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Dynamic processes leading to natural disasters often translate into random walks in state spaces which are inhomogeneous in transport characteristics. In other words, such random walks will behave differently in different parts of a network which would have different values for transport properties (\(d\!\!f\) and dw if using methods from Chap. 4). Thus, for such networks the application of \(\textit{MFPT}\) calculation methods introduced in Chap. 4 along with many other methods described in literature are not straight forward. This chapter proposes that using the novel concept of dividing the node distribution into patches/clusters known as network primitives (NPs) where all nodes within each primitive share common transport variables, and adopting a ‘hop-wise’ approach to calculate \(\textit{MFPT}\) between any source and target pair as an extension to the methods described under Chap. 4, can be a viable solution for predicting \(\textit{MFPT}\) for random walks in inhomogeneous networks. This methodology’s potential is demonstrated through simulated random walks and with a case study using the dataset of past cyclone tracks over the North Atlantic Ocean. The predictions using the presented method are compared to real data averages and predictions assuming homogeneous transport properties.

This chapter consists of three major subsections. Following a discussion of the ability to model a large class of environmental processes governing natural disasters as random walks, this chapter provides the theoretical background for this book which will be useful throughout the rest of the chapters. The first subsection will summarise the main notion used throughout the book and the adaptation of the theoretical background for natural disaster prediction in an engineering applications perspective. The second subsection will review recent research advancements in \(M\!F\!P\!T\) prediction methods. The final subsection will briefly describe some mathematical tools used in the proceeding chapters.

This book discusses natural disaster prediction methods by analysing the state changes of the dynamics behind natural disasters. It shows that many such state changes behave as random walks in their respective state spaces. Therefore, this book mainly focusses on methods that predict the \(\textit{MFPT}\) for random walks resulting from natural processes leading to natural disasters viewed in an engineering perspective. The main objective of this book is to address network inhomogeneity and directional bias that affect random walks’ \(\textit{MFPT}\) calculations. The motivation behind this discussion is the large class of process behind natural disaster dynamics of which the state transitions translate to random walks where the non-linear interactions with the environment result in biased random walks showing inhomogeneous transport characteristics.

Taking forward the concept of calculating the mean first passage time for natural disasters, this chapter focusses on discussing generic methods for calculations in directionally biased networks. Following a short discussion on the current state of art in \(M\!F\!P\!T\) prediction for biased random walks, a case invariant modification of transport variables is described as a viable solution to the problem of addressing directional bias. The empirical equation that modifies the walk dimension leads the way to significantly better \(M\!F\!P\!T\) estimations when compared to the use of conventional transport variables. A range of simulated random walks on hypothetical networks are used to describe this model. Following this, a comprehensive case study is presented to validate the model with the real dataset of archived cyclone tracks over the North Atlantic Ocean. The comparative studies of prediction results and the generic nature of model development indicate wider applicability of the presented methods.

The prediction of \({\textit{MFPT}}\) in the propagation of cyclone induced flood is comprehensively discussed in this chapter as a special case study to gain an entrance to a more generic study in the proceeding chapters. The inhomogeneity in terrain is addressed using a novel concept of dividing the terrain into geographic primitives (GPs) identified through flood propagation vector fields. A modularised approach including a Bayesian framework is followed allowing easy and flexible assimilation of data and integration of outputs from other models when they become available. Simultaneous events distributed in the environment leading to biased propagation are encapsulated into transition probability matrices (TPM) allowing efficient real time predictions via computation through a Markov chain. Comparative studies using real datasets show the models ability of predicting up to 3 h ahead of official forecasts with a 33 % improvement of accuracy when compared to other models currently used.