Functions on metric graphs

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# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = FALSE,    
  # Disable warnings printed by R code chunks
  warning = FALSE,    
  # Show R code within code chunks in output
  echo = TRUE,        
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  eval = TRUE,       
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  # Align figures in the center of the output
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  retina = 2,
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  error = TRUE,
  # Do not collapse code and output into a single block
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)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}

Import libraries

library(MetricGraph)
library(dplyr)
library(plotly)

library(here)
library(rmarkdown)
library(grateful) # to cite

Building the graph object

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)

Geodesic distance on a metric graph

Figure 1 below shows an illustration of a compact and connected graph \(\Gamma = (\mathcal{V},\mathcal{E})\), where \(\mathcal{E}= \{e_1,\dots,e_8\}\) (black lines) and \(\mathcal{V} = \{v_1,\dots,v_7\}\) (blue points).

---

Press the Show button below to reveal the code.

---

df = data.frame(x = c(0.5,0), y = c(-1,-1), z = c(0,0))
df2 = data.frame(x = c(0,0), y = c(-1,0), z = c(0,0))
df3 = data.frame(x = c(0,-0.5), y = c(0,0), z = c(0,0))

graph$build_mesh(h = 1/40)
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
f <- x^2 - y^2
graph$plot(plotly = TRUE, vertex_size = 5, vertex_color = "blue",
           edge_color = "black", edge_width = 2) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\text{Geodesic distance on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"), 
    showlegend = FALSE,
                    font = list(family = "Palatino"),
                    scene = list(
                      aspectratio = list(x = 1.6, y = 1.6, z = 1),
                      annotations = list(
                        list(
                          x = -1, y = 0, z = 0,
                          text = TeX("v_1"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 0, z = 0,
                          text = TeX("v_2"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = -1, z = 0,
                          text = TeX("v_3"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = -1, z = 0,
                          text = TeX("v_4"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 0, z = 0,
                          text = TeX("v_5"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 1, z = 0,
                          text = TeX("v_6"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 1, z = 0,
                          text = TeX("v_7"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = -1, z = 0,
                          text = TeX("s_1"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = -0.5, y = 0, z = 0,
                          text = TeX("s_2"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 20),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = -0.5, y = 0, z = 0,
                          text = TeX("e_1"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = -0.5, z = 0,
                          text = TeX("e_2"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = -1, z = 0,
                          text = TeX("e_3"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = -0.5, z = 0,
                          text = TeX("e_4"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = 0, z = 0,
                          text = TeX("e_5"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 0.5, z = 0,
                          text = TeX("e_6"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = 1, z = 0,
                          text = TeX("e_7"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 0.5, z = 0,
                          text = TeX("e_8"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 20),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
                        ))) %>% 
  add_trace(x = 0.5, y = -1, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 5, color = "red", symbol = 104)) %>% 
  add_trace(x = -0.5, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 5, color = "red", symbol = 104)) %>% 
  add_trace(data = df, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = df2, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = df3, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104))

Figure 1: Illustration of a compact and connected graph \(\Gamma = \left(\mathcal{V},\mathcal{E}\right)\), where \(\mathcal{E}= \left\{e_1,\dots,e_8\right\}\) (black) and \(\mathcal{V}= \left\{v_1,\dots,v_7\right\}\) (blue). Notice that \(\ell_{e_i} = 1\) for \(i = 1,\dots,8\). Note also that, for example, \(\mathcal{E}_{v_2} = \left\{e_1,e_2,e_5,e_8\right\}\) and therefore \(d_{v_2} = 4.\) Given two points \(s_1 = (e_3, 0.5)\) and \(s_2 = (e_1, 0.5)\), we have that \(d(s_1,s_2) = 2\) (red). Following the convention mentioned in the text, we may write, for example, \(e_8 = (v_2,v_7) \cong [0,\ell_{e_8}]\).

Function on a metric graph

Figure 2 below provides an illustration of a function defined on a graph.

---

Press the Show button below to reveal the code.

---

graph$build_mesh(h = 1/40)
x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
f <- x^2 - y^2

df = data.frame(x = c(0.5,0), y = c(-1,-1), z = c(0,0))
df2 = data.frame(x = c(0,0), y = c(-1,0), z = c(0,0))
df3 = data.frame(x = c(0,-0.5), y = c(0,0), z = c(0,0))


graph$plot_function(f,plotly = TRUE, vertex_size = 5, vertex_color = "blue", line_color = "blue",
           edge_color = "black", edge_width = 2) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\text{Function on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"), 
    showlegend = FALSE,
                    font = list(family = "Palatino"),
                    scene = list(
                      aspectratio = list(x = 1.6, y = 1.6, z = 1),
                      annotations = list(
                        list(
                          x = -1, y = 0, z = 0,
                          text = TeX("v_1"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 0, z = 0,
                          text = TeX("v_2"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = -1, z = 0,
                          text = TeX("v_3"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = -1, z = 0,
                          text = TeX("v_4"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 0, z = 0,
                          text = TeX("v_5"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 1, z = 0,
                          text = TeX("v_6"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 1, z = 0,
                          text = TeX("v_7"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = -1, z = 0,
                          text = TeX("s_1"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = -0.5, y = 0, z = 0,
                          text = TeX("s_2"),
                          textangle = 0, ax = 0, ay = -25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = -0.5, y = 0, z = 0,
                          text = TeX("e_1"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = -0.5, z = 0,
                          text = TeX("e_2"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = -1, z = 0,
                          text = TeX("e_3"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = -0.5, z = 0,
                          text = TeX("e_4"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = 0, z = 0,
                          text = TeX("e_5"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 0.5, z = 0,
                          text = TeX("e_6"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.5, y = 1, z = 0,
                          text = TeX("e_7"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 0.5, z = 0,
                          text = TeX("e_8"),
                          textangle = 0, ax = 0, ay = 10,
                          font = list(color = "black", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = -1, z = 1,
                          text = TeX("f=\\gamma\\big|_\\Gamma"),
                          textangle = 0, ax = 60, ay = 0,
                          font = list(color = "blue", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
                        ))) %>% 
  add_trace(x = 0.5, y = -1, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 5, color = "red", symbol = 104)) %>% 
  add_trace(x = -0.5, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 5, color = "red", symbol = 104)) %>% 
  add_trace(data = df, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = df2, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = df3, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104))

Figure 2: Illustration of a function \(f\) (in blue) defined on the graph \(\Gamma= (\mathcal{V},\mathcal{E})\), where \(\mathcal{E}= \left\{e_1,\dots,e_8\right\}\) and \(\mathcal{V}= \left\{v_1,\dots,v_7\right\}\). Function \(f:\Gamma\longrightarrow\mathbb{R}\) is the restriction of the surface \(\gamma:\mathbb{R}^2\longrightarrow\mathbb{R}\) to \(\Gamma\), where \(\gamma(x,y) = x^2-y^2\). That is, \(f= \gamma\big|_\Gamma\).

References

cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.4.1 (R Core Team 2024a) and the following R packages: cowplot v. 1.1.3 (Wilke 2024), ggmap v. 4.0.0.900 (Kahle and Wickham 2013), ggpubr v. 0.6.0 (Kassambara 2023), ggtext v. 0.1.2 (Wilke and Wiernik 2022), grid v. 4.4.1 (R Core Team 2024b), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 24.12.11 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.12.0.9002 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.48 (Xie 2014, 2015, 2024), latex2exp v. 0.9.6 (Meschiari 2022), Matrix v. 1.6.5 (Bates, Maechler, and Jagan 2024), MetricGraph v. 1.4.0.9000 (Bolin, Simas, and Wallin 2023b, 2023a, 2023c, 2024; Bolin et al. 2024), OpenStreetMap v. 0.4.0 (Fellows and JMapViewer library by Jan Peter Stotz 2023), osmdata v. 0.2.5 (Mark Padgham et al. 2017), patchwork v. 1.2.0 (Pedersen 2024), plotly v. 4.10.4 (Sievert 2020), plotrix v. 3.8.4 (J 2006), reshape2 v. 1.4.4 (Wickham 2007), rmarkdown v. 2.28 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.4.0.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.3.0 (Wickham, Pedersen, and Seidel 2023), sf v. 1.0.19 (E. Pebesma 2018; E. Pebesma and Bivand 2023), sp v. 2.1.4 (E. J. Pebesma and Bivand 2005; Bivand, Pebesma, and Gomez-Rubio 2013), tidyverse v. 2.0.0 (Wickham et al. 2019), viridis v. 0.6.4 (Garnier et al. 2023), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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