</a>Challenge of network connectivity and networking efficiencies

Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance (Part #2)

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"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.

Schläfli symbol: Michael J. Bucknum and Eduardo A. Castro (Geometrical-Topological Correlation in Structures. Nature Precedings, March 2008) describe the topological indexes of polygonality, n, and connectivity, p, which can be identified for various structures, including the 2-dimensional (2D) tessellations and the 3- dimensional (3D) crystalline patterns. The ordered pair (n, p), called the Schläfli symbol, that is descriptive of the topology of each and every distinct structure, is identified and used to illustrate a Schläfli-space, entries of which have the coordinates n and p, that can be employed to map the innumerable structures, and to identify relations between and among these structures graphically, so that absolute identities and locations can be ascribed to them.

Wells fundamental polyhedra metric: Bucknum and Castro use the work of A.F. Wells (Three Dimensional Nets and Polyhedra, 1977) -- his polyhedral metric, structural correspondence principle and morphological principle to derive the polyhedral and 2D and 3D metrics. They note that Wells, as "a master of applied topology" derived more than 100 novel networks.

Wythoff construction: In geometry, a Wythoff construction is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. It is based on the idea of tiling a sphere, with spherical triangles. If three mirrors were to be arranged so that their planes intersected at a single point, then the mirrors would enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections would produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. A Wythoff symbol is a short-hand notation for naming the regular and semiregular polyhedra using a kaleidoscopic construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane (see examples).

Greg Egan (Wythoff, 2002) provides an applet that displays uniform polyhedra, using Wythoff's kaleidoscopic construction to compute the locations of the vertices. By clicking on the applet, 74 of the 80 possible uniform polyhedra (including single examples from each of the five infinite classes of prisms and antiprisms) are displayed.

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:

• node connectivity = minimum number of nodes needed to remove to obtain a disconnected network
• link connectivity = minimum number of links needed to remove to obtain a disconnected network.

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.

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