knitr::opts_chunk$set(echo = TRUE)

Graph metrics are quantitative measures that provide insights into the structural
properties of pathway graphs, playing a crucial role in understanding the topology
of biological networks and revealing key characteristics. Through the integration
of WikiPathways and R's `igraph`

package, `WayFindR`

provides a suite of functions
that enables researchers to compute graph metrics on biological pathways. In this
vignette, we demonstrate how to compute some of these metrics on the pathway named
"Factors and pathways influencing insulin-like growth factor (IGF1)-Akt signaling
(WP3850)" accessible at https://www.wikipathways.org/pathways/WP3850.html. This
GPML file is included in the package as a system file.

First, we load the required libraries:

library(WayFindR) suppressMessages( library(igraph) )

Now we are ready to load our GPML file and convert it into an igraph object:

xmlfile <- system.file("pathways/WP3850.gpml", package = "WayFindR") G <- GPMLtoIgraph(xmlfile) class(G)

After obtaining an `igraph`

object, we can use functions from the `igraph`

package to compute its structural properties.

Users have the flexibility to choose which metrics to calculate for their research purposes. However, for the exploration of cycles in graphs, we will concentrate on a selection of global metrics that are potentially intriguing:

- Number of vertices
- Number of edges
- Number of negative interactions (inhibition processes or edges)
- Presence/absence of loops
- Presence/absence of multiple edges
- Presence/absence of Eulerian path or cycle in the input graph
- Number of clusters
- Density
- Radius of the graph
- Diameter of the graph
- Girth
- Global efficiency of the graph
- Average path length in the graph
- Number of cliques
- Reciprocity

We refer readers to the `igraph`

package tutorial for more detailed explanations
of these metrics.

Next, let's create a table summarizing all the metrics of interest.

# Calculate metrics metrics <- data.frame(nVertices = length(V(G)), nEdges = length(E(G)), nNegative = sum(edge_attr(G, "MIM") == "mim-inhibition"), hasLoop = any_loop(G), hasMultiple = any_multiple(G), hasEuler = has_eulerian_cycle(G) | has_eulerian_path(G), nComponents = count_components(G), density = edge_density(G), diameter = diameter(G), radius = radius(G), girth = ifelse(is.null(girth(G)), NA, girth(G)$girth), nTriangles = sum(count_triangles(G)), efficiency = global_efficiency(G), meanDistance = mean_distance(G), cliques = clique_num(G), reciprocity = reciprocity(G)) metrics

We can find cycles and analyze cycle subgraph (i.e., the subgraph defined by
including only the nodes that re presen tin at least one cycle.
Here, `nCyVert`

is the number of vertices in the cycle subgraph,
`nCyEdge`

is the number of edges in the cycle subgraph,
`nCyNeg`

is the number of edges in the cycle subgraph with the attribute "MIM"
equal to "mim-inhibition". You can visually confirm the cunts in the
plot below.

cy <- findCycles(G) length(cy) S <- cycleSubgraph(G, cy) cymetrics <- data.frame(nCycles = length(cy), nCyVert = length(V(S)), nCyEdge = length(E(S)), nCyNeg = sum(edge_attr(S, "MIM") == "mim-inhibition")) cymetrics

set.seed(93217) plot(S) nodeLegend("topleft", S) edgeLegend("bottomright", S)

In addition to numerous "global" graph metrics, the `igraph`

package includes
tools to compute numerous "local" metrics, which describe the properties of
individual nodes or edges. In many networks (which, like pathways, can be
represented by mathematical graphs), highly connected nodes are often viewed as
"hubs" that play a more important role. The simplest such metric is the "degree",
which counts the number of edges connected to the node. (In directed graphs, which
we use to instantiate pathway, one can talk about both inbound and outbound edges
and degrees.)

Here we compute the total degree of each edge

deg <- degree(G) summary(deg) tail(sort(deg))

We see that the largest degree is equal to 12. To find our which node that is, we peek into the graph.

w <- which(deg == 12) V(G)[w]$label

So, the highest degree belongs to the group representing the mTORC1 complex. We know that the in-degree for this graph node is artificially inflated by the "contained" arrows defining the group members. So, it may be worth exploring how many such arrows there are, and how many are actual interactions.

Here are the genes that are the source of inbound arrows.

A <- adjacent_vertices(G, w, "in") A

By default, we only see the cryptic alphanumeric identifiers. By extracting the IDs, we can find the gene names.

ids <- as_ids(A[[1]]) V(G)$label[as_ids(V(G)) %in% ids]

Knowing the IDs, we can also determine the edge type.

Earg <- as.vector(t(as.matrix(data.frame(Source = ids, Target = names(w))))) E(G, P = Earg)$MIM

Now we can plot the subgraph that connects directly to the mTORC1 complex.

B <- adjacent_vertices(G, w, "out") subg <- subgraph(G, c(names(w), ids, as_ids(B[[1]]))) plot(subg, lwd=3)

We see that five of the inbound arrows are for the genes that are "contained" in the complex, leaving seven arrows rhat have biological meaning for the pathway.

We saw above that there is another node in this pathway that has 7 connected edges. It may be worth looking more closely at that node.

w <- which(deg == 7) V(G)[w]$label

This time, we get an actual gene, `FoxO`

.

B <- adjacent_vertices(G, w, "all") subg <- subgraph(G, c(names(w), as_ids(B[[1]]))) plot(subg, lwd=3)

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