Generalized Nonlinear Yule Models

Abstract

With the aim of considering models related to random graphs growth exhibiting persistent memory, we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macroevolution. Here the model is analyzed and interpreted in the framework of the development of networks such as the World Wide Web. Nonlinearity is introduced by replacing the linear birth process governing the growth of the in-links of each specific webpage with a fractional nonlinear birth process with completely general birth rates. Among the main results we derive the explicit distribution of the number of in-links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction. The mean value of the latter distribution is also calculated explicitly in the most general case. Furthermore, in order to show the usefulness of our results, we particularize them in the case of specific birth rates giving rise to a saturating behaviour, a property that is often observed in nature. The further specialization to the non-fractional case allows us to extend the Yule model accounting for a nonlinear growth.

Appendix 1

For the sake of self-containedness we give here a brief description of the fractional nonlinear birth process. For a quick comparison with the classical nonlinear birth process see Table 1.

Let us consider a population of individuals developing with continuous time t initiated by one single initial progenitor at time $t=0$. We indicate the random number of individuals in the population for any fixed time t with the random variable $\mathfrak {N}^\nu (t)$. It is known [32] that the state probabilities $p_n^\nu (t) = \mathbb {P}(\mathfrak {N}^\nu (t)=n)$, $n \ge 1$, satisfy the system of difference-differential equations

$$\begin{aligned} \frac{d ^\nu p_n^\nu }{d t^\nu } = - \lambda _n p_n^\nu + \lambda _{n-1} p^\nu _{n-1}, \qquad n \ge 1, \end{aligned}$$

(5.1)

where $p_0^\nu (s)=0$. Moreover, $p_n^\nu (0) =\delta _{n,1}$, that is the process starts with only one initial progenitor, and the fractional derivative is the Caputo derivative (see e.g. [14, 21]). Briefly, the Caputo derivative is an integral operator of convolution-type with a singular power-law kernel. The Caputo derivative can be defined in several equivalent ways. We consider here the following form:

Definition 5.1

(Caputo derivative) Let $\alpha >0$, $m = \lceil \alpha \rceil$, and $f \in AC^m[a,b]$. The Caputo derivative of order $\alpha >0$ is defined as

$$\begin{aligned} \frac{d ^\alpha }{d t^\alpha } f(t)= \frac{1 }{\Gamma (m-\alpha )}\int _a^{t}(t-s)^{m-1-\alpha }\frac{d ^m}{d s^m}f(s) \, ds. \end{aligned}$$

(5.2)

In our case we have $\alpha =\nu \in (0,1)$, $m=1$, $a=0$, obtaining

$$\begin{aligned} \frac{d ^\nu }{d t^\nu } p_n^\nu (t)= \frac{1}{\Gamma \left( 1- \nu \right) } \int _0^t \frac{ \frac{d }{d s} p_n^\nu \left( s \right) }{\left( t-s \right) ^\nu } \, d s, \qquad 0< \nu < 1. \end{aligned}$$

(5.3)

It is evident from the above Definition 5.1 that the Caputo derivative is a non-local operator in the sense that the integration over the interval (0, t) furnishes the system with a persistent memory. Roughly speaking the first order derivative $\frac{d }{d s}p_n^\nu (s)$ is evaluated along the whole time interval (0, t) and weighted by means of the power-law kernel.

The state probabilities $p_n^\nu (t)$ of the fractional birth process can be explicitly determined and (with the convention that empty products equal unity) have the form (for the nonlinear rates $\lambda _j$, $j \ge 1$, all different) [32]

$$\begin{aligned} p_n^\nu (t) = \mathbb {P}(\mathfrak {N}^\nu (t)=n)= \prod _{j=1}^{n-1} \lambda _j \sum _{m=1}^n \frac{ E_{\nu } (- \lambda _m t^\nu )}{\prod _{ l=1,l \ne m}^n \left( \lambda _l - \lambda _m \right) }, \qquad n \ge 1, \, t\ge 0, \end{aligned}$$

(5.4)

where $E_{\nu } (\zeta )$ is the so-called Mittag–Leffler function, a special function defined as

$$\begin{aligned} E_{\nu } \left( \zeta \right) = \sum _{h=0}^\infty \frac{\zeta ^h}{\Gamma \left( \nu h+1 \right) }, \qquad \zeta \in \mathbb {R}, \, \nu > 0, \end{aligned}$$

(5.5)

and having Laplace transform

$$\begin{aligned} \mathcal {L} \bigl ( E_\nu (-\lambda t^\nu ) \bigr ) (z) = \int _0^\infty e^{- z t} E_{\nu } \left( - \xi t^\nu \right) d t = \frac{z^{\nu -1}}{z^\nu + \xi }, \qquad \nu >0, \, \xi \in \mathbb {R}. \end{aligned}$$

(5.6)

The state probabilities (5.4) can be actually derived by means of an iterated application of the Laplace transform on the equations (5.1) starting from $n=1$. For details on this point see [32], Sect. 2. The Mittag–Leffler function (5.5) is in practice a generalization of the exponential function in the sense that $E_{1} \left( \zeta \right) = \exp (\zeta )$. General properties of the Mittag–Leffler functions are contained in many classical reference books and articles (see e.g. the very recent monograph [19] and the references listed therein).

In the present paper we will often make use of the Laplace transform of the state probabilities (5.4) of the fractional nonlinear birth process. From [32, 34] we we can easily check that that

$$\begin{aligned} \mathbb {L}_n(z) = \int _0^\infty e^{-z t} p_n^\nu (t)\, d t = z^{\nu -1} \frac{\prod _{r=1}^{n-1}\lambda _r}{\prod _{r=1}^n(z^\nu +\lambda _r)}, \qquad n \ge 1. \end{aligned}$$

(5.7)

If the rates are all different, equation (5.7) can be written as

$$\begin{aligned} \mathbb {L}_n(z) = \prod _{r=1}^{n-1}\lambda _r \sum _{m=1}^n \frac{1}{\prod _{l=1,l\ne m}^n(\lambda _l-\lambda _m)} \frac{z^{\nu -1}}{z^\nu +\lambda _m}, \qquad n \ge 1, \end{aligned}$$

(5.8)

that is a more manageable form for the purpose of specializing the rates.

For more insights on the properties of the fractional nonlinear birth process see [32, 34]. Here we conclude this section by recalling an interesting representation of the fractional nonlinear birth process as a time-changed process and by giving some details of the specific case in which the rates are linear. Regarding the first point the fractional nonlinear birth process can be constructed as a classical birth process stopped at an independent random time given by the inverse process to an independent $\nu$-stable subordinator. Notice that stable subordinators are increasing spectrally positive Lévy processes with Lévy measure given by $m(d x)= \left[ \nu /\Gamma (1-\nu ) \right] x^{-1-\nu }d x$. For more details on this last point see [6, 24].

An interesting particular case is when the rates are linear, i.e. $\lambda _r=\lambda r$. Here the state probability distribution (5.4) specializes in the rather simple form

$$\begin{aligned} p_n^\nu (t) = \sum _{j=1}^n \left( {\begin{array}{c}n-1\\ j-1\end{array}}\right) (-1)^{j-1} E_\nu (-\lambda jt^\nu ), \qquad t \ge 0, \, n \ge 1. \end{aligned}$$

(5.9)

The geometric distribution of the linear birth process (also known as Yule or Yule–Furry process and indicated in the paper with $\mathfrak {N}^\nu _{\text {lin}}(t)$, $t \ge 0$) is retrieved from (5.9) if the parameter $\nu$ is taken equal to unity. Properties of the fractional Yule process have been studied in [32, 41] while estimators for the intensity $\lambda$ and the fractional parameter $\nu$ have been derived in [9, 11].