May 25, 2009
Costa et al.  provide possibly the most comprehensive survey of measurements used in complex networks analysis [Arxiv]. Starting from well known measures such as degree distribution, centrality, modularity etc., they move on to spectral measures and fractal dimensionality. (I am not sure what some of these measures are!) They finally discuss statistical methods for measurement selection, including Principal Component Analysis. I believe they are using PCA like analysis to select “features” of networks based on which they can classify them. If you are doing network analysis, I suggest you keep this survey handy.
 Luciano da F. Costa, Francisco A. Rodrigues, Gonzalo Travieso, P. R. Villas Boas. Characterization of complex networks: A survey of measurements. Advances in Physics, Volume 56, pages 167 – 242, Issue 1 (2007)
May 25, 2009
Over the last few days I received several spam comments on this blog (written using Cyrillic alphabet, incidentally). That made me visit this blog for the first time in several months, to turn off comments on old posts. Inadvertently, that also made me consider rebooting this. (Spam can be useful!) At the least, I can use this as a space where I put down comments about interesting papers I read. I might also write about some of the things I am working on.
Let me begin by discussing a paper of mine that just got accepted at SMC 2009. The title of the paper is ‘Classes of Optimal Network Topologies under Multiple Efficiency and Robustness Constraints’. In this paper, we have looked at the problem of designing optimal network topologies under efficiency, robustness and cost trade-offs. Efficiency, robustness and cost are defined in terms of structural properties: diameter, average path length, closeness centrality (efficiency); degree centrality, node betweenness, connectivity (robustness); number of edges (cost). A slider $\alpha$ is used to indicate the emphasis on robustness (on a scale of [0, 1]), and thus decides the trade-off between efficiency and robustness. Another parameter $\beta$ (in [0, 1]) decides how cheap or expensive edges are. For example, if $\beta$ is 0, we might make use of the full budget allocated, whereas if $\beta$ is 1, we want to “squeeze out” the most cost-effective topology by removing superfluous edges. Finally, there is also a constraint on the maximum permissible degree (indegree and outdegree, in case of digraphs) on a node in the resulting graph. Maximum degree can be thought of as a constraint on the “bookkeeping” (for example, size of DHT finger tables) a node has to do, or as constraints on maximum inflow and outflow through a node.
Different efficiency and robustness metrics are useful in different application contexts. For example, in case of traffic flow networks, congestion is a major challenge. Hence, node betweenness is a useful robustness metric, along with diameter as the efficiency metric which relates to upper bounds on the communication delay. In a Network Centric Warfare (NCW) scenario, it might be important to have alternate communication paths in case targeted attacks occur on nodes or links. Connectivity is a useful robustness metric there. And so on. Thus, what we do next is an ‘exploration of the parameter space’ involving efficiency and robustness metrics, $\alpha$, $\beta$, cost and maximum degree. We conduct genetic algorithm based experiments to evolve the “fittest” or the “most optimal” topologies under given trade-offs.
The main results in the paper are the following:
- Two prominent classes of topologies that emerge as optimal: (1) star-like or scale-free networks with small diameters, low cost, high resilience to random failures and very low resilience to targeted attacks (2) Circular skip lists (CSL) which are highly resilient in the face of both random failures and targeted attacks, have small diameters under moderate costs.
- Further, we observe that star-like networks are optimal only when both the following design requirements are satisfied: (1) very low emphasis on robustness (or very high emphasis on efficiency) and (2) severe restrictions on cost. In all other cases, CSL are optimal topologies in terms of balancing efficiency, robustness and cost.
Thus, we observe a sharp transition from star-like networks to CSL as emphasis on robustness ($\alpha$) increases or cost restrictions become less severe.
- Circular skip lists are highly homogeneous wrt many structural properties such as degree, betweenness, closeness, pairwise connectivity, pairwise path lengths. Further, they have structural motifs such as Hamiltonicity, (near) optimal connectivity, low diameter, which are optimal under varied application requirements. We argue that CSLs are potential underpinnings for optimal network design in terms of balancing efficiency, robustness and cost.