Stochastic and deterministic approaches for modeling complex networks are presented. The methodology combines analysis of the structure formed by the interconnections among the elements of a network with an assessment of the vulnerability towards the propagation of cascading failures. The goal is to understand the mutual interplay between the structure of the network connections and the propagation of cascading failures.

Two fundamental issues related to the optimal design and operation of complex networks are addressed. The first concerns the impact that cascading failures have on networks due to the connectivity pattern linking their components. If the state of load on the network components is high, the risk of cascade spreadings becomes significant. In this case, the needed reduction of the connectivity efficiency to prevent the propagation of failures affecting the entire system is quantified. The second issue concerns the realization of the most efficient connectivity in a network that minimizes the propagations of cascading failures. It is found that a system that routinely approaches the critical load for the onset of cascading failures during its operation should have a larger efficiency value. This allows for a smoother transition to the cascade region and for a reasonable reaction time to counteract the onset of significant cascading failures.

The interplay between the structure of the network connections and the propagation of cascading failures is assessed also in interdependent networks. In these systems, the linking among several network infrastructures is necessary for their optimal and economical operation. Yet, the interdependencies introduce weaknesses due to the fact that failures may cascade from one system to other interdependent systems, possibly affecting their overall functioning. Inspired by the global efficiency, a measure of the communication capabilities among interdependent systems, i.e. the interdependency efficiency, is defined. The relations between the structural parameters, i.e. the system links and the interdependency links, and the interdependency efficiency, are also quantified, as well as the relations between the structural parameters and the vulnerability towards the propagation of cascading failures. Resorting to this knowledge, the optimal interdependency connectivity is identified.

Similar to the spreading of failures, the formation of a giant component is a critical phenomenon emerging as a result of the connectivity pattern in a network. This structural transition is exploited to identify the formation of macrometastases in the developed model for metastatic colonization in tumor growth. The methods of network theory proves particularly suitable to reproduce the local interactions among tumor cells that lead to the emergent global behavior of the metastasis as a community. This model for intercellular sensing reproduces the stepwise behavior characteristic of metastatic colonization. Moreover, it prompts the consideration of a curative intervention that hinders intercellular communication, even in the presence of a significant tumor cell population.