A Formal Model for Path Set Attribute Calculation in Network Systems
Giovanni Fiaschi, Carlo Vitucci, Thomas Westerbäck, Daniel Sundmark, Thomas Nolte
TL;DR
The paper addresses the challenge of evaluating attributes for sets of paths in networks, proposing a formal functional model that generalizes single-path routing to path sets. It introduces a framework where path-set attributes are computed via $Psi(P) = OP_s ∘ OP_p ∘ T$, with path-set representations built on vertex-weighted hypergraphs and r-incidence matrices. The method yields explicit formulations for Delay, Cost, Capacity, UnavailProb, and FaultProb, and analyzes their (poly)modularity properties, showing that Delay and Cost are polymatroidal while Capacity, UnavailProb, and FaultProb require further treatment. This formalism supports automated, multi-path routing decisions and lays groundwork for future implementation, complexity analysis, and extended polymatroidal analyses for Availability and Serviceability.
Abstract
In graph theory and its practical networking applications, e.g., telecommunications and transportation, the problem of finding paths has particular importance. Selecting paths requires giving scores to the alternative solutions to drive a choice. While previous studies have provided comprehensive evaluation of single-path solutions, the same level of detail is lacking when considering sets of paths. This paper emphasizes that the path characterization strongly depends on the properties under consideration. While property-based characterization is also valid for single paths, it becomes crucial to analyse multiple path sets. From the above consideration, this paper proposes a mathematical approach, defining a functional model that lends itself well to characterizing the path set in its general formulation. The paper shows how the functional model contextualizes specific attributes.