The aim of this first chapter is to motivate why stochastic processes, probability theory and graph theory are useful to solve problems in network science.

In any system or node in a network, there is always a non-zero probability of failure or of error penetration. A lot of problems in quantifying the failure rate, bit error rate or the computation of redundancy to recover from hazards are successfully treated by probability theory. Often we deal in communications with a large variety of signals, calls, source-destination pairs, messages, the number of customers per region, and so on. Often, precise information at any time is not available or, if it is available, deterministic studies or simulations are simply not feasible due to the large number of different parameters involved. For such problems, a stochastic approach is often a powerful vehicle, as has been demonstrated in the field of statistical physics or thermodynamics. Failure or attacks at the network level have reestablished the interest in network robustness analyses in relation to network security. In spite of the intuitively easy concept, a globally accepted definition as well as a framework to compute the robustness of a network is still lacking. Graph and probability theory are essential to address questions like: “Which are the vulnerable nodes?”, “Is this network robust?”, “Where do we need to add, remove or rewire links at minimum cost in order to maximize the network robustness?”