A cooperative game model for the multimodality of coauthorship networks

Abstract

This study provided a game model to simulate the evolution of coauthorship networks, which is a geometric hypergraph built on a circle. A fraction of nodes are randomly selected to attach an arc to express their reputation. The cooperation condition of a new node and existing nodes depends on their distance and the existing nodes’ reputation. The condition gives an expression of kin selection and network reciprocity, two typical mechanisms of cooperation. The size of a node’s reputation is expressed by the length of its arc, which is defined by a function of time and hyperdegree. The function describes the heterogeneity in the size of reputation on nodes and that in the fading speed of reputation on hyperdegrees. The model reveals that the heterogeneities can reproduce the dichotomy of node clustering and degree assortativity, as well as the trichotomy of degree and hyperdegree distributions: generalized Poisson, power-law, and exponential cutoff.

Appendices

Appendix A

Ambiguities exist in coauthorship data, known as merging and splitting errors. Since there are many nodes with a small degree or hyperdegree, these errors do not change the feature of the heads of degree and hyperdegree distributions, as well as the heads of C(k) and N(k). Merging errors can ruin the exponential cutoffs of degree and hyperdegree distributions, because they would generate nodes with a degree far more than ground truth. Splitting errors generate nodes with a degree smaller than ground truth; thus would generate the exponential cutoffs.

Consider a dataset suffering heavy merging errors and free of splitting errors. If the dataset still has exponential cutoffs in its degree and hyperdegree distributions, we could say these distributions of the corresponding ground-truth dataset have an exponential cutoff. In fact, the density function of the summation of two random variables drawn from an exponential distribution still has an exponential factor. Generate entities through short name (surname and the initial of the first given name) and use them to construct networks (PNAS-s, PRE-s). These networks are almost free of splitting errors (unless an author provides a wrong name), but heavily suffer merging errors. Table 1 shows certain statistical indexes of these networks. Figure 6 shows the degree and hyperdegree distributions still have an exponential cutoff, which means the cutoff can be regarded as a feature of the corresponding ground-truth distributions.

Merging errors satisfying specific distributions can generate a power law. Therefore, our conclusions are drawn based on the assumption that the ground-truth distributions of degrees and hyperdegrees have a power-law part, because the considered data are not free of merging errors. The explanation for the features of the tails of C(k) and N(k) is also dependent on this assumption.

To show the reasonability of this assumption, we computed the proportion of short names in the author names of papers, and that of the short names appeared in more than one paper, because using short names will generate a lot of merging errors. Chinese names were also found to account for merging errors (Kim and Diesner 2016). For the names on papers, we calculated the proportion of the names consisting of a surname among major 100 Chinese surnames and a given name less than six characters. The small proportion of such names and the much small proportion of such names appeared in more than one paper imply the impact of merging errors is limited on the considered datasets, especially on the dataset PNAS (Table 3).

Table 3 Specific indexes of empirical data

Fig. 6

The multimodality of the networks generated by using short names. Panels show the multimodality appeared in C(k), N(k), degree and hyperdegree distributions. It means the multimodality is robust to merging errors at certain levels

Appendix B

The hyperedge-size distributions can be fitted by a mixture of a generalized Poisson and a power-law distribution. Let the domains of generalized Poisson $f_1(x)= {a (a+bx)^{x-1}}{ \mathrm {e}^{-a-bx }/{ x!}}$, cross-over, and power-law $f_2(x)=cx^{-d}$ be $[\min (x),E]$, [B, E], and $[B,\max (x)]$ respectively. The fitting function is

$$\begin{aligned} f(x)= {\left\{ \begin{array}{ll} sf_1(x), & \hbox { if}\ x\in [\min (x),B), \\ q(x) s f_1(x) +(1-q(x))f_2(x), & \hbox { if}\ x\in [B,E],\\ f_2(x), & \hbox { if}\ x\in (E,\max (x)], \end{array}\right. } \end{aligned}$$

(6)

where $q(x)=\mathrm {e}^{ - (x -B)/(E-x ) }$. The fitting process is: calculate parameters of $sf_1(x)$ and $f_2(x)$ by regressing the head and tail of a given distribution respectively; find B and E through exhaustion to make f(x) pass the KS test, namely p-value $>0.05$. This value larger than 0.05 means the test cannot reject the hypothesis of following the same distribution at significance level $5\%$. The fitting results are listed in Table 4.

Table 4 The fitting parameters of hyperedge-size distributions

Appendix C

Table 5 shows the boundary detection algorithm for generalized Poisson distributions. The observations $\{D_s, s=1,\ldots ,n\}$ are nodes’ degree or hyperdegree. The input of the algorithm is $h(x)=a(a+bx)^{x-1}\mathrm {e}^{-a-bx}/x!$, where $a, b, x\in {{\mathbb {R}}}$. Changing the algorithm as follows, we can obtain the boundary of power-law distributions with an exponential cutoff. For k from 1 to $\max (D_1,D_2,\ldots ,D_n)$, we fit the density $h(x) \propto x^{-\mu }\mathrm {e}^{-{\nu }{x}}$ (where $\mu , \nu \in {{\mathbb {R}}}$) to the observations not smaller than k until the KS test accepts the null hypothesis that these observations follow the distribution with density h(x). Table 6 shows the fitting results for the heads and tails of degree and hyperdegree distributions.

Table 5 A detection algorithm for the boundary of generalized Poisson distribution

Table 6 Fitting results for the heads and tails of degree and hyperdegree distributions

Table 7 shows the boundary detection algorithm for the average local clustering coefficient of k-degree nodes C(k) and the average degree of k-degree nodes’ neighbors N(k) (Xie et al. 2018). The inputs are $g(\cdot )=\log (\cdot )$, $h(s)=a_1 \mathrm {e}^{-((s-a_2)/a_3)^2}$ for C(k), and $h(s)= a_1 s^3 + a_2 s^2 + a_3 s + a_4$ for N(k), where s, $a_i\in {{\mathbb {R}}}$ ($i=1,\ldots ,4$). Using these inputs is based on the observations on C(k) and N(k).

Table 7 A detection algorithm for the tipping points of C(k) and N(k)

Appendix D

We compared the provided model with two typical models of coauthorship networks (Barabási et al. 2002; Catanzaro et al. 2004), the pseudo codes of which are listed in Tables 8 and 9. The first model captures node clustering, but cannot predict degree assortativity. And the second model captures degree assortativity, but cannot predict node clustering (Table 10). Both of them neglect the aging of nodes, and cannot capture the multimodality of the empirical coauthorship networks (Fig. 7).

Table 8 The pseudo code of the collaboration model in Reference (Barabási et al. 2002)

Table 9 The pseudo code of the assortative model in Reference (Catanzaro et al. 2004)

Table 10 Statistical indexes of two reference networks

Fig. 7

Reference networks’ degree distribution, C(k), and N(k). Panels show that the models in References (Barabási et al. 2002; Catanzaro et al. 2004) cannot predict the multimodality in terms of degree distribution, node clustering, and degree assortativity