Buidling Networks from Spatial Data

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class: center middle main-title section-title-4

# Building Networks from Spatial Data

.class-info[

**Session 9**

.light[HES597: Introduction to Spatial Data in R<br>
Boise State University Human-Environment Systems<br>
Fall 2021]

]

---

# Outline

- Introducing networks

- Applications of networks

- Building networks (in `R`)

- (Some) Network properties of interest

---

# Defining Networks

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- Network theory is a subset of __graph theory__

- _Graphs_: mathematical structures used to model the relations between objects
  
  - Comprised of _nodes_ (vertices, points) and _edges_ (links)
  
- In network analysis, our _nodes_ and/or _edges_ have __names__ (or attributes)
]

.pull-right[
<img src="05-slides_files/figure-html/simpnet-1.png" width="360" style="display: block; margin: auto;" />
]
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## What sorts of things might comprise nodes and edges in your work?
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# Types of Networks
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- Undirected

<img src="05-slides_files/figure-html/undigraph-1.png" width="504" style="display: block; margin: auto;" />
]
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- Directed
<img src="05-slides_files/figure-html/digraph-1.png" width="504" style="display: block; margin: auto;" />
]
---
# Types of Networks
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- Uweighted

<img src="05-slides_files/figure-html/unwtgraph-1.png" width="504" style="display: block; margin: auto;" />
]

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- Weighted

]
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# Types of Graphs

- Many additional common types of graphs (beyond the scope of this course)

- Connected graphs, bipartite graphs, acyclic graphs are common in our field(s)

- Check out Vicken Hillis' HES 597: Social Network Analysis course for more
---
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# Common of Applications of Networks

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# Transportation and infrastructure

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<figure>
  <img src="img/05/airports-map-small.png" alt="ZZZ" title="From [Martin Grandjean](http://www.martingrandjean.ch/connected-world-air-traffic-network/)" width="100%">
</figure>
.caption[
From [Martin Grandjean](http://www.martingrandjean.ch/connected-world-air-traffic-network/)]
]

.pull-right[

<figure>
  <img src="img/05/powernetwork.png" alt="ZZZ" title="From Jasny et al. 2015" width="100%">
</figure>
.caption[
Paul Cuffe, [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0), via Wikimedia Commons
]
]

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# Social Sciences

.pull-left[

<figure>
  <img src="img/05/socialnetwork.png" alt="ZZZ" title="From Buyalskaya et al. 2021" width="100%">
</figure>
.captiona[
From Buyalskaya et al. 2021
]

]

.pull-right[

<figure>
  <img src="img/05/ClimateEgo_jasny.png" alt="ZZZ" title="From Jasny et al. 2015" width="100%">
</figure>
.caption[
From Jasny et al. 2015
]
]

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# Ecology

.pull-left[
<figure>
  <img src="img/05/burgessshale.png" alt="ZZZ" title="From Jasny et al. 2015" width="100%">
</figure>
.caption[
Jennifer Dunne, Santa Fe Institute
]
]

.pull-right[
<figure>
  <img src="img/05/pa_network.png" alt="ZZZ" title="From Jasny et al. 2015" width="100%">
</figure>
.caption[
Adapted from Palomo et. al 2014
]
]
---
# Constructing networks

## Adjacency matrices

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- unweighted: 1 if 'adjacent' 0 if not
- undirected: matrix is symmetric

```
##   A B C D E F
## A 0 0 1 0 1 0
## B 0 0 0 0 0 0
## C 1 0 0 0 1 0
## D 0 0 0 0 1 0
## E 1 0 1 1 0 1
## F 0 0 0 0 1 0
```
]
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- weighted: value reflects the strength of connections
- directed: not all connections are reciprocal

```
##      A    B    C    D    E    F
## A 0.00 0.00 0.00 0.67 0.95 0.87
## B 0.00 0.68 0.84 0.00 0.00 0.53
## C 0.66 0.00 0.83 0.00 0.91 0.58
## D 0.77 0.00 0.00 0.85 0.78 0.00
## E 0.00 0.51 0.00 0.00 0.91 0.72
## F 0.85 0.00 0.60 0.00 0.82 0.00
```
]
---
# We can use the `igraph` package to convert these to graphs
.pull-left[

```r
plot(igraph::graph_from_adjacency_matrix(mtx, mode='undirected'))
```

<img src="05-slides_files/figure-html/adjnetwork-1.png" width="360" style="display: block; margin: auto;" />
]
.pull-right[

```r
g <- igraph::graph_from_adjacency_matrix(adjm, mode='directed', weighted = TRUE)
  
plot(g, edge.label=E(g)$weight)
```

<img src="05-slides_files/figure-html/wtdnetwork-1.png" width="360" style="display: block; margin: auto;" />
]
---
# Constructing Networks

## Edge lists

- Adjacency matrices become LARGE for relatively simple networks

- For many networks, connections are limited (i.e., matrices are sparse)

- Often easier to provide a list of connections

- Can also provide weights in a dataframe (see the transportation example)
---

# Characterizing networks

- Degree distribution: the number of edges between each node and other nodes
- Clustering coefficient: provides an indication of local spatial structure
- Motifs: repeating subgraph shapes in a netwrok
- Connectivity: minimum # of elements to remove to generate isolated subraphs
- Centrality: characterize the network position of nodes or edges
- LOTS of others

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# Spatial Networks
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# Spatial Networks

- Space is often _implicit_ in network analyses

- Probability of finding a link decreases with distance

- Different edge weights may alter these probabilities

- Generally planar

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# Generating Spatial Networks in R

## Defining adjacency
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```r
p = st_sfc(st_point(c(0,2)), st_point(c(0,1)), st_point(c(0,4)), st_point(c(1,2)), st_point(c(3,1)), st_point(c(2,4)))
round(st_distance(p), digits = 2)
```

```
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.00 1.00 2.00 1.00 3.16 2.83
## [2,] 1.00 0.00 3.00 1.41 3.00 3.61
## [3,] 2.00 3.00 0.00 2.24 4.24 2.00
## [4,] 1.00 1.41 2.24 0.00 2.24 2.24
## [5,] 3.16 3.00 4.24 2.24 0.00 3.16
## [6,] 2.83 3.61 2.00 2.24 3.16 0.00
```
]

.pull-right[
<img src="05-slides_files/figure-html/pdist-1.png" width="360" style="display: block; margin: auto;" />
]
---
# Generating Spatial Networks in R

## Defining adjacency
 - With a distance threshold
 
.pull-left[

```r
adjm <-  ifelse(st_distance(p) <1.5, 1,0)  
adjm
```

```
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    1    0    1    0    0
## [2,]    1    1    0    1    0    0
## [3,]    0    0    1    0    0    0
## [4,]    1    1    0    1    0    0
## [5,]    0    0    0    0    1    0
## [6,]    0    0    0    0    0    1
```
]
.pull-right[
<img src="05-slides_files/figure-html/netplot-1.png" width="360" style="display: block; margin: auto;" />
]
---

# Generating Spatial Networks in R
- for rasters
- `directions` define window for connected cells
- important for connectivity analyses

```r
r <- rast(nrows=10, ncols=10)
values(r) <- round(runif(ncell(r)) * 5)
a <- adjacent(r, 1:ncell(r), directions="queen") 
```