Stochastic Dynamics of the Multi-State Voter Model Over a Network Based on Interacting Cliques and Zealot Candidates

Abstract

The stochastic dynamics of the multi-state voter model is investigated on a class of complex networks made of non-overlapping cliques, each hosting a political candidate and interacting with the others via Erdős–Rényi links. Numerical simulations of the model are interpreted in terms of an ad-hoc mean field theory, specifically tuned to resolve the inter/intra-clique interactions. Under a proper definition of the thermodynamic limit (with the average degree of the agents kept fixed while increasing the network size), the model is found to display the empirical scaling discovered by Fortunato and Castellano (Phys Rev Lett 99(13):138701, 2007) , while the vote distribution resembles roughly that observed in Brazilian elections.

Appendix: On Polynomial Approximations of \({\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })\)

Equations (4.21)–(4.22) show that the coefficient functions $A^{(i)}_\ell$ and $B^{(i)}_{\ell m}$ of the FP equation are polynomials of respectively first and second degree in $\bar{\phi }$. Since the drift and the diffusion terms involve respectively one and two differentiations, it makes sense to expand ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ in polynomial functions. The polynomial coefficients should be such that the FP equation is fulfilled order by order. Could the maximum degree of the polynomials be finite? To answer this question, it should not be forgotten that the VM collapses to intra-clique consensus as $\omega _2\rightarrow 0$. Although we did not perform numerical integrations of the FP equation in this limit, it is reasonable to expect that

$$\begin{aligned} \lim _{\omega _2\rightarrow 0} {\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi }) \simeq \prod _{i=1}^Q\prod _{\ell \ne i}^{1\ldots Q}\delta (\phi ^{(i)}_\ell ). \end{aligned}$$

(7.1)

If this is correct, the maximum degree of an approximating polynomial should depend on the model parameters and rapidly increase as $\omega _2\rightarrow 0$. Developing polynomial approximations of ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ is a hard task, although polynomials are the most elementary functions one can deal with. Here, we make do with discussing some theoretical aspects of the problem and refer the reader to a future publication for quantitative results.

1.1 Projecting ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ Onto a Basis of Dirichlet Distributions

We have seen in Sect. 3 that the marginal distributions of $\phi ^{(i)}_\ell$ can be empirically modelled by Beta distributions with very good match. In this respect, Eqs. (3.12) and (3.13) are highly suggestive. Recall that a Dirichlet distribution Dir($\alpha$) of order $d+1$ with parameters $\alpha =\{\alpha _k\}_{k=1}^{d+1}$ has probability density

$$\begin{aligned}&\mathcal{D}_\alpha (x) = \frac{\Gamma \left( |\alpha |_1\right) }{\prod _{k=1}^{d+1}\Gamma (\alpha _k)} \left( \prod _{k=1}^d x_k^{\alpha _k-1}\right) (1-|x|_1)^{\alpha _{d+1}-1}\,\!,\qquad x\in {\bar{T}}_d\,,\end{aligned}$$

(7.2)

$$\begin{aligned}&{\bar{T}}_d = \left\{ y\in \mathbb {R}^d_+: |y|_1\le 1\right\} . \end{aligned}$$

(7.3)

It is well known that if $X\sim \text {Dir}(\alpha )$, then $X_i\sim \text {Beta}(\alpha _i,|\alpha |_1-\alpha _i)$. Inferring from Eqs. (3.12) and (3.13) that the clique vector variable $\bar{\phi }^{(i)}\equiv \{\phi ^{(i)}_k\}_{k\ne i}^{1\ldots , Q}\sim \text {Dir}(\alpha ,\ldots ,\alpha ,\bar{\alpha },\alpha ,\ldots ,\alpha )$, with $\alpha ,\bar{\alpha }$ set according to Table 3 and $\bar{\alpha }$ being the $i^\text {th}$ component of the parameter vector, could be not far from being correct: at least the argument suggests that Dirichlet distributions might play a relevant rôle in analytic representations of ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$. On a more sound basis, we observe that

$$\begin{aligned} \text {span}\left\{ \mathcal{D}_\alpha : \alpha \in \mathbb {N}^{d+1},|\alpha |_1=n+d+1\right\} = \text {span}\left\{ x_1^{\alpha _1}\cdot \ldots \cdot x_d^{\alpha _d}:\alpha \in \mathbb {N}_0^d,|\alpha |_1\le n\right\} \,\!, \end{aligned}$$

(7.4)

i.e. Dirichlet distributions with positive integer parameters form a basis of the vector space of multivariate polynomials. Thus, a representation of ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ as a polynomial series can be certainly given in terms of Dirichlet distributions as

$$\begin{aligned}&{\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi }) = \sum _{\underline{\alpha }} c_{\underline{\alpha }}\,\mathcal{D}_{\underline{\alpha }}\left( \bar{\phi }\right) \,, \qquad \mathcal{D}_{\underline{\alpha }}(\bar{\phi })=\mathcal{D}^{(1)}_{\alpha ^{(1)}}\left( \bar{\phi }^{(1)}\right) \,\ldots \, \mathcal{D}^{(Q)}_{\alpha ^{(Q)}}\left( \bar{\phi }^{(Q)}\right) \,\!, \end{aligned}$$

(7.5)

$$\begin{aligned}&\mathcal{D}^{(i)}_{\alpha ^{(i)}}\left( \bar{\phi }^{(i)}\right) \equiv \frac{\Gamma \left( |\alpha ^{(i)}|_1\right) }{\prod _{k=1}^{Q}\Gamma \left( \alpha ^{(i)}_k\right) } \left( \prod _{k\ne i}^{1\ldots Q} \left[ \phi ^{(i)}_k\right] ^{\alpha ^{(i)}_k-1}\right) \left( 1-\sum _{k\ne i }^{1\ldots Q}\phi ^{(i)}_{k} \right) ^{\alpha ^{(i)}_{i}-1}\!,\end{aligned}$$

(7.6)

$$\begin{aligned}&\sum _{\underline{\alpha }} c_{\underline{\alpha }}=1\,\!, \end{aligned}$$

(7.7)

where the sum in Eq. (7.5) extends to all combinations $\underline{\alpha }=\{\alpha ^{(1)},\ldots ,\alpha ^{(Q)}\}\in \mathbb {N}^{Q\times Q}$ of positive integers. In order to avoid complications, we have deliberately neglected the simplex cut off  $\sum _{k\ne i}\phi ^{(i)}_k<1-\omega _1^{-1}$ in Eq. (7.5): we shall assume for the time being that the support of ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ is $\mathcal{T}_Q(0)$ instead of $\mathcal{T}_Q(\omega _1^{-1})$ and come back to this point later on. From Eq. (7.5) it follows

$$\begin{aligned} {\mathcal{F}_{\scriptscriptstyle \mathrm{FP}}}(x)&=\mathop \int \limits _{\mathcal{T}_Q(0)}\!\!\delta (\phi _\ell -x)\,{\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })\,\text {d}\bar{\phi }\, = \sum _{\underline{\alpha }} c_{\underline{\alpha }}\, \mathcal{F}_{\underline{\alpha }}(x)\,\!, \end{aligned}$$

(7.8)

with

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }}(x) = \mathop \int \limits _{\mathcal{T}_Q(0)}\delta (\phi _\ell -x) \,\mathcal{D}^{(1)}_{\alpha ^{(1)}}\left( \bar{\phi }^{(1)}\right) \,\ldots \, \mathcal{D}^{(Q)}_{\alpha ^{(Q)}}\left( \bar{\phi }^{(Q)}\right) \,\text {d}\bar{\phi }\,\!. \end{aligned}$$

(7.9)

Before embarking on a determination of the coefficients $c_{\underline{\alpha }}$ based on the FP equation (which is anyway beyond the scope of this appendix), we should investigate the analytic structure of the integrals $\mathcal{F}_{\underline{\alpha }}(x)$. To this aim, we make use of a well known formal trick based on the Fourier representation of the Dirac delta function

$$\begin{aligned} \delta (x) = \frac{1}{2\pi }\mathop \int \limits _{-\infty }^{+\infty }\text {d}\lambda \ \text {e}^{\text {i}\lambda x}\,\!, \end{aligned}$$

(7.10)

and the formalism of the Laplace transform

$$\begin{aligned} \mathcal{L}\{f\}(p) = \mathop \int \limits _0^\infty \text {e}^{-p t}f(t)\,\text {d}t\,\!. \end{aligned}$$

(7.11)

To start with, we observe that by virtue of Eq. (2.26) many variables in Eq. (7.9) can be integrated out. Accordingly, $\mathcal{F}_{\underline{\alpha }}(x)$ boils down to

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }}(x)&= \int \delta \left( 1-x + \sum _{k\ne \ell }^{1\ldots Q}\left[ \phi ^{(k)}_\ell -\phi ^{(\ell )}_k\right] \right) \nonumber \\&\qquad \times \left\{ \prod _{k\ne \ell }^{1\ldots Q} \beta _{\alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }\left( \phi ^{(k)}_\ell \right) \right\} \mathcal{D}^{(\ell )}_{\alpha ^{(\ell )}}\left( \bar{\phi }^{(\ell )}\right) \prod _{k\ne \ell }^{1\ldots Q}\text {d}\phi ^{(\ell )}_k\text {d}\phi ^{(k)}_\ell \,\!. \end{aligned}$$

(7.12)

We process separately the three terms under the integral sign. First, we have

$$\begin{aligned} \delta \left( 1-x+\sum _{k\ne \ell }^{1\ldots Q}\left[ \phi ^{(k)}_\ell - \phi ^{(\ell )}_k\right] \right) = \frac{1}{2\pi }\mathop \int \limits _{\infty }^{\infty }\text {d}q \,\text {e}^{\text {i}q\left\{ 1-x+\sum _{k\ne \ell }\left[ \phi ^{(k)}_\ell - \phi ^{(\ell )}_k\right] \right\} }\,\!, \end{aligned}$$

(7.13)

We then represent $\mathcal{D}^{(\ell )}_{\alpha ^{(\ell )}}(\bar{\phi }^{(\ell )})$ in the equivalent form

$$\begin{aligned} \mathcal{D}^{(\ell )}_{\alpha ^{(\ell )}}(\bar{\phi }^{(\ell )})&= \frac{\Gamma \left( |\alpha ^{(\ell )}|_1\right) }{\prod _{k=1}^{Q}\Gamma (\alpha ^{(\ell )}_k)} \left( \prod _{k\ne \ell }^{1\ldots Q} \left[ \phi ^{(\ell )}_k\right] ^{\alpha ^{(\ell )}_k-1}\right) \mathop \int \limits _0^\infty \, \left[ \phi ^{(\ell )}_i\right] ^{\alpha ^{(\ell )}_{\ell }-1} \delta \left( 1-\sum _{k=1}^Q\phi ^{(\ell )}_{k} \right) \,\text {d}\phi ^{(\ell )}_\ell \nonumber \\&= \frac{\Gamma \left( |\alpha ^{(\ell )}|_1\right) }{\prod _{k\ne \ell }^{1\ldots Q}\Gamma \left( \alpha ^{(\ell )}_k\right) } \left( \prod _{k\ne \ell }^{1\ldots Q} \left[ \phi ^{(\ell )}_k\right] ^{\alpha ^{(\ell )}_k-1}\right) \frac{1}{2\pi \text {i}}\mathop \int \limits _{-\text {i}\infty }^{\text {i}\infty } \,\frac{\text {e}^{p_0\left( 1-\sum _{k\ne \ell }^{1\ldots Q}\phi ^{(\ell )}_k\right) }}{p_0^{\alpha ^{(\ell )}_\ell }}\,\text {d}p_0\,\!.\nonumber \\ \end{aligned}$$

(7.14)

In particular, the second line of Eq. (7.14) follows from (i) a rotation of the Fourier integral representing the Dirac delta to the imaginary axis, (ii) a subsequent exchange of integrals and (iii) the Laplace transform of monomials. Similarly, we represent the marginal distribution $\beta _{\alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }\left( \phi ^{(k)}_\ell \right)$ as

$$\begin{aligned} \beta _{\alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }\left( \phi ^{(k)}_\ell \right) = \frac{\Gamma (|\alpha ^{(k)}|_1)}{\Gamma (\alpha ^{(k)}_\ell )} \left( \phi ^{(k)}_\ell \right) ^{\alpha ^{(k)}_\ell -1}\frac{1}{2\pi \text {i}} \mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\text {d}p\, \frac{\text {e}^{p\left( 1-\phi ^{(k)}_\ell \right) }}{p^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }}. \end{aligned}$$

(7.15)

In particular, from Eq. (7.15) it follows

$$\begin{aligned} \prod _{k\ne \ell }^{1\ldots Q}\beta _{\alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell } (\phi ^{(k)}_\ell )&= \prod _{k\ne \ell }^{1\ldots Q}\left[ \frac{\Gamma (|\alpha ^{(k)}|_1)}{\Gamma (\alpha ^{(k)}_\ell )} \left( \phi ^{(k)}_\ell \right) ^{\alpha ^{(k)}_\ell -1}\right] \nonumber \\&\qquad \times \frac{1}{(2\pi \text {i})^{Q-1}}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\prod _{k\ne \ell }^{1\ldots Q} \left[ \frac{\text {d}p_k}{p_k^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }}\right] \text {e}^{\sum _{k\ne \ell } p_k\left[ 1-\phi ^{(k)}_\ell \right] }\!.\nonumber \\ \end{aligned}$$

(7.16)

The next step is to insert Eqs. (7.13), (7.14) and (7.16) into Eq. (7.12). Internal integrals over the $\phi$’s read

$$\begin{aligned}&\left\{ \prod _{k\ne \ell }^{1\ldots Q}\mathop \int \limits _0^\infty \text {d}\phi ^{(\ell )}_k\,\left[ \phi ^{(\ell )}_k\right] ^{\alpha ^{(\ell )}_k-1} \text {e}^{-(\text {i}q+p_0)\phi ^{(\ell )}_k}\right\} \left\{ \prod _{k\ne \ell }^{1\ldots Q}\mathop \int \limits _0^\infty \text {d}\phi ^{(k)}_\ell \,\left[ \phi ^{(k)}_\ell \right] ^{\alpha ^{(k)}_\ell -1} \text {e}^{-(p_k-\text {i}q)\phi ^{(k)}_\ell }\right\} \nonumber \\&\quad = \prod _{k\ne \ell }^{1\ldots Q} \left[ \frac{\Gamma (\alpha ^{(\ell )}_k)}{(p_0+\text {i}q)^{\alpha ^{(\ell )}_k}}\right] \prod _{k\ne \ell }^{1\ldots Q}\left[ \frac{\Gamma (\alpha ^{(k)}_\ell )}{(p_k-\text {i}q)^{\alpha ^{(k)}_\ell }}\right] \,\!\!. \end{aligned}$$

(7.17)

Thus, we have

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }}(x)&= \frac{\prod _{k=1}^Q\Gamma \left( |\alpha ^{(k)}|_1\right) }{2\pi } \mathop \int \limits _{-\infty }^{+\infty } \text {d}q\, \text {e}^{\text {i}q(1-x)}\nonumber \\&\qquad \times \frac{1}{2\pi \text {i}}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\,\text {d}p_0\,\frac{1}{p_0^{\alpha ^{(\ell )}_\ell }}\prod _{k\ne \ell }^{1\ldots Q}\left[ \frac{1}{(p_0+\text {i}q)^{\alpha ^{(\ell )}_k}}\right] \text {e}^{p_0s}\biggr |_{s=1}\nonumber \\&\qquad \times \frac{1}{(2\pi \text {i})^{Q-1}}\prod _{k\ne \ell }^{1\ldots Q}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\,\text {d}p_k\, \frac{\text {e}^{p_k s}}{p_k^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }(p_k-\text {i}q)^{\alpha ^{(k)}_\ell }}\biggr |_{s=1}\,\!. \end{aligned}$$

(7.18)

The second and third line of Eq. (7.18) are inverse Laplace transforms of products of functions. In order to solve the integrals, one could work out the products in partial fractions or make use of the Laplace convolution theorem, while recalling that

$$\begin{aligned} \frac{1}{p_0^{\alpha ^{(\ell )}_\ell }}&= \mathcal{L}\left\{ f(s)=\frac{s^{\alpha ^{(\ell )}_\ell -1}}{\Gamma (\alpha ^{(\ell )}_\ell )}\right\} (p_0)\,\!,\end{aligned}$$

(7.19)

$$\begin{aligned} \frac{1}{(p_0+iq)^{\sum _{k\ne l}\alpha ^{(\ell )}_k}}&= \mathcal{L}\left\{ f(s)=\frac{s^{\sum _{k\ne \ell }\alpha ^{(\ell )}_k-1}\text {e}^{-iqs}}{\Gamma \left( \sum _{k\ne \ell }\alpha ^{(\ell )}_k\right) }\right\} (p_0)\,\!,\end{aligned}$$

(7.20)

$$\begin{aligned} \frac{1}{p_k^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }}&= \mathcal{L}\left\{ f(s)=\frac{s^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell -1}}{\Gamma \left( |\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell \right) }\right\} (p_k)\,\!,\end{aligned}$$

(7.21)

$$\begin{aligned} \frac{1}{(p_k-iq)^{\alpha ^{(k)}_\ell }}&= \mathcal{L}\left\{ f(s)=\frac{s^{\alpha ^{(k)}_\ell -1}\text {e}^{iqs}}{\Gamma \left( \alpha ^{(k)}_\ell \right) }\right\} (p_k)\,\!. \end{aligned}$$

(7.22)

From the Laplace convolution theorem, it follows

$$\begin{aligned}&\frac{1}{2\pi \text {i}}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\text {d}p_0\,\frac{\text {e}^{p_0s}}{p_0^{\alpha ^{(\ell )}_\ell }(p_0+\text {i}q)^{\sum _{k\ne \ell }\alpha ^{(\ell )}_k}}\biggr |_{s=1} \nonumber \\&=\frac{1}{\Gamma (\alpha ^{(\ell )}_\ell )\Gamma \left( \sum _{k\ne \ell }\alpha ^{(\ell )}_k\right) }\mathop \int \limits _0^1\text {d}\sigma \, \sigma ^{\sum _{k\ne \ell }\alpha ^{(\ell )}_k-1}(1-\sigma )^{\alpha ^{(\ell )}_\ell -1}\text {e}^{-\text {i}q\sigma }\nonumber \\&= \frac{1}{\Gamma \left( |\alpha ^{(\ell )}|_1\right) }M\left( \sum _{k\ne \ell }\alpha ^{(\ell )}_k,|\alpha ^{(\ell )}|_1,-\text {i}q\right) \!, \end{aligned}$$

(7.23)

with $M(a,b,z)={}_1F_1(a;b;z)$ being the Kummer function. Analogously, we have

$$\begin{aligned}&\frac{1}{(2\pi \text {i})^{Q-1}}\prod _{k\ne \ell }^{1\ldots Q}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\text {d}p_k\, \frac{\text {e}^{p_ks}}{p_k^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell }(p_k-\text {i}q)^{\alpha ^{(\kappa )}_\ell }}\biggr |_{s=1}\nonumber \\&\quad =\prod _{k\ne \ell }^{1\ldots Q}\left\{ \frac{1}{\Gamma (\alpha ^{(k)}_\ell )\Gamma \left( |\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell \right) }\mathop \int \limits _0^1\text {d}\sigma \,\sigma ^{\alpha ^{(k)}_\ell -1}(1-\sigma )^{|\alpha ^{(k)}|_1-\alpha ^{(k)}_\ell -1}\,\text {e}^{\text {i}q\sigma }\right\} \nonumber \\&\quad = \prod _{k\ne \ell }^{1\ldots Q}\left\{ \frac{1}{\Gamma \left( |\alpha ^{(k)}|_1\right) }M\left( \alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1,\text {i}q\right) \right\} \,\!. \end{aligned}$$

(7.24)

Inserting the right-hand side of Eqs. (7.23) and (7.24) back into Eq. (7.18) and using the Kummer identity (cf. Eq. (13.1.27) of [26])

$$\begin{aligned} M\left( \sum _{k\ne \ell }\alpha ^{(\ell )}_k,|\alpha ^{(\ell )}|_1,-\text {i}q\right) = \text {e}^{-\text {i}q}M\left( \alpha ^{(\ell )}_\ell ,|\alpha ^{(\ell )}|_1,\text {i}q\right) \!, \end{aligned}$$

(7.25)

yields

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }}(x) = \frac{1}{2\pi }\mathop \int \limits _{-\infty }^{+\infty }\text {d}q\,\text {e}^{-\text {i}qx}\,\prod _{k=1}^QM\left( \alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1,\text {i}q\right) \!, \end{aligned}$$

(7.26)

or equivalently (by setting $p=-\text {i}q$),

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }}(x) = \frac{1}{2\pi \text {i}}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\text {d}p\ \text {e}^{px}\,\prod _{k=1}^Q M\left( \alpha ^{(k)}_\ell ,|\alpha ^{(k)}|_1,-p\right) \!. \end{aligned}$$

(7.27)

Thus, we see that the contributions to ${\mathcal{F}_{\scriptscriptstyle \mathrm{FP}}}(x)$ arising from a polynomial expansion of ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ based on Dirichlet distributions are higher transcendental functions. Alternatively, by using once again the Laplace convolution theorem, we can express $\mathcal{F}_{\underline{\alpha }}(x)$ as a nested integral

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }}(x)&= \mathop \int \limits _0^x\text {d}\tau _1\mathop \int \limits _0^{\tau _1}\text {d}\tau _2\ldots \mathop \int \limits _0^{\tau _{Q-2}}\text {d}\tau _{Q-1}\,W\left( \alpha ^{(\ell )}_\ell ,|\alpha ^{(\ell )}|_1,x-\tau _1\right) \nonumber \\&\qquad \times W\left( \alpha ^{(1)}_\ell ,|\alpha ^{(1)}|_1,\tau _1-\tau _2\right) \ldots W\left( \alpha ^{(Q)}_\ell ,|\alpha ^{(Q)}|_1,\tau _{Q-1}\right) \!, \end{aligned}$$

(7.28)

with

$$\begin{aligned} W(a,b,t) = \frac{\Gamma (b)}{\Gamma (a)\Gamma (b-a)}t^{a-1}(1-t)^{b-a-1}\theta (1-t) = \beta _{a,b-a}(t)\,\theta (1-t)\,\!, \end{aligned}$$

(7.29)

and $\theta (\cdot )$ being the Heaviside step function. Eq. (7.26) can be computed numerically once $\underline{\alpha }$ is assigned. As an example, we consider the subset of indices

$$\begin{aligned} \underline{\alpha } = \underline{\alpha }(\kappa ,\bar{\kappa })\,\equiv \, \left\{ \begin{array}{ll} \alpha ^{(i)}_j = \kappa \,, &{} \quad i\ne j \text {, and }i,j=1,\ldots ,Q\,,\\ \alpha ^{(i)}_i=\bar{\kappa }\,, &{} \quad i=1,\ldots , Q\,, \end{array}\right. \end{aligned}$$

(7.30)

corresponding to the contributions

$$\begin{aligned} \mathcal{F}_{\underline{\alpha }(\kappa ,\bar{\kappa })}(x) = \frac{1}{2\pi }\mathop \int \limits _{-\infty }^{+\infty }\text {d}q\, \text {e}^{-\text {i}qx}\,\left[ M(\kappa ,\bar{\kappa }+(Q-1)\kappa ,\text {i}q)\right] ^{Q-1}\,M(\bar{\kappa },\bar{\kappa }+(Q-1)\kappa ,\text {i}q). \end{aligned}$$

(7.31)

In Fig. 8 we show $\mathcal{F}_{\underline{\alpha }(\kappa ,\bar{\kappa })}$ for some choices of $(\kappa ,\bar{\kappa })$. We see that all curves lie below ${\mathcal{F}_{\scriptscriptstyle \mathrm{FP}}}(x)$ by orders of magnitude along the tails, thus suggesting that contributions $\mathcal{F}_{\underline{\alpha }}(x)$, where the indices $\alpha ^{(i)}_j$ are not the same for $i\ne j$, cannot be neglected.

Fig. 8

Some contributions to ${\mathcal{F}_{\scriptscriptstyle \mathrm{FP}}}(x)$ arising from a polynomial expansion of ${\mathcal{P}_{\scriptscriptstyle \mathrm{FP}}}(\bar{\phi })$ based on Dirichlet distributions

1.2 Cutting Off the Simplex

We finally explore how the Dirichlet distribution changes upon cutting off its defining domain. Specifically, we consider

$$\begin{aligned} \mathcal{D}_\alpha (x) = Z_\alpha (\omega _1^{-1}) \left( \prod _{k=1}^d x_k^{\alpha _k-1}\right) (1-|x|_1)^{\alpha _{d+1}-1}\,, \end{aligned}$$

(7.32)

as a distribution with support in

$$\begin{aligned} {\bar{T}}_d(\omega _1^{-1}) = \{y\in \mathbb {R}^d_+: |y|_1\le 1-\omega _1^{-1}\}. \end{aligned}$$

(7.33)

The normalization constant $Z_\alpha (\omega _1^{-1})$ can be calculated analytically according to the same technique used for $\mathcal{F}_{\underline{\alpha }}(x)$ by imposing that the integral of $\mathcal{D}_\alpha (x)$ over ${\bar{T}}_d(\omega _1^{-1})$ equals one. A little algebra yields

$$\begin{aligned}{}[Z_\alpha (\omega _1^{-1})]^{-1} = \frac{1}{2\pi \text {i}}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\text {d}\lambda \ \text {e}^{\lambda t}\ \left( \prod _{i=1}^{d} \mathop \int \limits _0^{+\infty }\text {d}x_i\ x_i^{\alpha _i-1} \text {e}^{-\lambda x_i}\right) \mathop \int \limits _{\omega _1^{-1}}^{+\infty }\text {d}x_{d+1}\ x_{d+1}^{\alpha _{d+1}-1} \text {e}^{-\lambda x_{d+1}}\biggr |_{t=1}\,, \end{aligned}$$

(7.34)

We notice that

$$\begin{aligned} \mathop \int \limits _{\omega _1^{-1}}^{+\infty }\text {d}z\ z^{\alpha _{d+1}-1} \text {e}^{-\lambda z} = \frac{\Gamma (\alpha _{d+1},\omega _1^{-1}\lambda )}{\lambda ^{\alpha _{d+1}}}\,, \end{aligned}$$

(7.35)

with $\Gamma (\alpha _{d+1},\omega _1^{-1}\lambda )$ denoting the lower incomplete Gamma function. Hence, it follows

$$\begin{aligned}{}[Z_{\alpha }(\omega _1^{-1})]^{-1} = \dfrac{\prod _{i=1}^d\Gamma (\alpha _i)}{2\pi \text {i}}\mathop \int \limits _{-\text {i}\infty }^{+\text {i}\infty }\text {d}\lambda \ \text {e}^{\lambda t}\ \frac{1}{\lambda ^{\psi }}\frac{\Gamma (\alpha _{d+1},\omega _1^{-1}\lambda )}{\lambda ^{\alpha _{d+1}}}\biggr |_{t=1}\,\!, \qquad \psi = \sum _{i=1}^d \alpha _i\,\!. \end{aligned}$$

(7.36)

Once more, the latter integral is the Laplace antitransform of a product of functions. According to the rules of the Laplace transform, it amounts to the convolution of the antitransform of the single functions. Therefore, we have

$$\begin{aligned}{}[Z_\alpha (\omega _1^{-1})]^{-1} = \dfrac{\prod _{i=1}^{d+1}\Gamma (\alpha _i)}{\Gamma (|\alpha |_1)}I_{\psi ,\alpha _{d+1}}(1-\omega _1^{-1}) = [Z_{{\alpha }}(0)]^{-1} I_{\psi ,\alpha _{d+1}}(1-\omega _1^{-1})\,\!, \end{aligned}$$

(7.37)

where

$$\begin{aligned} I_{a,b}(z) = \mathop \int \limits _0^z \text {d}\tau \ \beta _{a,b}(\tau )\,\!, \end{aligned}$$

(7.38)

denotes the regularized incomplete beta function. Since this has an asymptotic expansion at $z=1$

$$\begin{aligned} I_{a,b}(z) = 1 - \frac{(1-z)^b z^a}{bB_{a,b}}\sum _{k=0}^\infty \frac{(-1)^k (a+b)_k (z-1)^k}{(b+1)_k}\,,\qquad b\ne -1\,, \end{aligned}$$

(7.39)

with $B_{a,b}=\Gamma (a)\Gamma (b)/\Gamma (a+b)$, we conclude that

$$\begin{aligned}{}[Z_{{\alpha }}(\omega _1^{-1})]^{-1} = [Z_{{\alpha }}(0)]^{-1} + \text {O}\left[ (\omega _1^{-1})^{\alpha _{d+1}}\right] \,\!. \end{aligned}$$

(7.40)

Thus, we see that depending on $\alpha _{d+1}$, the impact of cutting off the integration domain (the Cartesian product of simplices), at the boundary on the Dirichlet integrals could be very small.