Ex-ante measure of patent quality reveals intrinsic fitness for citation-network growth

We have constructed a fitness parameter, characterizing the intrinsic attractiveness for patents to be cited, from attributes of the associated inventions known at the time a patent is granted. This exogenously obtained fitness is shown to determine the temporal growth of the citation network in conjunction with mechanisms of preferential attachment and obsolescence-induced aging that operate without reference to characteristics of individual patents. Our study opens a window to understanding quantitatively the interplay of the rich-gets-richer and fit-gets-richer paradigms that have been suggested to govern the growth dynamics of real-world complex networks.

Figure 1

Distilling a fitness for citation-network growth from the $N_v=13$ quality indicators in Table 1 for the cohort of USPTO patents from CPC Section A with grant dates during 1999–2001. Factor analysis yields $N_u=6$ factors ${\overline{u}}^{\left(b\right)}$ representing minimally correlated quality-indicator variances across the patent cohort. Panel (a) illustrates the loadings $L_a^{\left(b\right)}$ that relate factor scores ${\overline{u}}_i^{\left(b\right)}$ for individual patents to the normalized raw quality-indicator values [cf. Eq. (2)] via ${\overline{u}}_i^{\left(b\right)}={\sum }_{a=1}^{N_v}{L}_a^{\left(b\right)}{\overline{v}}_i^{\left(a\right)}$. Panel (b) shows the relative weights $w_b$ for each factor in Eq. (5) for the fitness variable, which are obtained by a forward-stepwise regression to the distribution of citations accrued by the time of grant by patents from a small training set. Here factor ${\overline{u}}^{\left(5\right)}$ was eliminated according to the Bayes information criterion.

Figure 2

Aging and preferential attachment in the citation rate of USPTO patents from CPC Section A granted during 1999–2001. Panel (a) shows fits of exponential-aging functions to both the fitness-controlled citation rate ${\lambda }_i/{\eta }_i\equiv [\mathrm{\Delta }{k}_i(t+\mathrm{\Delta }t)/{\eta }_i]/\mathrm{\Delta }t$ (data points indicated by triangles) and the uncontrolled-for-fitness average citation rate ${\overline{\lambda }}_i\equiv \mathrm{\Delta }{k}_i(t+\mathrm{\Delta }t)/\mathrm{\Delta }t$ (data points indicated by circles). Here $t$ corresponds to the age of individual patents counted from the time of grant, and $\mathrm{\Delta }t=3\mathrm{months}$. An example for typical fits of preferential-attachment kernels to the data for both citation rates is provided in panel (b). Panel (c) collates values for the obsolescence time $\tilde{\tau }$ extracted from fits of the right-hand side of Eq. (6a) to ${\lambda }_i/{\eta }_i$ for patents with fixed number $k$ of citations. Similarly, panel (d) gives the exponents $\tilde{\alpha }$ obtained from fitting the right-hand side of Eq. (6b) to ${\lambda }_i/{\eta }_i$ for patents with fixed age $t$. In panels (c) and (d), circles are fitted parameter values, solid lines indicate their weighted averages, and the black dashed (red dotted) curves show the 95% confidence intervals for fit-parameter values (their weighted averages).

Figure 3

Comparing preferential attachment and aging exhibited in the fitness-controlled citation rate ${\lambda }_i/{\eta }_i$ and the uncontrolled-for-fitness average citation rate ${\overline{\lambda }}_i$, respectively. Panel (a) shows both the saturation value ${\tilde{\alpha }}_{\infty }$ [the weighted average of the latest third among the values obtained for $\tilde{\alpha }$ from fits of Eq. (6b) to ${\lambda }_i/{\eta }_i$ for patent cohorts with fixed age $t$; see Fig. 2] and the value ${\tilde{\alpha }}_0$ for $t=0$ (corresponding to the time of grant) alongside $\alpha$ extracted from fitting the uncontrolled-for-fitness average citation rate ${\overline{\lambda }}_i\propto (k_i^{\alpha }+f_0)$. In panel (b), the time $\tilde{\tau }$ ($\tau$) is the weighted average of obsolescence times extracted from fitting Eq. (6a) to fixed-$k$ bins of ${\lambda }_i/{\eta }_i$ as shown in Fig. 2 for Section A patents (fits of ${\overline{\lambda }}_i\propto A_{0}exp\{-t/\tau \}$). Citation data have been analyzed for USPTO patents granted during 1999–2001 in the CPC Sections indicated in the figure. Fitness values ${\eta }_i$ were estimated based on the quality-indicator variables from Table 1 using Eq. (5) with Eq. (2).