Temporal multilayer network with interlayer edges

Data set

A temporal multilayer network. Each layer is a host-parasite bipartite network. Intralayer edges between a parasite species and a host species are the number of parasite individuals divided by the number of host individuals. Interlayer coupling edges connect each physical node to itself in the next layer (e.g., host A in layer 1 to host A in layer 2), and are calculated as the number of individuals in layer l+1 divided by the number of individuals in layer l. They therefore represent population dynamics. Interlayer edges only go one way (l-->l+1) because time flow one way. We represented the undirected edges within each layer as directed edges that go both ways (with the same weight) to be able to have a directed flow. This does not affect the calculation of L. This data set was taken from Pilosof S, Porter MA, Pascual M, Kéfi S. The multilayer nature of ecological networks. Nature Ecology & Evolution. 2017;1: 0101.

Data sets in infomapecology:

# Get data
data("siberia1982_7_links")
data("siberia1982_7_nodes")

Input

A general multilayer link-list. This multilayer format gives full control of the dynamics, and no other movements are encoded. When using this input format, it is expected that interlayer edges will be provided, otherwise there will be no inter-layer links in the final state network.

*Multilayer
#layer node layer node weight
1 1 1 23 0.15853658536585366
1 1 1 25 0.024390243902439025
1 1 1 27 0.0975609756097561
... some more links...
4 61 4 17 0.009433962264150943
4 62 4 17 0.009433962264150943
4 63 4 17 0.03773584905660377
... some more links...
3 78 4 78 0
4 78 5 78 0
5 78 6 78 1.5

R Code

The description of functions create_multilayer_object and run_infomap_multilayer in the infomapecology package contains everything you need to know. Plotting is done using two dedicated functions: plot_multilayer_modules and plot_multilayer_alluvial.

# Get data
data("siberia1982_7_links")
data("siberia1982_7_nodes")
NEE2017 <- create_multilayer_object(extended = siberia1982_7_links, nodes = siberia1982_7_nodes, intra_output_extended = T, inter_output_extended = T)

#Run infomap
NEE2017_modules <- run_infomap_multilayer(M=NEE2017, relax = F, flow_model = 'directed', silent = T, trials = 100, seed = 497294, temporal_network = T)

#Module persistance
modules_persistence <- NEE2017_modules$modules %>%
  group_by(module) %>%
  summarise(b=min(layer_id), d=max(layer_id), persistence=d-b+1) %>%
  count(persistence) %>%
  mutate(percent=(n/max(NEE2017_modules$module$module))*100)

# 1. Modules' persistence
plot_multilayer_modules(NEE2017_modules, type = 'rectangle', color_modules = T)+
  scale_x_continuous(breaks=seq(0,6,1))+
  scale_y_continuous(breaks=seq(0,40,5))+
  scale_fill_viridis_c()+
  theme_bw()+
  theme(panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        axis.title = element_text(size=20),
        axis.text = element_text(size = 20),
        legend.text =  element_text(size=15),
        legend.title = element_text(size=20))

#2. Species flow through modules in time
plot_multilayer_alluvial(NEE2017_modules, module_labels = F)+
  scale_x_continuous(breaks=seq(0,6,1))+
  scale_y_continuous(breaks=seq(0,70,5))+
  labs(y='Number of species')+
  theme_bw()+
  theme(legend.position = "none",
        panel.grid = element_blank(),
        axis.text = element_text(color='black', size = 20),
        axis.title = element_text(size=20))

Infomap

Under the hood, the function run_infomap_multilayer runs:

./Infomap infomap_multilayer.txt . -2 --seed 497294 -N 100 -i multilayer -f directed --silent

Explanation of key arguments:

• -i multilayer indicates a multilayer input format, which is automatically recognised as a general multilayer link-list.
• -f directed indicates flow on a directed network. The visitation rates of nodes is obtained with a PageRank algorithm based on the direction and weight of edges. This includes interlayer edges.

Output

For multilayer network the output file has a _states suffix, with the following format. Note the state_id column. For example, node 6 in layer 5 has a state_id of 377 (last line). The state_ids are created by Infomap but not used in our R code.

# path flow name state_id node_id layer_id
1:1 0.000779105 "0" 0 1 1
1:2 0.000364122 "8" 8 44 1
1:3 0.00193867 "67" 67 1 2
1:4 0.00109331 "77" 77 44 2
1:5 0.00413471 "135" 135 1 3
1:6 0.000184892 "143" 143 36 3
1:7 0.00194127 "148" 148 44 3
         ...
38:2 0.0201909 "325" 325 34 6
38:3 0.0192017 "349" 349 6 6
38:4 0.000541133 "350" 350 38 6
39:1 0 "377" 377 6 5

This output is parsed by run_infomap_multilayer to obtain a table in which each state node (combination of a physical node in a layer) is assigned to a module. This can be obtained by:

NEE2017_modules$modules