Analytical Approaches to Agent-Based Models

Abstract

The aim of this article is to present an approach to the analysis of simple systems composed of a large number of units in interaction. Suppose to have a large number of agents belonging to a finite number of different groups: as the agents randomly interact with each other, they move from a group to another as a result of the interaction. The object of interest is the stochastic process describing the number of agents in each group. As this is generally intractable, it has been proposed in the literature to approximate it in several ways. We review these approximations and we illustrate them with reference to a version of the epidemic model. The tools presented in the paper should be considered as a complement rather than as a substitute of the classical analysis of ABMs through simulation.

Appendix: Technical Conditions

In this appendix, we discuss the technical conditions under which the results stated above hold true.

As concerns the deterministic approximation of Sect. 13.3, we follow Theorem 8.1 in Kurtz (1981) (similar results are Theorem 3.1 in Norman, 1968; Theorem (3.1) in Kurtz, 1970; Theorem 8.1.1 in Norman, 1972; Theorem (2.1) in Kurtz, 1976; Theorem 2.2 in Kurtz, 1978; Theorem (2.16) in Kurtz, 1980; Theorem 2.1 in Chap. 11 in Ethier & Kurtz, 1986).

Let K ⊂ E be a bounded and closed (i.e., compact) set. The first condition requires that, for each K:

$$\displaystyle{\sum _{\boldsymbol{\ell}}\left \vert \boldsymbol{\ell}\right \vert \cdot \sup _{\mathbf{x}\in K}\beta _{\boldsymbol{\ell}}\left (\mathbf{x}\right ) <\infty.}$$

The second condition requires that, for any K, there exists M _K such that:

$$\displaystyle{\left \vert \mathbf{f}\left (\mathbf{x}\right ) -\mathbf{f}\left (\mathbf{y}\right )\right \vert \leq M_{K} \cdot \left \vert \mathbf{x} -\mathbf{y}\right \vert,\quad \mathbf{x},\mathbf{y} \in K.}$$

At last, we require that the initial condition of the original process converges to the one of the deterministic one, i.e. $\lim _{N\rightarrow \infty }\mathbf{X}_{0}^{\left (N\right )} = \mathbf{x}_{0}$. By the way, under these conditions, the convergence of $\left \{\mathbf{X}_{t}^{\left (N\right )}\right \}_{t\in \mathbb{R}_{+}}$ to $\left \{\mathbf{X}_{t}^{{\prime}}\right \}_{t\in \mathbb{R}_{+}}$ is uniform for t belonging to bounded subsets of $\mathbb{R}_{+}$.

Exercise 8 (News Diffusion Model—Continued).

Consider the epidemic model seen in Exercise 1 in the second rewriting. Using the fact that $x \in \left [0,1\right ]$, it is possible to see that $x\left (1 - x\right ) \leq \frac{1} {4}$. Therefore, we have:

$$\displaystyle{\sum _{\ell}\left \vert \ell\right \vert \cdot \sup _{x\in K}\beta _{\ell}\left (x\right ) = p \cdot \sup _{x\in K}x\left (1 - x\right ) \leq \frac{p} {4} <\infty.}$$

As concerns the second hypothesis, we have:

$$\displaystyle\begin{array}{rcl} \left \vert f\left (x\right ) - f\left (y\right )\right \vert & =& p \cdot \left \vert x\left (1 - x\right ) - y\left (1 - y\right )\right \vert {}\\ &\leq & p \cdot \sup _{z\in \left [x,y\right ]}\left \vert \frac{\partial \left [z\left (1 - z\right )\right ]} {\partial z} \right \vert \cdot \left \vert x - y\right \vert {}\\ & =& p \cdot \sup _{z\in \left [x,y\right ]}\left \vert 1 - 2z\right \vert \cdot \left \vert x - y\right \vert \leq p \cdot \left \vert x - y\right \vert {}\\ \end{array}$$

where the second step derives from the mean value theorem. At last, we have supposed that $I_{0} = i_{0}$ so that the initial condition is trivially verified. □ 

The diffusion approximation of Sect. 13.4.1 holds under the following conditions (this is Theorem 8.4 in Kurtz, 1981; see Theorem (3.13) in Kurtz, 1976; Theorem 3.3 in Kurtz, 1978; Theorem 2.1 in Kurtz, 1983; Theorem 3.1 in Chap. 11 in Ethier & Kurtz, 1986 for alternative or more general conditions):

• for any index $\boldsymbol{\ell}$ but a finite number, $\beta _{\boldsymbol{\ell}}\left (\mathbf{x}\right ) \equiv 0$;

• for any index $\boldsymbol{\ell}$, $\overline{\beta }_{\boldsymbol{\ell}} =\sup _{\mathbf{x}}\beta _{\boldsymbol{\ell}}\left (\mathbf{x}\right ) <+\infty$;

• there exists M > 0 such that:

  $$\displaystyle{\left \vert \beta _{\boldsymbol{\ell}}\left (\mathbf{x}\right ) -\beta _{\boldsymbol{\ell}}\left (\mathbf{y}\right )\right \vert \leq M \cdot \overline{\beta }_{\boldsymbol{\ell}}\cdot \left \vert \mathbf{x} -\mathbf{y}\right \vert;}$$

• there exists M > 0 such that:

  $$\displaystyle{\left \vert \mathbf{f}\left (\mathbf{x}\right ) -\mathbf{f}\left (\mathbf{y}\right )\right \vert \leq M \cdot \left \vert \mathbf{x} -\mathbf{y}\right \vert.}$$

The rate on the approximation of $\left \{\mathbf{X}_{t}\right \}_{t\in \mathbb{R}_{+}}$ through $\left \{\mathbf{X}_{t}^{{\prime\prime}}\right \}_{t\in \mathbb{R}_{+}}$ at the end of Sect. 13.4.1 can be found in Theorem (3.13) in Kurtz (1976), Theorem 3.3 in Kurtz (1978), Theorem 8.4 in Kurtz (1981) and Theorem 3.1 in Chap. 11 in Ethier and Kurtz (1986). By the way, the coupling is uniform over bounded intervals of the real line.

Exercise 9 (News Diffusion Model—Continued).

There exists only one index ℓ, i.e. ℓ = 1, for which $\beta _{\ell}\not\equiv 0$. For this index, $\overline{\beta }_{1} = p \cdot \sup _{x\in \left [0,1\right ]}x\left (1 - x\right ) = p/4 <+\infty$. Now, from Exercise 8:

$$\displaystyle{\left \vert \beta _{1}\left (x\right ) -\beta _{1}\left (y\right )\right \vert \leq p \cdot \left \vert x - y\right \vert,}$$

i.e. one can take M = 4. On the other hand, always from Exercise 8:

$$\displaystyle{\left \vert f\left (x\right ) - f\left (y\right )\right \vert = \left \vert \beta _{1}\left (x\right ) -\beta _{1}\left (y\right )\right \vert \leq p \cdot \left \vert x - y\right \vert,}$$

i.e. one can take M = p. Therefore, any $M \geq \max \left \{4,p\right \}$ respects the conditions. □ 

The convergence in Sect. 13.4.2 holds under the following conditions (these are the ones stated in Theorem 8.2 in Kurtz, 1981; for related results, see Theorem 1.1 in Norman, 1968; Theorem (3.5) in Kurtz, 1971; Theorem 8.1.1 in Norman, 1972; Theorem 1 in Barbour, 1974; Theorem (2.3) in Kurtz, 1976; Theorem 2 in Allain, 1976a; Theorem 4.4 in Kurtz, 1978; Theorem 2.2 in Kurtz, 1983; Theorem 2.3 in Chap. 11 in Ethier & Kurtz, 1986):

• for each bounded closed set K, we have:

  $$\displaystyle{\sum _{\boldsymbol{\ell}}\left \vert \boldsymbol{\ell}\right \vert ^{2}\sup _{ \mathbf{x}\in K}\beta _{\boldsymbol{\ell}}\left (\mathbf{x}\right ) <\infty;}$$

• the functions ∂ f and $\beta _{\boldsymbol{\ell}}$, for each $\boldsymbol{\ell}$, are continuous;

• the initial conditions converge in such a way that $\lim _{N\rightarrow \infty }\sqrt{N}\left \vert \mathbf{X}_{0}^{\left (N\right )} -\mathbf{x}_{0}\right \vert = \mathbf{0}$.

Versions of this result holding uniformly for t > 0 have been stated in Theorem 3.2 (ii) in Norman (1974b), Theorem 1 in Norman (1974a), Theorem (2.7) in Kurtz (1976) and Theorem 8.5 in Kurtz (1981). Berry–Esséen-type theorems can be found in Theorem 1 in Barbour (1974), Theorem (2.5) in Kurtz (1976), Allain (1976b), Corollary 4.5 in Kurtz (1978) and Chapters 5 and 6 in Alm (1978).

The rate on the approximation of $\left \{\mathbf{X}_{t}\right \}_{t\in \mathbb{R}_{+}}$ through $\left \{\mathbf{X}_{t}^{{\prime\prime\prime}}\right \}_{t\in \mathbb{R}_{+}}$ at the end of Sect. 13.4.2 is uniform over bounded subsets of the real line and can be found in Theorem 4.4 in Kurtz (1978) and in Theorem 3.2 and following remarks in Chap. 11 in Ethier and Kurtz (1986).

Exercise 10 (News Diffusion Model—Continued).

Reasoning as in Exercise 8, we have:

$$\displaystyle{\sum _{\ell}\left \vert \ell\right \vert ^{2} \cdot \sup _{ x\in K}\beta _{\ell}\left (x\right ) \leq \frac{p} {4} <\infty.}$$

As concerns $\partial f\left (x\right ) = p \cdot \left (1 - 2x\right )$ and $\beta _{1}\left (x\right ) = p \cdot x\left (1 - x\right )$, they are clearly continuous. □