Working with metric graphs

Introduction

Networks such as street or river networks are examples of metric graphs. A compact metric graph \(\Gamma\) consists of a set of finitely many vertices \(\mathcal{V}=\{v_i\}\) and a finite set \(\mathcal{E}=\{e_j\}\) of edges connecting the vertices. Each edge \(e\) is a curve of finite length \(l_e\) that connects two vertices. These curves are parameterized by arc length and a location \(s\in \Gamma\) is a position on an edge, and can thus be represented as a touple \((e,t)\) where \(t\in[0,l_e]\). Compared to regular graphs, where one typically defines functions on the vertices, we are for metric graphs interested in function that are defined on both the vertices and the edges.

In this vignette we will introduce the metric_graph class of the MetricGraph package. This class provides a user friendly representation of metric graphs, and we will show how to use the class to construct and visualize metric graphs, add data to them, and work with functions defined on the graphs.

For details about Gaussian processes and inference on metric graphs, we refer to the Vignettes

Constructing metric graphs

Basic constructions and properties

A metric graph can be constructed in two ways. The first is to specify all edges in the graph as a list object, where each entry is a matrix. To illustrate this, we first construct the following edges

edge1 <- rbind(c(0,0),c(1,0))
edge2 <- rbind(c(0,0),c(0,1))
edge3 <- rbind(c(0,1),c(-1,1))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
edge4 <- cbind(sin(theta),1+ cos(theta))
edges = list(edge1, edge2, edge3, edge4)

We can now create the graph based on the edges object as follows

graph <- metric_graph$new(edges = edges)
graph$plot()

The plot function that was used to create the plot above has various parameters to set the sizes and colors of the vertices and edges, and it has a plotly argument to visualize the graph in 3D. For this to work, the plotly library must be installed.

graph$plot(plotly = TRUE, vertex_size = 5, vertex_color = "blue",
           edge_color = "red", edge_width = 2)

It is also important to know that the 2d version of the plot() method returns a ggplot2 object and can be modified as such. For instance:

p <- graph$plot()
p + ggplot2::labs(title = "Metric graph",
          x = "long", y = "lat")

Similarly, the 3d version of the plot() method returns a plotly object that can also be modified. For instance:

p <- graph$plot(plotly = TRUE)
p <- plotly::layout(p, title = "Metric graph", 
              scene = list(xaxis=
              list(title = "Long"),yaxis=list(title = "Lat")))
p

We can now obtain various properties of the graph: The vertex matrix, which specifies the Euclidian coordinates of the vertices is

graph$V

##      [,1] [,2]
## [1,]    0    0
## [2,]    1    0
## [3,]    0    1
## [4,]   -1    1

and the edge matrix that specified the edges of the graph (i.e., which vertices that are connected by edges) is

graph$E

##      [,1] [,2]
## [1,]    1    2
## [2,]    1    3
## [3,]    3    4
## [4,]    1    4

To obtain the geodesic (shortest path) distance between the vertices, we can use the function compute_geodist:

graph$compute_geodist()
graph$geo_dist

## $.vertices
##          [,1]     [,2] [,3]     [,4]
## [1,] 0.000000 1.000000    1 1.570729
## [2,] 1.000000 0.000000    2 2.570729
## [3,] 1.000000 2.000000    0 1.000000
## [4,] 1.570729 2.570729    1 0.000000

The second option it to construct the graph using two matrices V and E that specify the locations (in Euclidean space) of the vertices and the edges. In this case, it is assumed that the graph only has straight edges:

V <- rbind(c(0, 0), c(1, 0), c(0, 1), c(-1, 1))
E <- rbind(c(1, 2), c(1, 3), c(3, 4), c(4, 1))
graph2 <- metric_graph$new(V = V, E = E)

## Starting graph creation...

## LongLat is set to FALSE

## The current tolerances are:

##   Vertex-Vertex 0.001

##   Vertex-Edge 0.001

##   Edge-Edge 0

## Setup edges and merge close vertices

## Snap vertices to close edges

## Total construction time: 0.69 secs

graph2$plot()

A third option is to create a graph from a SpatialLines object:

library(sp)
line1 <- Line(rbind(c(0,0),c(1,0)))
line2 <- Line(rbind(c(0,0),c(0,1)))
line3 <- Line(rbind(c(0,1),c(-1,1)))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
line4 <- Line(cbind(sin(theta),1+ cos(theta)))
Lines = sp::SpatialLines(list(Lines(list(line1),ID="1"),
                              Lines(list(line2),ID="2"),
                              Lines(list(line3),ID="3"),
                              Lines(list(line4),ID="4")))

graph <- metric_graph$new(edges = Lines)                    
graph$plot()

A final option is to create from a MULTILINESTRING object:

library(sf)
line1 <- st_linestring(rbind(c(0,0),c(1,0)))
line2 <- st_linestring(rbind(c(0,0),c(0,1)))
line3 <- st_linestring(rbind(c(0,1),c(-1,1)))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
line4 <- st_linestring(cbind(sin(theta),1+ cos(theta)))
multilinestring = st_multilinestring(list(line1, line2, line3, line4))

graph <- metric_graph$new(edges = multilinestring)                    
graph$plot()

Tolerances for merging vertices and edges

The constructor of the graph has one argument tolerance which is used for connecting edges that are close in Euclidean space. Specifically, the tolerance argument is given as a list with three elements:

vertex_vertex vertices that are closer than this number are merged (the default value is 1e-7)
vertex_edge if a vertex at the end of one edge is closer than this number to another edge, this vertex is connected to that edge (the default value is 1e-7)
edge_edge if two edges at some point are closer than this number, a new vertex is added at that point and the two edges are connected (the default value is 0 which means that the option is not used)

These options are often needed when constructing graphs based on real data, for example from OpenStreetMap as we will see later. To illustrate these options, suppose that we want to construct a graph from the following three edges:

edge1 <- rbind(c(0,0),c(1,0))
edge2 <- rbind(c(0,0.03),c(0,0.75))
edge3 <- rbind(c(-1,1),c(0.5,0.03))
edge4 <- rbind(c(0,0.75), c(0,1))
edges =  list(edge1, edge2, edge3, edge4)
graph3 <- metric_graph$new(edges = edges)
graph3$plot(degree = TRUE)
print(graph3$nV)

## [1] 7

We added the option degree=TRUE to the plot here to visualize the degrees of each vertex. As expected, one sees that all vertices have degree 1, and none of the three edges are connected. If these are streets in a street network, one might suspect that the two vertices at \((0,0)\) and \((0,0.03)\) really should be the same vertex so that the two edges are connected. This can be adjusted by increasing the vertex_vertex tolerance:

graph3 <- metric_graph$new(edges = edges, tolerance = list(vertex_vertex = 0.05))
graph3$plot(degree = TRUE)

One might also want to add the vertex at \((0.5, 0.03)\) as a vertex on the first edge, so that the two edges there are connected. This can be done by adjusting the vertex_edge tolerance:

graph3 <- metric_graph$new(edges = edges, tolerance = list(vertex_vertex = 0.05,
                                                           vertex_edge = 0.1))
graph3$plot(degree = TRUE)

We can see that the vertex at \((0.5,0)\) was indeed connected with the edge from \((0,0)\) to \((1,0)\) and that vertex now has degree 3 since it is connected with three edges. One can also note that the edges object that was used to create the graph is modified internally in the metric_graph object so that the connections are visualized correctly in the sense that all edges are actually shown as connected edges.

Finally, to add a vertex at the intersection between edge2 and edge3 we can adjust the edge_edge tolerance:

graph3 <- metric_graph$new(edges = edges, tolerance = list(vertex_vertex = 0.2,
                                                           vertex_edge = 0.1,
                                                           edge_edge = 0.001))
graph3$plot(degree = TRUE)

Now, the structure of the metric graph does not change if we add or remove vertices of degree 2. Because of this,one might want to remove vertices of degree 2 since this can reduce computational costs. This can be done by setting the remove_deg2 argument while creating the graph:

graph3 <- metric_graph$new(edges = edges, tolerance = list(vertex_vertex = 0.2,
                                                           vertex_edge = 0.1,
                                                           edge_edge = 0.001),
                           remove_deg2 = TRUE)
graph3$plot(degree = TRUE)

Observe that one vertex of degree 2 was not removed. This is due to the fact that if we consider the direction, this vertex has directions incompatible for removal. We can see this by setting the argument direction=TRUE in the plot() method:

graph3$plot(direction = TRUE)

Characteristics of the graph

A brief summary of the main characteristics of the graph can be obtained by the compute_characteristics() method:

graph3$compute_characteristics()

We can, then, view those characteristics by simply printing the metric_graph object:

graph3

## A metric graph with  6  vertices and  6  edges.
## 
## Vertices:
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Lengths: 
##       Min: 0.3533333  ; Max: 1.190873  ; Total: 3.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: unknown

## To check if the graph satisfies the distance consistency, run the `check_distance_consistency()` method.

##   Has Euclidean edges: unknown

## To check if the graph has Euclidean edges, run the `check_euclidean()` method.

Observe that we do not know that the graph has Euclidean edges. We can check by using the check_euclidean() method:

graph3$check_euclidean()

Let us view the characteristics again:

graph3

## A metric graph with  6  vertices and  6  edges.
## 
## Vertices:
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Lengths: 
##       Min: 0.3533333  ; Max: 1.190873  ; Total: 3.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE

It can happen that we have that the graph does not have Euclidean edges without the need to check the distance consistency. However, if one wants to check if the graph satisfies the distance consistency assumption, one can run the check_distance_consistency() function:

graph3$check_distance_consistency()

Further, the individual characteristics can be accessed (after running the compute_characteristics() method) through the characteristics component of the graph object:

graph3$characteristics

## $has_loops
## [1] FALSE
## 
## $connected
## [1] TRUE
## 
## $has_multiple_edges
## [1] FALSE
## 
## $is_tree
## [1] FALSE
## 
## $distance_consistency
## [1] TRUE
## 
## $euclidean
## [1] TRUE

Summaries of metric graphs

We can also obtain a summary of the informations contained on a metric graph object by using the summary() method. Let us obtain a summary of informations of graph3:

summary(graph3)

## A metric graph object with:
## 
## Vertices:
##   Total: 6 
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 6 
##   Lengths: 
##       Min: 0.3533333  ; Max: 1.190873  ; Total: 3.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: The graph has no mesh! 
## 
## Data: The graph has no data!
## 
## Tolerances: 
##   vertex-vertex:  0.2 
##   vertex-edge:  0.1 
##   edge-edge:  0.001

Observe that there are some quantities that were not computed on the summary. We can see how to compute them by setting the argument messages to TRUE:

summary(graph3, messages = TRUE)

## A metric graph object with:
## 
## Vertices:
##   Total: 6 
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 6 
##   Lengths: 
##       Min: 0.3533333  ; Max: 1.190873  ; Total: 3.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE

## To compute the Laplacian, run the 'compute_laplacian()' method.

##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE

## To compute the resistance distances, run the 'compute_resdist()' method.

## To compute the finite element matrices, run the 'compute_fem()' method.

## 
## Mesh: The graph has no mesh!

## To build the mesh, run the 'build_mesh()' method.

## 
## Data: The graph has no data!

## To add observations, use the 'add_observations()' method.

## 
## Tolerances: 
##   vertex-vertex:  0.2 
##   vertex-edge:  0.1 
##   edge-edge:  0.001

Finally, the summary() can also be accessed from the metric graph object:

graph3$summary()

## A metric graph object with:
## 
## Vertices:
##   Total: 6 
##   Degree 1: 3;  Degree 2: 1;  Degree 3: 1;  Degree 4: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 6 
##   Lengths: 
##       Min: 0.3533333  ; Max: 1.190873  ; Total: 3.788107 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: The graph has no mesh! 
## 
## Data: The graph has no data!
## 
## Tolerances: 
##   vertex-vertex:  0.2 
##   vertex-edge:  0.1 
##   edge-edge:  0.001

Vertices and edges

The metric graph object has the edges and vertices elements in its list. We can get a quick summary from them by calling these elements:

graph3$vertices

## Vertices of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Summary: 
##      x         y Degree Indegree Outdegree Problematic
## 1  0.0 0.0000000      2        0         2        TRUE
## 2  1.0 0.0000000      1        1         0       FALSE
## 3 -1.0 1.0000000      1        0         1       FALSE
## 4  0.5 0.0300000      3        2         1       FALSE
## 5  0.0 1.0000000      1        1         0       FALSE
## 6  0.0 0.3533333      4        2         2       FALSE

and

graph3$edges

## Edges of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Summary: 
## 
## Edge 1 (first and last coordinates): 
##    x    y
##  0.0 0.00
##  0.5 0.03
## Total number of coordinates: 2 
## Edge length: 0.5008992 
## Weight: 1 
## 
## Edge 2 (first and last coordinates): 
##  x         y
##  0 0.0000000
##  0 0.3533333
## Total number of coordinates: 2 
## Edge length: 0.3533333 
## Weight: 1 
## 
## Edge 3 (first and last coordinates): 
##   x         y
##  -1 1.0000000
##   0 0.3533333
## Total number of coordinates: 2 
## Edge length: 1.190873 
## Weight: 1 
## 
## Edge 4 (first and last coordinates): 
##  x         y
##  0 0.3533333
##  0 1.0000000
## Total number of coordinates: 4 
## Edge length: 0.6466667 
## Weight: 1

## # 2 more edges

## # Use `print(n=...)` to see more edges

We can also look at individual edges and vertices. For example, let us look at vertice number 2:

graph3$vertices[[2]]

## Vertex 2 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Summary: 
##  x y Degree Indegree Outdegree Problematic
##  1 0      1        1         0       FALSE

Similarly, let us look at edge number 4:

graph3$edges[[4]]

## Edge 4 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##  x         y
##  0 0.3533333
##  0 1.0000000
## 
## Coordinates of the edge:
##  x         y
##  0 0.3533333
##  0 0.7500000
##  0 0.7500000
##  0 1.0000000

## Relative positions of the coordinates on the graph edges were not computed.

## To compute them, run the `compute_PtE_edges()` method.

## 
## Total number of coordinates: 4 
## Edge length: 0.6466667 
## Weight: 1

We have a message informing us that the relative positions of the edges were not computed. They are used to produce better plots when using the plot_function() method, see the section “Improving the plot obtained by plot_function” below. To compute the relative positions, we run the compute_PtE_edges() method:

graph3$compute_PtE_edges()

Now, we take a look at edge number 4 again:

graph3$edges[[4]]

## Edge 4 of the metric graph
## 
## Longitude and Latitude coordinates: FALSE 
## 
## Coordinates of the vertices of the edge: 
##  x         y
##  0 0.3533333
##  0 1.0000000
## 
## Coordinates of the edge:
##  x         y
##  0 0.3533333
##  0 0.7500000
##  0 0.7500000
##  0 1.0000000
## 
## Relative positions of the edge:
##  Edge number Distance on edge
##            2        1.0000000
##            4        0.6134021
##            4        0.6134021
##            4        1.0000000
## 
## Total number of coordinates: 4 
## Edge length: 0.6466667 
## Weight: 1

Understanding coordinates on graphs

The locations of the vertices are specified in Euclidean coordinates. However, when specifying a position on the graph, it is not practical to work with Euclidean coordinates since not all locations in Euclidean space are locations on the graph. It is instead better to specify a location on the graph by the touple \((i, t)\), where \(i\) denotes the number of the edge and \(t\) is the location on the edge. The location \(t\) can either be specified as the distance from the start of the edge (and then takes values between 0 and the length of the edge) or as the normalized distance from the start of the edge (and then takes values between 0 and 1). The function coordinates can be used to convert between coordinates in Euclidean space and locations on the graph. For example the location at distance 0.2 from the start of the second edge is:

graph$coordinates(PtE = matrix(c(2, 0.2), 1,2), normalized = FALSE)

##      [,1] [,2]
## [1,]    0  0.2

In this case, since the edge has length 1, the location of the point at normalized distance 0.2 from the start of the edge is the same:

graph$coordinates(PtE = matrix(c(2, 0.2), 1,2), normalized = TRUE)

##      [,1] [,2]
## [1,]    0  0.2

The function can also be used to find the closest location on the graph for a location in Euclidean space:

graph$coordinates(XY = matrix(c(0, 0.2), 1,2))

##      [,1] [,2]
## [1,]    2  0.2

In this case, the normalized argument decides whether the returned value should be given in normalized distance or not.

Methods for working with real data

To illustrate the useage of metric_graph on some real data, we use the osmdata package to download data from OpenStreetMap. In the following code, we extract highways in the city of Copenhagen:

library(osmdata)
set_overpass_url("https://maps.mail.ru/osm/tools/overpass/api/interpreter")
call <- opq(bbox = c(12.4,55.5,12.65,55.9))
call <- add_osm_feature(call, key = "highway",value=c("motorway", "primary",
                                                      "secondary"))
data <- osmdata_sp(call)

graph5 <- metric_graph$new(SpatialLines(data$osm_lines@lines))
graph5$plot(vertex_size = 0)

There are a few things to note about data like this. The first is that the coordinates are given in Longitude and Latitude. Because of this, the edge lengths are by default given in degrees, which may result in very small numbers:

range(graph5$get_edge_lengths())

## [1] 0.001000842 0.111550344

This may cause numerical instabilities when dealing with random fields on the graph, and it also makes it difficult to interpret results (unless one has a good intuition about distances in degrees). To avoid such problems, it is better to set the longlat argument when constructing the graph:

graph5 <- metric_graph$new(SpatialLines(data$osm_lines@lines), longlat = TRUE)

This tells the constructor that the coordinates are given in Longitude and Latitude and that distances should be calculated in km. So if we now look at the edge lengths, they are given in km:

range(graph5$get_edge_lengths())

## Units: [km]
## [1]  0.001652242 11.209690354

The second thing to note is that the constructor gave a warning that the graph is not connected. This might not be ideal for modeling and we may want to study the different connected components separately. If this is not a concern, one can set the argument check_connected = FALSE while creating the graph. In this case the check is not done and no warning message is printed. To construct all connected components, we can create a graph_components object:

graphs <- graph_components$new(SpatialLines(data$osm_lines@lines), 
                                                    longlat = TRUE)

The graph_components class contains a list of metric_graph objects, one for each connected component. In this case, we have

graphs$n

## [1] 12

components in total, and their total edge lengths in km at

graphs$lengths

## Units: [km]
##  [1] 179.416486 113.343274  36.634315  10.747573  10.678542   9.410596
##  [7]   8.960023   8.722649   5.458977   4.246385   1.907717   1.082217

To plot all of them, we can use the plot command of the class:

graphs$plot(vertex_size = 0)

One reason for having multiple components here might be that we set the tolerance for merging nodes too low. In fact, by looking at the edge lengths we see that we have vertices that are as close as

min(graph5$get_edge_lengths(unit='m'))

## 1.652242 [m]

meters. Let us increase the tolerances so that vertices that are at most a distance of 100 meters (0.1km) apart should be merged:

graphs <- graph_components$new(SpatialLines(data$osm_lines@lines), longlat = TRUE,
                               tolerance = list(vertex_vertex = 0.1))
graphs$plot(vertex_size = 0)

With this choice, let us check the number of connected components:

graphs$n

## [1] 3

We can retrieve the graph as a standard metric_graph object to work with in further analysis via the get_largest command:

graph5 <- graphs$get_largest()

Let us take a quick look at its vertices and edges:

graph5$vertices

## Vertices of the metric graph
## 
## Longitude and Latitude coordinates: TRUE 
## Coordinate reference system: +proj=longlat +datum=WGS84 +no_defs 
## 
## Summary: 
##    Longitude Latitude Degree Indegree Outdegree Problematic
## 1   12.52171 55.70606      4        2         2       FALSE
## 2   12.52309 55.70762      6        3         3       FALSE
## 3   12.58734 55.68087      3        1         2       FALSE
## 4   12.58839 55.68207      2        1         1       FALSE
## 5   12.59058 55.75007     10        5         5       FALSE
## 6   12.50931 55.65054      4        2         2       FALSE
## 7   12.51405 55.65039      6        3         3       FALSE
## 8   12.54080 55.65389      8        4         4       FALSE
## 9   12.43157 55.62945      4        2         2       FALSE
## 10  12.44046 55.62304      3        2         1       FALSE

## # 516 more rows

## # Use `print(n=...)` to see more rows

and

graph5$edges

## Edges of the metric graph
## 
## Longitude and Latitude coordinates: TRUE 
## Coordinate reference system: +proj=longlat +datum=WGS84 +no_defs 
## 
## Summary: 
## 
## Edge 1 (first and last coordinates): 
##  Longitude Latitude
##   12.52171 55.70606
##   12.52309 55.70762
## Total number of coordinates: 9 
## Edge length: 0.201448 km 
## Weight: 1 
## 
## Edge 2 (first and last coordinates): 
##  Longitude Latitude
##   12.58734 55.68087
##   12.58839 55.68207
## Total number of coordinates: 4 
## Edge length: 0.1494666 km 
## Weight: 1 
## 
## Edge 3 (first and last coordinates): 
##  Longitude Latitude
##   12.59058 55.75007
##   12.59058 55.75007
## Total number of coordinates: 4 
## Edge length: 0.1031872 km 
## Weight: 1 
## 
## Edge 4 (first and last coordinates): 
##  Longitude Latitude
##   12.50931 55.65054
##   12.51405 55.65039
## Total number of coordinates: 5 
## Edge length: 0.2991002 km 
## Weight: 1

## # 1000 more edges

## # Use `print(n=...)` to see more edges

It is also relevant to obtain a summary of this graph:

summary(graph5)

## A metric graph object with:
## 
## Vertices:
##   Total: 526 
##   Degree 1: 18;  Degree 2: 199;  Degree 3: 49;  Degree 4: 150;  Degree 5: 14; 
##   Degree 6: 44;  Degree 7: 9;  Degree 8: 15;  Degree 9: 5;  Degree 10: 13; 
##   Degree 11: 1;  Degree 12: 4;  Degree 14: 1;  Degree 16: 1;  Degree 18: 2; 
##   Degree 28: 1; 
##   With incompatible directions:  1 
## 
## Edges: 
##   Total: 1004 
##   Lengths: 
##       Min: 0.1002405  ; Max: 11.20537  ; Total: 424.5386 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  216 
## 
## Graph units: 
##   Vertices unit:  degrees  ; Lengths unit:  km 
## 
## Longitude and Latitude coordinates:  TRUE
##   Which spatial package:  sp 
##   CRS:  +proj=longlat +datum=WGS84 +no_defs
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: TRUE
##   Has multiple edges: TRUE
##   Is a tree: FALSE
##   Distance consistent: FALSE
##   Has Euclidean edges: FALSE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: The graph has no mesh! 
## 
## Data: The graph has no data!
## 
## Tolerances: 
##   vertex-vertex:  0.1 
##   vertex-edge:  1e-07 
##   edge-edge:  0

We have that this graph does not have Euclidean edges. Also observe that we have the units for the vertices and lengths, and also the coordinate reference system being used by this object.

Adding data to the graph

Given that we have constructed the metric graph, we can now add data to it. For further details on data manipulation on metric graphs, see Data manipulation on metric graphs. As an example, let us consider the first graph again and suppose that we have observations at a distance 0.5 from the start of each edge. One way of specifying this is as follows

obs.loc <- cbind(1:4, rep(0.5, 4))
obs <- c(1,2,3,4)
df_graph <- data.frame(y = obs, edge_number = obs.loc[,1], 
                        distance_on_edge = obs.loc[,2])
graph$add_observations(data = df_graph)

## Adding observations...

graph$plot(data = "y", data_size = 2)

In certain situations, it might be easier to specify the relative distances on the edges, so that 0 represents the start and 1 the end of the edge (instead of the edge length). To do so, we can simply specify normalized = TRUE when adding the observations. For example, let us add one more observation at the midpoint of the fourth edge:

obs.loc <- matrix(c(4, 0.5),1,2)
obs <- c(5)
df_new <- data.frame(y=obs, edge_number = obs.loc[,1], 
                          distance_on_edge = obs.loc[,2])
graph$add_observations(data=df_new, normalized = TRUE)

## Adding observations...

graph$plot(data = "y")

An alternative method is to specify the observations as spatial points objects, where the locations are given in Euclidean coordinates. In this case the observations are added to the closes location on the graph:

obs.loc <- rbind(c(0.7, 0), c(0, 0.2))
obs <- c(6,7)
points <- SpatialPointsDataFrame(coords = obs.loc,
                                 data = data.frame(y = obs))
graph$add_observations(points)

## Adding observations...

## Converting data to PtE

## [1] y        .coord_x .coord_y
## <0 rows> (or 0-length row.names)

graph$plot(data = "y")

If we want to replace the data in the object, we can use clear_observations() to remove all current data:

graph$clear_observations()

The metric_graph object can hold multiple variables of data, and one can also specify a group argument that it useful for specifying that data is grouped, which for example is the case with data observed at different time points. As an example, let us add two variables observed at two separate time points:

obs.loc <- cbind(c(1:4, 1:4), c(rep(0.5, 4), rep(0.7, 4)))
df_rep <- data.frame(y = c(1, 2, NA, 3, 4, 6, 5, 7),
                     x = c(NA, 8, 9, 10, 11, 12, 13, NA),
                     edge_number = obs.loc[,1], 
                     distance_on_edge = obs.loc[,2],
                     time = c(rep(1, 4), rep(2, 4)))
graph$add_observations(data = df_rep, group = "time")

## Adding observations...

If NA is given in some variable, this indicates that this specific variable is not measured at that location and group. We can now plot the individual variables by specifying their names in the plot function together with the group number that we want to see. By default the first group is shown.

graph$plot(data = "y", group = 2)

In some cases, we might want to add the observation locations as vertices in the graph. This can be done as follows:

graph$observation_to_vertex()
graph$plot(data = "x", group = 1)

One can note that the command adds all observation locations, from all groups as vertices.

Finally, let us obtain a summary of the graph containing data:

summary(graph)

## A metric graph object with:
## 
## Vertices:
##   Total: 12 
##   Degree 1: 1;  Degree 2: 10;  Degree 3: 1; 
##   With incompatible directions:  2 
## 
## Edges: 
##   Total: 12 
##   Lengths: 
##       Min: 0.2  ; Max: 0.8707291  ; Total: 4.570729 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  None  ; Lengths unit:  None 
## 
## Longitude and Latitude coordinates:  FALSE
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: FALSE
##   Is a tree: FALSE
##   Distance consistent: TRUE
##   Has Euclidean edges: TRUE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: The graph has no mesh! 
## 
## Data: 
##   Columns:  y x time 
##   Groups:  time 
## 
## Tolerances: 
##   vertex-vertex:  0.001 
##   vertex-edge:  0.001 
##   edge-edge:  0

Working with functions on metric graphs

When working with data on metric graphs, one often wants to display functions on the graph. The best way to visualize functions on the graph is to evaluate them on a fine mesh over the graph and then use plot_function. To illustrate this procedure, let us consider the following graph:

V <- rbind(c(0, 0), c(1, 0), c(1, 1), c(0, 1), c(-1, 1), c(-1, 0), c(0, -1))
E <- rbind(c(1, 2), c(2, 3), c(3, 4), c(4, 5),
           c(5, 6), c(6, 1), c(4, 1),c(1, 7))
graph <- metric_graph$new(V = V, E = E)
graph$build_mesh(h = 0.5)
graph$plot(mesh=TRUE)

In the command build_mesh, the argument h decides the largest spacing between nodes in the mesh. As can be seen in the plot, the mesh is very coarse, so let’s reduce the value of h and rebuild the mesh:

  graph$build_mesh(h = 0.01)

Suppose now that we want to display the function \(f(x, y) = x^2 - y^2\) on this graph. We then first evaluate it on the vertices of the mesh and then use the function plot_function to display it:

x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
f <- x^2 - y^2
graph$plot_function(X = f)

Alternatively, we can set plotly = TRUE in the plot command to get a 3D visualization of the function:

graph$plot_function(X = f, plotly = TRUE)

When the first argument of plot_function is a vector, the function assumes that the values in the vector are the values of the function evaluated at the vertices of the mesh. As an alternative, one can also provide a new dataset. To illustrate this, let us first construct a set of locations that are evenly spaced over each edge. Then, we will convert the data to Euclidean coordinates so that we can evaluate the function above. Then, process the data and finally plot the result:

n.e <- 30
PtE <- cbind(rep(1:graph$nE, each = n.e), 
             rep(seq(from = 0, to = 1, length.out = n.e), graph$nE))        
XY <- graph$coordinates(PtE)
f <- XY[, 1]^2 - XY[, 2]^2
data_f <- data.frame(f = f, "edge_number" = PtE[,1], 
            "distance_on_edge" = PtE[,2])
data_f <- graph$process_data(data = data_f, normalized=TRUE)            
graph$plot_function(data = "f", newdata = data_f, plotly = TRUE)

Alternatively, we can add the data to the graph, and directly plot by using the data argument. To illustration purposes, we will build this next plot as a 2d plot:

graph$add_observations(data = data_f, clear_obs = TRUE, normalized = TRUE)

## Adding observations...

graph$plot_function(data = "f")

Improving the plot obtained by `plot_function`

If we compute the relative positions of the edge coordinates, we can improve the quality of the plot obtained by using plot_function. Observe that depending on the size of the graph, this might be costly, since we will need to do this conversion.

We begin by recreating the first graph we used in this vignette:

edge1 <- rbind(c(0,0),c(1,0))
edge2 <- rbind(c(0,0),c(0,1))
edge3 <- rbind(c(0,1),c(-1,1))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
edge4 <- cbind(sin(theta),1+ cos(theta))
edges = list(edge1, edge2, edge3, edge4)

graph <- metric_graph$new(edges = edges)

Now, let us compute the relative positions of the coordinates of the edge by using the compute_PtE_edges() method:

graph$compute_PtE_edges()

Let us create a coarse mesh, so the improvement can be easily seen.

graph$build_mesh(h=0.8)
graph$plot(mesh=TRUE)

Let us compute the function the same function as before on this mesh:

XY <- graph$mesh$V
f <- XY[, 1]^2 - XY[, 2]^2

We can directly plot from the mesh values. Let us start with the default, non-improved, plot:

graph$plot_function(f)

We can, then, plot the improved version by setting improve_plot to TRUE:

graph$plot_function(f, improve_plot = TRUE)

Alternatively, we can supply the data through the newdata argument. To such an end, we will need to supply the data.frame. So let us build it:

df_f <- data.frame(edge_number = graph$mesh$VtE[,1],
                    distance_on_edge = graph$mesh$VtE[,2],
                    f = f)

Let us now process the data.frame to the graph’s internal data format using the process_data() method:

df_f <- graph$process_data(data = df_f, normalized=TRUE)

We will now plot this function. First, let us do the default plot from plot_function:

graph$plot_function(data = "f", newdata= df_f)

Let us now, create the same plot with improve_plot=TRUE:

graph$plot_function(data = "f", newdata= df_f, improve_plot=TRUE)

The reason it refined a lot on the circular part is that it interpolates the values of the function at the edge coordinates, and the circular part has the most coordinates. All the other edges only have two coordinates, the start and end.

Let us now obtain a 3d plot. First, of the default version:

graph$plot_function(data = "f", newdata= df_f, plotly=TRUE)

Now, of the improved version:

graph$plot_function(data = "f", newdata= df_f, plotly=TRUE,
                    improve_plot = TRUE)

Note about continuity

However, it is important to observe that plot_function assumes continuity of the function by default. Let us see an example of a discontinuous function, and observe the effect of plot_function. In such a case, improve_plot does not affect the action on the discontinuity points.

For example, let us consider the function that takes the edge number (normalized). Let us also refine the mesh a bit, so it is easier to observe what is happening.

graph$build_mesh(h=0.2)
f <- graph$mesh$VtE[,1]/4

Let us proceed with the creation of the data.frame and processing of the data:

df_f <- data.frame(edge_number = graph$mesh$VtE[,1],
                    distance_on_edge = graph$mesh$VtE[,2],
                    f = f)
df_f <- graph$process_data(data = df_f, normalized=TRUE)

We will now plot this function with the default plot from plot_function:

graph$plot_function(data = "f", newdata= df_f)

Let us now, create the same plot with improve_plot=TRUE:

graph$plot_function(data = "f", newdata= df_f, improve_plot=TRUE)

Let us now obtain a 3d plot. First, of the default version:

graph$plot_function(data = "f", newdata= df_f, plotly=TRUE)

Now, of the improved version:

graph$plot_function(data = "f", newdata= df_f, plotly=TRUE,
                    improve_plot = TRUE)

Observe that for both versions we have the same behavior around the vertices (which are the discontinuity points).

To work with discontinuous functions, the simplest option is to create a mesh that does not assume continuity:

graph$build_mesh(h=0.8, continuous = FALSE)

In this mesh, each vertex is split into one vertex per edge connecting to the vertex, which allows for discontinuous functions.

f <- graph$mesh$PtE[,1]/4
graph$plot_function(f)

Similarly to the case for continuous meshes, we have the improve_plot option:

graph$plot_function(f, improve_plot = TRUE)

Now, the 3d plot:

graph$plot_function(f, plotly=TRUE, improve_plot = TRUE)

Alternatively, we can similarly plot by setting the data.frame and passing the newdata argument:

df_f <- data.frame(edge_number = graph$mesh$VtE[,1],
                    distance_on_edge = graph$mesh$VtE[,2],
                    f = f)
df_f <- graph$process_data(data = df_f, normalized=TRUE)

Let us now build the plots. In this case, we need to set the continuous argument to FALSE. First the 2d plot:

graph$plot_function(data = "f", newdata= df_f, 
                      improve_plot = TRUE,  continuous = FALSE)

Now, the 3d plot:

graph$plot_function(data = "f", newdata= df_f, plotly=TRUE,
                    improve_plot = TRUE, continuous = FALSE)

David Bolin, Alexandre B. Simas, and Jonas Wallin

Created: 2022-11-23. Last modified: 2024-05-08.

Introduction

Constructing metric graphs

Basic constructions and properties

Tolerances for merging vertices and edges

Characteristics of the graph

Summaries of metric graphs

Vertices and edges

Understanding coordinates on graphs

Methods for working with real data

Adding data to the graph

Working with functions on metric graphs

Improving the plot obtained by `plot_function`

Note about continuity

Working with metric graphs

David Bolin, Alexandre B. Simas, and Jonas Wallin

Created: 2022-11-23. Last modified: 2024-05-08.

Introduction

Constructing metric graphs

Basic constructions and properties

Tolerances for merging vertices and edges

Characteristics of the graph

Summaries of metric graphs

Vertices and edges

Understanding coordinates on graphs

Methods for working with real data

Adding data to the graph

Working with functions on metric graphs

Improving the plot obtained by plot_function

Note about continuity

Improving the plot obtained by `plot_function`