4 Complex Networks

To complete these assignments you need these packages:

library(igraph)
library(qgraph)
library(casnet)

There are many frameworks for analysing and plotting graphs and they are implemented on different platforms often in more-or-less the same way. We’ll use the igraph framework: https://igraph.org for R, but there many more options, see e.g. https://github.com/briatte/awesome-network-analysis#r
The manual pages of package igraph in R provide a lot of information, but not everything the package can do. One thing that is missing are the parameters available to customise the graphs for plotting. If you can’t find something in the R manual, have a look at the online documentation: https://igraph.org/r/doc/plot.common.html
Package qgraph was developped to fit and analyse the so-called Gaussian Graph Model, e.g in the symptom network studies. It is built on the igraph framework and you can transform qgraph output to an igraph object using the function qgraph::as.igraph.qgraph()
The functions in package casnet developped to create, analyse and plot recurrencde networks are also based on the igraph framework.

4.1 Creating, plotting and describing simple graphs

We’ll start using igraph and create some simple graphs.

To create a small ring graph and plot it run the code below.

gR <- graph.ring(20)

# This will display the igraph structure
gR

## IGRAPH 35bfcbe U--- 20 20 -- Ring graph
## + attr: name (g/c), mutual (g/l), circular (g/l)
## + edges from 35bfcbe:
##  [1]  1-- 2  2-- 3  3-- 4  4-- 5  5-- 6  6-- 7  7-- 8  8-- 9  9--10 10--11
## [11] 11--12 12--13 13--14 14--15 15--16 16--17 17--18 18--19 19--20  1--20

# To plot, simply call:
plot(gR)

You can request to represent the graph as a so-called edge-list which just lists the edges in a graph by reporting the nodes the edge connects from - to

get.edgelist(gR)

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

Other useful functions are E() and V() for manipulating attributes of Edges (connections) and Vertices (nodes). See https://igraph.org/r/doc/plot.common.html for all available attributes.

# List vertices
V(gR)

## + 20/20 vertices, from 35bfcbe:
##  [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20

# List edges
E(gR)

## + 20/20 edges from 35bfcbe:
##  [1]  1-- 2  2-- 3  3-- 4  4-- 5  5-- 6  6-- 7  7-- 8  8-- 9  9--10 10--11
## [11] 11--12 12--13 13--14 14--15 15--16 16--17 17--18 18--19 19--20  1--20

# Set vertex color
V(gR)$color <- "red3"
# set vertex label
V(gR)$label.color <- "white"
# Set vertex shape
V(gR)$shape <- "square"
# Set edge color
E(gR)$color <- "steelblue"
# Set efge size
E(gR)$width <- 4

# Plot
plot(gR)

Questions

What will be the degree of each node in the ring graph?
- To check your answer, look in the manual pages of the igraph package for a function that will get you the degree of each node
- Also, get the average path length.
Create a “small world” graph like the one descibed by Watts & Strogatz (1998) and plot it, using the igraph function watts.strogatz.game().
- Use dimension dim = 1, number of nodes size = 20, neighbourhood nei = 5, and connection probability p = .05
- Get the degree, average path length and transitivity of this graph.
- qgraph has a function called smallworldness() that allows you to calculate the so-called Small Worls Index (Humphries & Gurney, 2008). Calulate the smallwordlness of the graph, a value > 1 implies a small world network.
Create a directed scale-free graph using igraph function barabasi.game() with 100 nodes n=100.
- Get the degree, average path length and transitivity
- There should be a power-law relation between the number og nodes that have a specific degree… (but this is a very small network)

Answers

# Ring graph degree
degree(gR)

##  [1] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

# avergage path length
average.path.length(gR)

## [1] 5.263158

# Small world network
gSW <- igraph::watts.strogatz.game(dim = 1, size = 20, nei = 5, p =0.05)

# Plot
plot(gSW)

# Degree
cbind(node = V(gSW), connections = degree(gSW))

##       node connections
##  [1,]    1          10
##  [2,]    2          10
##  [3,]    3          10
##  [4,]    4          10
##  [5,]    5           9
##  [6,]    6          10
##  [7,]    7          12
##  [8,]    8          10
##  [9,]    9          10
## [10,]   10          11
## [11,]   11          11
## [12,]   12          12
## [13,]   13          10
## [14,]   14           9
## [15,]   15          10
## [16,]   16          10
## [17,]   17          10
## [18,]   18           7
## [19,]   19          10
## [20,]   20           9

# Average path length
igraph::average.path.length(gSW)

## [1] 1.473684

# Transitivity
igraph::transitivity(gSW)

## [1] 0.6026345

# Complute the Small World Index, aka 'smallworldness'
qgraph::smallworldness(gSW)

##       smallworldness         trans_target averagelength_target 
##            1.2435897            0.6026345            1.4736842 
##          trans_rnd_M         trans_rnd_lo         trans_rnd_up 
##            0.4846169            0.4445664            0.5203074 
##  averagelength_rnd_M averagelength_rnd_lo averagelength_rnd_up 
##            1.4737579            1.4736842            1.4789474

# Scale-free network
gSF <- barabasi.game(100)

## Change some aesthetics
# No numbers in nodes
V(gSF)$name <- ""
# Change node size
V(gSF)$size = 5
# Change Edge size
E(gSF)$arrow.size = .5
E(gSF)$arrow.width = .5


# Plot
plot(gSF)

# Degree
rbind(node = V(gSF), connections = degree(gSF))

##                                                                          
## node         1 2 3 4  5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
## connections 12 4 3 1 10 5 7 1 4  5  1  5  1  3  2  3  6  5  1  1  1  2  6
##                                                                           
## node        24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
## connections  2  2  1  4  2  1  2  2  3  3  2  1  1  1  1  2  3  2  1  2  1
##                                                                           
## node        45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
## connections  1  1  2  2  3  1  1  4  1  1  1  1  1  1  1  1  2  1  1  2  1
##                                                                           
## node        66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
## connections  1  1  1  2  1  1  1  1  3  1  1  1  2  1  1  1  1  1  1  1  1
##                                                       
## node        87 88 89 90 91 92 93 94 95 96 97 98 99 100
## connections  2  1  1  1  1  1  1  1  1  1  1  1  1   1

# Average path length
average.path.length(gSF)

## [1] 2.374214

# Transitivity
transitivity(gSF)

## [1] 0

## Power-law?
# Get degree distribution
dist <- degree_distribution(gSF)
# Note nonzero elements for log trasnsform
ID <- dist>0
# Get an index for node groups
nodes <- seq_along(dist)

# Fit a line in log-log space
lf  <- lm(log(dist[ID])~log(nodes[ID]))
lfit <- predict(lf)

# Plot it!
plot(nodes,dist,pch=16,log = "xy")

## Warning in xy.coords(x, y, xlabel, ylabel, log): 4 y values <= 0 omitted
## from logarithmic plot

lines(x = nodes[ID],y = exp(lfit), col="red")

4.2 Social Networks

The package igraph contains data on a Social network of friendships between 34 members of a karate club at a US university in the 1970s.

See W. W. Zachary, An information flow model for conflict and fission in small groups, Journal of Anthropological Research 33, 452-473 (1977).

Get a graph of the community structure within the social network.
- There are many procedures for finding communities, or, subgraphs, in graphs, try to understand how thewalktrap procedure finds them.
- Check the manual pages to figure out what kind of information the functions sizes, communities, membership and modularity provide

# Community membership
karate <- graph.famous("Zachary")
wc     <- cluster_walktrap(karate)
plot(wc, karate)

# Measures
modularity(wc)

## [1] 0.3532216

sizes(wc)

## Community sizes
## 1 2 3 4 5 
## 9 7 9 4 5

communities(wc)

## $`1`
## [1]  1  2  4  8 12 13 18 20 22
## 
## $`2`
## [1]  3  9 10 14 29 31 32
## 
## $`3`
## [1] 15 16 19 21 23 27 30 33 34
## 
## $`4`
## [1] 24 25 26 28
## 
## $`5`
## [1]  5  6  7 11 17

cbind(node = V(karate) , community = membership(wc))

##       node community
##  [1,]    1         1
##  [2,]    2         1
##  [3,]    3         2
##  [4,]    4         1
##  [5,]    5         5
##  [6,]    6         5
##  [7,]    7         5
##  [8,]    8         1
##  [9,]    9         2
## [10,]   10         2
## [11,]   11         5
## [12,]   12         1
## [13,]   13         1
## [14,]   14         2
## [15,]   15         3
## [16,]   16         3
## [17,]   17         5
## [18,]   18         1
## [19,]   19         3
## [20,]   20         1
## [21,]   21         3
## [22,]   22         1
## [23,]   23         3
## [24,]   24         4
## [25,]   25         4
## [26,]   26         4
## [27,]   27         3
## [28,]   28         4
## [29,]   29         2
## [30,]   30         3
## [31,]   31         2
## [32,]   32         2
## [33,]   33         3
## [34,]   34         3

# Is the structure a hierarchy?
if(is_hierarchical(wc)){
  plot(as.dendrogram(wc))
}

It is also possible to get the adjacency matrix of a graph
- What do the columns, rows and 0s and 1s stand for?
- Looks a bit like a recurrence matrix, except, the rows and columns don’t represent an (embedded) time series

(am <-get.adjacency(karate))

## 34 x 34 sparse Matrix of class "dgCMatrix"
##                                                                          
##  [1,] . 1 1 1 1 1 1 1 1 . 1 1 1 1 . . . 1 . 1 . 1 . . . . . . . . . 1 . .
##  [2,] 1 . 1 1 . . . 1 . . . . . 1 . . . 1 . 1 . 1 . . . . . . . . 1 . . .
##  [3,] 1 1 . 1 . . . 1 1 1 . . . 1 . . . . . . . . . . . . . 1 1 . . . 1 .
##  [4,] 1 1 1 . . . . 1 . . . . 1 1 . . . . . . . . . . . . . . . . . . . .
##  [5,] 1 . . . . . 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . .
##  [6,] 1 . . . . . 1 . . . 1 . . . . . 1 . . . . . . . . . . . . . . . . .
##  [7,] 1 . . . 1 1 . . . . . . . . . . 1 . . . . . . . . . . . . . . . . .
##  [8,] 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
##  [9,] 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 1
## [10,] . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
## [11,] 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## [12,] 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## [13,] 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## [14,] 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
## [15,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
## [16,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
## [17,] . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
## [18,] 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## [19,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
## [20,] 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
## [21,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
## [22,] 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
## [23,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
## [24,] . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . 1 . . 1 1
## [25,] . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . 1 . .
## [26,] . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . 1 . .
## [27,] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1
## [28,] . . 1 . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . 1
## [29,] . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1
## [30,] . . . . . . . . . . . . . . . . . . . . . . . 1 . . 1 . . . . . 1 1
## [31,] . 1 . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 1
## [32,] 1 . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . . . 1 1
## [33,] . . 1 . . . . . 1 . . . . . 1 1 . . 1 . 1 . 1 1 . . . . . 1 1 1 . 1
## [34,] . . . . . . . . 1 1 . . . 1 1 1 . . 1 1 1 . 1 1 . . 1 1 1 1 1 1 1 .

rp_plot(am)

4.3 Recurrence Networks

Recurrence networks are graphs created from a recurrence matrix. This means the nodes of the graph represent time points and the connections between nodes represent a recurrence relation betwen the values observed at those time points. That is, often the matrix represents recurrences in a reconstructed state space, the values are coordinates and therefore we would say the edges of a recurrence network represent a temporal relation between recurring states. The ultimate reference for learning everything about recurrence networks is:

Zou, Y., Donner, R. V., Marwan, N., Donges, J. F., & Kurths, J. (2018). Complex network approaches to nonlinear time series analysis. Physics Reports

Package casnet has some functions to create recurrence networks, they are similar to the functions used for CRQA: * rn() is very similar to rp(), it will create a matrix based on embedding parameters. One difference is the option to create a weighted matrix. This is a matrix in which non-recurring values are set to 0, but the recurring values are not replaced by a 1, the distance value is retained and acts as an edge-weight * rn_plot() will produce the same as rp_plot()

We can turn the recurrence matrix into an adjecency matrix, an igraph object. This means we can use all the igraph functions to calculate network measures.

# Reload the data we used earlier
series <- rio::import("https://github.com/complexity-methods/CSA-assignments/raw/master/assignment_data/BasicTSA_arma/series.xlsx")

# Lets use a shorter dataset to speed things up
series <- series[1:500,]

We’ll analyse the three time series as a recurrence network: * Compare the results, look at the SMall World Index and other measures - Remember: TS_1 was white noise, TS_2 was a sine with added noise, TS_3 was the logistic map in the chaotic regime. * Note that some of the RQA measures can be exactly calculated from the measures of the network representation. - Try to understand why the Recurrence is reprented as the *degree centrality() of the network (igraph::centr_degree())

TS 1

#----------------------
# Adjacency matrix TS_1
#----------------------
plot(ts(series$TS_1))

arcs=6

crqa_parameters(series$TS_1)

# By passing emRad = NA, a radius will be calculated
RN1 <- casnet::rn(y1 = series$TS_1, emDim = 7, emLag = 2, emRad = NA, targetValue = .01)

## 
## Auto-recurrence: Setting diagonal to (1 + max. distance) for analyses
## 
## Searching for a radius that will yield 0.01 for RR

casnet::rn_plot(RN1)

# Get RQA measures
rqa1 <- crqa_rp(RN1)
knitr::kable(rqa1,digits = 2)

emRad	RP_N	RR	SING_N	SING_rate	DIV_dl	REP_av	ANI	N_dl	N_dlp	DET	MEAN_dl	MAX_dl	ENT_dl	ENTrel_dl	CoV_dl	N_vl	N_vlp	LAM_vl	TT_vl	MAX_vl	ENT_vl	ENTrel_vl	CoV_vl	REP_vl	N_hlp	N_hl	LAM_hl	TT_hl	MAX_hl	ENT_hl	ENTrel_hl	CoV_hl	REP_hl
0.55	2368	0.01	2348	0.99	0.33	4.65	1	8	20	0.01	2.5	3	0.69	0.11	0.21	45	93	0.04	2.07	3	0.24	0.04	0.12	4.65	93	45	0.04	2.07	3	0.24	0.04	0.12	4.65

g1  <- igraph::graph_from_adjacency_matrix(RN1, mode="undirected", diag = FALSE)
V(g1)$size <- degree(g1)
g1r <- casnet::make_spiral_graph(g1,arcs = arcs,type = "Euler" ,epochColours = getColours(arcs))

# Network measures
igraph::average.path.length(g1)

## [1] 4.242546

igraph::transitivity(g1)

## [1] 0.3158716

qgraph::smallworldness(g1)

##       smallworldness         trans_target averagelength_target 
##           6.59400692           0.31587164         215.96216865 
##          trans_rnd_M         trans_rnd_lo         trans_rnd_up 
##           0.04091995           0.03304293           0.04917736 
##  averagelength_rnd_M averagelength_rnd_lo averagelength_rnd_up 
##         184.48094592         181.75063490         194.23956359

recs1 <- igraph::centr_degree(g1)

(RP_N <- sum(recs1$res))

## [1] 2368

rqa1$RP_N

## [1] 2368

RP_N / recs1$theoretical_max

## [1] 0.01000499

rqa1$RR

## [1] 0.01000499

TS 2

#----------------------
# Adjacency matrix TS_2
#----------------------
plot(ts(series$TS_2))

crqa_parameters(series$TS_2)

RN2 <- rn(y1 = series$TS_2, emDim = 7, emLag = 2, emRad = NA,  targetValue = 0.01)

## 
## Auto-recurrence: Setting diagonal to (1 + max. distance) for analyses
## 
## Searching for a radius that will yield 0.01 for RR

rn_plot(RN2)

# Get RQA measures
rqa2 <- crqa_rp(RN2)
knitr::kable(rqa2,digits = 2)

emRad	RP_N	RR	SING_N	SING_rate	DIV_dl	REP_av	ANI	N_dl	N_dlp	DET	MEAN_dl	MAX_dl	ENT_dl	ENTrel_dl	CoV_dl	N_vl	N_vlp	LAM_vl	TT_vl	MAX_vl	ENT_vl	ENTrel_vl	CoV_vl	REP_vl	N_hlp	N_hl	LAM_hl	TT_hl	MAX_hl	ENT_hl	ENTrel_hl	CoV_hl	REP_hl
0.32	2368	0.01	2270	0.96	0.14	1.99	1	28	98	0.04	3.5	7	1.23	0.2	0.57	93	195	0.08	2.1	4	0.33	0.05	0.16	1.99	195	93	0.08	2.1	4	0.33	0.05	0.16	1.99

g2 <- graph_from_adjacency_matrix(RN2, mode="undirected", diag = FALSE)
V(g2)$size <- degree(g2)
g2r <- casnet::make_spiral_graph(g2,arcs = arcs,type = "Euler" ,epochColours = getColours(arcs))

# Network measures
average.path.length(g2)

## [1] 9.696495

transitivity(g2)

## [1] 0.3422773

smallworldness(g2)

##       smallworldness         trans_target averagelength_target 
##          10.96191871           0.34227726          77.69775479 
##          trans_rnd_M         trans_rnd_lo         trans_rnd_up 
##           0.02344651           0.01685415           0.03061576 
##  averagelength_rnd_M averagelength_rnd_lo averagelength_rnd_up 
##          58.34386230          55.86626786          68.86170647

recs2 <- centr_degree(g2)

(RP_N <- sum(recs2$res))

## [1] 2368

rqa2$RP_N

## [1] 2368

RP_N / recs2$theoretical_max

## [1] 0.01000499

rqa2$RR

## [1] 0.01000499

TS 3

#----------------------
# Adjacency matrix TS_3
#----------------------
plot(ts(series$TS_3))

crqa_parameters(series$TS_3)

RN3 <- rn(y1 = series$TS_3, emDim = 7, emLag = 8, emRad = NA, targetValue = 0.01)

## 
## Auto-recurrence: Setting diagonal to (1 + max. distance) for analyses
## 
## Searching for a radius that will yield 0.01 for RR

rn_plot(RN3)

# Get RQA measures
rqa3 <- crqa_rp(RN3)
knitr::kable(rqa3,digits = 2)

emRad	RP_N	RR	SING_N	SING_rate	DIV_dl	REP_av	ANI	N_dl	N_dlp	DET	MEAN_dl	MAX_dl	ENT_dl	ENTrel_dl	CoV_dl	N_vl	N_vlp	LAM_vl	TT_vl	MAX_vl	ENT_vl	ENTrel_vl	CoV_vl	REP_vl	N_hlp	N_hl	LAM_hl	TT_hl	MAX_hl	ENT_hl	ENTrel_hl	CoV_hl	REP_hl
0.57	2030	0.01	1852	0.91	0.25	0.09	1	84	178	0.09	2.12	4	0.37	0.06	0.19	8	16	0.01	2	2	0	0	0	0.09	16	8	0.01	2	2	0	0	0	0.09

g3 <- graph_from_adjacency_matrix(RN3, mode="undirected", diag = FALSE)
V(g3)$size <- degree(g3)
g3r <- casnet::make_spiral_graph(g3,arcs = arcs,type = "Euler" ,epochColours = getColours(arcs))

average.path.length(g3)

## [1] 6.094221

transitivity(g3)

## [1] 0.3300908

smallworldness(g3)

##       smallworldness         trans_target averagelength_target 
##         29.023059466          0.330090791         13.959812762 
##          trans_rnd_M         trans_rnd_lo         trans_rnd_up 
##          0.010338521          0.005188067          0.017509728 
##  averagelength_rnd_M averagelength_rnd_lo averagelength_rnd_up 
##         12.689596344         12.063925795         19.894706430

recs3 <- centr_degree(g3)

(RP_N <- sum(recs3$res))

## [1] 2030

rqa3$RP_N

## [1] 2030

RP_N / recs3$theoretical_max

## [1] 0.01000246

rqa3$RR

## [1] 0.01000246

4.4 Multiplex Recurrence Networks

Consider the three time series to be part of a multi-layer recurrence network. Common properties of the multiplex network are multi-layer mutual information and edge overlap.

# Inter-layer Mutual Information
(interlayer_mi12 <- mif_interlayer(g1,g2))

## [1] 2.20549
## attr(,"miType")
## [1] "inter-layer mutual information"

(interlayer_mi13 <- mif_interlayer(g1,g3))

## [1] 1.666615
## attr(,"miType")
## [1] "inter-layer mutual information"

(interlayer_mi23 <- mif_interlayer(g2,g3))

## [1] 2.155163
## attr(,"miType")
## [1] "inter-layer mutual information"

# mean I-L MI  
mean(c(interlayer_mi12,interlayer_mi13,interlayer_mi23))

## [1] 2.009089

# Edge Overlap
(edge_overlap12 <- length(E(g1 %s% g2)) / (length(E(g1))+length(E(g2))))

## [1] 0.003800676

(edge_overlap13 <- length(E(g1 %s% g3)) / (length(E(g1))+length(E(g3))))

## [1] 0.005457026

(edge_overlap23 <- length(E(g2 %s% g3)) / (length(E(g2))+length(E(g3))))

## [1] 0.004092769

# mean EO 
mean(c(edge_overlap12,interlayer_mi13,interlayer_mi23))

## [1] 1.275193

# Overall Edge Overlap
(eo_all  <- length(E(intersection(g1,g2,g3))))

## [1] 0

(eo_mean <- eo_all / (edge_overlap12+edge_overlap13+edge_overlap23))

## [1] 0

4.5 Symptom Networks: Package `qgraph`

Learn about the functionality of the qgraph frm its author Sacha Epskamp.

Try running the examples, e.g. of the Big 5:

http://sachaepskamp.com/qgraph/examples

4.5.1 `qgraph` tutorials / blog

Great site by Eiko Fried: http://psych-networks.com

DCS Assignments Day 4

Fred Hasselman

1/14/2019

4 Complex Networks

4.1 Creating, plotting and describing simple graphs

Questions

Answers

4.3 Recurrence Networks

TS 1

TS 2

TS 3

4.4 Multiplex Recurrence Networks

4.5 Symptom Networks: Package `qgraph`

4.5.1 `qgraph` tutorials / blog

DCS Assignments Day 4

Fred Hasselman

1/14/2019

4 Complex Networks

4.1 Creating, plotting and describing simple graphs

Questions

Answers

4.2 Social Networks

4.3 Recurrence Networks

TS 1

TS 2

TS 3

4.4 Multiplex Recurrence Networks

4.5 Symptom Networks: Package qgraph

4.5.1 qgraph tutorials / blog

4.5 Symptom Networks: Package `qgraph`

4.5.1 `qgraph` tutorials / blog