</a>Polyhedral networks: designing for robustness and survivability

Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance (Part #4)

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Curiously 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:

To aid in this understanding, operations research, sociological, and behavioral theory relevant to the study of social networks are applied, thereby providing theoretical foundations for new and useful methodologies to analyze non-cooperative organizations. Such organizations are defined as those trying to hide their structures or are unwilling to provide information regarding their operations; examples include criminal networks, secret societies, and, most importantly, clandestine terrorist organizations.

As noted by Anthony H. Dekker and Bernard D. Colbert (Network Robustness and Graph Topology, 2004; Network Robustness for Critical Infrastructure Networks, 2008):

Two important recent trends in military and civilian communications have been the increasing tendency to base operations around an internal network, and the increasing threats to communications infrastructure. This combination of factors makes it important to study the robustness of network topologies. We use graph-theoretic concepts of connectivity to do this, and argue that node connectivity is the most useful such measure. We examine the relationship between node connectivity and network symmetry, and describe two conditions which robust networks should satisfy. To assist with the process of designing robust networks, we have developed a powerful network design and analysis tool called CAVALIER, which we briefly describe.

After reviewing a range of polyhedra, the authors conclude:

We have discussed the graph-theoretic concepts of node connectivity and link connectivity as measures of network robustness, and argued that node connectivity is most appropriate for modelling the robustness of network topologies in the face of possible node destruction. This is important both for military networks and for civilian networks facing possible terrorist activity.... We therefore suggest that military networks, or civilian communications backbones, be node-similar and optimally connected, with degree as high as feasible, diameter as low as feasible, symmetric if possible, and containing no large subrings.

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.

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