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sna_eg_stanford:lab02

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Lab 02

####################################################################
# LAB 2: Methodological beginnings - Density, Reciprocity, Triads, #
# Transitivity, and heterogeneity. Node and network statistics.    #
####################################################################


# NOTE: if you have trouble because some packages are not installed, 
# see lab 1 for instructions on how to install all necessary packages. 
# Also see Lab 1 for prior functions.


##############################################################
# 
# Lab 2 
#
# The purpose of this lab is to acquire basic cohesion 
# metrics of density, reciprocity, reach, path distance, 
# and transitivity. In addition, we'll develop triadic 
# analyses and a measure of ego-network heterogenity. 
#
##############################################################



### 
# 1. SET UP SESSION
###
install.packages("NetData")

library(igraph)
library(NetData)


### 
# 2. LOAD DATA
###

# We would ordinarily need to follow the same proceedure we did for the Krackhardt data
# as we did in lab 1; see that lab for detail.

data(kracknets, package = "NetData")

# Reduce to non-zero edges and build a graph object
krack_full_nonzero_edges <- subset(krack_full_data_frame, (advice_tie > 0 | friendship_tie > 0 | reports_to_tie > 0))
head(krack_full_nonzero_edges)

krack_full <- graph.data.frame(krack_full_nonzero_edges) 
summary(krack_full)

# Set vertex attributes
for (i in V(krack_full)) {
	for (j in names(attributes)) {
		krack_full <- set.vertex.attribute(krack_full, j, index=i, attributes[i+1,j])
	}
}
summary(krack_full)

# Create sub-graphs based on edge attributes
krack_advice <- delete.edges(krack_full, E(krack_full)[get.edge.attribute(krack_full,name = "advice_tie")==0])
summary(krack_advice)

krack_friendship <- delete.edges(krack_full, E(krack_full)[get.edge.attribute(krack_full,name = "friendship_tie")==0])
summary(krack_friendship)

krack_reports_to <- delete.edges(krack_full, E(krack_full)[get.edge.attribute(krack_full,name = "reports_to_tie")==0])
summary(krack_reports_to)


### 
# 3. NODE-LEVEL STATISTICS
###

# Compute the indegree and outdegree for each node, first in the 
# full graph (accounting for all tie types) and then in each 
# tie-specific sub-graph. 
deg_full_in <- degree(krack_full, mode="in") 
deg_full_out <- degree(krack_full, mode="out") 
deg_full_in
deg_full_out

deg_advice_in <- degree(krack_advice, mode="in") 
deg_advice_out <- degree(krack_advice, mode="out") 
deg_advice_in
deg_advice_out

deg_friendship_in <- degree(krack_friendship, mode="in") 
deg_friendship_out <- degree(krack_friendship, mode="out") 
deg_friendship_in
deg_friendship_out

deg_reports_to_in <- degree(krack_reports_to, mode="in") 
deg_reports_to_out <- degree(krack_reports_to, mode="out") 
deg_reports_to_in
deg_reports_to_out

# Reachability can only be computed on one vertex at a time. To
# get graph-wide statistics, change the value of "vertex"
# manually or write a for loop. (Remember that, unlike R objects,
# igraph objects are numbered from 0.)

reachability <- function(g, m) {
	reach_mat = matrix(nrow = vcount(g), 
                       ncol = vcount(g))
	for (i in 1:vcount(g)) {
		reach_mat[i,] = 0
		this_node_reach <- subcomponent(g, (i), mode = m)

		for (j in 1:(length(this_node_reach))) {
			alter = this_node_reach[j]
			reach_mat[i, alter] = 1
		}
	}
	return(reach_mat)
}

reach_full_in <- reachability(krack_full, 'in')
reach_full_out <- reachability(krack_full, 'out')
reach_full_in
reach_full_out

reach_advice_in <- reachability(krack_advice, 'in')
reach_advice_out <- reachability(krack_advice, 'out')
reach_advice_in
reach_advice_out

reach_friendship_in <- reachability(krack_friendship, 'in')
reach_friendship_out <- reachability(krack_friendship, 'out')
reach_friendship_in
reach_friendship_out

reach_reports_to_in <- reachability(krack_reports_to, 'in')
reach_reports_to_out <- reachability(krack_reports_to, 'out')
reach_reports_to_in
reach_reports_to_out


# Often we want to know path distances between individuals in a network. 
# This is often done by calculating geodesics, or shortest paths between
# each ij pair. One can symmetrize the data to do this (see lab 1), or 
# calculate it for outward and inward ties separately. Averaging geodesics 
# for the entire network provides an average distance or sort of cohesiveness
# score. Dichotomizing distances reveals reach, and an average of reach for 
# a network reveals what percent of a network is connected in some way.

# Compute shortest paths between each pair of nodes. 
sp_full_in <- shortest.paths(krack_full, mode='in')
sp_full_out <- shortest.paths(krack_full, mode='out')
sp_full_in
sp_full_out

sp_advice_in <- shortest.paths(krack_advice, mode='in')
sp_advice_out <- shortest.paths(krack_advice, mode='out')
sp_advice_in
sp_advice_out

sp_friendship_in <- shortest.paths(krack_friendship, mode='in')
sp_friendship_out <- shortest.paths(krack_friendship, mode='out')
sp_friendship_in
sp_friendship_out

sp_reports_to_in <- shortest.paths(krack_reports_to, mode='in')
sp_reports_to_out <- shortest.paths(krack_reports_to, mode='out')
sp_reports_to_in
sp_reports_to_out


# Assemble node-level stats into single data frame for export as CSV.

# First, we have to compute average values by node for reachability and
# shortest path. (We don't have to do this for degree because it is 
# already expressed as a node-level value.)
reach_full_in_vec <- vector()
reach_full_out_vec <- vector()
reach_advice_in_vec <- vector()
reach_advice_out_vec <- vector()
reach_friendship_in_vec <- vector()
reach_friendship_out_vec <- vector()
reach_reports_to_in_vec <- vector()
reach_reports_to_out_vec <- vector()

sp_full_in_vec <- vector()
sp_full_out_vec <- vector()
sp_advice_in_vec <- vector()
sp_advice_out_vec <- vector()
sp_friendship_in_vec <- vector()
sp_friendship_out_vec <- vector()
sp_reports_to_in_vec <- vector()
sp_reports_to_out_vec <- vector()

for (i in 1:vcount(krack_full)) {
	reach_full_in_vec[i] <- mean(reach_full_in[i,])
	reach_full_out_vec[i] <- mean(reach_full_out[i,])
	reach_advice_in_vec[i] <- mean(reach_advice_in[i,])
	reach_advice_out_vec[i] <- mean(reach_advice_out[i,])
	reach_friendship_in_vec[i] <- mean(reach_friendship_in[i,])
	reach_friendship_out_vec[i] <- mean(reach_friendship_out[i,])
	reach_reports_to_in_vec[i] <- mean(reach_reports_to_in[i,])
	reach_reports_to_out_vec[i] <- mean(reach_reports_to_out[i,])

	sp_full_in_vec[i] <- mean(sp_full_in[i,])
	sp_full_out_vec[i] <- mean(sp_full_out[i,])
	sp_advice_in_vec[i] <- mean(sp_advice_in[i,])
	sp_advice_out_vec[i] <- mean(sp_advice_out[i,])
	sp_friendship_in_vec[i] <- mean(sp_friendship_in[i,])
	sp_friendship_out_vec[i] <- mean(sp_friendship_out[i,])
	sp_reports_to_in_vec[i] <- mean(sp_reports_to_in[i,])
	sp_reports_to_out_vec[i] <- mean(sp_reports_to_out[i,])
}

# Next, we assemble all of the vectors of node-levelvalues into a 
# single data frame, which we can export as a CSV to our working
# directory.
node_stats_df <- cbind(deg_full_in,
                       deg_full_out,
                       deg_advice_in,
                       deg_advice_out,
                       deg_friendship_in,
                       deg_friendship_out,
                       deg_reports_to_in,
                       deg_reports_to_out, 

                       reach_full_in_vec, 
                       reach_full_out_vec, 
                       reach_advice_in_vec, 
                       reach_advice_out_vec, 
                       reach_friendship_in_vec, 
                       reach_friendship_out_vec, 
                       reach_reports_to_in_vec, 
                       reach_reports_to_out_vec, 

                       sp_full_in_vec, 
                       sp_full_out_vec, 
                       sp_advice_in_vec, 
                       sp_advice_out_vec, 
                       sp_friendship_in_vec, 
                       sp_friendship_out_vec, 
                       sp_reports_to_in_vec, 
                       sp_reports_to_out_vec)

write.csv(node_stats_df, 'krack_node_stats.csv')

# Question #1 - What do these statistics tell us about
# each network and its individuals in general? 

### 
# 3. NETWORK-LEVEL STATISTICS
###

# Many initial analyses of networks begin with distances and reach, 
# and then move towards global summary statistics of the network. 
#
# As a reminder, entering a question mark followed by a function 
# name (e.g., ?graph.density) pulls up the help file for that function.
# This can be helpful to understand how, exactly, stats are calculated.

# Degree
mean(deg_full_in)
sd(deg_full_in)
mean(deg_full_out)
sd(deg_full_out)

mean(deg_advice_in)
sd(deg_advice_in)
mean(deg_advice_out)
sd(deg_advice_out)

mean(deg_friendship_in)
sd(deg_friendship_in)
mean(deg_friendship_out)
sd(deg_friendship_out)

mean(deg_reports_to_in)
sd(deg_reports_to_in)
mean(deg_reports_to_out)
sd(deg_reports_to_out)


# Shortest paths
# ***Why do in and out come up with the same results?
# In and out shortest paths are simply transposes of one another; 
# thus, when we compute statistics across the whole network they have to be the same.

mean(sp_full_in[which(sp_full_in != Inf)])
sd(sp_full_in[which(sp_full_in != Inf)])
mean(sp_full_out[which(sp_full_out != Inf)])
sd(sp_full_out[which(sp_full_out != Inf)])

mean(sp_advice_in[which(sp_advice_in != Inf)])
sd(sp_advice_in[which(sp_advice_in != Inf)])
mean(sp_advice_out[which(sp_advice_out != Inf)])
sd(sp_advice_out[which(sp_advice_out != Inf)])

mean(sp_friendship_in[which(sp_friendship_in != Inf)])
sd(sp_friendship_in[which(sp_friendship_in != Inf)])
mean(sp_friendship_out[which(sp_friendship_out != Inf)])
sd(sp_friendship_out[which(sp_friendship_out != Inf)])

mean(sp_reports_to_in[which(sp_reports_to_in != Inf)])
sd(sp_reports_to_in[which(sp_reports_to_in != Inf)])
mean(sp_reports_to_out[which(sp_reports_to_out != Inf)])
sd(sp_reports_to_out[which(sp_reports_to_out != Inf)])

# Reachability
mean(reach_full_in[which(reach_full_in != Inf)])
sd(reach_full_in[which(reach_full_in != Inf)])
mean(reach_full_out[which(reach_full_out != Inf)])
sd(reach_full_out[which(reach_full_out != Inf)])

mean(reach_advice_in[which(reach_advice_in != Inf)])
sd(reach_advice_in[which(reach_advice_in != Inf)])
mean(reach_advice_out[which(reach_advice_out != Inf)])
sd(reach_advice_out[which(reach_advice_out != Inf)])

mean(reach_friendship_in[which(reach_friendship_in != Inf)])
sd(reach_friendship_in[which(reach_friendship_in != Inf)])
mean(reach_friendship_out[which(reach_friendship_out != Inf)])
sd(reach_friendship_out[which(reach_friendship_out != Inf)])

mean(reach_reports_to_in[which(reach_reports_to_in != Inf)])
sd(reach_reports_to_in[which(reach_reports_to_in != Inf)])
mean(reach_reports_to_out[which(reach_reports_to_out != Inf)])
sd(reach_reports_to_out[which(reach_reports_to_out != Inf)])

# Density 
graph.density(krack_full)
graph.density(krack_advice)
graph.density(krack_friendship)
graph.density(krack_reports_to)

# Reciprocity
reciprocity(krack_full)
reciprocity(krack_advice)
reciprocity(krack_friendship)
reciprocity(krack_reports_to)

# Transitivity
transitivity(krack_full)
transitivity(krack_advice)
transitivity(krack_friendship)
transitivity(krack_reports_to)

# Triad census. Here we'll first build a vector of labels for 
# the different triad types. Then we'll combine this vector
# with the triad censuses for the different networks, which 
# we'll export as a CSV.

census_labels = c('003',
                  '012',
                  '102',
                  '021D',
                  '021U',
                  '021C',
                  '111D',
                  '111U',
                  '030T',
                  '030C',
                  '201',
                  '120D',
                  '120U',
                  '120C',
                  '210',
                  '300')
tc_full <- triad.census(krack_full)
tc_advice <- triad.census(krack_advice)
tc_friendship <- triad.census(krack_friendship)
tc_reports_to <- triad.census(krack_reports_to)

triad_df <- data.frame(census_labels,
                       tc_full, 
                       tc_advice, 
                       tc_friendship,
                       tc_reports_to)
triad_df

# To export any of these vectors to a CSV for use in another program, simply
# use the write.csv() command:
write.csv(triad_df, 'krack_triads.csv')

# Question #2 - (a) How do the three networks differ on network statictics? 
# (b) What does the triad census tell us? Can you calculate the likelihood of
# any triad's occurrence? (c) See the back of Wasserman and Faust and its section
# on triads. Calculate the degree of clustering and hierarchy in Excel. 
# What do we learn from that?




### 
# 4. HETEROGENEITY 
###

# Miller and McPherson write about processes of homophily and
# here we take a brief look at one version of this issue. 
# In particular, we look at the extent to which each actor's
# "associates" (friend, advisor, boos) are heterogenous or not.

# We'll use a statistic called the IQV, or Index of Qualitative
# Variation. This is just an implementation of Blau's Index of
# Heterogeneity (known to economists as the Herfindahl-Hirschman
# index), normalized so that perfect heterogeneity (i.e., equal 
# distribution across categories) equals 1.

# NOTE that this code only works with categorical variables that 
# have been numerically coded to integer values that ascend
# sequentially from 0; you may have to recode your data to get this
# to work properly.
# We are interested in many of the attributes of nodes.  To save 
# time and to make our lives better we are going to create a function
# that will provide an IQV statistic for any network and for
# any categorical variable.  A function is a simple way to
# create code that is both reusable and easier to edit.

# Functions have names and receive arguments.  For example,
# anytime you call table() you are calling the table function.
# We could write code to duplicate the table function for each
# of our variables, but it is faster to write a single general tool
# that will provide frequencies for any variable. If I have
# a dataframe with the variable gender and I want to see the
# split of males and females I would pass the argument
# "dataframe$gender" to the table function. We follow a
# similar model here. Understanding each step is less important
# than understanding the usefulness and power of functions.

get_iqvs <- function(graph, attribute) {

#we have now defined a function, get_iqvs, that will take the
# graph "graph" and find the iqv statistic for the categorical
# variable "attribute." Within this function whenever we use the 
#variables graph or attribute they correspond to the graph and
# variable we passed (provided) to the function

mat <- get.adjacency(graph)
				
# To make this function work on a wide variety of variables we
# find out how many coded levels (unique responses) exist for
# the attribute variable programatically

	attr_levels = get.vertex.attribute(graph,
	                                   attribute,
	                                   V(graph))
	
	num_levels = length(unique(attr_levels))
	iqvs = rep(0, nrow(mat))

# Now that we know how many levels exist we want to loop
# (go through) each actor in the network. Loops iterate through
# each value in a range.  Here we are looking through each ego
# in the range of egos starting at the first and ending at the
# last.  The function nrow provides the number of rows in an
# object and the ":" opperand specifies the range.  Between
# the curly braces of the for loop ego will represent exactly
# one value between 1 and the number of rows in the graph
# object, iterating by one during each execution of the loop.

	for (ego in 1:nrow(mat)) {
		
		# initialize actor-specific variables
		alter_attr_counts = rep(0, num_levels)
		num_alters_this_ego = 0
		sq_fraction_sum = 0

# For each ego we want to check each tied alter for the same
# level on the variable attribute as the ego.
	
		for (alter in 1:ncol(mat)) {
			
			# only examine alters that are actually tied to ego
			if (mat[ego, alter] == 1) {
				
				num_alters_this_ego = num_alters_this_ego + 1

				# get the alter's level on the attribute 
				alter_attr = get.vertex.attribute(graph, 
				    attribute, (alter - 1))

				# increment the count of alters with this level
				# of the attribute by 1
				alter_attr_counts[alter_attr + 1] =
				    alter_attr_counts[alter_attr + 1] + 1
			}
		}

		# now that we're done looping through all of the alters,
		# get the squared fraction for each level of the attribute
		# out of the total number of attributes
		for (i in 1:num_levels) {
			attr_fraction = alter_attr_counts[i] /
			    num_alters_this_ego
			sq_fraction_sum = sq_fraction_sum + attr_fraction ^ 2
		}
		
		# now we can compute the ego's blau index...
		blau_index = 1 - sq_fraction_sum
		
		# and the ego's IQV, which is just a normalized blau index
		iqvs[ego] = blau_index / (1 - (1 / num_levels))
	}

# The final part of a function returns the calculated value.
#  So if we called get_iqvs(testgraph, gender) return would
# provide the iqvs for gender in the test graph.  If we are also
# intersted in race we could simply change the function call
# to get_iqvs(testgraph, race).  No need to write all this
# code again for different variables.

	return(iqvs)
}



# For this data set, we'll look at homophily across departments, 
# which is already coded 0-4, so no recoding is needed. 

advice_iqvs <- get_iqvs(krack_advice, 'DEPT')
advice_iqvs

friendship_iqvs <- get_iqvs(krack_friendship, 'DEPT')
friendship_iqvs

reports_to_iqvs <- get_iqvs(krack_reports_to, 'DEPT')
reports_to_iqvs

# Question #3 - What does the herfindahl index reveal about 
# attribute sorting in networks? What does it mean for each network?


#####
# Extra-credit: What might be a better way to test the occurrence 
# of homophily or segregation in a network? How might we code that in R?
#####

#####
# Tau statistic (code by Sam Pimentel)
#####


#R code for generating random graphs:
#requires packages ergm, intergraph

#set up weighting vectors for clustering and hierarchy
clust.mask <- rep(0,16)
clust.mask[c(1,3,16)] <- 1
hier.mask <- rep(1,16)
hier.mask[c(6:8,10:11)]  <- 0

#compute triad count and triad proportion for a given weighting vector
mask.stat <- function(my.graph, my.mask){
    n.nodes <- vcount(my.graph)
    n.edges <- ecount(my.graph)
    #set probability of edge formation in random graph to proportion of possible edges present in original
    p.edge <- n.edges/(n.nodes*(n.nodes +1)/2)
    r.graph <- as.network(n.nodes, density = p.edge)
    ## r.igraph <- as.igraph(r.graph)
    r.igraph <- asIgraph(r.graph)
    ##################################
    tc.graph <- triad.census(r.igraph)
    clust <- sum(tc.graph*my.mask)
    clust.norm <- clust/sum(tc.graph)
    return(c(clust,clust.norm))
}

#build 100 random graphs and compute their clustering and hierarchy measurements to create an empirical null distribution
emp.distro <- function(this.graph){
  clust <- matrix(rep(0,200), nrow=2) 
  hier <- matrix(rep(0,200),nrow=2)
  for(i in c(1:100)){
     clust[,i] <- mask.stat(this.graph, clust.mask)
     hier[,i] <- mask.stat(this.graph, hier.mask)
  }
  my.mat <- rbind(clust, hier)
  rownames(my.mat) <- c("clust.ct", "clust.norm", "hier.ct", "hier.ct.norm")
  return(my.mat)
}

#fix randomization if desired so results are replicable
#set.seed(3123)
#compute empirical distributions for each network
hc_advice <- emp.distro(krack_advice)
hc_friend <- emp.distro(krack_friendship)
hc_report <- emp.distro(krack_reports_to)

#find empirical p-value
get.p <- function(val, distro)
{
	distro.n <- sort(distro)
	distro.n <- distro.n - median(distro.n)
	val.n <- val - median(distro.n)
	p.val <- sum(abs(distro.n) > abs(val.n))/100
	return(p.val)
}
get.p(198, hc_full[1,])
get.p(194, hc_advice[1,])
get.p(525, hc_friend[1,])
get.p(1003, hc_report[1,])
get.p(979, hc_full[3,])
get.p(1047, hc_advice[3,])
get.p(1135, hc_friend[3,])
get.p(1314, hc_report[3,])

#generate  95% empirical confidence intervals for triad counts

#clustering
c(sort(hc_advice[1,])[5], sort(hc_advice[1,])[95])
c(sort(hc_friend[1,])[5], sort(hc_friend[1,])[95])
c(sort(hc_report[1,])[5], sort(hc_report[1,])[95])

#hierarchy
c(sort(hc_advice[3,])[5], sort(hc_advice[3,])[95])
c(sort(hc_friend[3,])[5], sort(hc_friend[3,])[95])
c(sort(hc_report[3,])[5], sort(hc_report[3,])[95])

no pdf

####################################################################
# LAB 2: Methodological beginnings - Density, Reciprocity, Triads, #
# Transitivity, and heterogeneity. Node and network statistics.    #
####################################################################


# NOTE: if you have trouble because some packages are not installed, 
# see lab 1 for instructions on how to install all necessary packages. 
# Also see Lab 1 for prior functions.


##############################################################
# 
# Lab 2 
#
# The purpose of this lab is to acquire basic cohesion 
# metrics of density, reciprocity, reach, path distance, 
# and transitivity. In addition, we'll develop triadic 
# analyses and a measure of ego-network heterogenity. 
#
##############################################################



### 
# 1. SET UP SESSION
###
install.packages("NetData")

library(igraph)
library(NetData)


### 
# 2. LOAD DATA
###

# We would ordinarily need to follow the same proceedure we did for the Krackhardt data
# as we did in lab 1; see that lab for detail.

data(kracknets, package = "NetData")

# Reduce to non-zero edges and build a graph object
krack_full_nonzero_edges <- subset(krack_full_data_frame, (advice_tie > 0 | friendship_tie > 0 | reports_to_tie > 0))
head(krack_full_nonzero_edges)

krack_full <- graph.data.frame(krack_full_nonzero_edges) 
summary(krack_full)

# Set vertex attributes
for (i in V(krack_full)) {
	for (j in names(attributes)) {
		krack_full <- set.vertex.attribute(krack_full, j, index=i, attributes[i+1,j])
	}
}
summary(krack_full)

# Create sub-graphs based on edge attributes
krack_advice <- delete.edges(krack_full, E(krack_full)[get.edge.attribute(krack_full,name = "advice_tie")==0])
summary(krack_advice)

krack_friendship <- delete.edges(krack_full, E(krack_full)[get.edge.attribute(krack_full,name = "friendship_tie")==0])
summary(krack_friendship)

krack_reports_to <- delete.edges(krack_full, E(krack_full)[get.edge.attribute(krack_full,name = "reports_to_tie")==0])
summary(krack_reports_to)


### 
# 3. NODE-LEVEL STATISTICS
###

# Compute the indegree and outdegree for each node, first in the 
# full graph (accounting for all tie types) and then in each 
# tie-specific sub-graph. 
deg_full_in <- degree(krack_full, mode="in") 
deg_full_out <- degree(krack_full, mode="out") 
deg_full_in
deg_full_out

deg_advice_in <- degree(krack_advice, mode="in") 
deg_advice_out <- degree(krack_advice, mode="out") 
deg_advice_in
deg_advice_out

deg_friendship_in <- degree(krack_friendship, mode="in") 
deg_friendship_out <- degree(krack_friendship, mode="out") 
deg_friendship_in
deg_friendship_out

deg_reports_to_in <- degree(krack_reports_to, mode="in") 
deg_reports_to_out <- degree(krack_reports_to, mode="out") 
deg_reports_to_in
deg_reports_to_out

# Reachability can only be computed on one vertex at a time. To
# get graph-wide statistics, change the value of "vertex"
# manually or write a for loop. (Remember that, unlike R objects,
# igraph objects are numbered from 0.)

reachability <- function(g, m) {
	reach_mat = matrix(nrow = vcount(g), 
                       ncol = vcount(g))
	for (i in 1:vcount(g)) {
		reach_mat[i,] = 0
		this_node_reach <- subcomponent(g, (i), mode = m)

		for (j in 1:(length(this_node_reach))) {
			alter = this_node_reach[j]
			reach_mat[i, alter] = 1
		}
	}
	return(reach_mat)
}

reach_full_in <- reachability(krack_full, 'in')
reach_full_out <- reachability(krack_full, 'out')
reach_full_in
reach_full_out

reach_advice_in <- reachability(krack_advice, 'in')
reach_advice_out <- reachability(krack_advice, 'out')
reach_advice_in
reach_advice_out

reach_friendship_in <- reachability(krack_friendship, 'in')
reach_friendship_out <- reachability(krack_friendship, 'out')
reach_friendship_in
reach_friendship_out

reach_reports_to_in <- reachability(krack_reports_to, 'in')
reach_reports_to_out <- reachability(krack_reports_to, 'out')
reach_reports_to_in
reach_reports_to_out


# Often we want to know path distances between individuals in a network. 
# This is often done by calculating geodesics, or shortest paths between
# each ij pair. One can symmetrize the data to do this (see lab 1), or 
# calculate it for outward and inward ties separately. Averaging geodesics 
# for the entire network provides an average distance or sort of cohesiveness
# score. Dichotomizing distances reveals reach, and an average of reach for 
# a network reveals what percent of a network is connected in some way.

# Compute shortest paths between each pair of nodes. 
sp_full_in <- shortest.paths(krack_full, mode='in')
sp_full_out <- shortest.paths(krack_full, mode='out')
sp_full_in
sp_full_out

sp_advice_in <- shortest.paths(krack_advice, mode='in')
sp_advice_out <- shortest.paths(krack_advice, mode='out')
sp_advice_in
sp_advice_out

sp_friendship_in <- shortest.paths(krack_friendship, mode='in')
sp_friendship_out <- shortest.paths(krack_friendship, mode='out')
sp_friendship_in
sp_friendship_out

sp_reports_to_in <- shortest.paths(krack_reports_to, mode='in')
sp_reports_to_out <- shortest.paths(krack_reports_to, mode='out')
sp_reports_to_in
sp_reports_to_out


# Assemble node-level stats into single data frame for export as CSV.

# First, we have to compute average values by node for reachability and
# shortest path. (We don't have to do this for degree because it is 
# already expressed as a node-level value.)
reach_full_in_vec <- vector()
reach_full_out_vec <- vector()
reach_advice_in_vec <- vector()
reach_advice_out_vec <- vector()
reach_friendship_in_vec <- vector()
reach_friendship_out_vec <- vector()
reach_reports_to_in_vec <- vector()
reach_reports_to_out_vec <- vector()

sp_full_in_vec <- vector()
sp_full_out_vec <- vector()
sp_advice_in_vec <- vector()
sp_advice_out_vec <- vector()
sp_friendship_in_vec <- vector()
sp_friendship_out_vec <- vector()
sp_reports_to_in_vec <- vector()
sp_reports_to_out_vec <- vector()

for (i in 1:vcount(krack_full)) {
	reach_full_in_vec[i] <- mean(reach_full_in[i,])
	reach_full_out_vec[i] <- mean(reach_full_out[i,])
	reach_advice_in_vec[i] <- mean(reach_advice_in[i,])
	reach_advice_out_vec[i] <- mean(reach_advice_out[i,])
	reach_friendship_in_vec[i] <- mean(reach_friendship_in[i,])
	reach_friendship_out_vec[i] <- mean(reach_friendship_out[i,])
	reach_reports_to_in_vec[i] <- mean(reach_reports_to_in[i,])
	reach_reports_to_out_vec[i] <- mean(reach_reports_to_out[i,])

	sp_full_in_vec[i] <- mean(sp_full_in[i,])
	sp_full_out_vec[i] <- mean(sp_full_out[i,])
	sp_advice_in_vec[i] <- mean(sp_advice_in[i,])
	sp_advice_out_vec[i] <- mean(sp_advice_out[i,])
	sp_friendship_in_vec[i] <- mean(sp_friendship_in[i,])
	sp_friendship_out_vec[i] <- mean(sp_friendship_out[i,])
	sp_reports_to_in_vec[i] <- mean(sp_reports_to_in[i,])
	sp_reports_to_out_vec[i] <- mean(sp_reports_to_out[i,])
}

# Next, we assemble all of the vectors of node-levelvalues into a 
# single data frame, which we can export as a CSV to our working
# directory.
node_stats_df <- cbind(deg_full_in,
                       deg_full_out,
                       deg_advice_in,
                       deg_advice_out,
                       deg_friendship_in,
                       deg_friendship_out,
                       deg_reports_to_in,
                       deg_reports_to_out, 

                       reach_full_in_vec, 
                       reach_full_out_vec, 
                       reach_advice_in_vec, 
                       reach_advice_out_vec, 
                       reach_friendship_in_vec, 
                       reach_friendship_out_vec, 
                       reach_reports_to_in_vec, 
                       reach_reports_to_out_vec, 

                       sp_full_in_vec, 
                       sp_full_out_vec, 
                       sp_advice_in_vec, 
                       sp_advice_out_vec, 
                       sp_friendship_in_vec, 
                       sp_friendship_out_vec, 
                       sp_reports_to_in_vec, 
                       sp_reports_to_out_vec)

write.csv(node_stats_df, 'krack_node_stats.csv')

# Question #1 - What do these statistics tell us about
# each network and its individuals in general? 

### 
# 3. NETWORK-LEVEL STATISTICS
###

# Many initial analyses of networks begin with distances and reach, 
# and then move towards global summary statistics of the network. 
#
# As a reminder, entering a question mark followed by a function 
# name (e.g., ?graph.density) pulls up the help file for that function.
# This can be helpful to understand how, exactly, stats are calculated.

# Degree
mean(deg_full_in)
sd(deg_full_in)
mean(deg_full_out)
sd(deg_full_out)

mean(deg_advice_in)
sd(deg_advice_in)
mean(deg_advice_out)
sd(deg_advice_out)

mean(deg_friendship_in)
sd(deg_friendship_in)
mean(deg_friendship_out)
sd(deg_friendship_out)

mean(deg_reports_to_in)
sd(deg_reports_to_in)
mean(deg_reports_to_out)
sd(deg_reports_to_out)


# Shortest paths
# ***Why do in and out come up with the same results?
# In and out shortest paths are simply transposes of one another; 
# thus, when we compute statistics across the whole network they have to be the same.

mean(sp_full_in[which(sp_full_in != Inf)])
sd(sp_full_in[which(sp_full_in != Inf)])
mean(sp_full_out[which(sp_full_out != Inf)])
sd(sp_full_out[which(sp_full_out != Inf)])

mean(sp_advice_in[which(sp_advice_in != Inf)])
sd(sp_advice_in[which(sp_advice_in != Inf)])
mean(sp_advice_out[which(sp_advice_out != Inf)])
sd(sp_advice_out[which(sp_advice_out != Inf)])

mean(sp_friendship_in[which(sp_friendship_in != Inf)])
sd(sp_friendship_in[which(sp_friendship_in != Inf)])
mean(sp_friendship_out[which(sp_friendship_out != Inf)])
sd(sp_friendship_out[which(sp_friendship_out != Inf)])

mean(sp_reports_to_in[which(sp_reports_to_in != Inf)])
sd(sp_reports_to_in[which(sp_reports_to_in != Inf)])
mean(sp_reports_to_out[which(sp_reports_to_out != Inf)])
sd(sp_reports_to_out[which(sp_reports_to_out != Inf)])

# Reachability
mean(reach_full_in[which(reach_full_in != Inf)])
sd(reach_full_in[which(reach_full_in != Inf)])
mean(reach_full_out[which(reach_full_out != Inf)])
sd(reach_full_out[which(reach_full_out != Inf)])

mean(reach_advice_in[which(reach_advice_in != Inf)])
sd(reach_advice_in[which(reach_advice_in != Inf)])
mean(reach_advice_out[which(reach_advice_out != Inf)])
sd(reach_advice_out[which(reach_advice_out != Inf)])

mean(reach_friendship_in[which(reach_friendship_in != Inf)])
sd(reach_friendship_in[which(reach_friendship_in != Inf)])
mean(reach_friendship_out[which(reach_friendship_out != Inf)])
sd(reach_friendship_out[which(reach_friendship_out != Inf)])

mean(reach_reports_to_in[which(reach_reports_to_in != Inf)])
sd(reach_reports_to_in[which(reach_reports_to_in != Inf)])
mean(reach_reports_to_out[which(reach_reports_to_out != Inf)])
sd(reach_reports_to_out[which(reach_reports_to_out != Inf)])

# Density 
graph.density(krack_full)
graph.density(krack_advice)
graph.density(krack_friendship)
graph.density(krack_reports_to)

# Reciprocity
reciprocity(krack_full)
reciprocity(krack_advice)
reciprocity(krack_friendship)
reciprocity(krack_reports_to)

# Transitivity
transitivity(krack_full)
transitivity(krack_advice)
transitivity(krack_friendship)
transitivity(krack_reports_to)

# Triad census. Here we'll first build a vector of labels for 
# the different triad types. Then we'll combine this vector
# with the triad censuses for the different networks, which 
# we'll export as a CSV.

census_labels = c('003',
                  '012',
                  '102',
                  '021D',
                  '021U',
                  '021C',
                  '111D',
                  '111U',
                  '030T',
                  '030C',
                  '201',
                  '120D',
                  '120U',
                  '120C',
                  '210',
                  '300')
tc_full <- triad.census(krack_full)
tc_advice <- triad.census(krack_advice)
tc_friendship <- triad.census(krack_friendship)
tc_reports_to <- triad.census(krack_reports_to)

triad_df <- data.frame(census_labels,
                       tc_full, 
                       tc_advice, 
                       tc_friendship,
                       tc_reports_to)
triad_df

# To export any of these vectors to a CSV for use in another program, simply
# use the write.csv() command:
write.csv(triad_df, 'krack_triads.csv')

# Question #2 - (a) How do the three networks differ on network statictics? 
# (b) What does the triad census tell us? Can you calculate the likelihood of
# any triad's occurrence? (c) See the back of Wasserman and Faust and its section
# on triads. Calculate the degree of clustering and hierarchy in Excel. 
# What do we learn from that?

omitted

### 
# 4. HETEROGENEITY 
###

# Miller and McPherson write about processes of homophily and
# here we take a brief look at one version of this issue. 
# In particular, we look at the extent to which each actor's
# "associates" (friend, advisor, boos) are heterogenous or not.

# We'll use a statistic called the IQV, or Index of Qualitative
# Variation. This is just an implementation of Blau's Index of
# Heterogeneity (known to economists as the Herfindahl-Hirschman
# index), normalized so that perfect heterogeneity (i.e., equal 
# distribution across categories) equals 1.

# NOTE that this code only works with categorical variables that 
# have been numerically coded to integer values that ascend
# sequentially from 0; you may have to recode your data to get this
# to work properly.
# We are interested in many of the attributes of nodes.  To save 
# time and to make our lives better we are going to create a function
# that will provide an IQV statistic for any network and for
# any categorical variable.  A function is a simple way to
# create code that is both reusable and easier to edit.

# Functions have names and receive arguments.  For example,
# anytime you call table() you are calling the table function.
# We could write code to duplicate the table function for each
# of our variables, but it is faster to write a single general tool
# that will provide frequencies for any variable. If I have
# a dataframe with the variable gender and I want to see the
# split of males and females I would pass the argument
# "dataframe$gender" to the table function. We follow a
# similar model here. Understanding each step is less important
# than understanding the usefulness and power of functions.

get_iqvs <- function(graph, attribute) {

#we have now defined a function, get_iqvs, that will take the
# graph "graph" and find the iqv statistic for the categorical
# variable "attribute." Within this function whenever we use the 
#variables graph or attribute they correspond to the graph and
# variable we passed (provided) to the function

mat <- get.adjacency(graph)
				
# To make this function work on a wide variety of variables we
# find out how many coded levels (unique responses) exist for
# the attribute variable programatically

	attr_levels = get.vertex.attribute(graph,
	                                   attribute,
	                                   V(graph))
	
	num_levels = length(unique(attr_levels))
	iqvs = rep(0, nrow(mat))

# Now that we know how many levels exist we want to loop
# (go through) each actor in the network. Loops iterate through
# each value in a range.  Here we are looking through each ego
# in the range of egos starting at the first and ending at the
# last.  The function nrow provides the number of rows in an
# object and the ":" opperand specifies the range.  Between
# the curly braces of the for loop ego will represent exactly
# one value between 1 and the number of rows in the graph
# object, iterating by one during each execution of the loop.

	for (ego in 1:nrow(mat)) {
		
		# initialize actor-specific variables
		alter_attr_counts = rep(0, num_levels)
		num_alters_this_ego = 0
		sq_fraction_sum = 0

# For each ego we want to check each tied alter for the same
# level on the variable attribute as the ego.
	
		for (alter in 1:ncol(mat)) {
			
			# only examine alters that are actually tied to ego
			if (mat[ego, alter] == 1) {
				
				num_alters_this_ego = num_alters_this_ego + 1

				# get the alter's level on the attribute 
				alter_attr = get.vertex.attribute(graph, 
				    attribute, (alter - 1))

				# increment the count of alters with this level
				# of the attribute by 1
				alter_attr_counts[alter_attr + 1] =
				    alter_attr_counts[alter_attr + 1] + 1
			}
		}

		# now that we're done looping through all of the alters,
		# get the squared fraction for each level of the attribute
		# out of the total number of attributes
		for (i in 1:num_levels) {
			attr_fraction = alter_attr_counts[i] /
			    num_alters_this_ego
			sq_fraction_sum = sq_fraction_sum + attr_fraction ^ 2
		}
		
		# now we can compute the ego's blau index...
		blau_index = 1 - sq_fraction_sum
		
		# and the ego's IQV, which is just a normalized blau index
		iqvs[ego] = blau_index / (1 - (1 / num_levels))
	}

# The final part of a function returns the calculated value.
#  So if we called get_iqvs(testgraph, gender) return would
# provide the iqvs for gender in the test graph.  If we are also
# intersted in race we could simply change the function call
# to get_iqvs(testgraph, race).  No need to write all this
# code again for different variables.

	return(iqvs)
}



# For this data set, we'll look at homophily across departments, 
# which is already coded 0-4, so no recoding is needed. 

advice_iqvs <- get_iqvs(krack_advice, 'DEPT')
advice_iqvs

friendship_iqvs <- get_iqvs(krack_friendship, 'DEPT')
friendship_iqvs

reports_to_iqvs <- get_iqvs(krack_reports_to, 'DEPT')
reports_to_iqvs

# Question #3 - What does the herfindahl index reveal about 
# attribute sorting in networks? What does it mean for each network?


#####
# Extra-credit: What might be a better way to test the occurrence 
# of homophily or segregation in a network? How might we code that in R?
#####

#####
# Tau statistic (code by Sam Pimentel)
#####


#R code for generating random graphs:
#requires packages ergm, intergraph

#set up weighting vectors for clustering and hierarchy
clust.mask <- rep(0,16)
clust.mask[c(1,3,16)] <- 1
hier.mask <- rep(1,16)
hier.mask[c(6:8,10:11)]  <- 0

#compute triad count and triad proportion for a given weighting vector
mask.stat <- function(my.graph, my.mask){
    n.nodes <- vcount(my.graph)
    n.edges <- ecount(my.graph)
    #set probability of edge formation in random graph to proportion of possible edges present in original
    p.edge <- n.edges/(n.nodes*(n.nodes +1)/2)
    r.graph <- as.network(n.nodes, density = p.edge)
    ## r.igraph <- as.igraph(r.graph)
    r.igraph <- asIgraph(r.graph)
    ##################################
    tc.graph <- triad.census(r.igraph)
    clust <- sum(tc.graph*my.mask)
    clust.norm <- clust/sum(tc.graph)
    return(c(clust,clust.norm))
}

#build 100 random graphs and compute their clustering and hierarchy measurements to create an empirical null distribution
emp.distro <- function(this.graph){
  clust <- matrix(rep(0,200), nrow=2) 
  hier <- matrix(rep(0,200),nrow=2)
  for(i in c(1:100)){
     clust[,i] <- mask.stat(this.graph, clust.mask)
     hier[,i] <- mask.stat(this.graph, hier.mask)
  }
  my.mat <- rbind(clust, hier)
  rownames(my.mat) <- c("clust.ct", "clust.norm", "hier.ct", "hier.ct.norm")
  return(my.mat)
}

#fix randomization if desired so results are replicable
#set.seed(3123)
#compute empirical distributions for each network
hc_advice <- emp.distro(krack_advice)
hc_friend <- emp.distro(krack_friendship)
hc_report <- emp.distro(krack_reports_to)

#find empirical p-value
get.p <- function(val, distro)
{
	distro.n <- sort(distro)
	distro.n <- distro.n - median(distro.n)
	val.n <- val - median(distro.n)
	p.val <- sum(abs(distro.n) > abs(val.n))/100
	return(p.val)
}
get.p(198, hc_full[1,])
get.p(194, hc_advice[1,])
get.p(525, hc_friend[1,])
get.p(1003, hc_report[1,])
get.p(979, hc_full[3,])
get.p(1047, hc_advice[3,])
get.p(1135, hc_friend[3,])
get.p(1314, hc_report[3,])

#generate  95% empirical confidence intervals for triad counts

#clustering
c(sort(hc_advice[1,])[5], sort(hc_advice[1,])[95])
c(sort(hc_friend[1,])[5], sort(hc_friend[1,])[95])
c(sort(hc_report[1,])[5], sort(hc_report[1,])[95])

#hierarchy
c(sort(hc_advice[3,])[5], sort(hc_advice[3,])[95])
c(sort(hc_friend[3,])[5], sort(hc_friend[3,])[95])
c(sort(hc_report[3,])[5], sort(hc_report[3,])[95])
sna_eg_stanford/lab02.txt · Last modified: 2019/12/11 08:52 by hkimscil