Benchmarking the Leiden algorithm with R and Python

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  eval = module
)
print(module)

Benchmarking the Leiden Algorithm

In this guide we will run the Leiden algorithm in both R and Python to benchmark performance and demonstrate how the algorithm is called with reticulate.

We are testing this in the following environment:

paste(Sys.info()[c(4, 2, 1)])
R.version$version.string

Clustering with the Leiden Algorithm in R

This package allows calling the Leiden algorithm for clustering on an igraph object from R. See the Python and Java implementations for more details:

https://github.com/CWTSLeiden/networkanalysis

https://github.com/vtraag/leidenalg

It calls the Python functions to run the algorithm and passes all arguments need to them.

Set up the python version to be called in R

Python implementation

The python version can be installed with pip or conda:

pip uninstall -y igraph
pip install -U -q leidenalg
conda install -c vtraag leidenalg

It is also possible to install the python dependencies with reticulate in R.

library("reticulate")
py_install("python-igraph")
py_install("leidenalg")

Running in Python

We are using the following version of Python:

import sys
print(sys.version)

First we load the packages:

import igraph as ig
print("igraph", ig.__version__)
import leidenalg as la
print("leidenalg", la.version)

Then we load the Zachary karate club example data from igraph.

G = ig.Graph.Famous('Zachary')
G.summary()
partition = la.find_partition(G, la.ModularityVertexPartition)
print(partition)
partition
partition.membership
partition <- py$partition$membership + 1
table(partition)

We can plot the result in R to show it in the network. This reproduces the example in the Python leidenalg documentation.

library("igraph")
library("reticulate")
library("RColorBrewer")
graph_object <- graph.famous("Zachary")
node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(graph_object, vertex.color = node.cols, layout=layout_with_kk)

We can reproduce passing arguments in this manner as well.

partition = la.find_partition(G, la.CPMVertexPartition, resolution_parameter = 0.05)
print(partition)
partition
partition.membership
partition <- py$partition$membership + 1
table(partition)

We can plot the result in R to show it in the network. This reproduces the example in the Python leidenalg documentation.

graph_object <- graph.famous("Zachary")
node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(graph_object, vertex.color = node.cols, layout=layout_with_kk)

We can run the RBC vertex method which generalises the modularity vertex partition.

partition = la.find_partition(G, la.RBConfigurationVertexPartition, resolution_parameter = 1.5)
print(partition)
partition
partition.membership
partition <- py$partition$membership + 1
table(partition)

We can plot the result in R to show it in the network.

graph_object <- graph.famous("Zachary")
node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(graph_object, vertex.color = node.cols, layout=layout_with_kk)

Benchmarking the Python version with reticulate

Now we can time how long the computation of the algorithm takes (for 1000 runs) running within python:

import time
G = ig.Graph.Famous('Zachary')
G.summary()
start = time.time()
for ii in range(100):
    partition = la.find_partition(G, la.ModularityVertexPartition)

end = time.time()
partition.membership
py_time = end - start
print("leiden time:", py_time, "seconds")
bash_py_time=`python -c 'import igraph as ig
import leidenalg as la
import time
G = ig.Graph.Famous("Zachary")
G.summary()
start = time.time()
for ii in range(100):
    partition = la.find_partition(G, la.ModularityVertexPartition)

end = time.time()
partition.membership
py_time = end - start
print(py_time)'`
echo $bash_py_time > bash_py_time
echo "leiden time:" $bash_py_time "seconds"
bash_py_time <- as.numeric(readLines("bash_py_time"))

We can also run the leiden algorithm in python by calling functions with reticulate:

leidenalg <- import("leidenalg", delay_load = TRUE)
ig <- import("igraph", delay_load = TRUE)
G = ig$Graph$Famous('Zachary')
G$summary()
partition = leidenalg$find_partition(G, leidenalg$ModularityVertexPartition)
partition$membership
leidenalg <- import("leidenalg", delay_load = TRUE)
ig <- import("igraph", delay_load = TRUE)
G = ig$Graph$Famous('Zachary')
G$summary()
start <- Sys.time()
for(ii in 1:100){
  partition = leidenalg$find_partition(G, leidenalg$ModularityVertexPartition)
}
end <- Sys.time()
partition$membership
reticulate_time <- difftime(end, start)[[1]]
print(paste(c("leiden time:", reticulate_time, "seconds"), collapse = " "))

R implementation

The R version can be installed with devtools or from CRAN:

install.packages("leiden")
install.packages("leiden")

Note that these require the Python version as a dependency.

Running in R via reticulate and igraph

We can reproduce these by running the Leiden algorithm in R using the functions in the leiden package.

We are using the following version of R:

R.version.string

First we load the packages:

library("igraph")
library("leiden")

Then we load the Zachary karate club example data from igraph.

G <- graph.famous("Zachary")
summary(G)

Calling python in R via reticulate

Here run “legacy” mode to call “leidenalg” in python with the R reticulate package.

partition <- leiden(G, "ModularityVertexPartition", legacy = TRUE)
partition
table(partition)

We can plot the result in R to show it in the network. This reproduces the example in the Python leidenalg documentation.

library("igraph")
library("reticulate")
library("RColorBrewer")
node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(G, vertex.color = node.cols, layout=layout_with_kk)

We can reproduce passing arguments in this manner as well.

partition <- leiden(G, "CPMVertexPartition", resolution_parameter = 0.05, legacy = TRUE)
partition
table(partition)

We can plot the result in R to show it in the network. This reproduces the example in the Python leidenalg documentation.

node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(G, vertex.color = node.cols, layout=layout_with_kk)

We can run the RBC vertex method which generalises the modularity vertex partition.

partition <- leiden(G, "RBConfigurationVertexPartition", resolution_parameter = 1.5)
partition
table(partition)

We can plot the result in R to show it in the network.

node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(G, vertex.color = node.cols, layout=layout_with_kk)

Calling C in R with igraph

We can improve performance for undirected graphs for Modularity and CPM cost functions by calling C in igraph.

G <- as.undirected(G, mode = "each")
is.directed(G)
partition <- leiden(G, "ModularityVertexPartition", legacy = FALSE)
partition
table(partition)

We can plot the result in R to show it in the network. This reproduces the example in the Python leidenalg documentation.

library("igraph")
library("reticulate")
library("RColorBrewer")
node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(G, vertex.color = node.cols, layout=layout_with_kk)

We check here that it returns the same results as in igraph.

partition <- membership(cluster_leiden(G, objective_function = "modularity"))
partition
table(partition)

We can also run CPM cost functions.

partition <- leiden(G, "CPMVertexPartition", resolution_parameter = 0.1, legacy = FALSE)
partition
table(partition)

We can plot the result in R to show it in the network. This reproduces the example in the Python leidenalg documentation.

node.cols <- brewer.pal(max(c(3, partition)),"Pastel1")[partition]
plot(G, vertex.color = node.cols, layout=layout_with_kk)

Benchmarking the R version with reticulate

Now we can time how long the computation of the algorithm takes (for 1000 runs) calling with R on a graph object:

G <- graph.famous('Zachary')
summary(G)
start <- Sys.time()
for(ii in 1:100){
  partition <- leiden(G, "ModularityVertexPartition", legacy = TRUE)
}
end <- Sys.time()
table(partition)
R_graph_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", R_graph_time, "seconds"), collapse = " "))

We can see that the R reticualte implementation does not perform as well as the Python version but it is convenient for R users. Calling from a graph object avoids casting to a dense adjacency matrix which reduces memory load for large graph objects.

We can see that calling leiden in R on an adjacency matrix has faster performance but it does require more memory. For example, on a dense adjacency matrix:

G <- graph.famous('Zachary')
summary(G)

start <- Sys.time()
for(ii in 1:100){
  adj_mat <- as_adjacency_matrix(G, sparse = FALSE)
}
end <- Sys.time()
dim(adj_mat)
R_mat_cast_time = difftime(end, start)[[1]]
paste(print(c("cast time:", R_mat_cast_time, "seconds"), collapse = " "))

start <- Sys.time()
for(ii in 1:100){
  partition <- leiden(adj_mat, "ModularityVertexPartition")
}
end <- Sys.time()
table(partition)
R_mat_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", R_mat_time, "seconds"), collapse = " "))

For example, on a sparse dgCMatrix for the adjacency matrix:

G <- graph.famous('Zachary')
summary(G)

start <- Sys.time()
for(ii in 1:100){
  adj_mat <- as_adjacency_matrix(G, sparse = TRUE)
}
end <- Sys.time()
class(adj_mat)
dim(adj_mat)
R_sparse_mat_cast_time = difftime(end, start)[[1]]
paste(print(c("cast time:", R_sparse_mat_cast_time, "seconds"), collapse = " "))

start <- Sys.time()
for(ii in 1:100){
  partition <- leiden(adj_mat, "ModularityVertexPartition")
}
end <- Sys.time()
table(partition)
R_sparse_mat_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", R_mat_time, "seconds"), collapse = " "))

Large adjacency matrices

The difference between sparse and dense matrices is more pronounced for large matrices (with few edges):

adjacency_matrix <- rbind(cbind(matrix(round(rbinom(1000000, 1, 0.008)), 1000, 1000),
                                matrix(round(rbinom(1000000, 1, 0.003)), 1000, 1000),
                                matrix(round(rbinom(1000000, 1, 0.001)), 1000, 1000)),
                          cbind(matrix(round(rbinom(1000000, 1, 0.003)), 1000, 1000),
                                matrix(round(rbinom(1000000, 1, 0.008)), 1000, 1000),
                                matrix(round(rbinom(0000000, 1, 0.002)), 1000, 1000)),
                          cbind(matrix(round(rbinom(1000000, 1, 0.003)), 1000, 1000),
                                matrix(round(rbinom(1000000, 1, 0.001)), 1000, 1000),
                                matrix(round(rbinom(1000000, 1, 0.009)), 1000, 1000)))
rownames(adjacency_matrix) <- 1:3000
colnames(adjacency_matrix) <- 1:3000
G <- graph_from_adjacency_matrix(adjacency_matrix)

start <- Sys.time()
for(ii in 1:10){
  adj_mat <- as_adjacency_matrix(G, sparse = FALSE)
}
end <- Sys.time()
class(adj_mat)
dim(adj_mat)
R_mat_large_cast_time = difftime(end, start)[[1]]
paste(print(c("cast time:", R_mat_large_cast_time, "seconds"), collapse = " "))

start <- Sys.time()
for(ii in 1:10){
  partition <- leiden(adj_mat, "ModularityVertexPartition")
}
end <- Sys.time()
table(partition)
R_mat_large_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", R_mat_large_time, "seconds"), collapse = " "))

For example, on a sparse adjacency matrix:

start <- Sys.time()
for(ii in 1:100){
  adj_mat <- as_adjacency_matrix(G, sparse = TRUE)
}
end <- Sys.time()
class(adj_mat)
dim(adj_mat)
R_mat_large_cast_time = difftime(end, start)[[1]]
paste(print(c("cast time:", R_mat_large_cast_time, "seconds"), collapse = " "))

start <- Sys.time()
for(ii in 1:10){
  partition <- leiden(adj_mat, "ModularityVertexPartition")
}
end <- Sys.time()
table(partition)
R_mat_large_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", R_mat_large_time, "seconds"), collapse = " "))

Comparing the adjacency matrix calling

We compare the processing of adjaceny matrices in the leiden.matrix method to casting to graph in python with reticulate. The current implementation of the R function works as follows. The adjacency matrix is passed to python and the graph object is create in the python-igraph:

partition_type <- "RBConfigurationVertexPartition"
initial_membership <- NULL
weights <- NULL
node_sizes = NULL
resolution_parameter = 1

G <- graph.famous('Zachary')
summary(G)
time1 <- Sys.time()
object <- as.matrix(as_adjacency_matrix(G))
time2 <- Sys.time()
timing = difftime(time2, time1)[[1]]
print(paste(c("cast to adjacent:", timing, "seconds"), collapse = " "))

#run matrix method
leidenalg <- import("leidenalg", delay_load = TRUE)
ig <- import("igraph", delay_load = TRUE)

#convert matrix input (corrects for sparse matrix input)
if(is.matrix(object) || is(adj_mat_sparse, "Matrix")){
  adj_mat <- object
} else{
  adj_mat <- as.matrix(object)
}

#compute weights if non-binary adjacency matrix given
is_pure_adj <- all(as.logical(adj_mat) == adj_mat)
if (is.null(weights) && !is_pure_adj) {
  #assign weights to edges (without dependancy on igraph)
  t_mat <- t(adj_mat)
  weights <- t_mat[t_mat!=0]
  #remove zeroes from rows of matrix and return vector of length edges
}

time3 <- Sys.time()
##convert to python numpy.ndarray, then a list
adj_mat_py <- r_to_py(adj_mat)
adj_mat_py <- adj_mat_py$tolist()
time4 <- Sys.time()
timing = difftime(time4, time3)[[1]]
print(paste(c("pass to python matrix:", timing, "seconds"), collapse = " "))


#convert graph structure to a Python compatible object
GraphClass <- if (!is.null(weights) && !is_pure_adj){
  ig$Graph$Weighted_Adjacency
} else {
  ig$Graph$Adjacency
}
time5 <- Sys.time()
snn_graph <- GraphClass(adj_mat_py)
time6 <- Sys.time()
timing = difftime(time6, time5)[[1]]
reticulate_create_time = difftime(time6, time5)[[1]]
print(paste(c("generate graph in python:", timing, "seconds"), collapse = " "))


# test performance for computing matrix to graph in R
# other option is to passing snn_graph to Python

time7 <- Sys.time()
#compute partitions
source("../R/find_partition.R")

partition <- find_partition(snn_graph, partition_type = partition_type,
                            initial_membership = initial_membership ,
                            weights = weights,
                            node_sizes = node_sizes,
                            resolution_parameter = resolution_parameter
)
time8 <- Sys.time()
timing = difftime(time8, time7)[[1]]
print(paste(c("compute partitions:", timing, "seconds"), collapse = " "))
timing = difftime(time8, time1)[[1]]
print(paste(c("total:", timing, "seconds"), collapse = " "))
partition

Is it more efficent to pass to create a graph object in R and pass this to python?

partition_type <- "RBConfigurationVertexPartition"
initial_membership <- NULL
weights <- NULL
node_sizes = NULL
resolution_parameter = 1

G <- graph.famous('Zachary')
summary(G)
time1 <- Sys.time()
object <- as.matrix(as_adjacency_matrix(G))
time2 <- Sys.time()
timing = difftime(time2, time1)[[1]]
print(paste(c("cast to adjacent:", timing, "seconds"), collapse = " "))

#run matrix method
leidenalg <- import("leidenalg", delay_load = TRUE)
ig <- import("igraph", delay_load = TRUE)

time3 <- Sys.time()
##convert to python numpy.ndarray, then a list
object <- graph_from_adjacency_matrix(adj_mat)
time4 <- Sys.time()
timing = difftime(time4, time3)[[1]]
print(paste(c("generate graph in R:", timing, "seconds"), collapse = " "))

#convert graph structure to a Python compatible object
time5 <- Sys.time()
##convert to list for python input
    if(!is.named(object)){
        vertices <- as.list(as.character(V(object)))
    } else {
        vertices <- as.list(names(V(object)))
    }

    edges <- as_edgelist(object)
    dim(edges)
    edgelist <- list(rep(NA, nrow(edges)))
    for(ii in 1:nrow(edges)){
        edgelist[[ii]] <- as.character(edges[ii,])
    }

    snn_graph <- ig$Graph()
    snn_graph$add_vertices(r_to_py(vertices))
    snn_graph$add_edges(r_to_py(edgelist))
time6 <- Sys.time()
timing = difftime(time6, time5)[[1]]
print(paste(c("pass to python graph:", timing, "seconds"), collapse = " "))



# test performance for computing matrix to graph in R
# other option is to passing snn_graph to Python

time7 <- Sys.time()
#compute partitions
partition <- find_partition(snn_graph, partition_type = partition_type,
                            initial_membership = initial_membership ,
                            weights = weights,
                            node_sizes = node_sizes,
                            resolution_parameter = resolution_parameter
)
time8 <- Sys.time()
timing = difftime(time8, time7)[[1]]
print(paste(c("compute partitions:", timing, "seconds"), collapse = " "))
timing = difftime(time8, time1)[[1]]
print(paste(c("total:", timing, "seconds"), collapse = " "))
partition

Another approach is to generate a graph in R and pass it to the leiden.igraph method.

partition_type <- "RBConfigurationVertexPartition"
initial_membership <- NULL
weights <- NULL
node_sizes = NULL
resolution_parameter = 1

G <- graph.famous('Zachary')
summary(G)
time1 <- Sys.time()
object <- as.matrix(as_adjacency_matrix(G))
time2 <- Sys.time()
timing = difftime(time2, time1)[[1]]
print(paste(c("cast to adjacent:", timing, "seconds"), collapse = " "))

time3 <- Sys.time()
##convert to python numpy.ndarray, then a list
object <- graph_from_adjacency_matrix(adj_mat)
time4 <- Sys.time()
timing = difftime(time4, time3)[[1]]
R_graph_create_time = difftime(time4, time3)[[1]]
print(paste(c("generate graph in R:", timing, "seconds"), collapse = " "))


#convert graph structure to a Python compatible object
time5 <- Sys.time()
##convert to list for python input
   snn_graph <- object
time6 <- Sys.time()
timing = difftime(time6, time5)[[1]]
print(paste(c("pass to R graph:", timing, "seconds"), collapse = " "))



# test performance for computing matrix to graph in R
# other option is to passing snn_graph to Python

time7 <- Sys.time()
#compute partitions
partition <- leiden(snn_graph, partition_type = partition_type,
                            initial_membership = initial_membership ,
                            weights = weights,
                            node_sizes = node_sizes,
                            resolution_parameter = resolution_parameter
)
time8 <- Sys.time()
timing = difftime(time8, time7)[[1]]
print(paste(c("compute partitions:", timing, "seconds"), collapse = " "))
timing = difftime(time8, time1)[[1]]
print(paste(c("total:", timing, "seconds"), collapse = " "))
partition

Here we can see that the current approach to pass adjacency matrices to Python and generate graphs in Python is more efficient for a dense matrix than computing the graph in R. Therefore the leiden.matrix method will not call the leiden.igraph method and they will remain distinct.

Calling C in R with igraph

Here we compare calling modularity clustering in igraph (R and C) to calling Python via reticulate. Note this is only available for undirected graphs with modularity or CPM. Calling igraph in R does not have a python dependency.

time9 <- Sys.time()
partition <- membership(cluster_leiden(G, objective_function = "modularity"))
partition
table(partition)
time10 <- Sys.time()
timing = difftime(time10, time9)[[1]]
print(paste(c("run with igraph:", timing, "seconds"), collapse = " "))

The updated leiden package calls this implementation when available. We can see this is considerably faster and may be faster than call leidenalg in Python.

time11 <- Sys.time()
partition <- leiden(G, "ModularityVertexPartition", legacy = FALSE)
partition
table(partition)
time12 <- Sys.time()
timing = difftime(time12, time11)[[1]]
print(paste(c("run with leiden in igraph:", timing, "seconds"), collapse = " "))

This is considerably faster than the reticulate implementation, especially for larger matrices.

time13 <- Sys.time()
partition <- leiden(G, "ModularityVertexPartition", legacy = TRUE)
partition
table(partition)
time14 <- Sys.time()
timing = difftime(time14, time13)[[1]]
print(paste(c("run with leiden with reticulate:", timing, "seconds"), collapse = " "))

Matrix methods in R are significantly slower than the updated igraph version.

library("Matrix")
adj_mat <- as(as(as(as_adjacency_matrix(G), Class = "CsparseMatrix"), "generalMatrix"), "dMatrix")
time15 <- Sys.time()
partition <- leiden(adj_mat, "ModularityVertexPartition", legacy = TRUE)
partition
table(partition)
time16 <- Sys.time()
timing = difftime(time16, time15)[[1]]
print(paste(c("run with leiden with reticulate:", timing, "seconds"), collapse = " "))
adj_mat <- as_adjacency_matrix(G)
time15 <- Sys.time()
partition <- leiden(adj_mat, "ModularityVertexPartition", legacy = TRUE)
partition
table(partition)
time16 <- Sys.time()
timing = difftime(time16, time15)[[1]]
print(paste(c("run with leiden with reticulate:", timing, "seconds"), collapse = " "))

Benchmarking C in R with igraph

For comparison with other methods we compute multiple iterations.

G <- graph.famous('Zachary')
summary(G)
start <- Sys.time()
for(ii in 1:100){
  partition <- membership(cluster_leiden(G, objective_function = "modularity"))
}
end <- Sys.time()
table(partition)
igraph_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", igraph_time, "seconds"), collapse = " "))
G <- graph.famous('Zachary')
summary(G)
start <- Sys.time()
for(ii in 1:100){
  partition <- leiden(G, "ModularityVertexPartition", legacy =  FALSE)
}
end <- Sys.time()
table(partition)
R_cigraph_time = difftime(end, start)[[1]]
print(paste(c("leiden time:", R_cigraph_time, "seconds"), collapse = " "))

Summary

Here we compare the compute time for the Zachary datasets between each method for computing paritions from the leiden clustering algorithm in R or Python.

barplot(c(bash_py_time, py$py_time, reticulate_time, R_graph_time,
          R_cigraph_time, igraph_time, R_mat_time, R_sparse_mat_time), 
        names = c("Python (shell)", "Python (Rmd)", "Reticulate",
                  "R igraph reticulate", "R igraph (C)", "R igraph cluster_leiden",
                  "R matrix","R dgCMatrix"), 
        col = brewer.pal(9,"Pastel1"), las = 2, srt = 45,
        ylab = "time (seconds)", main = "benchmarking 100 computations")
abline(h=0)

If we account for time to cast matrices from graph objects. Then these are the time taken to compute partitions from a graph in R.

barplot(c(bash_py_time, py$py_time, reticulate_time, R_graph_time,
          R_cigraph_time, igraph_time, R_mat_time+R_mat_cast_time, 
          R_sparse_mat_time+R_sparse_mat_cast_time), 
        names = c("Python (shell)", "Python (Rmd)", "Reticulate",
                  "R igraph reticulate", "R igraph (C)", "R igraph cluster_leiden",
                  "R matrix","R dgCMatrix"), 
        col = "grey80", las = 2, srt = 45,
        ylab = "time (seconds)", main = "benchmarking 100 computations")
barplot(c(bash_py_time, py$py_time, reticulate_time, R_graph_time,
          R_cigraph_time, igraph_time, R_mat_time,  R_sparse_mat_time), 
        names = c("Python (shell)", "Python (Rmd)", "Reticulate",
                  "R igraph reticulate", "R igraph (C)", "R igraph cluster_leiden",
                  "R matrix","R dgCMatrix"),
        col = brewer.pal(9,"Pastel1"), las = 2, srt = 45,
        ylab = "time (seconds)", main = "benchmarking 100 computations", add = TRUE)
abline(h=0)

Similarly, if we account for time to generate graph from an adjaceny matrix. Then these are the time taken to compute partitions from a matrix in R.

R_graph_create_time = difftime(time4, time3)[[1]]
barplot(c(bash_py_time, py$py_time+reticulate_create_time*100, reticulate_time+reticulate_create_time*100, R_graph_time+R_graph_create_time*100,
        R_cigraph_time, igraph_time, R_mat_time, R_sparse_mat_time), 
        names = c("Python (shell)", "Python (Rmd)", "Reticulate",
                  "R igraph reticulate", "R igraph (C)", "R igraph cluster_leiden",
                  "R matrix","R dgCMatrix"), 
        col = "grey80", las = 2, srt = 45,
        ylab = "time (seconds)", main = "benchmarking 100 computations")
barplot(c(bash_py_time, py$py_time, reticulate_time, R_graph_time, 
        R_cigraph_time, igraph_time, R_mat_time,  R_sparse_mat_time), 
        names = c("Python (shell)", "Python (Rmd)", "Reticulate",
                  "R igraph reticulate", "R igraph (C)", "R igraph cluster_leiden",
                  "R matrix","R dgCMatrix"), 
        col = brewer.pal(9,"Pastel1"), las = 2, srt = 45,
        ylab = "time (seconds)", main = "benchmarking 100 computations", add = TRUE)
abline(h=0)