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--- sna_eg_stanford:lab02 [2019/11/29 09:33] – hkimscil
+++ sna_eg_stanford:lab02 [2019/12/11 08:52] (current) – [no pdf] hkimscil
@@ Line 614: / Line 614: @@
 </code>
+====== no pdf ======
+<code>
+####################################################################
+# 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?
+</code>
+====== omitted ======
+<code>
+###
+# 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])
+</code>