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## Notes

- 1.
Note that simulations of stochastic models are actually using pseudo-random number generators, which are deterministic algorithms that require a seed as an input.

- 2.
A formal model is a model expressed in a formal system (Cutland 1980). A formal system consists of a formal language and a deductive apparatus (a set of axioms and inference rules). Formal systems are used to derive new expressions by applying the inference rules to the axioms and/or previously derived expressions in the same system.

- 3.
The mere fact that the model has been implemented and can be run in a computer is a proof that the model is formal (Suber 2002).

- 4.
As a matter of fact, strictly speaking, inputs and outputs in a computer model are

*never*numbers. We may interpret strings of bits as numbers, but we could equally well interpret the same strings of bits as e.g. letters. More importantly, a bit itself is already an abstraction, an interpretation we make of an electrical pulse that can be above or below a critical voltage threshold. - 5.
A sufficient condition for a programming language to be “sophisticated enough” is to allow for the implementation of the following three control structures:

• Sequence (i.e. executing one subprogram, and then another subprogram),

• Selection (i.e. executing one of two subprograms according to the value of a boolean variable, e.g. IF[boolean == true]-THEN[subprogram1]-ELSE[subprogram2]), and

• Iteration (i.e. executing a subprogram until a boolean variable becomes false, e.g. WHILE[boolean == true]-DO[subprogram]).

Any programming language that can combine subprograms in these three ways can implement any computable function; this statement is known as the “structured program theorem” (Böhm and Jacopini 1966; Harel 1980; Wikipedia 2007).

- 6.
Note that statistics extracted from the model can be of any nature, as long as they are unambiguously defined. For example, they can refer to various time-steps, and only to certain agents (e.g. “average wealth of female agents in odd time-steps from 1 to 99”).

- 7.
We use the term “mathematical analysis” in its broadest sense, i.e. we do not refer to any particular branch of mathematics, but to the general use of (any type of) mathematical technique to analyse a system.

- 8.
Unless, of course, all possible particular instances are explored.

- 9.
The frequency of the event “there are

*i*walkers in a patch with a house” calculated over*n*simulation runs can be seen as the mean of a sample of*n*i.i.d. Bernouilli random variables where success denotes that the event occurred and failure denotes that it did not. Thus, the frequency*f*is the maximum likelihood (unbiased) estimator of the exact probability with which the event occurs. The standard error of the calculated frequency*f*is the standard deviation of the sample divided by the square root of the sample size. In this particular case, the formula reads:$$ \rm Std. error(\it f,\it n)=(\it f(\rm 1-\it f)/(\it n-\rm 1))^{1/2}$$Where

*f*is the frequency of the event,*n*is the number of samples, and the standard deviation of the sample has been calculated dividing by (*n*– 1). - 10.
The term ‘Markov chain’ allows for countably infinite state spaces too (Karr 1990).

- 11.
Formally, the occupancy of state

*i*is defined as:$$ \pi_i^{*}=\mathop{\lim}\limits_{{n\to \infty }}\frac{{E({N_i}(n))}}{n+1 } $$where

*N*_{ i }(*n*) denotes the number of times that the THMC visits state*i*over the time span {0, 1,…,*n*}. - 12.
Given that the system has entered the absorbing class

*C*_{ v }. - 13.
This finding does not refute some of the most important conclusions obtained by the authors of the original model.

- 14.
This is so because many assumptions we make in our models are, to some extent, for the sake of simplicity. As a matter of fact, in most cases the whole purpose of modelling is to build an abstraction of the world which is simpler than the world itself, so we can make inferences about the model that we cannot make directly from the real world (Edmonds 2001; Galán et al. 2009; Izquierdo et al. 2008a).

- 15.
This comment was added by the editors as the authors are too modest to so describe their own work.

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## Acknowledgements

The authors have benefited from the financial support of the Spanish Ministry of Education and Science (projects DPI2004-06590, DPI2005-05676 and TIN2008-06464-C03-02), the Spanish Ministry for Science and Innovation (CSD2010-00034), and the JCyL (projects VA006B09, BU034A08 and GREX251-2009). We are also very grateful to Nick Gotts, Bruce Edmonds, Gary Polhill, and Cesáreo Hernández for many extremely useful discussions.

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## Further Reading

### Further Reading

Firstly we suggest three things to read to learn more about Markov Chain models. Grinstead and Snell (1997) provides an excellent introduction to the theory of finite Markov Chains, with many examples and exercises. Häggström (2002) gives a clear and concise introduction to Probability theory and Markov Chain theory, and then illustrates the usefulness of these theories by studying a range of stochastic algorithms with important applications in optimisation and other problems in computing. One of the algorithms covered is the Markov chain Monte Carlo method. Finally, Kulkarni (1995) provides a rigorous analysis of many types of useful stochastic processes, e.g. discrete and continuous time Markov Chains, renewal processes, regenerative processes, and Markov regenerative processes.

The reader may find three other papers helpful. Izquierdo et al. (2009) analyses the dynamics of ten well-known models in the social simulation literature using the theory of Markov Chains, and is thus a good illustration of the approach in practice within the context of social simulation.^{Footnote 15} Epstein (2006) is a more general discussion, treating a variety of foundational and epistemological issues surrounding generative explanation in the social sciences, and discussing the role of agent-based computational models in generative social science. Finally, Leombruni and Richiardi (2005) usefully discusses several issues surrounding the interpretation of simulation dynamics and the generalisation of the simulation results. For a different approach to analysing the dynamics of a simulation model we refer the interested reader to Chap. 9 in this volume (Evans et al. 2013).

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Izquierdo, L.R., Izquierdo, S.S., Galán, J.M., Santos, J.I. (2013). Combining Mathematical and Simulation Approaches to Understand the Dynamics of Computer Models. In: Edmonds, B., Meyer, R. (eds) Simulating Social Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93813-2_11

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