• Adjacency matrix (squared)
  • Symmetric = Undirected
  • Asymmetric = Directed
• Incidence matrix (rectangular)
  • Bipartite / hypergraphs
• Edgelist
  
  
• Jargon check
  • Node = vertex
  • Link = edge = arc
  • Network = graph

\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \dots & a_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{d1} & a_{d2} & a_{d3} & \dots & a_{dn} \end{bmatrix} \]

\[A_{ij} = \] 1 if there is a link from j to i, 0 otherwise

statnet::network

• Social sciences and statistics
• Strong support for statistical modelling
• Network size = small - medium

igraph

• Computer science and math
• Strong support for algorithms
• Network size = large

Create a network

library(network)
mat <- matrix(rbinom(25,1,0.3), 5,5)
diag(mat) <- 0
net <- network(mat)
summary(net)

## Network attributes:
##   vertices = 5
##   directed = TRUE
##   hyper = FALSE
##   loops = FALSE
##   multiple = FALSE
##   bipartite = FALSE
##  total edges = 5 
##    missing edges = 0 
##    non-missing edges = 5 
##  density = 0.25 
## 
## Vertex attributes:
##   vertex.names:
##    character valued attribute
##    5 valid vertex names
## 
## No edge attributes
## 
## Network adjacency matrix:
##   1 2 3 4 5
## 1 0 0 0 0 0
## 2 1 0 0 0 1
## 3 0 0 0 0 1
## 4 0 0 0 0 0
## 5 0 0 1 1 0

Plot it

plot(net) 

Adjacency matrix are used in bipartite graphs

library(network)
mat <- matrix(rbinom(30,1,0.3), 6,5)
diag(mat) <- 0
net <- network(mat)
summary(net)

## Network attributes:
##   vertices = 11
##   directed = FALSE
##   hyper = FALSE
##   loops = FALSE
##   multiple = FALSE
##   bipartite = 6
##  total edges = 10 
##    missing edges = 0 
##    non-missing edges = 10 
##  density = 0.1818182 
## 
## Vertex attributes:
##   vertex.names:
##    character valued attribute
##    11 valid vertex names
## 
## No edge attributes
## 
## Network adjacency matrix:
##   7 8 9 10 11
## 1 0 0 0  0  0
## 2 1 0 0  1  1
## 3 1 0 0  0  0
## 4 1 1 0  0  0
## 5 1 0 1  0  0
## 6 1 0 1  0  0

Edgelists allow for more compact formats

library(tidyverse)
dat <- tribble(
    ~from, ~to, ~value,
    "Agnes", "Mary", 4,
    "John", "Andrea", 2,
    "Mary", "John", 1,
    "Ingo", "Mary", 6,
    "Stefan", "Ingo", 1
)
net <- network(dat)
summary(net)

## Network attributes:
##   vertices = 6
##   directed = TRUE
##   hyper = FALSE
##   loops = FALSE
##   multiple = FALSE
##   bipartite = FALSE
##  total edges = 5 
##    missing edges = 0 
##    non-missing edges = 5 
##  density = 0.1666667 
## 
## Vertex attributes:
##   vertex.names:
##    character valued attribute
##    6 valid vertex names
## 
## Edge attributes:
## 
##  value:
##    numeric valued attribute
##    attribute summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     1.0     1.0     2.0     2.8     4.0     6.0 
## 
## Network adjacency matrix:
##        Agnes John Mary Ingo Stefan Andrea
## Agnes      0    0    1    0      0      0
## John       0    0    0    0      0      1
## Mary       0    1    0    0      0      0
## Ingo       0    0    1    0      0      0
## Stefan     0    0    0    1      0      0
## Andrea     0    0    0    0      0      0

sna: social network analysis

library(sna) 
net <- network(
    rgraph(n = 30, tprob = 0.15, mode = "graph")) #digraph for directed
# number of connections
degree(net)

##  [1]  4 10 10  8  6  2  4  4  6  6 10 12  2 14  8  6  8  8  6  6 12 18 12 10 10
## [26] 12 10  6 10  4

On a directed network you will have in- and out- degree.

Butts, T. 2008. Social Network Analysis with sna. Journal of Statistical Software

Degree is the number of connections

plot.network(
    net, edge.col = "grey75", edge.lwd = 0.05,
    vertex.cex = degree(net)/10)

Betweenness measures the extent to which a vertex lie in the paths between other vertices

plot.network(
    net, edge.col = "grey75", edge.lwd = 0.05,
    vertex.cex =  betweenness(net, gmode = "graph")/10)

On a directed network direction is accounted for.

Closeness is the mean distance from a vertex to other vertices

plot.network(
    net, edge.col = "grey75", edge.lwd = 0.05,
    vertex.cex = closeness(net)*2)

• Degree
• Betweenness
• Closeness
• Eigenvector
• PageRank
• Katz
• Hubs
• Bonacich Power
• Cliques and cores

Check:

help(package = "sna")

Other interesting topics:

• Network partitions (clustering)
• Community detection
• Cycles and paths census
• Maximum flow

Most network operations are simply linear algebra on matrices

\[ A_{1} = BB^T\] \[A_{2} = B^TB\]

• “Zoonotic EID risk is elevated in forested tropical regions experiencing land-use changes and where wildlife biodiversity is high” Allen et al 2017
  

• Companies who extract natural resources
• Shareholders who invested in them
• Networks of shareholders with common investments

Galaz et al 2022 Beijer Discussion Papers

• Strong focus on algorithms
• Optimized to handle large objects
• Slightly different syntax

Be aware of conflicts between packages

detach(package:sna)
detach(package:network)
library(igraph)

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:lubridate':
## 
##     %--%, union

## The following objects are masked from 'package:dplyr':
## 
##     as_data_frame, groups, union

## The following objects are masked from 'package:purrr':
## 
##     compose, simplify

## The following object is masked from 'package:tidyr':
## 
##     crossing

## The following object is masked from 'package:tibble':
## 
##     as_data_frame

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

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

Create a graph

g <- graph_from_edgelist(
    as.matrix(dat[,1:2]))
g

## IGRAPH 578bfae DN-- 6 5 -- 
## + attr: name (v/c)
## + edges from 578bfae (vertex names):
## [1] Agnes ->Mary   John  ->Andrea Mary  ->John   Ingo  ->Mary   Stefan->Ingo

Other options

make_tree(10, 2)
graph_from_dataframe()
graph_from_incidence_matrix()
graph_from_adjancecy_matrix()

Networks are complex objects:

• Matrix or edgelist
• Node attributes
• Edge attributes

Useful for visualization and statistical modelling

Motifs

igraph::triad_census(g)

##  [1]  5 10  0  0  1  4  0  0  0  0  0  0  0  0  0  0

statnet

net %v% "sex" <- sample(c(0,1), 30, replace = TRUE)
set.vertex.attribute(net, "sex", value = sample(c(0,1), 30, replace = TRUE))
get.vertex.attribute(net, "sex")

igraph

V(g)

## + 6/6 vertices, named, from 578bfae:
## [1] Agnes  Mary   John   Andrea Ingo   Stefan

V(g)$sex <- c("F", "F", "M", "F", "M", "M")
vertex_attr(g)

## $name
## [1] "Agnes"  "Mary"   "John"   "Andrea" "Ingo"   "Stefan"
## 
## $sex
## [1] "F" "F" "M" "F" "M" "M"

statnet

net %e% "type" <- "co-workers"
set.vertex.attribute(net, "sex", value = sample(c(0,1), 30, replace = TRUE))
get.vertex.attribute(net, "sex")

igraph

E(g)

## + 5/5 edges from 578bfae (vertex names):
## [1] Agnes ->Mary   John  ->Andrea Mary  ->John   Ingo  ->Mary   Stefan->Ingo

E(g)$sport <- c("chess", "swimming", "swimming", "climbing", "fencing")
edge_attr(g)

## $sport
## [1] "chess"    "swimming" "swimming" "climbing" "fencing"

E(g)$sport

## [1] "chess"    "swimming" "swimming" "climbing" "fencing"

Mechanisms for network formation

• Random (Erdos-Renyi)
• Small-world (Watts-Strogatz)
• Preferential attachment (Barabasi-Albert)
• Homophily vs influence
• Assortativity
• Motifs mixture

You want to test if your observed network has features that are signature of an underlying process or theory (e.g. assortativity ~ triangles), and compare how your hypothesis differs from a random reference point.

detach(package:igraph)
library(ergm)
data(flo)
flomarriage <- network(flo,directed=FALSE)
flomarriage %v% "wealth" <- c(10,36,27,146,55,44,20,8,42,103,48,49,10,48,32,3)
fit1 <- ergm(flomarriage ~ edges + absdiff("wealth"))
summary(fit1)

## Call:
## ergm(formula = flomarriage ~ edges + absdiff("wealth"))
## 
## Maximum Likelihood Results:
## 
##                 Estimate Std. Error MCMC % z value Pr(>|z|)    
## edges          -1.457666   0.354532      0  -4.112   <1e-04 ***
## absdiff.wealth -0.004176   0.007387      0  -0.565    0.572    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##      Null Deviance: 166.4  on 120  degrees of freedom
##  Residual Deviance: 107.8  on 118  degrees of freedom
##  
## AIC: 111.8  BIC: 117.4  (Smaller is better. MC Std. Err. = 0)

• ergm: exponential random graph models
• ergm.multi: for multilayer networks
• tergm: time varying networks (with tsna)
• ergm.count: weighted networks
• ergm.rank: rank value modelling
• bergm: Bayessian ERGM (btergm for time)
• EpiModel
• hergm: hierarchical

fit2 <- ergm(flomarriage ~ edges + absdiff("wealth") + triangle)
summary(fit2)

## Call:
## ergm(formula = flomarriage ~ edges + absdiff("wealth") + triangle)
## 
## Monte Carlo Maximum Likelihood Results:
## 
##                 Estimate Std. Error MCMC % z value Pr(>|z|)   
## edges          -1.467774   0.503880      0  -2.913  0.00358 **
## absdiff.wealth -0.004324   0.007631      0  -0.567  0.57095   
## triangle        0.072487   0.602304      0   0.120  0.90421   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##      Null Deviance: 166.4  on 120  degrees of freedom
##  Residual Deviance: 107.8  on 117  degrees of freedom
##  
## AIC: 113.8  BIC: 122.1  (Smaller is better. MC Std. Err. = 0.00496)