Complex Networks Vulnerability: A Multiple-Objective Optimization Approach

This paper extends previous works developed for evaluating the security characteristics of a network system (e.g. for water supply, electric power or gas distribution) exposed to a terrorist "attack", by taking into account simultaneously several characteristics. Given a source node where the hazard is injected, two criteria are used to assess the vulnerability of the network: the time to reach all network destination nodes (TTRAD), a problem similar to the "all-terminal network" evaluation, often performed within network reliability analysis and a measure related to the damage it causes, like the average number of persons affected (ANPA) or other entities (ANEA), in the system. The use of simulation models allows the identification of the network most critical vulnerabilities, i.e. the nodes where an "attack" can cause the worst damage. To effectively handle attacks, several "immunization" schemata with different characteristics could be proposed in order to maximize TTRAD or to minimize ANAP. The single implementation of any of the immunization schemata could mitigate the effects of the attacks; however a more realistic formulation should consider the simultaneous optimization of two or more objectives, such as cost and vulnerability or vulnerability and reliability. Using a single-objective (SO) formulation, any designer or decision-maker (DM) must solve several problems by varying a group of constraints to obtain a set of alternatives from which to choose the final solution. On the contrary, a multiple objective (MO) approach allows determining directly the Pareto set of alternatives from which the DM can choose the preferred one. The MO problem is solved using multiple objective evolutionary algorithms (MOEA), a group of evolutionary algorithms tailored to cope with MO problems. This group of algorithms conjugates the basic concepts of dominance with the general characteristics of evolutionary algorithms. In this paper we propose a MO formulation, which is able to generate a set of alternatives, based on two or more conflicting objectives. The DM can perform a complete vulnerability study and, a posteriori, define a possible protective scheme. Using the MO approach, it is also possible to define a robust protective scheme. That is to select a set of nodes whose protection generates on average, the least damage. Numerical examples illustrate the approach