PathwaySpace: Spatial projection of network signals along geodesic paths.
In PathwaySpace: Spatial Projection of Network Signals along Geodesic Paths

knitr::opts_chunk$set(echo = TRUE)

Package: PathwaySpace r packageVersion('PathwaySpace')

Highlights

Produces landscape images representing graphs by geodesic paths.
Projects signals using a decay function to model signal attenuation.
Applies a convolution algorithm to combine signals from neighboring nodes.

Overview

For a given igraph object containing vertices, edges, and a signal associated with the vertices, PathwaySpace performs a convolution operation, which involves a weighted combination of neighboring signals on a 2D graph space. Figure 1 illustrates the convolution operation problem. Each vertex's signal is positioned on a grid at specific x and y coordinates, represented by cones (for available signals) or question marks (for null or missing values). Our model considers the vertex-signal positions as source points (or transmitters) and the null-signal positions as end points (or receivers). The signal values from vertex-signal positions are then projected to the null-signal positions according to a decay function, which will control how the signal values attenuate as they propagate across the 2D space. For a given null-signal position, the k-top signals are used to define the contributing vertices for signal convolution. The convolution operation aggregates the signals from these contributing vertices, considering their intensities reaching the end points. Users can adjust both the aggregation and decay functions; the aggregation function can be any arithmetic rule that reduces a numeric vector into a single scalar value (e.g., mean, weighted mean), while available decay functions include linear, exponential, and Weibull models (Fig.1B). Additionally, users can assign vertex-specific decay functions to model signal projections for subsets of vertices that may exhibit distinct behaviors. The resulting image forms geodesic paths in which the signal has been projected from vertex- to null-signal positions, using a density metric to measure the signal intensity along these paths.

knitr::include_graphics("figures/fig1.png")

Quick start

#--- Load required packages for this section
library(igraph)
library(ggplot2)
library(RGraphSpace)
library(PathwaySpace)

Setting basic input data

This section will create an igraph object containing a binary signal associated to each vertex. The graph layout is configured manually to ensure that users can easily view all the relevant arguments needed to prepare the input data for the PathwaySpace package. The igraph's make_star() function creates a star-like graph and the V() function is used to set attributes for the vertices. The PathwaySpace package will require that all vertices have x, y, and name attributes.

# Make a 'toy' igraph object, either a directed or undirected graph
gtoy1 <- make_star(5, mode="undirected")

# Assign 'x' and 'y' coordinates to each vertex
# ..this can be an arbitrary unit in (-Inf, +Inf)
V(gtoy1)$x <- c(0, 2, -2, -4, -8)
V(gtoy1)$y <- c(0, 0,  2, -4,  0)

# Assign a 'name' to each vertex (here, from n1 to n5)
V(gtoy1)$name <- paste0("n", 1:5)

Checking igraph validity

Next, we will create a GraphSpace-class object using the GraphSpace() constructor. This function will check the validity of the igraph object. For this example mar = 0.2, which sets the outer margins as a fraction of the 2D space on which the convolution operation will project the signal.

# Check graph validity
g_space1 <- GraphSpace(gtoy1, mar = 0.2)

Our graph is now ready for the PathwaySpace package. We can check its layout using the plotGraphSpace() function.

# Check the graph layout
plotGraphSpace(g_space1, add.labels = TRUE)

# gg <- plotGraphSpace(g_space1, add.labels = TRUE)
# ggsave(filename = "./figures/fig2.png", height=4, width=5,
#   units="in", device="png", dpi=250, plot=gg)

knitr::include_graphics("figures/fig2.png")

Creating a PathwaySpace

Next, we will create a PathwaySpace-class object using the buildPathwaySpace() constructor. This will calculate pairwise distances between vertices, subsequently required by the signal projection methods.

# Run the PathwaySpace constructor
p_space1 <- buildPathwaySpace(g_space1)

As a default behavior, the buildPathwaySpace() constructor initializes the signal of each vertex as 0. We can use the length(), names(), and vertexSignal() accessors to get and set vertex signals in the PathwaySpace object; for example, in order to get vertex names and signal values:

# Check the number of vertices in the PathwaySpace object
length(p_space1)
## [1] 5

# Check vertex names
names(p_space1)
## [1] "n1" "n2" "n3" "n4" "n5"

# Check signal (initialized with '0')
vertexSignal(p_space1)
## n1 n2 n3 n4 n5 
##  0  0  0  0  0

...and for setting new signal values in PathwaySpace objects:

# Set new signal to all vertices
vertexSignal(p_space1) <- c(1, 4, 2, 4, 3)

# Set a new signal to the 1st vertex
vertexSignal(p_space1)[1] <- 2

# Set a new signal to vertex "n1"
vertexSignal(p_space1)["n1"] <- 6

# Check updated signal values
vertexSignal(p_space1)
## n1 n2 n3 n4 n5 
##  6  4  2  4  3

Signal projection

Circular projection

Following that, we will use the circularProjection() function to project the network signals, using the signalDecay() function with default settings. We set k = 1, defining the contributing vertices for signal convolution. In this case, each null-signal position will receive the projection from a single vertex-signal position (i.e. the highest signal intensity in pathway space reaching that position). We then create a landscape image using the plotPathwaySpace() function.

# Run signal projection
p_space1 <- circularProjection(p_space1, k = 1, pdist = 0.4)

# Plot a PathwaySpace image
plotPathwaySpace(p_space1, add.marks = TRUE)

# gg <- plotPathwaySpace(p_space1, add.marks = TRUE)
# ggsave(filename = "./figures/fig3.png", height=3.5, width=5,
#   units="in", device="png", dpi=350, plot=gg)

knitr::include_graphics("figures/fig3.png")

The pdist term determines a distance unit for the signal convolution related to the pathway space. This distance unit will affect the extent over which the convolution operation projects the signal in the pathway space. Next, we reassess the same PathwaySpace object using k = 2. The user can also customize a few arguments in plotPathwaySpace() function, which is a wrapper to create ggplot graphics for PathwaySpace-class objects.

# Re-run signal projection with 'k = 2'
p_space1 <- circularProjection(p_space1, k = 2, pdist = 0.4)

# Plot the PathwaySpace image
plotPathwaySpace(p_space1, marks = c("n3","n4"), theme = "th2")

# gg <- plotPathwaySpace(p_space1, marks = c("n3","n4"), theme = "th2")
# ggsave(filename = "./figures/fig4.png", height=3.5, width=5,
#   units="in", device="png", dpi=350, plot=gg)

knitr::include_graphics("figures/fig4.png")

By default, the signal projection uses a Weibull decay function, which is passed to the circularProjection() function via decay.fun argument. The user can modify the decay function, for example, adjusting the Weibull's shape:

# Re-run signal projection, adjusting Weibull's shape
p_space1 <- circularProjection(p_space1, k = 2, pdist = 0.2, 
  decay.fun = signalDecay(shape = 2))

# Plot the PathwaySpace image
plotPathwaySpace(p_space1, marks = "n1", theme = "th2")

# gg <- plotPathwaySpace(p_space1, marks = "n1", theme = "th2")
# ggsave(filename = "./figures/fig5.png", height=3.5, width=5,
#   units="in", device="png", dpi=350, plot=gg)

knitr::include_graphics("figures/fig5.png")

In this case, we set shape = 2; this parameter allows a projection to take a variety of shapes. When shape = 1 the projection follows an exponential decay, and when shape > 1 the projection is first convex, then concave with an inflection point along the decay path.

Polar projection

In this section we will project the network signal using a polar coordinate system. This representation may be useful for certain types of data, for example, to highlight patterns of signal propagation on directed graphs, especially to explore the orientation aspect of signal flow. To demonstrate this feature we will used the gtoy2 directed graph, already available in the RGraphSpace package.

# Load a pre-processed directed igraph object
data("gtoy2", package = "RGraphSpace")
# Check graph validity
g_space2 <- GraphSpace(gtoy2, mar = 0.2)

# Check the graph layout
plotGraphSpace(g_space2, add.labels = TRUE)

# gg <- plotGraphSpace(g_space2, add.labels = TRUE)
# ggsave(filename = "./figures/fig6.png", height=4, width=5,
#   units="in", device="png", dpi=250, plot=gg)

knitr::include_graphics("figures/fig6.png")

# Build a PathwaySpace for the 'g_space2'
p_space2 <- buildPathwaySpace(g_space2)

# Set '1s' as vertex signal
vertexSignal(p_space2) <- 1

For fine-grained modeling of signal decay, the vertexDecay() accessor allows assigning decay functions at the level of individual vertices. For example:

# Modify the vertex 'decayFunction' attribute
vertexDecay(p_space2) <- signalDecay(shape = 2)
vertexDecay(p_space2)[["n1"]] <- signalDecay(shape = 3)

Next, we run signal projection using polar coordinates:

# Run signal projection using polar coordinates
p_space2 <- polarProjection(p_space2, k = 2, theta = 45)

# Plot the PathwaySpace image
plotPathwaySpace(p_space2, theme = "th2", add.marks = TRUE)

# gg <- plotPathwaySpace(p_space2, theme = "th2", add.marks = TRUE)
# ggsave(filename = "./figures/fig7.png", height=3.5, width=5,
#   units="in", device="png", dpi=350, plot=gg)

knitr::include_graphics("figures/fig7.png")

Note that this projection emphasizes signals along the edges of the network. In order to also consider the direction of edges, next we set directional = TRUE.

# Re-run signal projection using 'directional = TRUE'
p_space2 <- polarProjection(p_space2, k = 2, theta = 45, directional = TRUE)

# Plot the PathwaySpace image
plotPathwaySpace(p_space2, theme = "th2", marks = c("n1","n3","n4","n5"))

# gg <- plotPathwaySpace(p_space2, theme = "th2", marks = c("n1","n3","n4","n5"))
# ggsave(filename = "./figures/fig8.png", height=3.5, width=5,
#   units="in", device="png", dpi=350, plot=gg)

knitr::include_graphics("figures/fig8.png")

This updated PathwaySpace polar projection emphasizes the signal flow into a defined direction (see the directional pattern of the igraph plot at the top of this section). However, when interpreting the results, users must be aware that this method may introduce distortions. For example, depending on the network's structure, the polar projection may not capture all aspects of a directed graph, such as cyclic dependencies, feedforward and feedback loops, or other intricate edge interplays.

Signal types

The PathwaySpace accepts binary, integer, and numeric signal types, including NAs. If a vertex signal is assigned with NA, it will be ignored by the convolution algorithm. Logical values are also allowed, but it will be treated as binary. Next, we show the projection of a signal that includes negative values, using the p_space1 object created previously.

# Set a negative signal to vertices "n3" and "n4"
vertexSignal(p_space1)[c("n3","n4")] <- c(-2, -4)

# Check updated signal vector
vertexSignal(p_space1)
# n1 n2 n3 n4 n5 
#  6  4 -2 -4  3 

# Re-run signal projection
p_space1 <- circularProjection(p_space1, k = 2,
  decay.fun = signalDecay(shape = 2))

# Plot the PathwaySpace image
plotPathwaySpace(p_space1, bg.color = "white", font.color = "grey20", add.marks = TRUE, mark.color = "magenta", theme = "th2")

# gg <- plotPathwaySpace(p_space1, bg.color = "white", font.color = "grey20",
#   add.marks = TRUE, mark.color = "magenta", theme = "th2")
# ggsave(filename = "./figures/fig9.png", height=3.5, width=5,
#   units="in", device="png", dpi=350, plot=gg)

knitr::include_graphics("figures/fig9.png")

Note that the original signal vector was rescale to [-1, +1]. If the signal vector is >=0, then it will be rescaled to [0, 1]; if the signal vector is <=0, it will be rescaled to [-1, 0]; and if the signal vector is in (-Inf, +Inf), then it will be rescaled to [-1, +1]. To override this signal processing, simply set the rescale argument to FALSE in the projection functions.

PathwaySpace decoration

In order to enhance clarity and make it less likely for viewers to miss important details of large graphs, in this section we introduce visual elements to large PathwaySpace images. We will use an igraph object with n = 12990 vertices to create a large PathwaySpace object, upon which we will project binary signals from a relatively small number of vertices. This example will emphasize clusters of vertices forming summits, but it might also come at the cost of reduced clarity in displaying the graph's overall structure, particularly in regions far from the summit areas. In order to balance between emphasizing clusters and maintaining the visibility of the entire graph structure, we will outline graph silhouettes as decoration elements in the PathwaySpace image.

#--- Load required packages for this section
library(PathwaySpace)
library(RGraphSpace)
library(igraph)
library(ggplot2)

Loading a large graph

Next, we will load an igraph object with n = 12990 vertices, containing gene interaction data available from the Pathway Commons database (version 12) [@Rodchenkov2019].

# Load a large igraph object
data("PCv12_pruned_igraph", package = "PathwaySpace")

# Check number of vertices
length(PCv12_pruned_igraph)
# [1] 12990

# Check vertex names
head(V(PCv12_pruned_igraph)$name)
# [1] "A1BG" "AKT1" "CRISP3" "GRB2" "PIK3CA" "PIK3R1"

# Get top-connected nodes for visualization
top10hubs <- igraph::degree(PCv12_pruned_igraph)
top10hubs <- names(sort(top10hubs, decreasing = TRUE)[1:10])
head(top10hubs)
# [1] "GNB1" "TRIM28" "RPS27A" "CTNNB1" "TP53" "ACTB"

## Check graph validity
g_space_PCv12 <- GraphSpace(PCv12_pruned_igraph, mar = 0.1)

## Visualize the graph layout labeled with 'top10hubs' nodes
plotGraphSpace(g_space_PCv12, node.labels = top10hubs, label.color = "blue", theme = "th3")

# gg <- plotGraphSpace(g_space_PCv12, node.labels = top10hubs,
# label.color = "blue",  theme = "th3", label.size = 2)
# ggsave(filename = "./figures/fig10.png", height=3.5, width=4.5,
#   units="in", device="png", dpi=200, plot=gg)

knitr::include_graphics("figures/fig10.png")

We will also load gene sets from the MSigDB collection [@Liberzon2015], which are subsequently used to project a binary signal in the PathwaySpace image.

# Load a list with Hallmark gene sets
data("Hallmarks_v2023_1_Hs_symbols", package = "PathwaySpace")

# There are 50 gene sets in "hallmarks"
length(hallmarks)
# [1] 50

# We will use the 'HALLMARK_P53_PATHWAY' (n=200 genes) for demonstration
length(hallmarks$HALLMARK_P53_PATHWAY)
# [1] 200

Running PathwaySpace

We now follow the PathwaySpace pipeline as explained in the previous sections, that is, using the buildPathwaySpace() constructor to initialize a new PathwaySpace object with the Pathway Commons interactions.

# Run the PathwaySpace constructor
p_space_PCv12 <- buildPathwaySpace(gs=g_space_PCv12, nrc=500)
# Note: 'nrc' sets the number of rows and columns of the
# image space, which will affect the image resolution (in pixels)

...and now we mark the HALLMARK_P53_PATHWAY genes in the PathwaySpace object.

# Intersect Hallmark genes with the PathwaySpace
hallmarks <- lapply(hallmarks, intersect, y = names(p_space_PCv12) )

# After intersection, the 'HALLMARK_P53_PATHWAY' dropped to n=173 genes
length(hallmarks$HALLMARK_P53_PATHWAY)
# [1] 173

# Set a binary signal (1s) to 'HALLMARK_P53_PATHWAY' genes
vertexSignal(p_space_PCv12) <- 0
vertexSignal(p_space_PCv12)[ hallmarks$HALLMARK_P53_PATHWAY ] <- 1

...and run the circularProjection() function.

# Run signal projection
p_space_PCv12 <- circularProjection(p_space_PCv12)
plotPathwaySpace(p_space_PCv12, title="HALLMARK_P53_PATHWAY", marks = top10hubs, mark.size = 2, theme = "th3")

# gg <- plotPathwaySpace(p_space_PCv12, theme = "th3",
#   title="HALLMARK_P53_PATHWAY", marks = top10hubs,
#   mark.size = 2, font.size = 0.8)
# ggsave(filename = "./figures/fig11.png", plot=gg, height=4,
#   width=5, units="in", device="png", dpi=250)

knitr::include_graphics("figures/fig11.png")

Note that this image emphasizes groups of vertices forming summits, but it misses the outline of the graph structure, which faded with the signal that reaches the furthermost points of the network.

Mapping silhouettes

Next, we will decorate the PathwaySpace image with graph's silhouettes.

# Add silhouettes
p_space_PCv12 <- silhouetteMapping(p_space_PCv12)
plotPathwaySpace(p_space_PCv12, title="HALLMARK_P53_PATHWAY", marks = top10hubs, mark.size = 2, theme = "th3")

# gg <- plotPathwaySpace(p_space_PCv12, theme = "th3", 
#   title="HALLMARK_P53_PATHWAY", marks = top10hubs, 
#   mark.size = 2, font.size = 0.8)
# ggsave(filename = "./figures/fig12.png", plot=gg, height=4,
#   width=5, units="in", device="png", dpi=250)

knitr::include_graphics("figures/fig12.png")

Mapping summits

The summits represent regions within the graph that exhibit signal values that are notably higher than the baseline level. These regions may be of interest for downstream analyses. One potential downstream analysis is to determine which vertices projected the original input signal. This could provide insights into the communities within these summit regions. One may also wish to explore other vertices within the summits, by querying associations with the original input gene set. In order to extract vertices within summits, next we use the summitMapping() function, which also decorate summits with contour lines.

# Mapping summits
p_space_PCv12 <- summitMapping(p_space_PCv12, minsize = 50)
plotPathwaySpace(p_space_PCv12, title="HALLMARK_P53_PATHWAY", theme = "th3")

# gg <- plotPathwaySpace(p_space_PCv12, title="HALLMARK_P53_PATHWAY", 
#   theme = "th3", font.size = 0.8,)
# ggsave(filename = "./figures/fig13.png", plot=gg, height=4,
#   width=5, units="in", device="png", dpi=250)

knitr::include_graphics("figures/fig13.png")

# Extracting summits from a PathwaySpace
summits <- getPathwaySpace(p_space_PCv12, "summits")
class(summits)
# [1] "list"

Citation

If you use PathwaySpace, please cite:

Tercan & Apolonio et al. Protocol for assessing distances in pathway space for classifier feature sets from machine learning methods. STAR Protocols 6(2):103681, 2025. https://doi.org/10.1016/j.xpro.2025.103681
Ellrott et al. Classification of non-TCGA cancer samples to TCGA molecular subtypes using compact feature sets. Cancer Cell 43(2):195-212.e11, 2025. https://doi.org/10.1016/j.ccell.2024.12.002

Session information

sessionInfo()

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PathwaySpace documentation built on Aug. 8, 2025, 6:47 p.m.

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