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# PROPERTY GRAPHS – A STATISTICAL MODEL FOR FIRE AND EXPLOSION LOSSES BASED ON GRAPH THEORY

## Abstract

It is rare that the severity loss distribution for a specific line of business can be derived from first principles. One such example is the use of generalised Pareto distribution for losses above a large threshold (or more accurately: asymptotically), which is dictated by extreme value theory. Most popular distributions, such as the lognormal distribution or the Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD), are convenient heuristics with no underlying theory to back them. This paper presents a way to derive a severity distribution for property losses based on modelling a property as a weighted graph, that is, a collection of nodes and weighted arcs connecting these nodes. Each node v (to which a value can also be assigned) corresponds to a room or a unit of the property where a fire can occur, while an arc (v, v′; p) between vertices v and v′ signals that the probability of the fire propagating from v to v′ is p. The paper presents two simple models for fire propagation (the random graph approach and the random time approach) and a model for explosion risk that allow one to calculate the loss distribution for a given property from first principles. The MBBEFD model is shown to be a good approximation for the simulated distribution of losses based on property graphs for both the random graph and the random time approach.

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