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sna_eg_stanford:lab02

# 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,
# analyses and a measure of ego-network heterogenity.
#
##############################################################

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

library(igraph)
library(NetData)

###
###

# 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))

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_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_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_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_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_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_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_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_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_friendship_in,
deg_friendship_out,
deg_reports_to_in,
deg_reports_to_out,

reach_full_in_vec,
reach_full_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_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_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_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_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_friendship)
graph.density(krack_reports_to)

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

# Transitivity
transitivity(krack_full)
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,
tc_friendship,
tc_reports_to)

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

# 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

# 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.

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

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)
##################################
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)){
}
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_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(525, hc_friend[1,])
get.p(1003, hc_report[1,])
get.p(979, hc_full[3,])
get.p(1135, hc_friend[3,])
get.p(1314, hc_report[3,])

#generate  95% empirical confidence intervals for triad counts

#clustering