A game model for the multimodality phenomena of coauthorship networks
Zheng Xie






A game model for the multimodality phenomena of coauthorship networks





Zheng Xie^1, ♯


1   College of Science,   National University of Defense Technology, Changsha,   China

  ^♯ xiezheng81@nudt.edu.cn
 





§ ABSTRACT













We provided a game model to simulate the evolution of coauthorship networks,  a   geometric hypergraph built on a circle.
  The model
  expresses kin selection  and network reciprocity, two typically  cooperative  mechanisms,   through
 a  cooperation  condition  called  positive  benefit-minus-cost.
 The costs are modelled through space distances. The benefits are modelled through geometric zones that depend on node hyperdegree,
 which gives an expression of the cumulative advantage on
   the  reputations of authors.
   Our findings indicate that
the model gives a reasonable fitting to empirical coauthorship  networks on their
degree  distribution,     node clustering, and so on. It reveals two properties of node attractions,
namely node heterogeneity
  and   fading  with the growth of hyperdegrees, can deduce
 the  dichotomy of nodes'  clustering behavior and    assortativity, as well as
  the trichotomy of degree and hyperdegree distributions: generalized Poisson,
power-law and exponential cutoff.


























§ INTRODUCTION








Analyzing large-scale coauthorship data
  provides a macro-level view of collaboration patterns in
scientific  research, and has become an important topic of social sciences<cit.>.
 Coauthorship relationship can be expressed through  a  hypergraph, where nodes represent authors, and hyperedges express  the author groups  per paper.
The simple graph   extracted from the hypergraph  is termed    coauthorship network, where edges are generated  by connecting each pair of the  nodes
belonging to the same  hyperedge.
Empirical  coauthorship networks are featured by  specific    local   (degree assortativity, high clustering) and global features (fat-tail, small-world)<cit.>.

For general networks, these features  have been reproduced by a range of important models, such as modeling   fat-tail   through   preferential attachment or cumulative advantage<cit.>, modeling   degree assortativity   by connecting two    non-connected nodes that have  similar   degrees<cit.>.
 For coauthorship networks,  can these emerged features be deduced through
cooperative
mechanisms?
The  five typical  mechanisms of cooperations exist in the evolution of  coauthorship networks<cit.>.
 Coauthoring behavior  often occurs in a research group (kin   selection).
 Cooperation  contributes to achieve breakthroughs and reputations (direct and indirect reciprocity).
 Sociable researchers and famous  research teams are easy to attract collaborators (network reciprocity, group selection).









   The inhomogeneity of node influences is an alternative explanation for fat-tail<cit.>. Nodes with wider influences are likely to gain more connections.
   The idea    has been adopted   to simulate the evolution of  coauthorship
  networks,
   in which node influences are modelled by attaching   specific   geometric zones to nodes<cit.>.
  The corresponding models,   geometric hypergraphs,   provide   reasonable  model-data fittings.
             The work in Ref.<cit.>  gives an explanation for   the models' mechanism
    from the perspective  of   cooperative game.
The costs are modelled through  space distances, and the benefits are expressed through node
reputations.   Node
reputations cumulate over time.
The cooperation condition   is positive  benefit-minus-cost.



  We  provided  a game model for coauthorship networks based on   the  cumulative advantage on reputations: supposing reputation to be a function of node hyperdegree (a node's hyperdegree
   is the number of the hyperedges  containing the node).
 The function  has a flexibility to    reproduce  the  trichotomy of      degree and hyperdegree distributions, namely a generalized Poisson
head, a power-law in the middle part, and an exponential tail. In addition, nodes'  clustering behavior and   assortative  behavior are different form  small degree   nodes to large degree nodes.
These dichotomous phenomena are also predicted by the model.
  

This paper is organized as follows: The model and   data are described in Section 2 and 3 respectively;  the multimodality phenomena are discussed
   in Section 4 and 5; Conclusions are drawn in Section 6.





§ THE MODEL




A  cooperative game  consists  a set of players  and  a
characteristic function specifying the value (benefit-minus-cost) created by  subsets of
players in the game.  Coauthorship networks can be regarded as  the results of such a game.
Researchers    can be viewed as players.    Coauthoring a researcher with high    reputation is a kind of   benefits, contributing to  academic success.
The  investments of  manpower and material resources
 on   research  are the  costs of cooperations.
We  presented an implement of the game
of coauthoring behavior
through a geometric hypergraph,     expressing
 the    reputations and   costs     through a function of   hyperdegree  and   that of   geometric distance  respectively.





The model  is built   on  a circle S^1 as follows.
 Sprinkle   a set  of nodes  N_t
on S^1   uniformly and randomly at each time t∈ [1,T], where T∈ℤ^+.
  Select a subset  N^l_t from N_t randomly  as lead nodes.
  Each
 lead  node i   is assigned an arc with centre θ_i to    imitate  its    reputation. The length of the  arc is 
    r_i(t) = α/t(h_i(t)+1)^e^-β(k_i(t)+1),

where h_i(t) is the hyperdegree of node i at time t,  α and β∈ℝ^+.
 The group of the nodes covered by a lead node's   arc  expresses
a   research team.

Coauthoring behavior is expressed through
 hyperedges generated as follows.   For each new node j(θ_j,t_j)∈ N_t, select a lead node set M^l_j for which   ∀ i (θ_i,t_i)∈ M^l_j satisfies r(i)>π-|π-|θ_i-θ_j|| and t_i<t_j. It means node j's reputation is larger than the angular distance between i and j,
  an expression of  positive    benefit-minus-cost.
For  each  i∈ M^l_j, we append j to R_i (a set used to model  node i's  research team),
and  generate a hyperedge at probability p by grouping   i, j, (min(x,|R_i|)-2)  nodes of R_i   nearest to j, and
(x-min(x,|R_i|))     nodes  ∉R_i randomly,  where   x is the random variable  of an inputting distribution,    a fitting of   the empirical distributions of  hyperedge sizes.



The  way of generating hyperedges  describes a usual  scene of cooperation.
A researcher   wants to complete a  work, which needs several researchers to work together.
Then he   would ask  his team leader    for help.  The leader would  suggest some team members
  with similar interest to work together.
Such   behavior can be viewed as kin selection.
When finishing the  work is beyond the ability of  the   team, the researcher   would ask for external helpers. This  inspires the design of randomly choosing  (x-min(x, |R_j|)  nodes outside of R_j
 to cooperate.





The random variable x, used to model hyperedge size, is generated as follows.
Give   the upper bound of small research team  μ>0, and the lower bound   of large research team  ν>0.
 Denote the expected value of hyperedge sizes and the size of
R_i   to be η and λ.
Let  η    = min{λ  ,μ} if   λ≤ν, and
draw η  from a power law distribution with  an exponent γ and   domain [μ,λ] if   λ>ν.
Draw
  x   from a Poisson distribution with  the expected value η.










We   assumed each lead node has the same attraction to new nodes,  and so let the reputation of  a lead node i (θ_i,t_i)  be
     r( i)∝ 1/t_i (which is   inherited from the PSO  model<cit.>).
       The novel aspect we introduced is
 the fading   heterogeneity  over  hyperdegrees.
 When a node's   zonal size
  fades to a value free of its hyperdegree, the  probability  of receiving connections is the same as that of the standard random   graph,   which   has an
exponentially decaying hyperdegree distribution. In contrast,
if its hyperdegree is not   large enough, then cumulative advantage is significant, which gives rise to the power law
  part of  hyperdegree distributions.






 Compared with the model in Ref.<cit.>, our model removes the  hypothesis of  the arc-length being a power function of nodes' birth time,
  thus gives a   direct simulation  of the  cumulative advantage for   hyperdegrees and    degrees. It  expresses
  a typically  cooperative mechanism, namely  network reciprocity. In addition,  compared with that previous  model, our  model has the   ability to predict the the transition from  power law to exponential cutoff for     hyperdegree and degree
distributions, thus   fully reproduces the trichotomy of these  distributions.

















§ THE DATA



We analyzed two   empirical coauthorship datasets  collected  from Web of Science (www.webofscience.com).
  Dataset  PNAS  is composed of  36,732     papers published in   Proceedings of the National Academy of
Sciences during 2007–2015.
Dataset PRE comprises 24,079 papers published in Physical Review E    during  2007-2016.
The different collaboration level (reflected by the average number of the authors per paper) of the two datasets (PNAS 6.712, PRE 3.102)
helps to test  model flexibility.

 The entities of persons are identified by    authors' name on their  papers, which constitute
 the nodes of the empirical networks analyzed  here.  Table <ref> shows certain statistical indexes for these networks.
Note that these networks
suffer errors: one person is identified as two
or more entities (splitting error); two or more persons are identified as one entity (merging error).
The analysis in Appendix A shows    the multimodality phenomena considered here  are robust to these errors at certain levels.

 




We modelled two hypergraphs to reproduce   the multimodality phenomena  emerged in the empirical  networks.
The reason   of choosing their parameters (Table <ref>) is as follow.
 We valued μ through the expected value of the general Poisson part of an empirical distribution of hyperedge sizes (Fig. <ref>), and
  valued ν through  iteration from  the starting point   of the power-law part of the distribution until the distribution of modelled hyperedge sizes is similar to  the empirical
one.
We
 valued    α,  |N_t|, |N^l_t| and p to make  the hyperdegrees of substantial nodes be one.

 






  





 

§ THE TRICHOTOMOUS   DISTRIBUTIONS OF DEGREES AND   HYPERDEGREES

 








The trichotomy of  degree and  hyperdegree   distributions  comprises a generalized Poisson head, a power-law in the middle part, and an
exponential tail (Fig. <ref>).
 The power law parts are fitted through the  method of Clauset et al<cit.>.
  The generalized Poisson distributions and the   power law  functions   with an exponential cutoff are  fitted through  maximum-likelihood  estimation. The  fitting process and     goodness-of-fit
  are shown    in Appendix C.

 

Generalized Poisson distributions can be derived from  a range of “coin flipping" behaviors, where  the    probability of observing “head"     is dependent on observed events<cit.>.
The event of publishing a paper  is kind of observing “head" (reject or accept),
where the probability of publishing
  is also affected by previous
events. For example, an  author would easily write his second paper, compared with his first one.
There is a cumulative advantage in writing papers, writing
experiments   accumulating  in the process of publishing papers.
In hyperdegree distributions, the  process
reflects as the transition from  generated Poisson  distribution  to     power law.
Meanwhile, aging  is against cumulative advantage, which reflects as the transition from  power law    to    exponential cutoff.




Fig. <ref> shows our model can fully reproduce the trichotomy.
The modelled hyperedges are generated through an inhomogeneous  Poisson  point process (namely its density of points is not a constant), because
the probability  of new nodes fallen   in a lead node's zone   depends on the lead node's hyperdegree.
     At each time t, the expected number of new hyperedges    r_i(t)m/2π  is a function of h_i(t), the  real-time hyperdegree at time t.
     Therefore,
the hyperdegree h_i(T)    follows  a generalized Poisson distribution.
When h_i(T) is sufficiently large and β(h_i(t)+1)≪1, the formula (<ref>) gives rise to

    d/dth_i(t)=m/2πr_i(t) ≈λ/  t  (h_i(t)+1)   ,

  where  λ= α m/2π.
The solution to Eq. (<ref>)
gives node i's expected hyperdegree   h̅_i(T)=(T/t_i)^λ-1, which   gives rise to    p(h̅_i(T)≤ h) =  p (t_i ≥    T/(h+1)^1/λ  ).
Based on the mean-field theory, we   deduced its density function as follows.
 The probability
 of a  node  generated at t_i is 1/T. Hence
 p (t_i   ≥    T (h+1)^-1/λ ) =1-p (t_i  <    T (h+1)^ -1 / λ )=1-   (h+1)^- 1/λ, which gives rise to
a power-law distribution
 
    p(h)=d /d hp(h̃_i(T) ≤   h) ∝  (h+1)^-1-1/λ≈ h^-1-1/λ
 for  sufficiently large h.
It shows
the transition from   generalized Poisson distribution   to    power law.




When β(h_i(t)+1)≫1,  the formula (<ref>) gives rise to

    d/dth_i(t)=m/2πr_i(t) ≈λ/  t  .

The solution to Eq. (<ref>)
gives node i's expected hyperdegree  h̅_i(T)= λlog (T/t_i)+δ,
where δ is the    hyperdegree accumulated from  the process governed by Eq. (<ref>).
It   yields   p(h̅_i(T)≤ h) =
  p (t_i ≥    T e^-  (h-δ)/λ  ).
 Hence
 p (t_i  ≥ T e^- (h-δ)/λ  ) =1-p (t_i  <    T e^-  (h-δ)/λ )=1-  e^-  (h-δ)/λ. It gives rise to 
    p(h)=d/dh p(h̅_i(T) ≤   h) ∝1/λe^- h-δ/λ ,
 which  is an exponential distribution on the interval [δ,+∞).
 Therefore, we can expect  a   power law with an exponential cutoff  for  sufficiently large  hyperdegrees, namely p(h ) ∝  h^-1- 1/λe^-h/λ.




Now we turn to the trichotomy of degree distributions (Fig. <ref>).
The hyperdegrees of many nodes are equal to   one    (PNAS: 74.0%, PRE: 63.9%).
As Fig. <ref> shows, most of hyperedge sizes follow a generated Poisson distribution (PNAS: 99.9%,  PRE: 99.9%).
Those lead the  generalized Poisson parts of degree distributions.
Meanwhile,  nodes' hyperdegree  positively correlates to their degree (Fig. <ref>),
  the growing   process of hyperdegrees coupling  with that of degrees.
The positive correlation leads   the multimodality of degree distributions.
Note that  the nodes of  hyperedges with a large size   also have a large degree,
which leads    the outliers in the tails of degree distributions.







In synthetic data,
The hyperdegrees of a large fraction of   nodes are equal to one (Synthetic-1: 47.0%, Synthetic-2: 60.7%). The   sizes of most   modelled hyperedges follow  a generalized   Poisson distribution (Synthetic-1: 99.8%, Synthetic-2: 99.7%). Those yield the generalized   Poisson part of   modelled degree distributions.
Meanwhile, the
 model preserves  the positive correlation between   hyperdegrees and degrees, especially h(t)∝ k(t) for the nodes with a large hyperdegree.
Therefore, it can reproduce   the multimodality of degree distributions.





§ THE DICHOTOMOUS PHENOMENON  IN  CLUSTERING AND   ASSORTATIVITY




Coauthorship networks are found to have
two features: high   clustering  (a high probability of a node's two neighbors connecting) and   degree assortativity (nodes' degree positively correlates to their average degree of  neighbors), which are reflected through  the high values of their global clustering coefficient  and
 assortive coefficient (Table <ref>).

Observing
these features over degrees, we can find that they  differ from   small degree nodes   to
large degree nodes.
Consider   the  average local  clustering coefficient of k-degree nodes C(k) and  the average degree of k-degree nodes'  neighbors  N(k).
We can find the  dichotomy of  the functions C(k) and  N(k) (Fig. <ref>).




The dichotomy is due to
the positive correlation between hyperdegrees and degrees.
  A large fraction of  nodes with a small degree only belong to  a few hyperedges.
  Their neighbors    probably connect to  each other, and only a few
   of their neighbors have a large degree.
  Therefore, over small k,
  the value of C(k) is high,     and the slope of N(k) is positive.
    Consider the  small fraction of nodes  with a large degree.
These nodes   probably have a large hyperdegree.
Their neighbors in different hyperedges  probably do not connect with each other and have a small degree. Otherwise, there will be many nodes with a large degree, out of touch with reality.
  Therefore,  the slopes   of C(k)      and   N(k) over large k  are negative.
 Our
model   captures   the positive correlation, thus gives a reasonable fit to the dichotomy.





§ DISCUSSION AND CONCLUSIONS





 A   hypergraph model is provided to
simulate the evolution of coauthorship networks from the perspective of    cooperative game, which
expresses two   cooperative mechanisms,   kin selection
and network reciprocity, in a geometric way.
It predicts  a
 range of  features of coauthorship networks, such as fat-tail, small-world, etc.
Especially,
it reproduces the trichotomous distributions for degrees and  hyperdegrees,
as well as the dichotomous phenomena in clustering and  assortativity.
 It overcomes the  weakness of the   model in Ref.<cit.>, providing a direct simulation of  the cumulative
 advantage on writing experiments.





The model provides an example of how
individual  strategies based on
positive benefit-minus-cost and on specific randomness
generate the   complexity emerged
in coauthorship networks. It  can be extended to analyze the coevolution with citation behavior<cit.>.
 It also has the potential to be a null model in the empirical analysis
of social affiliation networks.
 Whereas   other typically  cooperative mechanisms should be addressed, such as group selection, direct and indirect reciprocity.








§ ACKNOWLEDGMENTS
  This work is supported by    National   Science Foundation of China (Grant No. 61773020).


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§ 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 cutoff, because they generate  nodes with a  degree
far more   than   ground truth.
Splitting errors generate nodes with a degree smaller than ground truth, thus would generate exponential cutoff.

Consider   a network suffering heavy merging errors and free of splitting errors.
If the network
  has an exponential cutoff in its degree and hyperdegree distribution,
we could say
these   distributions of the corresponding ground-truth network has
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 <ref> shows certain statistical indexes of these networks.
   Fig. <ref> shows the degree   and hyperdegree   distribution
  still have an exponential cutoff, which means    the  cutoff exists in the corresponding ground-truth
    distributions.



Merging errors satisfying specific  distribution can   generate  a power law. Therefore,
our conclusions are drawn under the assumption that the ground-truth coauthorship networks have
a power law part in their degree and hyperdegree distribution,
 because the considered data  are not free 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 names on papers,
and that of the short   names   appearing in  more than one paper, because
  using short names will   generate  a lot of merging errors<cit.>.
 Chinese names were also found to
account for merging errors<cit.>. For the names on papers, we counted the
proportion of the names  consisting of  a surname
among major 100 Chinese surnames and a given name less than six characters.   The small proportions of such   names
and
those of such names   appearing in  more than one paper
  imply
  the impacts of merging errors are limited, especially for the dataset PNAS (Table <ref>).


 









§ APPENDIX B


The empirical distribution of hyperedge sizes can be fitted by a mixture  of a generalized Poisson distribution   and a power law.
Let
the domains of  generalized Poisson  f_1(x)= a (a+bx)^x-1e^-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 defined on [min(x),max(x)] is f(x)=q(x)      s f_1(x) +(1-q(x))f_2(x), where
  q(x)=e^ - (x -B)/(E-x ).    The fitting processes are:
  calculate parameters of sf_1(x)   and f_2(x)  by regressing  the head and tail of  empirical  distribution  respectively;      find  B and E through exhaustion    to  make f(x) pass  the Kolmogorov-Smirnov (KS)   test (p-value>0.05).  The fitting results are listed in Table <ref>.











§ APPENDIX C


Table <ref> shows the
  boundary    detection  algorithm for   generalized Poisson distributions.
  The  observations  {D_s, s=1,...,n} are nodes' degree  or hyperdegree.
 Changing the algorithm as follows, we can obtain the
  boundary of  the power law functions
with an exponential cutoff. For   k from 1 to max(D_1,...,D_n),
we fit   h(·)  to   the density h_0(·) of    {D_s, s=1,...,n|D_s ≥ k},
until    the KS test accepts the null hypothesis.
  Table <ref> shows the corresponding goodness-of-fit.
















Table <ref> 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).
The inputs are g(·)=log(·), h(s)=a_1 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∈ℝ (i=1,...,4).
  Using  these inputs is based on   the observations on C(k) and N(k).






