Network Analysis Tutorial

With Applications in R  

Bruce A. Desmarais

Pennsylvania State University

1 Introduction

• Provide a broad introduction to many concepts/methods (nothing in depth)

• Present technical intuition and some essential math (no derivations)

• Comments describing functions generally (see help for explanations of all options)

2 Introduction to R

2.1 Programming in R: First Steps

• R  is a Command-line interpreted programming language

• Commands executed sequentially by return (i.e., `enter’) or separated by ‘;’

• Script files are formatted in plain text files (e.g. UTF-8) with extension “.R”

• Comment heavily using ‘#’

# In R, functions are executed as '<function.name>(<input>)'
# <input> is a comma-separated list of arguments
# The exception is the 'print()' function, which can 
# be executed by typing the name of the object to 
# print and hitting enter
# Try
print(x='Hello World')

## [1] "Hello World"

# x is the only argument

2.2 Objects: Vectors and Matrices

• In R  everything is an object.

• R  environment - collection of objects accessible to R  in RAM

• Vector - column of nubmers, characters, logicals (T/F)

# Vectors contain data of the same type
# Create a character vector
char_vec <- c('a','b','c')
# Look at it
char_vec

## [1] "a" "b" "c"

# Create a numeric vector
num_vec <- numeric(5)
num_vec

## [1] 0 0 0 0 0

# Change Values
num_vec[1] <- 4
num_vec[2:4] <- c(3,2,1) 
num_vec

## [1] 4 3 2 1 0

num_vec[5] <- '5'
num_vec

## [1] "4" "3" "2" "1" "5"

# Reference all but 3
num_vec[-3]

## [1] "4" "3" "1" "5"

num_vec

## [1] "4" "3" "2" "1" "5"

• Matrix

# Create a matrix
MyMat <- matrix(1:25,nrow=5,ncol=5)
MyMat

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    6   11   16   21
## [2,]    2    7   12   17   22
## [3,]    3    8   13   18   23
## [4,]    4    9   14   19   24
## [5,]    5   10   15   20   25

# Access (or change) a cell
MyMat[1,3] 

## [1] 11

MyMat[2,4] <- 200 
MyMat[2,4]

## [1] 200

# Rows then columns
MyMat[1,]

## [1]  1  6 11 16 21

MyMat[,3] <- c(1,1,1,1,1)
MyMat[,3]

## [1] 1 1 1 1 1

MyMat

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    6    1   16   21
## [2,]    2    7    1  200   22
## [3,]    3    8    1   18   23
## [4,]    4    9    1   19   24
## [5,]    5   10    1   20   25

# Multiple rows/columns and negation
MyMat[1:3,-c(1:3)]

##      [,1] [,2]
## [1,]   16   21
## [2,]  200   22
## [3,]   18   23

# The matrix (shortcut for network objects)
MyMat[,]

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    6    1   16   21
## [2,]    2    7    1  200   22
## [3,]    3    8    1   18   23
## [4,]    4    9    1   19   24
## [5,]    5   10    1   20   25

2.3 Objects: Data Frames

• Data Frames can hold columns of different types

# A Data Frame is the conventional object type for a dataset
## Create a data frame containing numbers and a character vector
## Construct a letter vector
let_vec <- c('a','b','c','d','e')

## Combine various objects into a data frame
dat <- data.frame(MyMat, num_vec,let_vec, stringsAsFactors=F)

## Create/override variable names
names(dat) <- c("mm1","mm2","mm3","mm4","mm5","nv","lv")

# Variables can be accessed with '/pre>
dat$lv

## [1] "a" "b" "c" "d" "e"

# Or with matrix-type column indexing
dat[,7]

## [1] "a" "b" "c" "d" "e"

2.4 R Packages

# Use install.packages() to install
# library() or require() to use the package
# install.packages('statnet') # - suite of great network analysis packages
# install.packages('igraph') # - other great network analysis package
library(statnet,quietly=T)

## network: Classes for Relational Data
## Version 1.13.0 created on 2015-08-31.
## copyright (c) 2005, Carter T. Butts, University of California-Irvine
##                     Mark S. Handcock, University of California -- Los Angeles
##                     David R. Hunter, Penn State University
##                     Martina Morris, University of Washington
##                     Skye Bender-deMoll, University of Washington
##  For citation information, type citation("network").
##  Type help("network-package") to get started.

## 
## ergm: version 3.7.1, created on 2017-03-20
## Copyright (c) 2017, Mark S. Handcock, University of California -- Los Angeles
##                     David R. Hunter, Penn State University
##                     Carter T. Butts, University of California -- Irvine
##                     Steven M. Goodreau, University of Washington
##                     Pavel N. Krivitsky, University of Wollongong
##                     Martina Morris, University of Washington
##                     with contributions from
##                     Li Wang
##                     Kirk Li, University of Washington
##                     Skye Bender-deMoll, University of Washington
## Based on "statnet" project software (statnet.org).
## For license and citation information see statnet.org/attribution
## or type citation("ergm").

## NOTE: Versions before 3.6.1 had a bug in the implementation of the
## bd() constriant which distorted the sampled distribution somewhat.
## In addition, Sampson's Monks datasets had mislabeled verteces. See
## the NEWS and the documentation for more details.

## 
## networkDynamic: version 0.9.0, created on 2016-01-12
## Copyright (c) 2016, Carter T. Butts, University of California -- Irvine
##                     Ayn Leslie-Cook, University of Washington
##                     Pavel N. Krivitsky, University of Wollongong
##                     Skye Bender-deMoll, University of Washington
##                     with contributions from
##                     Zack Almquist, University of California -- Irvine
##                     David R. Hunter, Penn State University
##                     Li Wang
##                     Kirk Li, University of Washington
##                     Steven M. Goodreau, University of Washington
##                     Jeffrey Horner
##                     Martina Morris, University of Washington
## Based on "statnet" project software (statnet.org).
## For license and citation information see statnet.org/attribution
## or type citation("networkDynamic").

## 
## tergm: version 3.4.0, created on 2016-03-28
## Copyright (c) 2016, Pavel N. Krivitsky, University of Wollongong
##                     Mark S. Handcock, University of California -- Los Angeles
##                     with contributions from
##                     David R. Hunter, Penn State University
##                     Steven M. Goodreau, University of Washington
##                     Martina Morris, University of Washington
##                     Nicole Bohme Carnegie, New York University
##                     Carter T. Butts, University of California -- Irvine
##                     Ayn Leslie-Cook, University of Washington
##                     Skye Bender-deMoll
##                     Li Wang
##                     Kirk Li, University of Washington
## Based on "statnet" project software (statnet.org).
## For license and citation information see statnet.org/attribution
## or type citation("tergm").

## 
## ergm.count: version 3.2.2, created on 2016-03-29
## Copyright (c) 2016, Pavel N. Krivitsky, University of Wollongong
##                     with contributions from
##                     Mark S. Handcock, University of California -- Los Angeles
##                     David R. Hunter, Penn State University
## Based on "statnet" project software (statnet.org).
## For license and citation information see statnet.org/attribution
## or type citation("ergm.count").

## NOTE: The form of the term 'CMP' has been changed in version 3.2
## of 'ergm.count'. See the news or help('CMP') for more information.

## sna: Tools for Social Network Analysis
## Version 2.4 created on 2016-07-23.
## copyright (c) 2005, Carter T. Butts, University of California-Irvine
##  For citation information, type citation("sna").
##  Type help(package="sna") to get started.

## 
## statnet: version 2016.9, created on 2016-08-29
## Copyright (c) 2016, Mark S. Handcock, University of California -- Los Angeles
##                     David R. Hunter, Penn State University
##                     Carter T. Butts, University of California -- Irvine
##                     Steven M. Goodreau, University of Washington
##                     Pavel N. Krivitsky, University of Wollongong
##                     Skye Bender-deMoll
##                     Martina Morris, University of Washington
## Based on "statnet" project software (statnet.org).
## For license and citation information see statnet.org/attribution
## or type citation("statnet").

## unable to reach CRAN

# R is OPEN SOURCE,  SO LOOK AT THE CODE !!!
network.size

## function (x) 
## {
##     if (!is.network(x)) 
##         stop("network.size requires an argument of class network.\n")
##     else get.network.attribute(x, "n")
## }
## <environment: namespace:network>

# And give credit to the authors
citation('statnet')

## 
## `statnet` is part of the Statnet suite of packages.  If you are
## using the `statnet` package for research that will be published,
## we request that you acknowledge this by citing the following. For
## BibTeX format, use toBibtex(citation("statnet")).
## 
## Handcock M, Hunter D, Butts C, Goodreau S, Krivitsky P,
## Bender-deMoll S and Morris M (2016). _statnet: Software Tools for
## the Statistical Analysis of Network Data_. The Statnet Project
## (<URL: http://www.statnet.org>). R package version 2016.9, <URL:
## CRAN.R-project.org/package=statnet>.
## 
## Handcock M, Hunter D, Butts C, Goodreau S and Morris M (2008).
## "statnet: Software Tools for the Representation, Visualization,
## Analysis and Simulation of Network Data." _Journal of Statistical
## Software_, *24*(1), pp. 1-11. <URL:
## http://www.jstatsoft.org/v24/i01>.
## 
## We have invested a lot of time and effort in creating the Statnet
## suite of packages for use by other researchers. Please cite it in
## all papers where it is used. The package statnet is made
## distributed under the terms of the license: GPL-3 + file LICENSE

# BibTeX Users
toBibtex(citation("statnet"))

## @Manual{,
##   author = {Mark S. Handcock and David R. Hunter and Carter T. Butts and Steven M. Goodreau and Pavel N. Krivitsky and Skye Bender-deMoll and Martina Morris},
##   title = {statnet: Software Tools for the Statistical Analysis of Network Data},
##   organization = {The Statnet Project (\url{http://www.statnet.org})},
##   year = {2016},
##   note = {R package version 2016.9},
##   url = {CRAN.R-project.org/package=statnet},
## }
## 
## @Article{,
##   title = {statnet: Software Tools for the Representation, Visualization, Analysis and Simulation of Network Data},
##   author = {Mark S. Handcock and David R. Hunter and Carter T. Butts and Steven M. Goodreau and Martina Morris},
##   journal = {Journal of Statistical Software},
##   year = {2008},
##   volume = {24},
##   number = {1},
##   pages = {1--11},
##   url = {http://www.jstatsoft.org/v24/i01},
## }

2.5 Interactions with the Hard Drive

# Saving dat as a .csv
write.csv(dat,'dat.csv', row.names=F)

# Loading it as such
dat2 <- read.csv('dat.csv',stringsAsFactors=F)

# Save to RData file and load
save(list=c("dat2","dat"),file="dat_and_dat2.RData")
load("dat_and_dat2.RData")
# understand that objects in the loaded objects will overwrite

# Save and load it all
save.image('ALL.RData')
load('ALL.RData')
# Beware of memory aggregation with save.image()!!

2.6 R can help

# When you know the function name exactly
help("evcent")
# or
?evcent

# Find help files containing a word
help.search("eigenvector")

R help files contain

• function usage arguments

• detailed description

• values (objects) returned

• links to related functions

• references on the methods

• error-free examples

2.7 Use R for graphics

# Initialize the plot
# first column of MyMat on x-axis
# second on y-axis
plot(MyMat[,1],MyMat[,2],pch=1:5,cex=3,col=1:5) 

# Draw a line
lines(MyMat[,1],MyMat[,2], lty =2,  col="grey45",lwd=3)

# Re-write the points
points(MyMat[,1],MyMat[,2],pch=1:5,cex=3, col=1:5)

# add a legend
legend("topleft", legend = c("Circle", "Triangle", "Plus", "Times", "Diamond"), col=1:5,pch=1:5)

# Check out all the other options
?par

# Save it this time (as a PDF)
pdf('myfirstplot.pdf')
plot(MyMat[,1],MyMat[,2],pch=1:5,cex=3,col=1:5)
lines(MyMat[,1],MyMat[,2], lty =2,  ,col="grey45",lwd=3)
points(MyMat[,1],MyMat[,2],pch=1:5,cex=3,col=1:5)
legend("topleft", legend = c("Circle", "Triangle", "Plus", "Times", "Diamond"), col=1:5,pch=1:5)
dev.off()

## quartz_off_screen 
##                 2

3 Introduction to Networks

3.1 Network Terminology and the Basics

• Units in the network: Nodes, actors, or vertices

• Relationships between nodes: edges, links, or ties

• Pairs of actors: Dyads

• Direction: Directed vs. Undirected (digraph vs. graph)

• Tie value: Dichotomous/Binary, Valued/Weighted

• Ties to Self: Loops

3.2 Network and Network Data Types

• Many Modes with unconnected nodes: Bi/Multipartite

  • Affiliation Networks

  • Association/correlation Networks

  • Beware of Collapsed Modes!

• Many relations among nodes: Multiplex

• Data types

  • Have all the data: Network Census

  • Have links data from a sample of nodes: Ego Network

  • Sample along links starting with Ego: link tracing, snowball, respondent-driven

3.3 Network Data

• Vertex-level Data: Vertex attributes (n rows and k columns)

• Adjacency Matrix Data for each relation (n by n) matrix

• Edgelist Data for each edge (e by p) matrix. p typically two—sender & receiver

# Read in adjacency matrices
## read.csv creates a data frame object from a CSV file
## Need to indicate that there's no header row in the CSV
advice <- read.csv("http://brucedesmarais.com/Advice.csv", header=F)

reportsto <- read.csv("http://brucedesmarais.com/ReportsTo.csv", header = F)

# Read in vertex attribute data
attributes <- read.csv("http://brucedesmarais.com/KrackhardtVLD.csv")

3.4.1 Creating Network Objects: Managers in a “Hi-Tech” Firm

# Read in the library for network analysis
library(network,quietly=T)

# Use the advice network dataset to create network object
adviceNet <- network(advice)

# Add the vertex attributes into the network
set.vertex.attribute(adviceNet,names(attributes),attributes)

# Add the organizational chart as a network variable
set.network.attribute(adviceNet,"reportsto",reportsto)

# Simple plot
## Set random number seed so the plot is replicable
set.seed(5)
## Plot the network
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Level"),vertex.cex=2,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col="lightblue")

# check out all the options with ?plot.network

3.4.2 Creating Network Objects: Defense Pacts (edgelist) (2000)

# Read in vertex dataset
allyV <- read.csv("http://brucedesmarais.com/allyVLD.csv",stringsAsFactors=F)

# Read in edgelist
allyEL <- read.csv("http://brucedesmarais.com/allyEL.csv", stringsAsFactors=F)

# Read in contiguity
contig <- read.csv("http://brucedesmarais.com/contiguity.csv",stringsAsFactors=F,row.names=1)

require(network)
# (1) Initialize network
# store number of vertices
n <- nrow(allyV)
AllyNet <- network.initialize(n,dir=F)

# (2) Set vertex labels
network.vertex.names(AllyNet)  <- allyV$stateabb

# (3) Add in the edges
# Note, edgelist must match vertex labels
AllyNet[as.matrix(allyEL)]  <- 1

# (4) Store country code attribute
set.vertex.attribute(x=AllyNet,             # Network in which to store
            "ccode",            # What to name the attribute
            allyV$ccode)            # Values to put in

# (5) Store year attribute
set.vertex.attribute(AllyNet,"created",allyV$styear)

# (6) Store network attribute
set.network.attribute(AllyNet,"contiguous",as.matrix(contig))

# Simple plot
plot(AllyNet,displaylabels=T,label.cex=.5,edge.col=rgb(150,150,150,100,maxColorValue=255))

# check out all the options with ?plot.network

4 The Individual Level: Actor Position Analysis

4.1 Connectedness: Degree Centrality

require(sna)
# (in-) Degree Centrality is the number of in-connections by node
dc <- degree(adviceNet, cmode="indegree")

# Store in vertex level data frame
attributes$dc <- dc

# Plot degree centrality against age
## Make a simple scatter plot
par(cex=2,las=1)
plot(attributes$Tenure,attributes$dc)
## Add a trend (i.e., regression) line
abline(lm(attributes$dc ~ attributes$Tenure))

# Plot network with node size proportional to Degree Centrality
## First normalize degree 
ndc <- dc/max(dc)
## Set random number seed so the plot is replicable
set.seed(5)
## Now plot
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Level"),vertex.cex=3*ndc,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col="lightblue")

4.2 Connectedness: Eigenvector Centrality

x = λ^-1Ax

# Eigenvector Centrality Recursively Considers Neighbors' Centrality
ec <- evcent(adviceNet)

# Store in vertex level data frame
attributes$ec <- ec

# Plot eigenvector centrality against age
## Make a simple scatter plot
par(cex=2,las=1)
plot(attributes$Tenure,attributes$ec)
## Add a trend (i.e., regression) line
abline(lm(attributes$ec ~ attributes$Tenure))

# Plot network with node size proportional to eigenvector centrality
## First normalize
nec <- ec/max(ec)
## Set random number seed so the plot is replicable
set.seed(5)
## Now plot
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Level"),vertex.cex=3*nec,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col="lightblue")

4.3 Connectedness: Betweenness Centrality

# Betweenness Centrality Considers unlikely connections
# Proportion of shortest paths that pass through a vertex
bc <- betweenness(adviceNet,rescale=T)

# Store in vertex level data frame
attributes$bc <- bc

# Plot eigenvector centrality against age
## Make a simple scatter plot
par(cex=2,las=1)
plot(attributes$Tenure,attributes$bc)
## Add a trend (i.e., regression) line
abline(lm(attributes$bc ~ attributes$Tenure))

# Plot network with node size proportional to betweenness centrality
## First normalize
nbc <- bc/max(bc)
## Set random number seed so the plot is replicable
set.seed(5)
## Now plot
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Level"),vertex.cex=3*nbc,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col="lightblue")

4.4 Comparing Centrality Measures

# DC vs. EC
par(cex=2,las=1)
plot(dc,ec)

# DC vs. BC
par(cex=2,las=1)
plot(dc,bc)

# BC vs. EC
par(cex=2,las=1)
plot(bc,ec)

# Correlations among all of them
cor(cbind(ec,bc,dc))

##           ec        bc         dc
## ec  1.000000 0.4320920 -0.3164510
## bc  0.432092 1.0000000  0.5436799
## dc -0.316451 0.5436799  1.0000000

4.5 Embeddedness: Clustering Coefficient

Clustering coefficient is the proportion of potential ties among a node’s neightbors that exist.

# Read in library for clustering coefficient
require(igraph,quietly=T)

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:sna':
## 
##     betweenness, bonpow, closeness, components, degree,
##     dyad.census, evcent, hierarchy, is.connected, neighborhood,
##     triad.census

## The following objects are masked from 'package:network':
## 
##     %c%, %s%, add.edges, add.vertices, delete.edges,
##     delete.vertices, get.edge.attribute, get.edges,
##     get.vertex.attribute, is.bipartite, is.directed,
##     list.edge.attributes, list.vertex.attributes,
##     set.edge.attribute, set.vertex.attribute

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

# Compute local transitivity, i.e., the clustering clef
anet <- graph.adjacency(adviceNet[,])
cc <- transitivity(anet,type="local")

# Store in data frame
attributes$cc <- cc
attributes

##    Age Tenure Level Department dc         ec           bc        cc
## 1   33      9     3          4 13 0.13887187 0.0511034401 0.5263158
## 2   42     20     2          4 18 0.04122224 0.0220658524 0.5666667
## 3   40     13     3          2  5 0.28317710 0.0245530182 0.5105263
## 4   33      8     3          4  8 0.21911327 0.0509618222 0.3473684
## 5   32      3     3          2  5 0.29140287 0.0188794477 0.4210526
## 6   59     28     3          1 10 0.02466730 0.0000000000 0.6909091
## 7   55     30     1          0 13 0.11196954 0.1026936921 0.3047619
## 8   34     11     3          1 10 0.16751801 0.0147754765 0.4967320
## 9   62      5     3          2  4 0.21796754 0.0146987667 0.5514706
## 10  37      9     3          3  9 0.34421295 0.0680179383 0.3162055
## 11  46     27     3          3 11 0.03538648 0.0044550658 0.7252747
## 12  34      9     3          1  7 0.03823355 0.0009441199 0.4722222
## 13  48      0     3          2  4 0.14344220 0.0033191715 0.5555556
## 14  43     10     2          2 10 0.09198919 0.0021891780 0.4835165
## 15  40      8     3          2  4 0.41860823 0.0227975453 0.4528986
## 16  27      5     3          4  8 0.11228627 0.0026022305 0.4848485
## 17  30     12     3          1  9 0.08660158 0.0094116953 0.4945055
## 18  33      9     2          3 15 0.40245199 0.3305452292 0.2016129
## 19  32      5     3          2  4 0.28747761 0.0028028560 0.4571429
## 20  38     12     3          2  8 0.21341504 0.0296630672 0.4842105
## 21  36     13     2          1 15 0.20359254 0.2235203871 0.2430769

# Remove igraph before using statnet functions
detach(package:igraph)

# Plot network with node size proportional to clustering clef
## First normalize
ncc <- cc/max(cc)
## Set random number seed so the plot is replicable
set.seed(5)
## Now plot
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Level"),vertex.cex=3*ncc,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col="lightblue")

# Correlations among all of them
cor(cbind(ec,bc,dc,cc))

##            ec         bc         dc         cc
## ec  1.0000000  0.4320920 -0.3164510 -0.5872217
## bc  0.4320920  1.0000000  0.5436799 -0.7536546
## dc -0.3164510  0.5436799  1.0000000 -0.2204078
## cc -0.5872217 -0.7536546 -0.2204078  1.0000000

5 Group-Level Analysis: Communities and Clusters

5.1 Clustering by Structural Equivalence

# Blockmodeling is the Classical SNA Approach
# Goal is to group nodes based on structural equivalence

## Create clusters based on structural equivalence
eclusts <- equiv.clust(adviceNet)

## First check out a dendrogram to eyeball the number of clusters
plot(eclusts)

# Run a block model identifying six groups
adviceBlockM <- blockmodel(adviceNet, eclusts, k=6)

# Create block membership vector and colors
## Extract block memberships
bmems <- adviceBlockM$block.membership[adviceBlockM$order.vec]
## Create group colors
colVec <- c("black","white","red","blue","yellow","gray60")
## Assign colors to individual nodes based on block membership
bcols <- colVec[bmems]

set.seed(5)
## Now plot
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Level"),vertex.cex=2,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col=bcols)

5.2 Clustering Based on Modularity: Community Detection

Modularity = 1/(2m)Σ_ij[A_ij-k_ik_j/(2m)]1(c_i=c_j)

• k is the degree

• m is the number of edges in the network

• c is the group index

• 1 is the indicator function (i.e., are the groups of i and j equal)

# Modularity-based community detection popular in physics
# Modularity = Dense within communities, sparse across 
library(igraph,quietly=T)

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:sna':
## 
##     betweenness, bonpow, closeness, components, degree,
##     dyad.census, evcent, hierarchy, is.connected, neighborhood,
##     triad.census

## The following objects are masked from 'package:network':
## 
##     %c%, %s%, add.edges, add.vertices, delete.edges,
##     delete.vertices, get.edge.attribute, get.edges,
##     get.vertex.attribute, is.bipartite, is.directed,
##     list.edge.attributes, list.vertex.attributes,
##     set.edge.attribute, set.vertex.attribute

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

# Convert into a graph
anet <- graph.adjacency(adviceNet[,])

## Use semi-greedy splitting and merging
mem <- spinglass.community(anet)$membership

# Check number of communities
max(mem)

## [1] 2

# Get memberships and plot
detach("package:igraph")
bcols <- c("lightblue","yellow")
set.seed(5)
## Now plot
plot(adviceNet,displaylabels=T,label=get.vertex.attribute(adviceNet,"Department"),vertex.cex=2,label.cex=1,edge.col=rgb(150,150,150,100,maxColorValue=255),label.pos=5,vertex.col=bcols)

6 Network-Level: Testing Structural Hypotheses

6.1 Introduction

• Now assume the network is stochastic

• We might have hypotheses about the stochastic process

  • Nodes that are similar are likely to form ties: Homophily

  • Directed edges are likely to be reciprocated: Reciprocity

  • A friend of a friend is a friend: Transitivity

6.3 μ=0, the z-test

Suppose x is a large univariate sample of size n, and

T(x) =  1 √n ^ σ n ∑ i=1  xi

Our null is that X has finite positive variance and a mean of zero.

–>

6.3 H₀ for a network…

Maximum Entropy Null Distribution

Uniform/Equal Probability of Every Network

6.5 Conditional Uniform Graph Testing: Comparing Observed to Null

# Conditional Uniform Graph Tests
# "CUG" tests allow you to control for features of the observed network in the null
# We should test for transitivity (is there even a reason for community detection?)
# gtrans function in sna package measures graph transitivity
ctDens <- cug.test(adviceNet,gtrans,cmode=c("edges"),reps=500)

# Check results
ctDens

## 
## Univariate Conditional Uniform Graph Test
## 
## Conditioning Method: edges 
## Graph Type: digraph 
## Diagonal Used: FALSE 
## Replications: 500 
## 
## Observed Value: 0.6639785 
## Pr(X>=Obs): 0 
## Pr(X<=Obs): 1

# Now lets look at something else
# Do more experienced managers have higher in-degree centrality?

# function to estimate correlation between in-degree and node attribute
indegCor <-  function(net,attr){
    require(sna)
    cor(degree(net,cmode="indegree"),attr)
}

# See the additional argument in cmode, now controlling for dyad census
ctDens2 <- cug.test(adviceNet,indegCor,cmode=c("dyad.census"),reps=500, FUN.args=list(attr = attributes$Tenure))

# Check results
ctDens2

## 
## Univariate Conditional Uniform Graph Test
## 
## Conditioning Method: dyad.census 
## Graph Type: digraph 
## Diagonal Used: FALSE 
## Replications: 500 
## 
## Observed Value: 0.5493177 
## Pr(X>=Obs): 0.002 
## Pr(X<=Obs): 0.998

## Can we incorporate both? 
## Need ERGM for that!

7 Independent exercise with the allliance network

• What are the 5 most central states, according to each centrality measure?

• Do community detection and blockmodeling result in substantially different partitions?

• Is the age of the state significantly correlated with the number of alliance ties?