The model w↦φ(w) can work reasonably well in scientometrics for social citation systems that are either sufficiently “ordered” or sufficiently “disordered”. In the limit of a large social citation system, we may at least assume that social citation system can be decomposed into a “structured” subsystem and a “stochastic” subsystem. For sake of concreteness, let us depart from the hypothesis that social citation system includes two types of subsystems whose nature is quite different. One of them could be identified as a social network, the other as a scientific market:

The social network (sufficiently structured subsystem) is a polycentric complex of interrelated scholars.

The scientific market (sufficiently stochastic subsystem) contains autonomous scholars who enter into the competition.

The social network is characterized by structural cohesion, while the scientific market is actually an amorphous medium for sharing information resources.

The evolution of the scientific market is of a stochastic nature.

The social network corresponds to the notion of a dynamic system.

The statistical properties of the citation distribution are partially determined by the nature of interactions between scientific market and social network.
Employing the previous notation, the postulated heuristic propositions, on the basis of which our model of the citation distribution is constructed, are as follows:

(1)
In the event horizon where the scientific market “lives”, it can be assumed that the function φ(w) in the expression (5) is invariant under translation of w
$$ \varphi(w+\cdot)=\varphi(w)\varphi(\cdot). $$
((6))

(2)
In the event horizon of the social network the function φ(w) may be intuitively considered as the positive contraction semigroup τ(w) on a real onedimensional Banach space generated by −β
$$ (\beta \in \mathbb{R})\colon \tau(w) = \exp( \beta w). $$
((7))

(3)
The social logic of the citation distribution is such that there is a twoway influence between the scientific market and the social network (Bourdieu 1975). However, in the limit of long time, social effects of the process of citation bring to screening “longrange” interactions. As a result, the subsystems in social citation system are almost independent and we obtain approximate translation invariance
$$ \varphi(w + \cdot) = \varphi(w)\varphi(\cdot) + r(w), $$
((8))
where r(w) indicates a remainder term.
In the framework of previously accepted propositions the following statements are considered:

The simplest and most intuitive general approach to translate invariance is via convolution. Let T
_{
a
} be the translation operator defined by T
_{
a
}
φ(w)=φ(w+a). Translation invariance of the convolution (φ∗χ) means that the convolution with a fixed function χ commutes with T
_{
a
}, i.e.,
$$ T_{a}\left(\varphi \ast \chi\right) = \left(T_{a}\varphi\right) \ast \chi = \varphi \ast \left(T_{a}\chi\right). $$
It can involve explicitly the wellknown Choquet–Deny convolution equation (CDCE for short, see (20)).

By virtue of formula (7), whatever the precise form of r(w) may be it will give to (8) a contribution of the form
$${\lim}_{w\uparrow{1}} r(w) = O\left(\exp( \beta w)\right). $$
This proposition corresponds to a functional equation that can be rewritten as the ICDCE (see (21)).
The translation invariance is an important concept, so it should be understood in a thorough manner. The probability W, of course, corresponds to terminal time, while the RV Z occurs at random in time. Since the RV Z in the scientific market should be independent of an arbitrary translation a, the constancy of termination rate of the process of citation take place in the scientific market. This is what we mean when we say that φ(w) has the translation invariant property (6) in the scientific market.
The motivation of the approximate translation invariance is to take the relation between the scientific market and the social network into consideration. In rough approximation, the scientific market and the social network can be considered as relatively independent. Consequently, their contributions to φ(w) are additive. Summing (6), and (7), we obtain (8), i.e., approximate translation invariance. The Eq. 8 therefore expresses some kind of linear superposition of the effect of the scientific market and the social network. This superposition is not valid in the general case.
To find the citation distribution that we seek, we will start off with certain wellknown mathematical constructions. Let \(\left (\Omega, \mathcal {F}, (\mathcal {F}_{t})_{t\in I}, \mathbf {P}\right)\) be a filtered probability space that satisfies the usual conditions (for details, see Chap. 1 of (Sharpe 1988)). In constructing a model of the citation distribution, we can imagine the social citation system as a normal Markov process X=(X
_{
t
})_{
t∈I
} in a state space \((S,\mathcal {S})\). Insofar as our interest in the social citation system is confined to a few of its features, the Markov processbased model may be relevant in explaining the citation distribution. Further, we shall suppose that the experimentally observed Markov process \(\tilde {X}\) is obtained from X by curtailment of its terminal time up to \(\tilde {\zeta }\)
$$(I \ni \tilde{\zeta}\colon\Omega \rightarrow \mathbb{R}_{+})(t<\tilde{\zeta})\colon \tilde{x}(t,\tilde{\omega}) = x(t,\omega). $$
Equivalently, the process \(\tilde {X}\) is given by a truncation of the duration of the original process X such that the trajectories of X are terminated in a random manner. One can easily see that, for a proper choice of the filtered probability space \(\left (\tilde {\Omega }, \tilde {\mathcal {F}}, {(\tilde {\mathcal {F}}_{t})}_{t\in I}, \tilde {\mathbf {P}}\right)\) and the state space \((S,\mathcal {S})\), the process \(\tilde {X}\) is a subprocess of the process X. The duration of the processes X and \(\tilde {X}\) are denoted by ζ and \(\tilde {\zeta }\), respectively, and
$$ \left(\forall (s, x)\right)\colon \mathbf{P}_{s,x}\left(\tilde{\zeta} \leq \zeta\right) = 1. $$
The construction of such a subprocess is minutely described in (Sharpe 1988 [p. 65–74]).
Under appropriate assumptions, we can represent the Markov process \(\tilde {X}\) using the concept of the multiplicative functional of the Markov process X. This approach is explained in detail in (Sharpe 1988, [p. 286–301]). Let us now introduce the contracting, multiplicative functional \((s\leq t\leq \infty)\colon {m_{t}^{s}}\colon I(\omega)\rightarrow (S,\mathcal {S})\) continuous from the right on X. It is proved (see Theorem 4 in (Gikhman and Skorokhod 2004, [p. 71–72])) that
$$ \begin{aligned} &\left({\Omega^{s}_{t}}=\left\{\omega\colon {s,t}\in I(\omega)\right\} \right) \left(\mathrm{a.s.}\ {\Omega^{s}_{t}}, \mathbf{P}_{s,x}\right)\colon \\ &{m_{t}^{s}} = \tilde{\mathbf{P}}_{s,x}\left(\tilde{\zeta} > t \ \left\vert\, \left(\tilde{\mathcal{F}}^{s} \right)_{s\in I}\right.\right). \end{aligned} $$
((9))
Let \({a_{t}^{s}}\colon I(\omega)\rightarrow (S,\mathcal {S})\) be an additive functional, continuous from the right on X. The formulae \({m_{t}^{s}}=\exp ({a_{t}^{s}})\), \({a_{t}^{s}}=\ln {{m_{t}^{s}}}\) establish a onetoone correspondence between \({a_{t}^{s}}\) and \({m_{t}^{s}}\) (Gikhman and Skorokhod 2004 [p. 64]). It follows in the usual way that,
$$ \left(\mathrm{a.s.}\ \Omega_{t}, \mathbf{P}_{t}\right) \colon \tilde{\mathbf{P}}_{s,x}\left(\tilde{\zeta} > t \ \left\vert\, \left(\tilde{\mathcal{F}}^{s}\right. \right)_{s\in I}\right) = \exp\left({a_{t}^{0}}\right). $$
((10))
In the expression (10), of the quantity
$$\tilde{\mathbf{P}}_{s,x}\left(\tilde{\zeta} > t \ \left\vert\, \left(\tilde{\mathcal{F}}^{s}\right. \right)_{s\in I}\right) $$
can be interpreted as the conditional probability W that the trajectory x(τ) does not terminate during the time interval [0,t]. Moreover, to simplify the argument, we set
$$\left(\forall t\in I(\omega)\right)\colon {a_{t}^{0}} = \vartheta t. $$
Then we immediately verify that,
$$ \tilde{\mathbf{P}}_{x}\left(\left. \tilde{\zeta} > t \right\vert (\mathcal{F}_{t})_{t\in I}\right) = \exp(\vartheta t), $$
((11))
where \(\tilde {\mathbf {P}}_{x}\left (\left. \tilde {\zeta } > t \right \vert (\mathcal {F}_{t})_{t\in I}\right)\) holds for the conditional probability W that the process of citation is of duration longer than t:
$$W \equiv \tilde{\mathbf{P}}_{x}\left(\left. \tilde{\zeta} > t \right\vert (\mathcal{F}_{t})_{t\in I}\right). $$
We assume without essential loss of generality that under a suitable normalization, the RV W has a standard exponential distribution. With the inverse method, we have
It follows from the above that the properties of the distribution P
_{
Z
}(z) depend on w. To be thorough, we must note that the distribution P
_{
Z
}(z) is defined on the probability space \(\left (\mathfrak {Z},\mathcal {B}(\mathfrak {Z}),\mathbf {P}_{Z}\right)\). Obviously, the connections between the Markov process X and the distribution P
_{
Z
}(z) may be based on the concept of the conditional probability W. A somewhat unrealistic, but simple, schematic idea of these connections is given by the equality
$$ \begin{array}{c} \left(\!\left(\forall{z}\in{\mathbb{R}_{+}}\right)\left({\left\{\mathfrak{z}\colon Z(\mathfrak{z})\leq z\right\}}\in{\mathcal{B}(\mathfrak{Z})}\right)\colon Z\colon \mathfrak{Z}\rightarrow\mathbb{R}_{+}\!\right) \\ \left(\varphi(w)\colon\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\right)\colon {z}={\varphi(w)}, \end{array} $$
((13))
where, as in Appendix 1, φ(w) is locally integrable (with respect to the Lebesgue measure Λ). However, the function φ(w) is not yet completely defined. In fact, the general problem of studying the form of φ(w) can be reduced to the case in which this function satisfies certain extra conditions. One can attempt to define φ(w) implicitly by some functional equation rather than by direct definitions. In particular, the general form of φ(w) may be derived uniquely from its invariance.
For the purpose of our study, based upon the denotation introduced in Appendix 1, let μ
^{n} be the nfold convolution of μ, and let φ(w) be a nontrivial positive solution of the ICDCE (21). Observe first that from the paper of (Gu and Lau 1984), we know that for a.a. (mod Λ)\(w\in \mathbb {R}_{+}\), we have the relation
$$\begin{aligned} \varphi(w) &= {\lim}_{n\to\infty}\int_{\mathbb{R}_{+}}\varphi(w + v)\, \mu^{n}\, (dv) \\ &\quad+ \sum_{n=0}^{\infty}\int_{\mathbb{R}_{+}}\varphi(w + v)r(w + v)\, \mu^{n}\, (dv). \end{aligned} $$
Suppose μ is a nonprobability measure. If we take μ without requiring \(\int _{\mathbb {R}_{+}}\mu (dv) = 1\) and, mutatis mutandis, use the arguments employed by (Gu and Lau 1984), we obtain the following expression for φ(w):
$$ \varphi(w) \propto \kappa_{1}\exp(\delta w) + \kappa_{2}\exp(\beta w), $$
((14))
where κ
_{1} and κ
_{2} are constants. It should be mentioned that the definition (13) allows us to write the function φ(w) in an explicit form of the RV Z
$$ Z \propto \kappa_{1}\exp(\delta w) + \kappa_{2}\exp(\beta w). $$
((15))
This expression is the relation we were seeking between the quantities we were interested in, Z and W. As could be expected, the RV Z contains two parts: one corresponds to the incident stream of citations, the other to the scattered stream of citations.
To extract the implications of (15), it is convenient to represent the RV W in terms of the uniform RV U. Now, if we recall the Eq. 12, the expression (15) can be straightforwardly rewritten as
$$ Z \propto \kappa_{1} U^{\delta} + \ \kappa_{2} U^{\beta}. $$
((16))
The study of the relation (16) makes it possible to obtain the PD of the RV Z. Motivated by the approximate translational invariance of z with respect to the probability w that the process of citation does not terminate, we suggest that this model is appropriate to provide a phenomenologically relevant picture of the citation distribution. Finally, starting from the statistical considerations connected with a common and convenient choice of distribution function (Johnson et al 2010, [Chap. 12]), a natural modification of the relation (16) can be written in the form
$$ Z = \upsilon(1  U)^{\delta}  \ \theta (1  U)^{\beta} + k. $$
((17))
The formula (17) defines the distribution, which is called the WD (Johnson et al. 2010, [p. 44–46]). This distribution was established by H. A. Thomas (Houghton 1978) (who lived on Wakeby pond on Cape Cod, Massachusetts) for hydrological data case studies (Griffiths 1989; Hosking and Wallis 2005). We stress that the explicit formula for the PDF of Z is not generally available.
For the sake of being definite, it would be better to rewrite (17) using the following notation
$$\upsilon = \gamma/\delta,\; \theta = \alpha/\beta,\; k = \xi + \theta  \phi. $$
Suppose all parameters α, β, γ, δ, ξ are continuous. Then, the WD becomes
$$ Z = \xi +\frac{\alpha}{\beta}\left(1  (1  U)^{\beta}\right)  \frac{\gamma}{\delta}\left(1  (1  U)^{\delta}\right). $$
((18))
It is readily seen that the WD has three disposable shape parameters, one location parameter and one scale parameter. Under the following conditions:
$$\begin{aligned} (\alpha \neq 0) & \vee (\gamma \neq 0), \\ (\beta + \delta > 0) & \vee (\beta = \gamma = \delta = 0), \\ (\alpha = 0) & \Rightarrow (\beta = 0), \\ (\gamma = 0) & \Rightarrow (\delta = 0), \\ (\gamma \geq 0) & \wedge (\alpha + \beta \geq 0) \end{aligned} $$
the Eq. 18 has a unique solution on dom Z; here
$$ \text{dom}~Z =\left\{ \begin{aligned} &[\xi, \infty)\qquad\qquad\qquad\,\, \text{if} \ (\delta \geq 0) \wedge (\gamma >0), \\ &\left[\xi, \xi + \frac{\alpha}{\beta}  \frac{\gamma}{\delta}\right] \qquad \text{if} \ (\delta > 0) \vee (\gamma = 0). \end{aligned} \right. $$
The WD in (18), when α=0 or γ=0 reduces to the GPD. The Eq. 18 is not very tractable for analysis but can yield efficient algorithms for the numerical simulation of the WD.
Nearly all the papers that deal with inference for the WD are based on the theory of Lmoments (Hosking 1990, 2006; Hosking and Wallis 2005). The free software statistical environment R contains functions to estimate the parameters of the WD from the data (see, e.g., (Asquith 2011), and packages ‘lmom’, ‘lmomco’).