This vignette serves the purpose to reproduce the results shown in
the section “Real data example” of the manuscript describing
*collaborative graphical lasso*. The computations carried out by
the following code are heavy and could take a considerable amount of
time. On our machine (RAM 48 Gb, CPU Intel(R) Xeon(R) W-2235 CPU @
3.80GHz) it took almost 3 hours. Hence, we refer the user who wants to
get started with the package to the introductory vignette:
`vignette("coglasso")`

. *Collaborative graphical
lasso* is an algorithm to reconstruct multi-omics networks based on
*graphical lasso* (Friedman, Hastie and
Tibshirani, 2008) and *collaborative regression* (Gross and Tibshirani, 2015). The coglasso package
implements and R interface to this algorithm. We will use it to
reproduce the results obtained in the original manuscript.

Let us first attach coglasso.

We will use the multi-omics data set provided by the package, in its
largest version: `multi_omics_sd`

. Let’s inspect its
description in the manual.

We notice that this data set has 162 transcripts and 76 metabolites. For further information on the data set sources and generation, we refer the user to the script multi_omics_sd.R on the GitHub repository of the package.

For the multi-omics network reconstruction carried out in the manuscript we explored 30 automatically generated \(λ_w\) and \(λ_b\) values and 9 given \(c\) values.

We can now build the networks, one per combination of the chosen
hyperparameters, with `coglasso()`

.

We proceed now to select the most stable, yet sparse network, using
an adaptation to coglasso of *StARS* (Liu, Roeder
and Wasserman, 2010). The function implementing this adapted version
is `stars_coglasso()`

.

We display now the selected network using the package
`igraph`

, reproducing the one displayed in Figure 3 of the
manuscript. The orientation of the network may vary.

```
# To create the igraph object from the selected adjacency matrix:
sel_graph <- igraph::graph.adjacency(sel_cg$sel_adj, mode = "undirected")
# Setting some graphical parameters
igraph::V(sel_graph)$label <- NA
igraph::V(sel_graph)$color <- c(rep("#00ccff", 162), rep("#ff9999", 76))
igraph::V(sel_graph)$frame.color <- c(rep("#002060", 162), rep("#800000", 76))
igraph::V(sel_graph)$frame.width <- 2
igraph::V(sel_graph)$size <- 4
igraph::E(sel_graph)$width <- 0.3
# Setting a force-based network layout and removing disconnected nodes from the graph
lo <- igraph::layout_with_fr(sel_graph)
diconnected <- which(igraph::degree(sel_graph) == 0)
sel_graph2 <- igraph::delete.vertices(sel_graph, diconnected)
lo2 <- lo[-diconnected, ]
# Plotting
plot(sel_graph2, layout = lo2)
```

We now plot the subnetwork of the gene *Cirbp* and of its
neighborhood. This reproduces the network displayed in Figure 4A.

```
igraph::V(sel_graph)$label<-colnames(sel_cg$data)
cirbp_index <- which(colnames(sel_cg$data) == "Cirbp")
subnetwork <- igraph::subgraph(
sel_graph,
c(cirbp_index, igraph::neighbors(sel_graph, cirbp_index))
)
plot(subnetwork)
```

A useful way to extract information from a network is community
discovery. This technique allows to simplify a network by breaking it in
functional units where nodes share several connections, enough to be
recognized as separate communities. In the manuscript we used the greedy
algorithm described in Clauset, Newman and Moore,
2004, part of the `igraph`

toolkit. We focused and
inspected the second largest community, shown in Figure 4B. Here is the
code to reproduce it.

```
communities <- igraph::cluster_fast_greedy(sel_graph)
community2 <- communities[[2]]
community2_graph<-igraph::subgraph(sel_graph, community2)
# Focusing on nodes of interest
fosjun_erg_AA <- c("Fos", "Fosb", "Junb", "Fosl2", "Egr1", "Egr2", "Egr3", "Ala", "Arg", "Asn","His","Ile","Leu","Lys","Met","Orn","Phe","Ser","Thr","Tyr","Val")
igraph::V(community2_igraph)[!(label %in% fosjun_erg_AA)]$color<-c(rep("#3bd8ff", 29), rep("#ffadad", 5))
igraph::V(community2_igraph)[!(label %in% fosjun_erg_AA)]$frame.color<-NA
igraph::V(community2_igraph)[!(label %in% fosjun_erg_AA)]$label<-NA
lo_community2 <- igraph::layout_with_fr(community2_graph)
plot(community2_graph, layout=lo_community2)
```

Friedman, J., Hastie, T., & Tibshirani, R. (2008). Sparse inverse
covariance estimation with the graphical lasso. *Biostatistics*,
9(3), 432–441. https://doi.org/10.1093/biostatistics/kxm045

Gross, S. M., & Tibshirani, R. (2015). Collaborative regression.
*Biostatistics*, 16(2), 326–338. https://doi.org/10.1093/biostatistics/kxu047

Liu, H., Roeder, K., & Wasserman, L. (2010). Stability Approach
to Regularization Selection (StARS) for High Dimensional Graphical
Models (arXiv:1006.3316). *arXiv.* https://doi.org/10.48550/arXiv.1006.3316

Clauset, A., Newman, M. E. J., & Moore, C. (2004). Finding
community structure in very large networks. *Physical Review E*,
70(6), 066111. https://doi.org/10.1103/PhysRevE.70.066111