Movement and Networks II

HES 505 Fall 2023: Session 26

Matt Williamson

Describing networks for analysis

Common measures

  • Graph-level: density, diameter, distance

  • Component-level: density, distribution

  • Node-level: centrality, degree-distribution

Common questions

  • What are the shortest paths across the network?

  • Where are the most important locations for maintaining the network?

  • How does the loss of a node alter the subsequent configuration of the network?

  • How do we translate typical movement paths into network structures?

An example

Sage Grouse in the West

Reading layer `GRSG_2015_USFWS_StatusReview_PACs' from data source 
  `/Users/mattwilliamson/Websites/isdrfall23/slides/data/GRSG_2015_USFWS_StatusReview_PACs.shp' 
  using driver `ESRI Shapefile'
Simple feature collection with 301 features and 9 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: -1361708 ymin: 381165.6 xmax: 147308.4 ymax: 1661769
Projected CRS: GRSG Range Wide

The Data

head(sg.pacs[,1:9])
Simple feature collection with 6 features and 9 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: -1183048 ymin: 446911.4 xmax: -1096618 ymax: 472038.5
Projected CRS: GRSG Range Wide
  OBJECTID UniqueID Population MgmtZone          ID_Name     Acres Shape_Leng
1        1      401   Bi-State      MZ3 401-Bi-State-MZ3  444.2940   5691.176
2        2      362   Bi-State      MZ3 362-Bi-State-MZ3 3553.0707  25207.100
3        3      400   Bi-State      MZ3 400-Bi-State-MZ3  494.7130   5958.419
4        4      399   Bi-State      MZ3 399-Bi-State-MZ3  995.5216  10911.550
5        5      398   Bi-State      MZ3 398-Bi-State-MZ3  395.0896   7904.861
6        6      361   Bi-State      MZ3 361-Bi-State-MZ3 8252.1309  60989.117
  Shape_Area Bi_State                       geometry
1    1797994      Yes POLYGON ((-1181624 457935.6...
2   14378767      Yes POLYGON ((-1102151 456210.8...
3    2002033      Yes POLYGON ((-1127306 459256.1...
4    4028733      Yes POLYGON ((-1122955 461942.7...
5    1598871      Yes POLYGON ((-1111653 471567.5...
6   33395189      Yes POLYGON ((-1100113 470204.4...

Preparing the adjacency matrix

sg.pacs.cent <- sg.pacs %>% 
  filter(., MgmtZone == "MZ3") %>% 
  st_centroid(sg.pacs, of_largest_polygon = TRUE)

Preparing the Adjacency Matrix

sg.pacs.dist <- st_distance(sg.pacs.cent) 

threshold <- units::as_units(50, "km") 

adj.mtx <- sg.pacs.dist < threshold 
      1     2     3     4     5
1  TRUE FALSE FALSE FALSE FALSE
2 FALSE  TRUE  TRUE  TRUE  TRUE
3 FALSE  TRUE  TRUE  TRUE  TRUE
4 FALSE  TRUE  TRUE  TRUE  TRUE
5 FALSE  TRUE  TRUE  TRUE  TRUE

Preparing the Adjacency Matrix

adj.mtx <- adj.mtx *1 
diag(adj.mtx) <- 0  

dimnames(adj.mtx) <- list(
  sg.pacs.cent$UniqueID, 
  sg.pacs.cent$UniqueID) 
    401 362 400 399 398
401   0   0   0   0   0
362   0   0   1   1   1
400   0   1   0   1   1
399   0   1   1   0   1
398   0   1   1   1   0

Graphing your network

sg.graph <- igraph::graph_from_adjacency_matrix(adj.mtx) %>% 
  as_tbl_graph(directed - FALSE,
               node_key = "UniqueID") %>% 
  left_join(., sg.pacs.cent, 
            by = c("name" = "UniqueID"))

Evaluating Network Metrics

Common Metrics

  • Betweenness Centrality: the sum of all of the shortest paths that flow through that node.

  • Degree reflects the number of connections a node has to other nodes.

  • Component size: the number of nodes in a group that is connected to each other, but disconnected from the rest of the graph

  • Degree Distribution: Describes the general connectedness of all the nodes in a network; vulnerability

Estimating metrics

sg.graph.mets <- sg.graph %>% 
  activate(nodes) %>% 
  mutate(., bet.centrality = centrality_betweenness(),
            deg = degree(.))

Estimating metrics: Betweeness

Estimating metrics: Degree

Estimating metrics: Component Size

comps <- components(sg.graph)
comps$csize
 [1] 61  1  1  1  1  1  4  1  1  1  1
comps$membership
401 362 400 399 398 361 360 397 359 396 395 358 353 394 393 392 357 354 355 352 
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1 
351 350 391 390 389 388 356 387 386 385 349 384 383 382 381 380 379 378 377 376 
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1 
375 374 239 347 345 373 372 348 371 346 370 369 368 344 367 366 365 364 343 242 
  1   1   2   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   3 
342 363 341 238 237 241 234 232 235 236 233 244 243 240 
  1   1   1   4   5   6   7   8   7   7   7   9  10  11

Estimating metrics: Degree distribution

[1] 26.10811
[1] 16.13943

Experimenting

Setting up the data

thresholds <- as_units(c(1, 10,25, 50, 100, 150, 250, 500), "km")
thresh.df <- data.frame(thresh.dist = rep(NA, length(thresholds)),
                        mean.between = rep(NA, length(thresholds)),
                        ncomps = rep(NA, length(thresholds)),
                        mean.deg = rep(NA, length(thresholds)))
sg.pacs.cent <- sg.pacs %>% 
  st_centroid(sg.pacs, of_largest_polygon = TRUE) 
sg.pacs.dist <- st_distance(sg.pacs.cent)

Iterating

for (i in 1:length(thresholds)){
  adj.mtx <- sg.pacs.dist < thresholds[i]
  adj.mtx <- adj.mtx *1
  diag(adj.mtx) <- 0
  dimnames(adj.mtx) <- list(sg.pacs.cent$UniqueID, sg.pacs.cent$UniqueID)
  
  sg.graph <- igraph::graph_from_adjacency_matrix(adj.mtx) %>% 
  as_tbl_graph(directed - FALSE,
               node_key = "UniqueID") %>% 
  left_join(., sg.pacs.cent, 
            by = c("name" = "UniqueID"))
  thresh.df$thresh.dist[i] <- thresholds[i]
  thresh.df$mean.between[i] <- mean(betweenness(sg.graph, directed = FALSE))
  thresh.df$ncomps[i] <- length(components(sg.graph)$csize)
  thresh.df$mean.deg[i] <- mean(degree(sg.graph))
}

Collecting Data

thresh.long <- thresh.df %>% 
  tidyr::pivot_longer(!thresh.dist, 
                      names_to = "metric", 
                      values_to = "estimate")

ggplot(data = thresh.long, aes(x= thresh.dist, y = estimate)) +
  geom_line()+
  facet_wrap(vars(metric)) +
  ggtitle("Whole Network")