# Introduction

If you have a vector dataset and suppose it has a power-law distribution you have to investigate it closer. Even in graph theory in the case of scale-free graphs, power-law distribution is a common theme.

The {poweRlaw} package gives really good functions to do that. In this post, an analysis pipe can be read begin with generate a random power law then look at the degree distribution of generated graph. The analysis continues with some bootstrap step which is same with any kind of dataset.

This post refer to Barabasi: Network science and Clauset,Shalizi, Newman: Power-Law Distributions in Empirical Data. The {poweRlaw} package is in JSS (Gillespie, 2015).

# Generate scale-free graph and dataset

## Generate graph

library(igraph)
set.seed(202)
g <- static.power.law.game(500, 1000, exponent.out= 2.2, exponent.in = -1, loops = FALSE, multiple = TRUE, finite.size.correction = TRUE)

plot(g, vertex.label= NA, edge.arrow.size=0.02,vertex.size = 0.5, xlab = "")


This graph contains multiple edges. There are nodes with the small and big degree as well.

## Dataset: degree of graph

data <- degree(g)
data 0]


# Fit power-law

The initial distribution of data can be seen in the figure below.

data.dist <- data.frame(k=0:max(data),p_k=degree_distribution(g))
data.dist 0,]

library(ggplot2)
ggplot(data.dist) + geom_point(aes(x=k, y=p_k)) + theme_bw()


It is very similar to lognormal distribution but let it see with {powerLaw}.

## Suggested steps

Authors of referred bibliographies suggested the following steps to analyze the dataset.

1. Initial estimation of $K_{min}$ and $\gamma$
2. Make a CDF (cumulative distribution function) with that
3. Kolgomorov-Smirnov test to determine minimum $D$
4. Repeat steps 1-3 to scanning whole $k_{min} - k_{max}$ range to idetify $K_{min}$ for which $D$ is minimal
5. Investigate goodness of fit with bootstrapping
• assign $D_{min}$ to $K_{min}$ which is $D^{real}$
• generate a data sequence of points with bootstrapping and determining $D^{syntetic}$
• determine $p$ value of null hypothesis that claims the distribution is power-law (if $p \geq 1$ than null hypothesis is true)
6. Fitting real distribution
• choose $k_{cut}$ and $k_{sat}$ to optimal cut
• estimate $\gamma$ exponent
• calculate $D$ with Kolgomorov-Smirnov test
• repeat steps 1-3 to determine real $k_{cut}$ and $k_{sat}$ for which $D$ is minimal

## Initial estimation

This estimation calcualate $K_{min}$, $\gamma$ and makes CDF and determine minimum $D$. With {poweRlaw} functions very easy to achive suggested steps 1-3.

library(poweRlaw)
m_pl <- displ$new(data) est_pl <- estimate_xmin(m_pl) est_pl$xmin #k_min

## [1] 5

est_pl$pars #gamma  ## [1] 3.175132  est_pl$gof #D

## [1] 0.03126577


## Scanning whole range

data.s <- unique(data)

d_est <- data.frame(K_min=sort(data.s)[1:(length(data.s)-2)], gamma=rep(0,length(data.s)-2), D=rep(0,length(data.s)-2))

for (i in d_est$K_min){ d_est[which(d_est$K_min == i),2] <- estimate_xmin(m_pl, xmins = i)$pars d_est[which(d_est$K_min == i),3] <- estimate_xmin(m_pl, xmins = i)$gof } K.min_D.min <- d_est[which.min(d_est$D), 1]

ggplot(data=d_est, aes(x=K_min, y=D)) + geom_line() + theme_bw() +
geom_vline(xintercept=K.min_D.min, colour="red") + annotate("text", x=K.min_D.min, y=max(d_est$D)/3*2, label=K.min_D.min)  ggplot(data=d_est, aes(x=K_min, y=gamma)) + geom_line() + theme_bw() + geom_vline(xintercept=K.min_D.min, colour="red") + annotate("text", x=K.min_D.min, y=max(d_est$gamma)/3*2, label=K.min_D.min)


And the fitted power-law on CDF curve.

m_pl$setXmin(est_pl) plot.data <- plot(m_pl, draw = F) fit.data <- lines(m_pl, draw = F)  ggplot(plot.data) + geom_point(aes(x=log(x), y=log(y))) + labs(x="log(k)", y="log(CDF)") + theme_bw() + geom_line(data=fit.data, aes(x=log(x), y=log(y)), colour="red")  ## Investigate goodness of fit bs_pl <- bootstrap_p(m_pl, no_of_sims=1000, threads=8, seed = 123) #threads=core number of processor that used by function #parallel::detectCores() determines how many cores in your computer plot(bs_pl)  df_bs_pl <- bs_pl$bootstraps

ggplot(data=df_bs_pl, aes(pars)) + geom_histogram() + labs(x="gamma", y="frequency") + theme_bw()


ggplot(data=df_bs_pl, aes(xmin)) + geom_histogram() + labs(x="K_min", y="frequency") + theme_bw()


gamma_D.min <- d_est[which.min(d_est$D), 2] ggplot(data=df_bs_pl, aes(x=xmin, y=pars)) + labs(x="K_min", y="gamma") + theme_bw() + geom_point(shape=21, colour="black", fill="red", size=0.5, stroke=2, position = position_jitter(), alpha=0.6) + geom_vline(xintercept=K.min_D.min, colour="blue") + geom_hline(yintercept=gamma_D.min, colour="blue") + annotate("text", x=K.min_D.min, y=min(df_bs_pl$pars), label=K.min_D.min, col="blue") +
annotate("text", x=min(df_bs_pl$xmin), y=gamma_D.min, label=round(gamma_D.min, digits=2), col="blue")  Initial estimated $\gamma$ and $K_{min}$ denoted with blue colour on the figure. D.min <- d_est[which.min(d_est$D), 3]

ggplot(data=df_bs_pl, aes(gof)) + geom_histogram() + labs(x="D", y="frequency") + geom_vline(xintercept=D.min, colour="red") + theme_bw()


bs_pl$p #p value  ## [1] 0.603  The $p$ value is 0.603. If $p$ value more than 1% means that null hypothesis cannot be rejected maybe it is a power-law distribution. ## Fitting real distribution #generate kmin & kmax pairs pairs <- as.data.frame(t(combn(sort(data.s), 2))) pairs$D <- rep(0, length(pairs$V1)) pairs$gamma <- rep(0, length(pairs$V1)) #scan D for all kmin-kmax pairs for (i in 1:length(pairs$D)){
m_pl$setXmin(pairs[i,1]) pairs[i,3]<- estimate_xmin(m_pl, xmin = pairs[i,1], xmax = pairs[i,2], distance = "ks")$gof
pairs[i,4]<- estimate_xmin(m_pl, xmin = pairs[i,1], xmax = pairs[i,2], distance = "ks")$pars } bs_pl_sat_cut <- bootstrap_p(m_pl, xmins = pairs[which.min(pairs$D), 1], xmax = pairs[which.min(pairs$D), 2], no_of_sims = 1000, threads = 8) pairs[which.min(pairs$D), 1] #k_{sat}

## [1] 5

pairs[which.min(pairs$D), 2] #k_{cut}  ## [1] 23  #in this range pairs[which.min(pairs$D), 3] #D

## [1] 0.03126577

pairs[which.min(pairs$D), 4] #gamma  ## [1] 3.175132  bs_pl_sat_cut$p #p-value

## [1] 0.931

pairs[which.min(pairs$D), 1] -> k_sat pairs[which.min(pairs$D), 2] -> k_cut
pairs[which.min(pairs$D), 4] -> gamma  • $k_{sat}$ = 5 • $k_{cut}$ = 23 • $D$ = 0.0312658 • $\gamma$ = 3.1751323 • $p$ = 0.931 (by bootstratpping) #powerlaw m_pl = displ$new(data)
est_pl <- estimate_xmin(m_pl, xmins = k_sat, xmax = k_cut, distance = "ks")
m_pl$setXmin(est_pl) #lognormal m_ln = dislnorm$new(data)
est_ln <- estimate_xmin(m_ln)
m_ln$setXmin(est_ln) #exponential m_exp = disexp$new(data)
est_exp <- estimate_xmin(m_exp)
m_exp$setXmin(est_exp) #poisson m_poi = dispois$new(data)
est_poi <- estimate_xmin(m_poi)
m_poi\$setXmin(est_poi)


This figure compares different distributions. (red: power-law, green: lognormal, blue: Poisson, magenta: exponential)

plot(m_pl)
lines(m_pl, col="red")
lines(m_ln, col="green")
lines(m_poi, col="blue")
lines(m_exp, col="magenta")


It seems that all distributions are fitted well except Poisson. And it is interesting that with static.power.law.game() funtion I wanted to generate graph with power-law degree distribution that has $\gamma$=2.2, and the resulted power-law degree distribution has $\gamma$ = 3.1751323 if it is power-law distribution.

Be happyR! đź™‚