## Abstract

Random graph theory was initially proposed by Paul Erdős and Alfred Rényi in the 1950s. A book of Béla Bollobás presented the first systematic and extensive group of results of random graphs. Associating to each edge of a random graph a real random variable, we obtain a probabilistic network. The determination of an optimal path between two nodes in a probabilistic network has first been studied by H. Frank in 1969. Since then few theoretical results have been found, even though there is a recognizable applicability of this type of networks to real problems, namely, social and telecommunication networks.

The mathematical model proposed in this chapter maximizes the expected value of a utility function over a directed random network, where the costs related to the arcs are real random variables following Gaussian distributions.

We consider the linear, quadratic, and exponential utility functions, presenting a theoretical formulation based on multicriterion models, as well as the resulting algorithms that may be applied to real-life problems. In the last section an application to a real problem that was developed to solve a criminality problem in the city of Lisbon is discussed. Due to its complexity these types of applications are suggested to use as project-based learning and not as an easy applied exercise to solve within a limited time.

The mathematical model proposed in this chapter maximizes the expected value of a utility function over a directed random network, where the costs related to the arcs are real random variables following Gaussian distributions.

We consider the linear, quadratic, and exponential utility functions, presenting a theoretical formulation based on multicriterion models, as well as the resulting algorithms that may be applied to real-life problems. In the last section an application to a real problem that was developed to solve a criminality problem in the city of Lisbon is discussed. Due to its complexity these types of applications are suggested to use as project-based learning and not as an easy applied exercise to solve within a limited time.

Original language | English |
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Title of host publication | Calculus for Engineering Students: Fundamentals, Real Problems, and Computers |

Editors | Jesús Martín-Vaquero, Michael Carr, Araceli Queiruga-Dios, Daniela Richtáriková |

Publisher | Academic Press |

Chapter | 10 |

Pages | 197-219 |

ISBN (Print) | 978-0-12-817210-0 |

DOIs | |

Publication status | Published - 2020 |

### Publication series

Name | Mathematics in Science and Engineering |
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Publisher | Academic Press |

## Keywords

- networks
- shortest path
- arc parameter
- probability distribution
- optimality principles
- applications in real problems