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Chapter 13
Combining Mathematical and Simulation
Approaches to Understand the Dynamics
of Computer Models

Luis R. Izquierdo, Segismundo S. Izquierdo, José M. Galán, and José I. Santos

Abstract This chapter shows how computer simulation and mathematical analysis
can be used together to understand the dynamics of computer models. For this
purpose, we show that it is useful to see the computer model as a particular
implementation of a formal model in a certain programming language. This formal
model is the abstract entity which is defined by the input–output relation that the
computer model executes and can be seen as a function that transforms probability
distributions over the set of possible inputs into probability distributions over the set
of possible outputs.
It is shown here that both computer simulation and mathematical analysis
are extremely useful tools to analyse this formal model, and they are certainly
complementary in the sense that they can provide fundamentally different insights
on the same model. Even more importantly, this chapter shows that there are plenty
of synergies to be exploited by using the two techniques together.
The mathematical analysis approach to analyse formal models consists in
examining the rules that define the model directly. Its aim is to deduce the logical
implications of these rules for any particular instance to which they can be applied.
Our analysis of mathematical techniques to study formal models is focused on the
theory of Markov Chains, which is particularly useful to characterise the dynamics
of computer models.
In contrast with mathematical analysis, the computer simulation approach does
not look at the rules that define the formal model directly but instead tries to
infer general properties of these rules by examining the outputs they produce when
applied to particular instances of the input space. Thus, conclusions obtained with
this approach may not be general. On a more positive note, computer simulation
enables us to explore formal models beyond mathematical tractability, and we can

L.R. Izquierdo ( ) • J.M. Galán • J.I. Santos
Departamento de Ingeniería Civil, Universidad de Burgos, E-09001, Burgos, Spain
e-mail: lrizquierdo@ubu.es; jmgalan@ubu.es; jisantos@ubu.es
S.S. Izquierdo
Departamento de Organización de Empresas y C.I.M., Universidad de Valladolid, E-47011,
Valladolid, Spain
e-mail: segis@eis.uva.es

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Understanding Complex Systems, https://doi.org/10.1007/978-3-319-66948-9_13