networks (From Networking to Tensegrity Organization, 1984; Documents relating to Networking, Tensegrity, Virtual Organization). Recent developments with respect to tensegrity as an extension of the focus on polyhedra, notably relevant software, are discussed elsewhere (Psycho-social operationalization of polyhedra through tensegrity representation, 2008). Network graphs as polyhedra: Just as three-dimensional polyhedra may be represented in two dimensions as a network -- a polyhedral net -- so there have been explorations of the value of representing networks by polyhedra (Branko Grünbauma, Graphs of Polyhedra; Polyhedra as Graphs. Discrete Mathematics, 2007). Of particular interest has been the generic challenge of network connectivity (M. Grötschel, C.L. Monma and M. Stoer, A Polyhedral Approach to Network Connectivity Problems, 1992). Optimal networks -- classification of polyhedra: As part of the quest for more useful networks, efforts have been made to classify polyhedra.
Such work focises the question of whether the range of polyhedra, especially with some degree of symmetry, constitute a rich repertoire for psycho-social organization appropriate to various conditions, as previously argued (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008). Optimal networks -- information traffic: There has naturally been a major interest in optimizing the flow of information through networks, notably in telecommunications. Raul J. Mondragon (Optimal Networks, Congestion and Braess’ Paradox, 2007), for example, explores how to deliver information efficiently in a communications network and how to build networks to perform this function -- notably by re-wiring them. Following Dekker and Colbert's work, they seek a compromise between robust networks and optimal networks:
which leads to a focus on regular and symmetric graphs in which the nodes are all similarly linked. They cite the work of L. Donetti et al. (Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that, 2006; Entangled Networks, Synchronization, and Optimal Network Topology, 2005) in discussing the network congestion under load in various configurations. Such considerations are especially relevant to consideration of information (if not "knowledge" or "wisdom") flows in psycho-social networks. Given the increasing degree of enablement offered by the internet and the web, to what extent could the challenges of information overload and information underuse be circumvented by a polyhedral approach to optimization of such networks? In principle this is highly relevant to the challenges of a learning society and the threats to collective memory (Societal Learning and the Erosion of Collective Memory a critique of the Club of Rome Report: No Limits to Learning, 1980). Optimal networks -- commodity distribution: Gábor Rétvári, József J. Bíró and Tibor Cinkler (Fairness in Capacitated Networks: a Polyhedral Approach) address the problem of allocating scarce resources in a network so that every user gets a fair share, for some reasonable definition of fairness. For example, a fair allocation would be such that every user gets the same share, and the allocation is maximal in the sense that there does not exist any larger, even and feasible allocation. We shall focus on the fair allocation problem that arises most often in networking: compute a fair rate at which users can send data in a telecommunications network, whose links are of limited capacity. The authors show that We show that the set of throughput configurations realizable in a capacitated network makes up a polyhedron, which gives rise to a max-min fair allocation completely analogous to the conventional one. An algorithm to compute this polyhedron is also presented, whose viability is demonstrated by comprehensive evaluation studies. Gabriella Muratore (Polyhedral approaches to survivable network design, 1999) studies the problem of designing a cost-efficient multicommodity flow network with survivability features and the geometrical structure of several polyhedra arising in this context. For some of these polyhedra it proved possible to give a complete description by extreme points and by facets, while for others the complete description was given by extreme points. Several classes of facet-defining inequalities were identified. It might be inferred that a more sophisticated approach to "fairness" is what is required in response to the variety of forms of "unfairness" that drive social unrest and cycles of violence -- whether collectively or within the interpersonal networks. Polyhedral approaches to social network analysisThe interest in the network structure of organizations, especially those of a potentially criminal or terrorist nature, has increased manifold as a result of the information revolution, as noted by B. Balasundaram, et al (Clique Relaxations in Social Network Analysis: The Maximum k-plex Problem, 2006) . Their paper introduces and studies the maximum k-plex problem. This arises in analysis of cohesion in social networks and is often used to explain and develop sociological theories. Members of a cohesive subgroup tend to share information, have homogeneity of thought, identity, beliefs, behavior, even food habits and illnesses. It is also believed to influence emergence of consensus among group members. The approach is relevant to the study of degrees of connectivity amongst sets of websites. It may be used in organizational management to study organizational structure to suggest better work practices and improve communication and work flow. Their study helpfully summarizes the distinction in the literature between three important structural properties expected of a cohesive subgroup that are idealized by models of clique models idealize:
Different models may relax relax different aspects of a cohesive subgroup:
This model relaxes familiarity within a cohesive subgroup and implicitly provides reachability and robustness. As the authors note:
It is curious that there are few memorable instances where such sophisticated tools have enabled better networks to form. The current explosion of "social networking" over the web, for example, does not appear to have benefitted from such insights. The relevant Wikipedia entry exemplifies the point with the statement: "Not to be confused with social network analysis, a type of social scientific model". It is however clear that the insights have been considered highly relevant in tracking criminal networks and terrorist suspects. Whereas they enable detection of problematic nodes against which security measures can be taken, they have not has yet enabled community-building in practice. Polyhedral networks: designing for robustness and survivabilityCuriously it would appear that the array of network analysis skills has been most significantly applied "defensively" in response to possible vulnerabilities of vital networks -- whether to protect against them or to exploit them. For example, the study by Jonathan T. Hamill (Analysis of Layered Social Networks, Air Force Institute of Technology, 2006) is concerned with prevention of near-term terrorist attacks:
As noted by Anthony H. Dekker and Bernard D. Colbert (Network Robustness and Graph Topology, 2004; Network Robustness for Critical Infrastructure Networks, 2008):
After reviewing a range of polyhedra, the authors conclude:
The problem of robustness is vital to telecommunications networks as explored by Bernard Fortz (Design of Survivable Networks with Bounded Rings, 2000) using polyhedral analysis. The results obtained demonstrate how to use polyhedral theory for practical network design problems. From a military perspective, a recurring theme in the literature is the "design of surivable communication networks" (M. Grötschel, C.L. Monma, M. Stoer, Polyhedral and Computational Investigations for Designing Communication Networks with High Survivability Requirements, 1992). This perspective is notably relevant to telecommunications networks (Arie Koster, Polyhedral Combinatorics to Solve Network Design Problems, Paper for 9th INFORMS Telecommunications Conference, 2008). In the increasing concerns with sustainability, and the longer-term viability of catalytic social projects, the issue of what makes for robustness would appear to be vital -- faced with the tendency of projects to collapse once seed funding ceases. Such robustness is clearly also of importance faced with the prospect of partial or complete social collapse, if only in the event of disasters. Recent disasters have indicated the vulnerability of food and utility supply networks, for example. Epistemic networks, simplicial complexes and polyhedraIn a society increasingly recognized to be significantly knowledge-based, social networks, concept networks and the manner of their apprehension are necessarily intimately intertwined -- beyond the challenge of the description of networks in the abstract and the flows of information through them. This interweaving has been remarkably explored by Camille Roth (Co-evolution in Epistemic Networks: reconstructing social complex systems, 2005). The author frames his discussion as folows:
In contrast to the graph theory tools typically deployed for social network analysis, Roth bases her analyses on the use of Galois lattices that may be termed “concept lattices” in other contexts (Wille, 1992; Stumme, 2002). With regard to the above objective, Roth notes:
With respect to polyhedra, Roth notes that the principles underlying use of Galois lattices (GLs) strongly relate to Q-analysis, notably that of Ron Atkin on simplicial complexes:
With respect to the structure of any knowledge community, Roth introduces a formal framework based on Galois lattices that categorizes epistemic communities automatically and hierarchically, rebuilding a whole community taxonomy in the form of a hypergraph of significant sub-communities. She argues that modeling social complex systems tends to require the introduction of co-evolutive frameworks.
Roth notes that beyond the profusion of community-finding methods, often leaning towards AI-oriented clustering, an interesting issue concerns the representation of communities in an ordered fashion. She cites as examples of different techniques for producing and representing categorical structures: hierarchical clustering, Q-analysis, formal concept analysis, information theory, blockmodeling graph theory-based techniques, neural networks, association mining, and dynamic exploration of taxonomies. In later work relevant to the co-evolution of social and knowledge networks, Camille Roth (Patterns and Processes in Socio-semantic Networks, 2007) notes:
Also of note is R. Cowan, et al. (The Joint Dynamics of Networks and Knowledge, 2002), as well as the literature on the diffusion of innovation through networks. It is however intriguing that Roth's central focus on "reconstructing social complex systems" might be understood purely as one of construcitng a more adequate model of such a system rather than responsing to the challenge of designing and facilitating the operation of more appropriate systems. This has long been a weakness of social network "analysis" which (as noted above) has not yet proven to be significant in designing better networks, whatever that might mean. Polyhedral dynamics and Q-analysisPolyhedral dynamics is a tool for representing network structure and behaviour. In introducing this field and its relevance to IIASA (International Institute for Applied Systems Analysis), J. Casti (Polyhedral Dynamics: the relevance of algebraic topology to human affairs, 1975) notes:
Casti notes that the approach was introduced by Ron Atkin (Mathematical Structure in Human Affairs, 1974) using ideas of algebraic topology.
Elsewhere (Topological Methods for Social and Behavioural Systems, 1982) Casti notes:
Atkin introduced polyhedral dynamics in terms of Q-analysis whereby patterns of q-connectivity are analyzed rather than connectivity (Combinatorial Connectivities in Social Systems; an application of simplicial complex structures to the study of large organizations, 1977). The approach has been applied with varied success as discussed by Jacky Legrand (How far can Q-analysis go into Social Systems Understanding?, 2002) and S. B. Seidman (Relational Models for Social Systems, 1987). Examples include: P. Doreian (Polyhedral Dynamics and Conflict Mobilization in Social Networks, 1981) and L. C. Freeman (Q-analysis and the Structure of Friendship Networks, 1980). In a later work (Ron Atkin, Multidimensional Man: can man live in three dimensions? 1981) he develops a theory of polyhedral events and makes a convincing argument that structural events are related to clock time in a nonlinear way related to their dimensions. He gives a convincing explanation why higher dimensional events take a lot longer to occur in clock time than simple events. Jeffrey Johnson (Hypernetworks for Reconstructing the Dynamics of Multilevel Systems, 2006) develops this argument that polyhedral dynamics form trajectories in a non-linear way in clock time. The formation of a simplex is a polyhedral event. Polyhedral events mark the passage of system time. Events occur at different levels on multilevel systems, and they have to be coordinated. Johnson's argument is that:
The relevance of Q-analysis to psycho-social issues is discussed in a commentary on the Global Strategies Project (Comprehension: social organization determined by incommunicability of insights). It is also discussed in Beyond Edge-bound Comprehension and Modal Impotence: combining q-holes through a pattern language (1981). Polyhedral theoryPolyhedra, linear inequalities and linear programming can be seen as three views of the same concept. Polyhedra represent a geometrical point of view, linear inequalities represent the algebraic point of view, and linear programming the optimization point of view. As noted by Geir Dahl (An Introduction to Convexity, Polyhedral Theory and Combinatorial Optimization. 1997)
It is the identification of the closure associated with appropriate "convex polyhedra" that is closely associated with identification of an "optimal subset". Most problems studied in combinatorial optimization involve looking for certain structures in graphs. The simplest symmetrical polyhedra (ocathedron, docdecahedron, etc) may therefore be understood as representing or "containing" the solutions to challenges of optimization in many fields. As Dahl further notes:
Polyhedral computing: optimizing responses to complexityThe set of feasible solutions of a linear optimization problem is a convex polyhedron. Specially structured variants of these problems define polytopes with special structures. In this way the theory and algorithms of linear optimization are inherently linked to polyhedral theory and properties of convex bodies. The generic issues posed by the above concerns were brought to a focus at the Centre de recherches mathématiques (University of Montreal) in the form of a semester on Polyhedral Computation Combinatorial Optimization (2006) presented as follows:
Polyhedral computation addresses the computational complexity of solving problems associated with convex polyhedra and search for efficient algorithms. One of the most fundamental problems is the vertex enumeration problem that is to list all vertices of a convex polytope given as the solution set to a system of m linear inequalities in d-variables -- as succinctly presented by Jakub Marecek (Polyhedral Approach to Multicriterial Optimization, 2006). It is in this sense that polyhedral approaches are fundamental to decision-making under uncertainty, if only in the field of economics (Andreas Eicchorn, et al. Polyhedral Risk Measures and Langrangian Relaxation in Electricity Portfoloio Optimization, 2005). More generally it is worth considering the possibility that an appropriate polyhedral configuration of criteria, opportunities and constraints constitutes a comprehensible, credible "solution" to any strategic dilemma. The argument here, notably in the light of the "epistemic" framing provided by Roth (above), is that it is precisely the polyhedral coherence (through symmetry effects) that renders a solution both understandable and memorable as a gestalt that "works". Thus what is otherwise understood through the intuitionist school of mathematics is echoed in how a viable solution is apprehended beyond that discipline. This possibility is clearly of significance to governance and to that to which the governed are asked to subscribe -- and to the manner in which a complex solution can be communicated (notably in the light of Atkin's arguments regarding the communicability of insights). How should governance envisage optimizing the multicriteria decision-making challenges of the future -- given the computing resources on which it can call -- in a manner to render the "solutions" comprehensible and credible to all concerned? In purely material terms, such arguments are of course consistent with the aesthetic appeal of sacred geometry as a response to architectural and design challenges. The question is how such polyhedral configuration might prove significant in psycho-social organization and knowledge structure design, notably when it takes virtual form on the web (cf Sacralization of Hyperlink Geometry, 1997; Spherical Configuration of Categories -- to reflect systemic patterns of environmental checks and balances, 1994; Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998). Polyhedral methods of conjoint analysisConjoint analysis, also called multi-attribute compositional models or stated preference analysis, is a statistical technique that originated in mathematical psychology. Today it is used in many of the social sciences and applied sciences including marketing, product management, and operations research. Olivier Toubia (New Approaches to Idea Generation and Consumer Input in the Product Development Process, 2001) describes a method, dependent on computer support, for an adaptive question design method that attempts to reduce respondent burden while simultaneously improving accuracy. For each respondent the question design method dynamically adapts the design of the next question using that respondent’s answers to previous questions. The adaptive method interprets question de-sign as a mathematical program and estimates the solution to the program using recent develop-ments based on the interior points of polyhedra. Toubia begins with a conceptual description that highlights the geometry of the conjoint-analysis parameter space, permitting analyses of decision challenges involving many "dimensions" -- understood as distinct features of a product (3 to 100, say) which respondents are called upon to evaluate. The polyhedral method is designed to "shrink" the feasible set of features -- reducing its dimensionality -- determining the key features of the product design. The respondent’s answers to the first q questions define a (p-q)-dimensional hy-perplane which intersects the initial p-dimensional polyhedron to give a (p-q)-dimensional polyhedron, namely one of lower dimensionality. The challenge is to select questions that reduce the dimensionality of the polyhedron as fast as possible. Conjoint analysis is one of a set of techniques of multivariate analysis in which, for example, potential customers are asked to compare pairs of products and make judgements about their similarity (or their dissimilarity in the case of ordination statistics). By contrast, whereas techniques such as factor analysis, discriminant analysis, and conjoint analysis obtain underlying dimensions from responses to product attributes identified by the researcher, multidimensional scaling obtains the underlying |