Generative Mechanism Estimation in Temporal Complex Networks

Description

A package for estimating preferential attachment and node fitness generative mechanisms in temporal complex networks. References: Thong Pham et al. (2015) <10.1371/journal.pone.0137796>, Thong Pham et al. (2016) <doi:10.1038/srep32558>, Thong Pham et al. (2020) <doi:10.18637/jss.v092.i03>, Thong Pham et al. (2021) <doi:10.1093/comnet/cnab024>.

Details

The PAFit package provides a comprehensive framework to deal with growth mechanisms of temporal complex networks. In particular, it implements functions to simulate various temporal network models, gather essential network statistics from raw input data, and use these summarized statistics in the estimation of the attachment function A_k and node fitnesses \eta_i. The heavy computational parts of the package are implemented in C++ through the use of the Rcpp package. Furthermore, users with a multi-core machine can enjoy a hassle-free speed up through OpenMP parallelization mechanisms implemented in the code. Apart from the main functions, the package also includes a real-world collaboration network dataset between scientists in the field of complex networks (coauthor.net). The main package functionalities are as follows.

Firstly, most well-known temporal network models based on the preferential attachment (PA) and node fitness mechanisms can be easily simulated using the package. PAFit implements generate_BA for the Barabási-Albert (BA) model, generate_ER for the growing Erdős–Rényi (ER) model, generate_BB for the Bianconi-Barabási (BB) model and generate_fit_only for the Caldarelli model. These functions have many customizable options, for example the number of new edges at each time-step are tunable stochastic variables. They are actually wrappers of the more powerful generate_net function, which simulates networks with more flexible attachment function and node fitness settings.

Secondly, the function get_statistics efficiently collects all temporal network summary statistics. We note that get_statistics automatically handles both directed and undirected networks. It returns a list containing many statistics that can be used to characterize the network growth process. Notable fields are m_tk containing the number of new edges that connect to a degree-k node at time-step t, and node_degree containing the degree sequence, i.e., the degree of each node at each time-step.

The most important functionality of the package is estimating the attachment function and node fitnesses of a temporal network. This is implemented through various methods. There are three usages: estimation of the attachment function in isolation, estimation of the node fitnesses in isolation, and the joint estimation of the attachment function and node fitnesses.

• The functions for estimating the attachment function in isolation are: Jeong for Jeong's method (Ref. 1), Newman for Newman's method (Ref. 2), and only_A_estimate for the PAFit method (Ref. 3).

• For estimation of node fitnesses in isolation, only_F_estimate implements a variant of the PAFit method (Ref. 4).

• For the joint estimation of the attachment function and node fitnesses, we implement the full version of the PAFit method in joint_estimate (Ref. 4).

• For estimating the nonparametric attachment function from a single snapshot, use PAFit_oneshot (Ref. 6).

Excluding PAFit_oneshot, the input of the remaining functions is the output object of the function get_statistics. The output object of these functions contains the estimation results as well as some additional information pertaining to the estimation process. The estimated attachment function and/or node fitnesses can be plotted by using the plot command directly on this output object. This will visualize not only the estimated results but also the remaining uncertainties when possible.

Author(s)

Thong Pham thongphamthe@gmail.com, Paul Sheridan, and Hidetoshi Shimodaira.

References

1. Jeong, H., Néda, Z. & Barabási, A. (2003). Measuring Preferential Attachment in Evolving Networks. Europhysics Letters 61(61):567-572. (doi:10.1209/epl/i2003-00166-9).

2. Newman, M. (2001). Clustering and Preferential Attachment in Growing Networks. Physical Review E 64(2):025102. (doi:10.1103/PhysRevE.64.025102).

3. Pham, T., Sheridan, P. & Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLOS ONE 10(9):e0137796. (doi:10.1371/journal.pone.0137796).

4. Pham, T., Sheridan, P. & Shimodaira, H. (2016). Joint Estimation of Preferential Attachment and Node Fitness in Growing Complex Networks. Scientific Reports 6, Article number: 32558. (doi:10.1038/srep32558).

5. Pham, T., Sheridan, P. & Shimodaira, H. (2020). PAFit: An R Package for the Non-Parametric Estimation of Preferential Attachment and Node Fitness in Temporal Complex Networks. Journal of Statistical Software 92 (3). (doi:10.18637/jss.v092.i03)

6. Pham, T., Sheridan, P. & Shimodaira, H. (2021). Non-parametric estimation of the preferential attachment function from one network snapshot. Journal of Complex Networks 9(5): cnab024. (doi:10.1093/comnet/cnab024).

See Also

See the accompanying vignette for a tutorial.

See also the GitHub page.

Examples

## Not run: 
  ### Jointly estimate the attachment function and node fitnesses
   library("PAFit")
   set.seed(1)
  # a Bianconi-Barabasi network 
  # size of initial network = 100
  # number of new nodes at each time-step = 100
  # Ak = k; inverse variance of distribution of fitness: s = 10
  net        <- generate_BB(N        = 1000 , m             = 10 , 
                            num_seed = 100  , multiple_node = 100,
                            s        = 10)
  net_stats  <- get_statistics(net)
  
  #Joint estimation of attachment function Ak and node fitness
  result     <- joint_estimate(net, net_stats)
  
  summary(result)
  
  # plot the estimated attachment function
  plot(result, net_stats)
  
  # true function
  true_A     <- pmax(result$estimate_result$center_k,1)
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  #plot distribution of estimated node fitnesses
  plot(result, net_stats, plot = "f")
  
  #plot the estimated node fitnesses and true node fitnesses
  plot(result, net_stats, true = net$fitness, plot = "true_f")

## End(Not run)

A collaboration network between authors of papers in the field of complex networks with article time-stamps

Description

The dataset is collaboration network of authors of network science articles with article time-stamps. An edge between two authors represents an article in common. Time stamps denote article publication dates. The network without time-stamps was compiled by Mark Newman in May 2006 from the bibliographies of two review articles on networks, M. E. J. Newman, SIAM Review 45, 167-256 (2003) and S. Boccaletti et al., Physics Reports 424, 175-308 (2006), with a few additional references added by hand. Paul Sheridan independently supplemented the network with time-stamps and some basic metadata in June 2015. The network is undirected with monthly resolution, and contains no duplicated edges. coauthor.net contains the network. coauthor.truetime contains the real times of processed time-stamps. Finally coauthor.author_id contains author names.

Reference: M. E. J. Newman, Finding community structure in networks using the eigenvectors of matrices, Preprint physics/0605087 (2006).

Usage

data(ComplexNetCoauthor)

Format

coauthor.net is a matrix with 2849 rows and 3 columns. Each row is an edge with the format (author id 1, author id 2, time_stamp). coauthor.truetime is a two-column matrix whose each row is (time_stamp, real time). coauthor.author_id is a two-column matrix whose each row is (author id, author name).

Source

https://www.paulsheridan.net/files/collabnet.zip

Jeong's method for estimating the preferential attachment function

Description

This function estimates the preferential attachment function by Jeong's method.

Usage

Jeong(net_object                               , 
      net_stat  = get_statistics(net_object)   , 
      T_0_start = 0                            ,
      T_0_end   = round(net_stat$T * 0.75)     ,
      T_1_start = T_0_end + 1                  ,
      T_1_end   = net_stat$T                   ,
      interpolate = FALSE)

Arguments

Value

Outputs an PA_result object which contains the estimated attachment function. In particular, it contains the following field:

• k and A: a degree vector and the estimated PA function.

• center_k and theta: when we perform binning, these are the centers of the bins and the estimated PA values for those bins.

• g: the number of bins used.

• alpha and ci: alpha is the estimated attachment exponenet \alpha (when assume A_k = k^\alpha), while ci is the confidence interval.

• loglinear_fit: this is the fitting result when we estimate \alpha.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Jeong, H., Néda, Z. & Barabási, A. . Measuring preferential attachment in evolving networks. Europhysics Letters. 2003;61(61):567–572. (doi:10.1209/epl/i2003-00166-9).

See Also

See get_statistics for how to create summerized statistics needed in this function.

See Newman and only_A_estimate for other methods to estimate the attachment function in isolation.

Examples

  library("PAFit")
  net        <- generate_net(N = 1000 , m = 1 , mode = 1 , alpha = 1 , s = 0)
  net_stats  <- get_statistics(net)
  result     <- Jeong(net, net_stats)
  # true function
  true_A     <- result$center_k
  #plot the estimated attachment function
  plot(result , net_stats)
  lines(result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")

Corrected Newman's method for estimating the preferential attachment function

Description

This function implements a correction proposed in [1] of the original Newman's method in [2] to estimate the preferential attachment function.

Usage

  Newman(net_object                              , 
         net_stat    = get_statistics(net_object), 
         start       = 1                         , 
         interpolate = FALSE)

Arguments

Value

Outputs an PA_result object which contains the estimated attachment function. In particular, it contains the following field:

• k and A: a degree vector and the estimated PA function.

• center_k and theta: when we perform binning, these are the centers of the bins and the estimated PA values for those bins.

• g: the number of bins used.

• alpha and ci: alpha is the estimated attachment exponenet \alpha (when assume A_k = k^\alpha), while ci is the mean plus/minus two-standard-deviation interval.

• loglinear_fit: this is the fitting result when we estimate \alpha.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE 10(9): e0137796. (doi:10.1371/journal.pone.0137796).

2. Newman, M.. Clustering and preferential attachment in growing networks. Physical Review E. 2001;64(2):025102 (doi:10.1103/PhysRevE.64.025102).

See Also

See get_statistics for how to create summerized statistics needed in this function.

See Jeong, only_A_estimate for other methods to estimate the attachment function in isolation.

Examples

  library("PAFit")
  net        <- generate_net(N = 1000 , m = 1 , mode = 1 , alpha = 1 , s = 0)
  net_stats  <- get_statistics(net)
  result     <- Newman(net, net_stats)
  summary(result)
  # true function
  true_A     <- result$center_k
  #plot the estimated attachment function
  plot(result , net_stats)
  lines(result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")

Estimating the nonparametric preferential attachment function from one single snapshot.

Description

This function estimates the attachment function A_k from one snapshot.

Usage

PAFit_oneshot(net_object, 
              M    = 10,
              S    = 5,
              loop = 5,
              G    = 1000)

Arguments

Value

Outputs a PAFit_result object.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2021). Non-parametric estimation of the preferential attachment function from one network snapshot. Journal of Complex Networks 9(5): cnab024. (doi:10.1093/comnet/cnab024).

Examples

## Not run: 
  library("PAFit")
  net_1    <- generate_BA(N = 10000, alpha = 1) # true attachment exponent = 1.0
  result_1 <- PAFit_oneshot(net_1)
  print(result_1)

  
  net_2    <- generate_BA(N = 10000, alpha = 0.5) # true attachment exponent = 0.5
  result_2 <- PAFit_oneshot(net_2)
  print(result_2)
  
## End(Not run)

Converting an edgelist matrix to a PAFit_net object

Description

This function converts a graph stored in an edgelist matrix format to a PAFit_net object.

Usage

as.PAFit_net(graph, type = "directed", PA = NULL, fitness = NULL)

Arguments

Value

An object of class PAFit_net

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

library("PAFit")
# a network from Bianconi-Barabasi model
net        <- generate_BB(N = 50 , m = 10 , s = 10)
as.PAFit_net(net$graph)

Convert an igraph object to a PAFit_net object

Description

This function converts an igraph object (of package igraph) to a PAFit_net object.

Usage

  from_igraph(net)

Arguments

Value

The function returns a PAFit_net object.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  # a network from Bianconi-Barabasi model
  net          <- generate_BB(N = 50 , m = 10 , s = 10)
  igraph_graph <- to_igraph(net)
  back         <- from_igraph(igraph_graph)

Convert a networkDynamic object to a PAFit_net object

Description

This function converts a networkDynamic object (of package networkDynamic) to a PAFit_net object.

Usage

  from_networkDynamic(net)

Arguments

Value

The function returns a PAFit_net object.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

library("PAFit")
# a network from Bianconi-Barabasi model
net          <- generate_BB(N = 50 , m = 10 , s = 10)
nD_graph     <- to_networkDynamic(net)
back         <- from_networkDynamic(nD_graph)

Simulating networks from the generalized Barabasi-Albert model

Description

This function generates networks from the generalized Barabási-Albert model. In this model, the preferential attachment function is power-law, i.e. A_k = k^\alpha, and node fitnesses are all equal to 1. It is a wrapper of the more powerful function generate_net.

Usage

generate_BA(N              = 1000, 
            num_seed       = 2   , 
            multiple_node  = 1   , 
            m              = 1   ,
            alpha          = 1)

Arguments

Value

The output is a PAFit_net object, which is a List contains the following four fields:

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Albert, R. & Barabási, A. (1999). Emergence of scaling in random networks. Science, 286,509–512 (https://www.science.org/doi/10.1126/science.286.5439.509).

See Also

For subsequent estimation procedures, see get_statistics.

For other functions to generate networks, see generate_net, generate_ER, generate_BB and generate_fit_only.

Examples

  library("PAFit")
  # generate a network from the BA model with alpha = 1, N = 100, m = 1
  net <- generate_BA(N = 100)
  str(net)
  plot(net)

Simulating networks from the Bianconi-Barabasi model

Description

This function generates networks from the Bianconi-Barabási model. It is a ‘preferential attachment with fitness’ model. In this model, the preferential attachment function is linear, i.e. A_k = k, and node fitnesses are sampled from some probability distribution.

Usage

generate_BB(N              = 1000   , 
            num_seed       = 2      , 
            multiple_node  = 1      , 
            m              = 1      ,
            mode_f         = "gamma", 
            s              = 10     )

Arguments

The parameters can be divided into two groups.

The first group specifies basic properties of the network:

The final group of parameters specifies the distribution from which node fitnesses are generated:

Value

The output is a PAFit_net object, which is a List contains the following four fields:

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Bianconni, G. & Barabási, A. (2001). Competition and multiscaling in evolving networks. Europhys. Lett., 54, 436 (doi:10.1209/epl/i2001-00260-6).

See Also

For subsequent estimation procedures, see get_statistics.

For other functions to generate networks, see generate_net, generate_BA, generate_ER and generate_fit_only.

Examples

  library("PAFit")
  # generate a network from the BB model with alpha = 1, N = 100, m = 1
  # The inverse variance of the Gamma distribution of node fitnesses is s = 10
  net <- generate_BB(N = 100,m = 1,mode = 1, s = 10)
  str(net)
  plot(net)

Simulating networks from the Erdos-Renyi model

Description

This function generates networks from the Erdős–Rényi model. In this model, the preferential attachment function is a constant function, i.e. A_k = 1, and node fitnesses are all equal to 1. It is a wrapper of the more powerful function generate_net.

Usage

  generate_ER(N              = 1000, 
              num_seed       = 2   , 
              multiple_node  = 1   , 
              m              = 1)

Arguments

Value

The output is a PAFit_net object, which is a List contains the following four fields:

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Erdös P. & Rényi A.. On random graphs. Publicationes Mathematicae Debrecen. 1959;6:290–297 (https://snap.stanford.edu/class/cs224w-readings/erdos59random.pdf).

See Also

For subsequent estimation procedures, see get_statistics.

For other functions to generate networks, see generate_net, generate_BA, generate_BB and generate_fit_only.

Examples

  library("PAFit")
  # generate a network from the ER model with N = 1000 nodes
  net <- generate_ER(N = 1000)
  str(net)
  plot(net)

Simulating networks from the Caldarelli model

Description

This function generates networks from the Caldarelli model. In this model, the preferential attachment function is constant, i.e. A_k = 1, and node fitnesses are sampled from some probability distribution.

Usage

generate_fit_only(N             = 1000   , 
                 num_seed       = 2      , 
                 multiple_node  = 1      , 
                 m              = 1      ,
                 mode_f         = "gamma", 
                 s              = 10     )

Arguments

The parameters can be divided into two groups.

The first group specifies basic properties of the network:

The final group of parameters specifies the distribution from which node fitnesses are generated:

Value

The output is a PAFit_net object, which is a List contains the following four fields:

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Caldarelli, G., Capocci, A. , De Los Rios, P. & Muñoz, M.A. (2002). Scale-Free Networks from Varying Vertex Intrinsic Fitness. Phys. Rev. Lett., 89, 258702 (doi:10.1103/PhysRevLett.89.258702).

See Also

For subsequent estimation procedures, see get_statistics.

For other functions to generate networks, see generate_net, generate_BA, generate_ER and generate_BB.

Examples

  library("PAFit")
  # generate a network from the Caldarelli model with alpha = 1, N = 100, m = 1
  # the inverse variance of distribution of node fitnesses is s = 10
  net <- generate_fit_only(N = 100,m = 1,mode = 1, s = 10)
  str(net)
  plot(net)

Simulating networks from preferential attachment and fitness mechanisms

Description

This function generates networks from the General Temporal model, a generative temporal network model that includes many well-known models such as the Erdős–Rényi model, the Barabási-Albert model or the Bianconi-Barabási model as special cases. This function also includes some flexible mechanisms to vary the number of new nodes and new edges at each time-step in order to generate realistic networks.

Usage

generate_net (N                 = 1000   , 
             num_seed           = 2      , 
             multiple_node      = 1      , 
             specific_start     = NULL   ,
             m                  = 1      ,
             prob_m             = FALSE  ,
             increase           = FALSE  , 
             log                = FALSE  , 
             no_new_node_step   = 0      ,
             m_no_new_node_step = m      ,
             custom_PA          = NULL   ,
             mode               = 1      , 
             alpha              = 1      , 
             beta               = 2      , 
             sat_at             = 100    ,
             offset             = 1      ,
             mode_f             = "gamma", 
             s                  = 10       )

Arguments

The parameters can be divided into four groups.

The first group specifies basic properties of the network:

The second group specifies the number of new edges at each time-step:

The third group of parameters specifies the preferential attachment function:

The final group of parameters specifies the distribution from which node fitnesses are generated:

Value

The output is a PAFit_net object, which is a List contains the following four fields:

Author(s)

Thong Pham thongphamthe@gmail.com

See Also

For subsequent estimation procedures, see get_statistics.

For simpler functions to generate networks from well-known models, see generate_BA, generate_ER, generate_BB and generate_fit_only.

Examples

library("PAFit")
#Generate a network from the original BA model with alpha = 1, N = 100, m = 1
net <- generate_net(N = 100,m = 1,mode = 1, alpha = 1, s = 0)
str(net)
plot(net)

Generating simulated data from a fitted model

Description

This function generates simulated networks from a fitted model and performs estimations on these simulated networks with the same setting used in the original estimation. Each simulated network is generated using parameters of the fitted model, while keeping other aspects of the growth process as faithfully as possible to the original observed network.

Usage

generate_simulated_data_from_estimated_model(net_object, net_stat, result, M = 5)

Arguments

Value

Outputs a Simulated_Data_From_Fitted_Model object, which is a list containing the following fields:

• graph_list: a list containing M simulated graphs.

• stats_list: a list containing M objects of class PAFit_data, which are the results of applying get_statistics on the simulated graphs.

• result_list: a list containing M objects of class Full_PAFit_result, which are the results of applying joint_estimate on the simulated graphs.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE 10(9): e0137796. (doi:10.1371/journal.pone.0137796).

2. Pham, T., Sheridan, P. & Shimodaira, H. (2016). Joint Estimation of Preferential Attachment and Node Fitness in Growing Complex Networks. Scientific Reports 6, Article number: 32558. (doi:10.1038/srep32558).

3. Pham, T., Sheridan, P. & Shimodaira, H. (2020). PAFit: An R Package for the Non-Parametric Estimation of Preferential Attachment and Node Fitness in Temporal Complex Networks. Journal of Statistical Software 92 (3). (doi:10.18637/jss.v092.i03).

4. Inoue, M., Pham, T. & Shimodaira, H. (2020). Joint Estimation of Non-parametric Transitivity and Preferential Attachment Functions in Scientific Co-authorship Networks. Journal of Informetrics 14(3). (doi:10.1016/j.joi.2020.101042).

See Also

get_statistics, joint_estimate, plot_contribution

Examples

## Not run: 
  
  library("PAFit")
  net_object     <- generate_net(N = 500, m = 10, s = 10, alpha = 0.5)
  net_stat       <- get_statistics(net_object) 
  result         <- joint_estimate(net_object, net_stat)
  simulated_data <- generate_simulated_data_from_estimated_model(net_object, net_stat, result)
  plot_contribution(simulated_data, result, which_plot = "PA")
  plot_contribution(simulated_data, result, which_plot = "fit")
  
## End(Not run)

Getting summarized statistics from input data

Description

The function summarizes input data into sufficient statistics for estimating the attachment function and node fitness, together with additional information about the data, such as total number of nodes, number of time-steps, maximum degree, and the final degree of the network, etc. . It also provides mechanisms to automatically deal with very large datasets by binning the degree, setting a degree threshold, or grouping time-steps.

Usage

get_statistics(net_object, only_PA  = FALSE , 
               only_true_deg_matrix = FALSE ,
               binning              = TRUE  , g              = 50    , 
               deg_threshold        = 0     , 
               compress_mode        = 0     , compress_ratio = 0.5   , 
               custom_time          = NULL)

Arguments

The parameters can be divided into four groups. The first group specifies input data and how the data will be summarized:

Second group of parameters specifies how to bin the degrees:

Third group contains a single parameter specifying how to reduce the number of node fitnesses:

Last group of parameters specifies how to group the time-stamps:

Value

An object of class PAFit_data, which is a list. Some important fields are:

Author(s)

Thong Pham thongphamthe@gmail.com

See Also

For creating the needed input for this function (a PAFit_net object), see as.PAFit_net, from_igraph, from_networkDynamic, and graph_from_file.

For the next step, see Newman, Jeong or only_A_estimate for estimating the attachment function in isolation, only_F_estimate for estimating node fitnesses in isolation, and joint_estimate for joint estimation of the attachment function and node fitnesses.

Examples

library("PAFit")
net        <- generate_BA(N = 100 , m = 1)
net_stats  <- get_statistics(net)
summary(net_stats)

Read file to a PAFit_net object

Description

This function reads an input file to a PAFit_net object. Accepted formats are the edgelist format or the gml format.

Usage

 graph_from_file(file_name, format = "edgelist", type = "directed")

Arguments

Value

An object of class PAFit_net containing the network.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  # a network from Bianconi-Barabasi model
  net        <- generate_BB(N = 50 , m = 10 , s = 10)
  
  #graph_to_file(net, file_name = "test.gml", format = "gml")
  #reread    <- graph_from_file(file_name = "test.gml", format = "gml")

Write the graph in a PAFit_net object to file

Description

This function writes a graph in a PAFit_net object to an output file. Accepted file formats are the edgelist format or the gml format.

Usage

graph_to_file(net_object, file_name, format = "edgelist")

Arguments

Value

The function writes directly to the output file.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

library("PAFit")
# a network from Bianconi-Barabasi model
net        <- generate_BB(N = 50 , m = 10 , s = 10)
#graph_to_file(net, file_name = "test.gml", format = "gml")

Joint inference of attachment function and node fitnesses

Description

This function jointly estimates the attachment function A_k and node fitnesses \eta_i. It first performs a cross-validation to select the optimal parameters r and s, then estimates A_k and eta_i using that optimal pair with the full data (Ref. 2).

Usage

joint_estimate(net_object                               , 
              net_stat      = get_statistics(net_object), 
              p             = 0.75                      ,
              stop_cond     = 10^-8                     ,
              mode_reg_A    = 0                         , 
              ...)

Arguments

Value

Outputs a Full_PAFit_result object, which is a list containing the following fields:

• cv_data: a CV_Data object which contains the cross-validation data. This is the testing data.

• cv_result: a CV_Result object which contains the cross-validation result. Normally the user does not need to pay attention to this data.

• estimate_result: this is a PAFit_result object which contains the estimated attachment function A_k, the estimated fitnesses \eta_i and their confidence intervals. In particular, the important fields are:

  • ratio: this is the selected value for the hyper-parameter r.

  • shape: this is the selected value for the hyper-parameter s.

  • k and A: a degree vector and the estimated PA function.

  • var_A: the estimated variance of A.

  • var_logA: the estimated variance of log A.

  • upper_A: the upper value of the interval of two standard deviations around A.

  • lower_A: the lower value of the interval of two standard deviations around A.

  • center_k and theta: when we perform binning, these are the centers of the bins and the estimated PA values for those bins. theta is similar to A but with duplicated values removed.

  • var_bin: the variance of theta. Same as var_A but with duplicated values removed.

  • upper_bin: the upper value of the interval of two standard deviations around theta. Same as upper_A but with duplicated values removed.

  • lower_bin: the lower value of the interval of two standard deviations around theta. Same as lower_A but with duplicated values removed.

  • g: the number of bins used.

  • alpha and ci: alpha is the estimated attachment exponent \alpha (when assume A_k = k^\alpha), while ci is the confidence interval.

  • loglinear_fit: this is the fitting result when we estimate \alpha.

  • f: the estimated node fitnesses.

  • var_f: the estimated variance of \eta_i.

  • upper_f: the estimated upper value of the interval of two standard deviations around \eta_i.

  • lower_f: the estimated lower value of the interval of two standard deviations around \eta_i.

  • objective_value: values of the objective function over iterations in the final run with the full data.

  • diverge_zero: logical value indicates whether the algorithm diverged in the final run with the full data.

• contribution: a list containing an estimate of the contributions of preferential attachment and fitness mechanisms in the growth process of the network. The calculation adapts a quantification method proposed in Section 3 of Ref. 4, which is for preferential attachment and transitivity, to preferential attachment and fitness.

  • PA_contribution: an array containing the contributions of preferential attachment at each time-step

  • fit_contribution: an array containing the contributions of the fitness mechanism at each time-step

  • mean_PA_contrib: the average contribution of preferential attachment through the whole growth process

  • mean_fit_contrib: the average contribution of the fitness mechanism through the whole growth process

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE 10(9): e0137796. (doi:10.1371/journal.pone.0137796).

2. Pham, T., Sheridan, P. & Shimodaira, H. (2016). Joint Estimation of Preferential Attachment and Node Fitness in Growing Complex Networks. Scientific Reports 6, Article number: 32558. (doi:10.1038/srep32558).

3. Pham, T., Sheridan, P. & Shimodaira, H. (2020). PAFit: An R Package for the Non-Parametric Estimation of Preferential Attachment and Node Fitness in Temporal Complex Networks. Journal of Statistical Software 92 (3). (doi:10.18637/jss.v092.i03).

4. Inoue, M., Pham, T. & Shimodaira, H. (2020). Joint Estimation of Non-parametric Transitivity and Preferential Attachment Functions in Scientific Co-authorship Networks. Journal of Informetrics 14(3). (doi:10.1016/j.joi.2020.101042).

See Also

See get_statistics for how to create summarized statistics needed in this function.

See Jeong, Newman and only_A_estimate for functions to estimate the attachment function in isolation.

See only_F_estimate for a function to estimate node fitnesses in isolation.

Examples

## Not run: 
  
  library("PAFit")
  #### Example 1: a linear preferential attachment kernel, i.e., A_k = k ############
  set.seed(1)
  # size of initial network = 100
  # number of new nodes at each time-step = 100
  # Ak = k; inverse variance of the distribution of node fitnesse = 5
  net        <- generate_BB(N        = 1000 , m             = 50 , 
                            num_seed = 100  , multiple_node = 100,
                            s        = 5)
  net_stats  <- get_statistics(net)
  
  # Joint estimation of attachment function Ak and node fitness
  result     <- joint_estimate(net, net_stats)
  
  summary(result)
  
  # plot the estimated attachment function
  true_A     <- pmax(result$estimate_result$center_k,1) # true function
  plot(result , net_stats, max_A = max(true_A,result$estimate_result$theta))
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  
  # plot the estimated node fitnesses and true node fitnesses
  plot(result, net_stats, true = net$fitness, plot = "true_f")
  
  #############################################################################
  #### Example 2: a non-log-linear preferential attachment kernel ############
  set.seed(1)
  # size of initial network = 100
  # number of new nodes at each time-step = 100
  # A_k = alpha* log (max(k,1))^beta + 1, with alpha = 2, and beta = 2
  # inverse variance of the distribution of node fitnesse = 10
  net        <- generate_net(N       = 1000 , m             = 50 , 
                            num_seed = 100  , multiple_node = 100,
                            s        = 10   , mode = 3, alpha = 2, beta = 2)
  net_stats  <- get_statistics(net)
  
  # Joint estimation of attachment function Ak and node fitness
  result     <- joint_estimate(net, net_stats)
  
  summary(result)
  
  # plot the estimated attachment function
  true_A     <- 2 * log(pmax(result$estimate_result$center_k,1))^2 + 1 # true function
  plot(result , net_stats, max_A = max(true_A,result$estimate_result$theta))
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  
  # plot the estimated node fitnesses and true node fitnesses
  plot(result, net_stats, true = net$fitness, plot = "true_f")
  #############################################################################
  #### Example 3: another non-log-linear preferential attachment kernel ############
  set.seed(1)
  # size of initial network = 100
  # number of new nodes at each time-step = 100
  # A_k = min(max(k,1),sat_at)^alpha, with alpha = 1, and sat_at = 100
  # inverse variance of the distribution of node fitnesse = 10
  net        <- generate_net(N       = 1000 , m             = 50 , 
                            num_seed = 100  , multiple_node = 100,
                            s        = 10   , mode = 2, alpha = 1, sat_at = 100)
  net_stats  <- get_statistics(net)
  
  # Joint estimation of attachment function Ak and node fitness
  result     <- joint_estimate(net, net_stats)
  
  summary(result)
  
  # plot the estimated attachment function
  true_A     <- pmin(pmax(result$estimate_result$center_k,1),100)^1 # true function
  plot(result , net_stats, max_A = max(true_A,result$estimate_result$theta))
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  
  # plot the estimated node fitnesses and true node fitnesses
  plot(result, net_stats, true = net$fitness, plot = "true_f")
  
## End(Not run)

Estimating the attachment function in isolation by PAFit method

Description

This function estimates the attachment function A_k by PAFit method. The method has a hyper-parameter r. It first performs a cross-validation step to select the optimal parameter r for the regularization of A_k, then uses that r to estimate the attachment function with the full data.

Usage

only_A_estimate(net_object                             , 
                net_stat   = get_statistics(net_object), 
                p          = 0.75                      ,
                stop_cond  = 10^-8                     , 
                mode_reg_A = 0                         ,
                MLE        = FALSE                     ,
               ...)

Arguments

Value

Outputs a Full_PAFit_result object, which is a list containing the following fields:

• cv_data: a CV_Data object which contains the cross-validation data. This is the final Normally the user does not need to pay attention to this data. NULL if MLE = TRUE.

• cv_result: a CV_Result object which contains the cross-validation result. Normally the user does not need to pay attention to this data. NULL if MLE = TRUE.

• estimate_result: this is a PAFit_result object which contains the estimated PA function and its confidence interval. It also includes the estimated attachment exponenent \alpha (assuming the model A_k = k^\alpha) in the field alpha, and the confidence interval of \alpha (in the field ci) when possible. In particular, the important fields are:

  • ratio: this is the selected value for the hyper-parameter r.

  • k and A: a degree vector and the estimated PA function.

  • var_A: the estimated variance of A.

  • var_logA: the estimated variance of log A.

  • upper_A: the upper value of the interval of two standard deviations around A.

  • lower_A: the lower value of the interval of two standard deviations around A.

  • center_k and theta: when we perform binning, these are the centers of the bins and the estimated PA values for those bins. theta is similar to A but with duplicated values removed.

  • var_bin: the variance of theta. Same as var_A but with duplicated values removed.

  • upper_bin: the upper value of the interval of two standard deviations around theta. Same as upper_A but with duplicated values removed.

  • lower_lower: the lower value of the interval of two standard deviations around theta. Same as lower_A but with duplicated values removed.

  • g: the number of bins used.

  • alpha and ci: alpha is the estimated attachment exponenet \alpha (when assume A_k = k^\alpha), while ci is the confidence interval.

  • loglinear_fit: this is the fitting result when we estimate \alpha.

  • objective_value: values of the objective function over iterations in the final run with the full data.

  • diverge_zero: logical value indicates whether the algorithm diverged in the final run with the full data.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE 10(9): e0137796. (doi:10.1371/journal.pone.0137796).

2. Pham, T., Sheridan, P. & Shimodaira, H. (2016). Joint Estimation of Preferential Attachment and Node Fitness in Growing Complex Networks. Scientific Reports 6, Article number: 32558. (doi:10.1038/srep32558).

See Also

See get_statistics for how to create summerized statistics needed in this function.

See Newman and Jeong for other methods to estimate the attachment function A_k in isolation.

Examples

## Not run: 
  library("PAFit")
  set.seed(1)
  #### Example 1: Linear preferential attachment  #########
  # a network from BA model
  net        <- generate_net(N = 1000 , m = 50 , mode = 1, alpha = 1, s = 0)
  
  net_stats  <- get_statistics(net, only_PA = TRUE)
  result     <- only_A_estimate(net, net_stats)
 
  # plot the estimated attachment function
  plot(result, net_stats)
  
  # true function
  true_A     <- result$estimate_result$center_k
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  
  #### Example 2: a non-log-linear preferential attachment  #########
  # A_k = alpha* log (max(k,1))^beta + 1, with alpha = 2, and beta = 2
  set.seed(1)
  net        <- generate_net(N = 1000 , m = 50 , mode = 3, alpha = 2, beta = 2, s = 0)
  
  net_stats  <- get_statistics(net,only_PA = TRUE)
  result     <- only_A_estimate(net, net_stats)
 
  # plot the estimated attachment function
  plot(result, net_stats)
  
  # true function
  true_A     <- 2 * log(pmax(result$estimate_result$center_k,1))^2 + 1 # true function
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  
  #############################################################################
  #### Example 3: another non-log-linear preferential attachment kernel ############
  set.seed(1)
  # A_k = min(max(k,1),sat_at)^alpha, with alpha = 1, and sat_at = 200
  # inverse variance of the distribution of node fitnesse = 10
  net        <- generate_net(N = 1000 , m = 50 , mode = 2, alpha = 1, sat_at = 200, s = 0)
  net_stats  <- get_statistics(net, only_PA = TRUE)
  
  result     <- only_A_estimate(net, net_stats)
  
  
  # plot the estimated attachment function
  true_A     <- pmin(pmax(result$estimate_result$center_k,1),200)^1 # true function
  plot(result , net_stats, max_A = max(true_A,result$estimate_result$theta))
  lines(result$estimate_result$center_k, true_A, col = "red") # true line
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
  
## End(Not run)

Estimating node fitnesses in isolation

Description

This function estimates node fitnesses \eta_i assusming either A_k = k (i.e. linear preferential attachment) or A_k = 1 (i.e. no preferential attachment). The method has a hyper-parameter s. It first performs a cross-validation to select the optimal parameter s for the prior of \eta_i, then estimates eta_i with the full data (Ref. 1).

Usage

only_F_estimate(net_object                             , 
               net_stat    = get_statistics(net_object), 
               p           = 0.75                      ,
               stop_cond   = 10^-8                     , 
               model_A     = "Linear"                  ,
               ...)

Arguments

Value

Outputs a Full_PAFit_result object, which is a list containing the following fields:

• cv_data: a CV_Data object which contains the cross-validation data. Normally the user does not need to pay attention to this data.

• cv_result: a CV_Result object which contains the cross-validation result. Normally the user does not need to pay attention to this data.

• estimate_result: this is a PAFit_result object which contains the estimated node fitnesses and their confidence intervals. In particular, the important fields are:

  • shape: this is the selected value for the hyper-parameter s.

  • g: the number of bins used.

  • f: the estimated node fitnesses.

  • var_f: the estimated variance of \eta_i.

  • upper_f: the estimated upper value of the interval of two standard deviations around \eta_i.

  • lower_f: the estimated lower value of the interval of two standard deviations around \eta_i.

  • objective_value: values of the objective function over iterations in the final run with the full data.

  • diverge_zero: logical value indicates whether the algorithm diverged in the final run with the full data.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2016). Joint Estimation of Preferential Attachment and Node Fitness in Growing Complex Networks. Scientific Reports 6, Article number: 32558. (doi:10.1038/srep32558).

2. Bianconni, G. & Barabási, A. (2001). Competition and multiscaling in evolving networks. Europhys. Lett., 54, 436 (doi:10.1209/epl/i2001-00260-6).

3. Caldarelli, G., Capocci, A. , De Los Rios, P. & Muñoz, M.A. (2002). Scale-Free Networks from Varying Vertex Intrinsic Fitness. Phys. Rev. Lett., 89, 258702 (doi:10.1103/PhysRevLett.89.258702).

See Also

See get_statistics for how to create summerized statistics needed in this function.

See joint_estimate for the method to jointly estimate the attachment function A_k and node fitnesses \eta_i.

Examples

## Not run: 
  library("PAFit")
  set.seed(1)
  # size of initial network = 100
  # number of new nodes at each time-step = 100
  # Ak = k; inverse variance of the distribution of node fitnesse = 10
  net        <- generate_BB(N        = 1000 , m             = 50 , 
                            num_seed = 100  , multiple_node = 100,
                            s        = 10)
                            
  net_stats  <- get_statistics(net)
  
  # estimate node fitnesses in isolation, assuming Ak = k
  result     <- only_F_estimate(net, net_stats)
 
  # plot the estimated node fitnesses and true node fitnesses
  plot(result, net_stats, true = net$fitness, plot = "true_f")
  
## End(Not run)

Plotting the estimated attachment function and node fitness

Description

This function plots the estimated attachment function A_k and node fitness eta_i, together with additional information such as their confidence intervals or the estimated attachment exponent (\alpha when assuming A_k = k^\alpha).

Usage

## S3 method for class 'Full_PAFit_result'
plot(x,
     net_stat                 ,
     true_f         = NULL    , plot             = "A"              , plot_bin   = TRUE ,
     line           = FALSE   , confidence       = TRUE             , high_deg_A = 1    ,
     high_deg_f     = 5       ,
     shade_point    = 0.5     , col_point        = "grey25"         , pch        = 16   ,
     shade_interval = 0.5     , col_interval     = "lightsteelblue" , label_x    = NULL , 
     label_y        = NULL    ,
     max_A          = NULL    , min_A            = NULL             , f_min      = NULL , 
     f_max          = NULL    , plot_true_degree = FALSE , 
     ...)

Arguments

Value

Outputs the desired plot.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

## Since the runtime is long, we do not let this example run on CRAN
## Not run: 
library("PAFit")
set.seed(1)
# a network from Bianconi-Barabasi model
net        <- generate_BB(N        = 1000 , m             = 50 , 
                          num_seed = 100  , multiple_node = 100,
                          s        = 10)
net_stats  <- get_statistics(net)
result     <- joint_estimate(net, net_stats)
#plot A
plot(result , net_stats , plot = "A")
true_A     <- c(1,result$estimate_result$center_k[-1])
lines(result$estimate_result$center_k + 1 , true_A , col = "red") # true line
legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
#plot true_f
plot(result, net_stats , net$fitness, plot = "true_f")

## End(Not run)

Plot a PAFit_net object

Description

This function plots a PAFit_net object. There are four options of plot to specify the type of plot.

The first two concern plotting the graph in $graph of the PAFit_net object. Option plot = "graph" plots the graph, while plot = "degree" plots the degree distribution. Option slice allows selection of the time-step at which the temporal graph is plotted.

The last two options concern plotting the PA function and node fitnesses (if they are not NULL).

Usage

## S3 method for class 'PAFit_net'
plot(x,
     plot = "graph"                         ,
     slice = length(unique(x$graph[,3])) - 1,
     ...)

Arguments

Value

Outputs the desired plot.

Author(s)

Thong Pham thongphamthe@gmail.com. When plot = "graph", the function uses plot.network.default in the network package.

Examples

    library("PAFit")
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N = 50 , m = 10 , s = 10)
    plot(net, plot = "graph")
    plot(net, plot = "degree")
    plot(net, plot = "PA")
    plot(net, plot = "fit")

Plotting the estimated attachment function and node fitness of a PAFit_result object

Description

This function plots the estimated attachment function A_k and node fitness eta_i, together with additional information such as their confidence intervals or the estimated attachment exponent (\alpha when assuming A_k = k^\alpha) of a PAFit_result object. This object is stored in the field $estimate_result of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'PAFit_result'
plot(x,
    net_stat       = NULL    ,
    true_f         = NULL    , plot             = "A"              , plot_bin   = TRUE ,
    line           = FALSE   , confidence       = TRUE             , high_deg_A = 1    ,
    high_deg_f     = 5       ,
    shade_point    = 0.5     , col_point        = "grey25"         , pch        = 16   ,
    shade_interval = 0.5     , col_interval     = "lightsteelblue" , label_x    = NULL , 
    label_y        = NULL    ,
    max_A          = NULL    , min_A            = NULL             , f_min      = NULL , 
    f_max          = NULL    , plot_true_degree = FALSE , 
    ...)

Arguments

Value

Outputs the desired plot.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    #plot A
    plot(result$estimate_result , net_stats , plot = "A")
    true_A     <- c(1,result$estimate_result$center_k[-1])
    lines(result$estimate_result$center_k + 1 , true_A , col = "red") # true line
    legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")
    #plot true_f
    plot(result, net_stats , net$fitness, plot = "true_f")
  
## End(Not run)

Plotting the estimated attachment function

Description

This function plots the estimated attachment function from the corrected Newman's method or the Jeong's method. Its also plots additional information such as the estimated attachment exponenent (\alpha when assuming A_k = k^\alpha).

Usage

## S3 method for class 'PA_result'
plot(x, 
     net_stat    = NULL,
     plot_bin    = TRUE   ,
     high_deg    = 1      ,  
     line        = FALSE  , 
     col_point   = "black",
     shade_point = 0.5    , 
     pch         = 16     ,
     max_A       = NULL   , 
     min_A       = NULL   , 
     label_x     = NULL   , 
     label_y     = NULL   ,
     ...)

Arguments

Value

Outputs the desired plot.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  net        <- generate_net(N = 1000 , m = 1 , mode = 1 , alpha = 1 , s = 0)
  net_stats  <- get_statistics(net)
  result     <- Newman(net, net_stats)
  # true function
  true_A     <- result$center_k
  # plot the estimated attachment function
  plot(result , net_stats)
  lines(result$center_k, true_A, col = "red") # true attachment function
  legend("topleft" , legend = "True function" , col = "red" , lty = 1 , bty = "n")

Plotting contributions calculated from the observed data and contributions calculated from simulated data

Description

This function extracts from a Simulated_Data_From_Fitted_Model object contributions of rich-get-richer and fit-get-richer effects calculated using simulated networks and plots these contributions versus the contributions calculated from the original observed network. See joint_estimate for a description of how the contributions are calculated.

Usage

plot_contribution(simulated_object,
                  original_result,
                  which_plot = "PA",
                  y_label = ifelse("PA" == which_plot,
                  "Contribution of the rich-get-richer effect",
                  "Contribution of the fit-get-richer effect"),
                  legend_pos_x = 0.75,
                  legend_pos_y = 0.9)

Arguments

Value

Output a plot.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Pham, T., Sheridan, P. & Shimodaira, H. (2015). PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE 10(9): e0137796. (doi:10.1371/journal.pone.0137796).

2. Pham, T., Sheridan, P. & Shimodaira, H. (2016). Joint Estimation of Preferential Attachment and Node Fitness in Growing Complex Networks. Scientific Reports 6, Article number: 32558. (doi:10.1038/srep32558).

3. Pham, T., Sheridan, P. & Shimodaira, H. (2020). PAFit: An R Package for the Non-Parametric Estimation of Preferential Attachment and Node Fitness in Temporal Complex Networks. Journal of Statistical Software 92 (3). (doi:10.18637/jss.v092.i03).

4. Inoue, M., Pham, T. & Shimodaira, H. (2020). Joint Estimation of Non-parametric Transitivity and Preferential Attachment Functions in Scientific Co-authorship Networks. Journal of Informetrics 14(3). (doi:10.1016/j.joi.2020.101042).

See Also

joint_estimate, plot_contribution

Examples

## Not run: 
  
  library("PAFit")
  net_object     <- generate_net(N = 500, m = 10, s = 10, alpha = 0.5)
  net_stat       <- get_statistics(net_object) 
  result         <- joint_estimate(net_object, net_stat)
  simulated_data <- generate_simulated_data_from_estimated_model(net_object, net_stat, result)
  plot_contribution(simulated_data, result, which_plot = "PA")
  plot_contribution(simulated_data, result, which_plot = "fit")
  
## End(Not run)

Printing simple information of the cross-validation data

Description

This function prints simple information of the cross-validation data stored in a CV_Data object. This object is the field $cv_data of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'CV_Data'
print(x,...)

Arguments

Value

Prints simple information of the cross-validation data.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    print(result$cv_data)
  
## End(Not run)

Printing simple information of the cross-validation result

Description

This function prints simple information of the cross-validation result stored in a CV_Result object. This object is the field $cv_result of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'CV_Result'
print(x,...)

Arguments

Value

Prints simple information of the cross-validation result.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    print(result$cv_result)
  
## End(Not run)

printing information on the estimation result

Description

This function outputs simple information of the estimation result.

Usage

  ## S3 method for class 'Full_PAFit_result'
print(x,...)

Arguments

Value

Outputs summary information on the estimation result.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    print(result)
  
## End(Not run)

Printing simple information on the statistics of the network stored in a PAFit_data object

Description

This function prints simple information of the statistics stored in a PAFit_data object. This object is the returning value of get_statistics.

Usage

  ## S3 method for class 'PAFit_data'
print(x,...)

Arguments

Value

Prints simple information of the network statistics.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    print(net_stats)
  
## End(Not run)

Printing simple information of a PAFit_net object

Description

This function outputs simple information of a PAFit_net object.

Usage

  ## S3 method for class 'PAFit_net'
print(x,
                            ...)

Arguments

Value

Outputs simple information of the network.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  # a network from Bianconi-Barabasi model
  net        <- generate_BB(N = 50 , m = 10 , s = 10)
  print(net)

printing information on the estimation result stored in a PAFit_result object

Description

This function outputs simple information of the estimation result stored in a PAFit_result object. This object is stored in the field $estimate_result of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'PAFit_result'
print(x,...)

Arguments

Value

Outputs summary information on the estimation result.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    print(result$estimate_result)
  
## End(Not run)

Printing information of the estimated attachment function

Description

This function outputs simple information of the estimated attachment function from the corrected Newman's method or the Jeong's method.

Usage

  ## S3 method for class 'PA_result'
print(x, 
                              ...)

Arguments

Value

Simple information of the estimated attachment function.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  net        <- generate_net(N = 1000 , m = 1 , mode = 1 , alpha = 1 , s = 0)
  net_stats  <- get_statistics(net)
  result     <- Newman(net, net_stats)
  print(result)

Printing summary information of the cross-validation data

Description

This function outputs summary information of the cross-validation data stored in a CV_Data object. This object is the field $cv_data of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'CV_Data'
summary(object,...)

Arguments

Value

Outputs summary information of the cross-validation data.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    summary(result$cv_data)
  
## End(Not run)

Output summary information of the cross-validation result

Description

This function outputs summary information of the cross-validation result stored in a CV_Result object. This object is the field $cv_result of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'CV_Result'
summary(object,...)

Arguments

Value

Outputs summary information of the cross-validation result.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    summary(result$cv_result)
  
## End(Not run)

Summary information on the estimation result

Description

This function outputs a summary on the estimation result.

Usage

  ## S3 method for class 'Full_PAFit_result'
summary(object,...)

Arguments

Value

Outputs summary information on the estimation result.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    summary(result)
  
## End(Not run)

Output summary information on the statistics of the network stored in a PAFit_data object

Description

This function outputs summary information of the statistics stored in a PAFit_data object. This object is the returning value of get_statistics.

Usage

  ## S3 method for class 'PAFit_data'
summary(object,...)

Arguments

Value

Outputs summary information of the network statistics.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    summary(net_stats)
  
## End(Not run)

Summary information of a PAFit_net object

Description

This function outputs summary information of a PAFit_net object.

Usage

  ## S3 method for class 'PAFit_net'
summary(object,
                           ...)

Arguments

Value

Outputs summary information of the network.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  # a network from Bianconi-Barabasi model
  net        <- generate_BB(N = 50 , m = 10 , s = 10)
  summary(net)

Output summary information on the estimation result stored in a PAFit_result object

Description

This function outputs summary information of the estimation result stored in a PAFit_result object. This object is stored in the field $estimate_result of a Full_PAFit_result object, which in turn is the returning value of only_A_estimate, only_F_estimate or joint_estimate.

Usage

  ## S3 method for class 'PAFit_result'
summary(object,...)

Arguments

Value

Outputs summary information on the estimation result.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  ## Since the runtime is long, we do not let this example run on CRAN
  ## Not run: 
    library("PAFit")
    set.seed(1)
    # a network from Bianconi-Barabasi model
    net        <- generate_BB(N        = 1000 , m             = 50 , 
                              num_seed = 100  , multiple_node = 100,
                              s        = 10)
    net_stats  <- get_statistics(net)
    result     <- joint_estimate(net, net_stats)
    summary(result$estimate_result)
  
## End(Not run)

Summary of the estimated attachment function

Description

This function outputs summary information of the estimated attachment function from the corrected Newman's method or the Jeong's method.

Usage

  ## S3 method for class 'PA_result'
summary(object, 
                           ...)

Arguments

Value

Summary information of the estimated attachment function.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  net        <- generate_net(N = 1000 , m = 1 , mode = 1 , alpha = 1 , s = 0)
  net_stats  <- get_statistics(net)
  result     <- Newman(net, net_stats)
  summary(result)

Fitting various distributions to a degree vector

Description

This function implements the method in Handcock and Jones (2004) to fit various distributions to a degree vector. The implemented distributions are Yule, Waring, Poisson, geometric and negative binomial. The Yule and Waring distributions correspond to a preferential attachment situation. In particular, the two distributions correspond to the case of A_k = k for k \ge 1 and \eta_i = 1 for all i (note that, the number of new edges and new nodes at each time-step are implicitly assumed to be 1).

Thus, if the best fitted distribution, which is chosen by either the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC), is NOT Yule or Waring, then the case of A_k = k for k \ge 1 and \eta_i = 1 for all i is NOT consistent with the observed degree vector.

The method allows the low-tail probabilities to NOT follow the parametric distribution, i.e., P(K = k) = \pi_k for all k \le k_{min} and P(K = k) = f(k,\theta) for all k > k_{min}. Here k_{min} is the degree threshold above which the parametric distribution holds, \pi_k are probabilities of the low-tail, f(.,\theta) is the parametric distribution with parameter vector \theta.

For fixed k_{min} and f, \pi_k and \theta can be estimated by Maximum Likelihood Estimation. We can choose the best k_{min} for each f by comparing the AIC (or BIC). More details can be founded in Handcock and Jones (2004).

Usage

 test_linear_PA(degree_vector)

Arguments

Value

Outputs a Linear_PA_test_result object which contains the fitting of five distributions to the degree vector: Yule (yule), Waring (waring), Poisson (pois), geometric (geom) and negative binomial (nb). In particular, for each distribution, the AIC and BIC are calcualted for each k_min.

Author(s)

Thong Pham thongphamthe@gmail.com

References

1. Handcock MS, Jones JH (2004). “Likelihood-based inference for stochastic models of sexual network formation.” Theoretical Population Biology, 65(4), 413 – 422. ISSN 0040-5809. doi:10.1016/j.tpb.2003.09.006. Demography in the 21st Century, https://www.sciencedirect.com/science/article/pii/S0040580904000310.

Examples

## Not run: 
  library("PAFit")
  set.seed(1)
  net   <- generate_BA(n = 1000)
  stats <- get_statistics(net, only_PA = TRUE)
  u     <- test_linear_PA(stats$final_deg)
  print(u)

## End(Not run)

Convert a PAFit_net object to an igraph object

Description

This function converts a PAFit_net object to an igraph object (of package igraph).

Usage

to_igraph(net_object)

Arguments

Value

The function returns an igraph object.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

library("PAFit")
# a network from Bianconi-Barabasi model
net          <- generate_BB(N = 50 , m = 10 , s = 10)
igraph_graph <- to_igraph(net)

Convert a PAFit_net object to a networkDynamic object

Description

This function converts a PAFit_net object to a networkDynamic object (of package networkDynamic).

Usage

  to_networkDynamic(net_object)

Arguments

Value

The function returns a networkDynamic object.

Author(s)

Thong Pham thongphamthe@gmail.com

Examples

  library("PAFit")
  # a network from Bianconi-Barabasi model
  net          <- generate_BB(N = 50 , m = 10 , s = 10)
  nD_graph     <- to_networkDynamic(net)