Graph theoretical analysis of the Dynamic Lines of Collaboration model for disruption response
The Dynamic Lines of Collaboration (DLOC) model was developed to address the Network-to-Network (N2N) service challenge found in e-Work networks with pervasive connectivity. A variant of the N2N service challenge found in emerging Cyber-Physical Infrastructures (CPI) networks is the collaborative disruption response (CDR) operation under cascading failures. The DLOC model has been validated as an appropriate modeling tool to aid the design of disruption responders in CPIs by eliciting the dynamic relation among the service team when handling service requests from clients in the CPI network. The DLOC model for CDR operation is conceptually an abstraction of the CPI network into two interdependent networks of client and service networks. No preliminary design guidelines have been devised for DLOC-CDR from a network perspective using graph properties. Previous results of graph theoretical analysis for network behaviors under disruption may also not apply to DLOC-CDR due to the intrinsic nature of the N2N service challenge. Previous research works in DLOC-CDR have also not taken into consideration in protecting vulnerable CPI network elements which can cause system collapse by a single failure. Based on these observations, this research is guided by the following questions: (1) What graph properties to be viable predictors for evaluating the reliability of N2N service designs in DLOC and (2) Where should the disruption responders (resource) to ensure timely disruption mitigation with regards to protection of vulnerable nodes? To answer question (1), it is found that resiliency of a CPI network, as measured in DLOC by the Recoverability metric (Precover), can be approximated by the proportion of vulnerable nodes (P vulnerable) as a function of average degree and cascade threshold (ϕ). Precover measures the probability of a network to fully recover from a cascading disruption. It is found that there lies a certain regime where Precover =1 as approximated by Pvulnerable. By means of graph property analysis, we initially proposed two heuristics, degG > 1/ϕ and ΣkPv(k)Pk < 0.70, to mark the regime where Precover is strictly 1. From numerical experiments, it was found that degG > 1/ϕ is over conservative, while the latter applies to all tested networks. This experiment result also supports the conclusion that the existence of a “small-world” phenomenon in networks can either inhibit or accelerate cascade, depending on the complexity of the propagation. To answer question (2), two heuristics protocols based on network centrality measures were initially proposed, namely the Bridge-Based Allocation (BBA) and Degree-Based Allocation (DBA). We initially hypothesized that the BBA would perform better in terms of preventing failures but with considerable trade-off in total response time compared to CBA, the existing resource allocation protocol of DLOC-CDR, in networks with high modularity. However, it was found that based on numerical experiments we concede that the BBA is not suitable to be applied in DLOC. The main advantage of the BBA is its ability to identify bridging elements which its removal will make the network disconnected. The current DLOC, on the other hand, does not take into consideration of the connectivity state of the CPI network. Thus, rendering the BBA to become less effective than CBA and DBA. We also found that both CBA and DBA can be used interchangeably. Given a simple propagation, DBA constantly performs better on networks displaying high affinity towards power-law degree distribution compared to CBA. This is due to the high correlation between both centralities in these networks. For small-world networks, the performance increment from CBA to DBA has a decreasing trend with increasing network size. CBA’s performance is relatively constant and outperforms DBA in large network size.
Nof, Purdue University.
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