Pyramid metanetwork example using 'metanetwork' • metanetwork

Preliminaries

This vignette illustrates ‘metanetwork’ through a pyramid network example. The packages required to run the vignette are the following:

library(igraph)
library(metanetwork)

Generating pyramid data set

Generating the metaweb an representing it using ‘ggnet2’

We first generate a pyramid network using ‘igraph’ and represent it using ‘ggnet2’

library(igraph)
library(network)
library(intergraph)
library(GGally)

n = 5
#generate a lattice 
g = igraph::make_lattice(dim = 2,length = n,directed = T)
#deleting nodes and edges
nodes_to_rm = c()
for (k in 1:(n-1)){
  nodes_to_rm = c(nodes_to_rm,((k-1)*n+1):(k*n - k))
}
g = delete_vertices(g,nodes_to_rm)
g = delete_edges(g,c("7|12","8|13","9|14","2|5"))
V(g)$name = LETTERS[1:vcount(g)]
#representing the lattice using ggnet package
network = asNetwork(g)
ggnet2(network, arrow.size = 7,size = 3 ,arrow.gap = 0.025, label = T)

plot of chunk pyramid_build

Notice that ‘ggnet2’ default layout algorithm (Fruchterman-Reingold algorithm, a force directed algorithm) is non-reproducible. Moreover, x-axis and y-axis do not have any ecological interpretation.

ggnet2(network, arrow.size = 7,size = 3 ,arrow.gap = 0.025, label = T)

plot of chunk unnamed-chunk-1

Generating abundance table

We now generate three local communities (presence/absence in our case)

#sampling a presence table
presence = rbind(c(1,1,1,1,1,1,0,0,1,0,0,0,0,0,1),
                 c(0,0,0,1,1,1,1,1,1,1,1,1,1,1,1),
                 c(0,0,1,1,1,1,1,1,1,1,1,1,1,0,0))

rownames(presence) = c('a','b','c')
colnames(presence) = V(g)$name

Building a metanetwork object

From the lattice metaweb and abundance table, build a S3 object of class ‘metanetwork’ using build_metanetwork

#building metanetwork object
meta0 = build_metanet(metaweb = g,abTable = presence)
class(g)

## [1] "igraph"

class(meta0)

## [1] "metanetwork"

method print prints a summary of the considered metanetwork.

print(meta0)

## metaweb has 15 nodes and 16 edges 
## 3 local networks 
## single resolution available

Handling metanetworks

the class ‘metanetwork’

A ‘metanetwork’ object consists in a list of ‘igraph’ objects:

metaweb, the metaweb used to build the metanetwork, an ‘igraph’ object with node attribute $ab indicating the local relative abundance of each node and graph attribute $nameindicating "metaweb"
local networks, a list of ‘igraph’ objects with node attribute $ab indicating the local relative abundance of each node in each network and graph attribute $name indicating local network names, that is rownames of the abundance table.

meta0$b$name

## [1] "b"

meta0$metaweb$name

## [1] "metaweb"

#abundances
table(V(meta0$b)$ab)

## 
## 0.0833333333333333 
##                 12

table(V(meta0$metaweb)$ab)

## 
##  0.032258064516129 0.0645161290322581 0.0967741935483871 
##                  3                  8                  4

Metaweb node relative abundances are the mean of the local relative abundances. Additional objects like abTable or trophicTable can be included in a ‘metanetwork’ object

Computing trophic levels

‘metanetwork’ package enables 2D network representation with x-axis equals to trophic levels. To compute trophic levels, ‘metanetwork’ implements the method, based on Laplacian matrix, described in MacKay et al. 2020.
The metaweb needs to be connected for trophic levels to be unique. Local networks can however be disconnected (see Ref). A method compute_TL allows computing trophic levels and storing it as node attribute $TL of each network.

#compute trophic levels for metaweb and local networks
meta0 = compute_TL(meta0)
V(meta0$metaweb)$name

##  [1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N" "O"

V(meta0$metaweb)$TL

##  [1] 0.000000e+00 0.000000e+00 1.000000e+00 4.440892e-16 1.000000e+00 2.000000e+00 4.440892e-16 1.000000e+00 2.000000e+00 3.000000e+00
## [11] 0.000000e+00 1.000000e+00 2.000000e+00 3.000000e+00 4.000000e+00

Representing metanetworks

‘metanetwork’ implements a layout algorithm, ‘TL-tsne’, specifically designed for trophic networks, based on trophic levels and on dimension reduction of graph diffusion kernel.

Diffusion graph kernel

Diffusion kernel is a similarity matrix between nodes according to a diffusion process. Let $G$ be a directed network, $\mathbf{A}$ its adjacency matrix and $\mathbf{D}$ its degree diagonal matrix. The laplacian matrix of $G$ is defined as:

\[\begin{equation} \mathbf{L} = \mathbf{D} - \mathbf{A} - \mathbf{A}^{T} \end{equation}\]

The diffusion kernel is then defined as (Kondor & Lafferty, 2002):

\[\begin{equation} \textbf{K} = \exp(-\beta\mathbf{L}) = \sum_{k \geq 0} \frac{(- \beta\mathbf{L})^k}{k!} \end{equation}\]

with $\beta$ a positive parameter. Diffusion kernel measures similarity between pairs of nodes by taking into account paths of arbitrary length. It does not restrict to direct neighbors.

beta parameter

$\beta$ is the single parameter of the diffusion kernel. It controls the weight given to the different paths in the diffusion kernel. It is also analogous to the diffusion constant in physics. We’ll see through examples its importance in squeezing networks.

`ggmetanet` function

The main metanetwork representation function is ggmetanet. It allows representing metaweb and local networks using ggnet and both layout algorithms. ggmetanet plots the metaweb of the current metanetwork by default.

#ggmetanet#
ggmetanet(metanetwork = meta0,beta = 0.1)

plot of chunk unnamed-chunk-7

#ggmetanet#
ggmetanet(metanetwork = meta0,beta = 0.45)

plot of chunk unnamed-chunk-8

ggmetanet can also represent local networks (with specific layout)

ggmetanet(g = meta0$b,beta = 0.1,metanetwork = meta0)

plot of chunk unnamed-chunk-9

Increasing beta squeeze y-axis

ggmetanet(g = meta0$b,beta = 1,metanetwork = meta0)

plot of chunk unnamed-chunk-10

Moreover, it clusters nodes belonging to different ‘branches’. They become more and more similar when beta is increased.

Representing disconnected networks

If the metaweb needs to be connected, local networks can be disconnected due to sampling effects. In that case, trophic levels are computed using metaweb trophic levels. The basal species of each connected component has a trophic level equals to its value in the metaweb.

ggmetanet(g = meta0$a,beta = 0.45,metanetwork = meta0)

plot of chunk unnamed-chunk-11

`diff_plot` function

In order to compare local networks, a diff_plot function is implemented. It colors nodes according to their presence/absence or variation in abundance in both networks.

diff_plot(g1 = meta0$a,g2 = meta0$b,beta = 0.1,mode = 'TL-tsne',metanetwork = meta0)

plot of chunk unnamed-chunk-12

Changing `ggnet` configuration parameters

In order to fine tune network plots, it is possible to modify ggnet parameters in metanetwork. An object ggnet.default is stored and wraps the different visualisation parameters. Change it to modify the plot.

ggnet.custom = ggnet.default
ggnet.custom$edge.size = 3*ggnet.default$edge.size
ggnet.custom$label.size = 7
ggmetanet(beta = 0.1,metanetwork = meta0,
          ggnet.config = ggnet.custom)

plot of chunk unnamed-chunk-13

Attaching layout

For network representation reproducibility and to gain computation time, we recommend to store ‘TL-tsne’ layout after computation. To do so, use the method attach_layout. Once the layout computed, it is attached to the network as node attribute.

#for the metaweb
meta0 = attach_layout(meta0,beta = 0.1)
V(meta0$metaweb)$layout_beta0.1

##  [1] -10.5245709   0.5715783  -4.6297208  30.2718230  18.7743114   7.3023306  43.1058371  36.1900901  23.8856698  12.6402267 -44.7744167
## [12] -38.1999701 -31.4062122 -24.7935220 -18.4134542

#for a local network
meta0 = attach_layout(metanetwork = meta0,g = meta0$a,beta = 0.1)
V(meta0$metaweb)$layout_beta0.1

##  [1] -10.5245709   0.5715783  -4.6297208  30.2718230  18.7743114   7.3023306  43.1058371  36.1900901  23.8856698  12.6402267 -44.7744167
## [12] -38.1999701 -31.4062122 -24.7935220 -18.4134542

Then, any call of ggmetanet or vismetaNetwork will use the computed layout for the desired $\beta$ value.

ggmetanet(meta0,beta = 0.1)

plot of chunk unnamed-chunk-15

Using that way, network representation is reproducible.

#calling again ggmetanet
ggmetanet(meta0,beta = 0.1)

plot of chunk unnamed-chunk-16 Once the layout computed for the metaweb, it can be used to represent local network or difference network using layout_metaweb = T