####################################################################
# 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 - 1), mode = m)
for (j in 1:(length(this_node_reach))) {
alter = this_node_reach[j] + 1
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.numeric(n.nodes, density = p.edge)
r.igraph <- as.igraph(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])