Preferential Attachment

Generating nodes and edges

Generate a network with 10,000 nodes with preferential attachment: \(\alpha, \beta, \gamma\) are the probabilities (weights) of adding a new node through an in-degree to one of the existing node, a new edge between two existing nodes, and a new node through an out-degree of an existing node. \(\delta_{\text{in}}\) and \(\delta_{\text{out}}\) are small constants to avoid zero weights in generating the edges.

Setting 1:

load("PA_10000_1.RData")
# g <- PA(10000, alpha =  10, beta = 5, gamma = 1, delta.in = 0.5, delta.out = 0.5)

Quantiles of in- and out-degrees

quantile(degrees[, "in"], probs = seq(0, 1, 0.05), names = TRUE, na.rm = T)

##   0%   5%  10%  15%  20%  25%  30%  35%  40%  45%  50%  55%  60%  65%  70% 
##    0    0    0    0    0    0    0    0    0    0    0    0    0    0    1 
##  75%  80%  85%  90%  95% 100% 
##    1    1    2    3    5 1446

quantile(degrees[, "out"], probs = seq(0, 1, 0.05), names = TRUE, na.rm = T)

##   0%   5%  10%  15%  20%  25%  30%  35%  40%  45%  50%  55%  60%  65%  70% 
##    0    0    1    1    1    1    1    1    1    1    1    1    1    1    1 
##  75%  80%  85%  90%  95% 100% 
##    2    2    2    3    3   24

Hill plots

Distribution of node types

## 
## both   in none  out 
##   64  234 6345  266

## 
##       both         in       none        out 
## 0.00926328 0.03386887 0.91836735 0.03850051

Plot conditional distributions of neighbors’ types

Step 0: randomly sample 20 nodes and neighbors they follow

## [1] "frequency and distribution of types"

## 
## both   in none  out 
##    1    0   47    2

## 
## both   in none  out 
## 0.02 0.00 0.94 0.04

## [1] "number of edges"

## [1] 69  2

## [1] "number of neighbors they follow"

## [1] 55

## [1] "frequency and distribution of types in neighbors"

## 
## both   in none  out 
##    8   23   22    2

## 
##       both         in       none        out 
## 0.14545455 0.41818182 0.40000000 0.03636364

## [1] "Distribution of types in all nodes"

## 
##       both         in       none        out 
## 0.08910891 0.22772277 0.65346535 0.02970297

Step 1: get all neighbors of the target nodes in step 0

## [1] "no. of initial target nodes that actually have a neighbor"

## [1] 17  3

## [1] "number of neighbors"

## [1] 24

## [1] "frequency and distribution of types of neighbors"

## 
## both   in none  out 
##    7   11    5    1

## 
##       both         in       none        out 
## 0.29166667 0.45833333 0.20833333 0.04166667

## [1] "distribution of types in all nodes involved"

## 
##       both         in       none        out 
## 0.17948718 0.66666667 0.12820513 0.02564103

Step 2: get all neighbors of the target neighbor nodes in step 1

## [1] "number of target neighbors"

## [1] 11

## [1] "number of neighbors"

## [1] 8

## [1] "frequency and distribution of types of neighbors"

## 
## both   in none  out 
##    3    3    2    0

## 
##  both    in  none   out 
## 0.375 0.375 0.250 0.000

## [1] "frequency and distribution of types in all nodes involved (including non-target neighbors reviewed)"

## 
## both   in none  out 
##    7   27    6    1

## 
##       both         in       none        out 
## 0.17073171 0.65853659 0.14634146 0.02439024

## [1] "number of all nodes involved (including step 0)"

## [1] 120   1

## [1] "Frequency and distribution of types"

## 
## both   in none  out 
##   14   32   71    3

## 
##      both        in      none       out 
## 0.1166667 0.2666667 0.5916667 0.0250000

Setting 2:

load("PA_10000_2.RData")
# g <- PA(10000, alpha =  10, beta = 1, gamma = 1, delta.in = 0.5, delta.out = 0.5)