class: middle background-image: url("background.png") background-position: right background-size: auto # .orange[Network Models] ### .fancy[The *One True* Representation] --- class: center, middle background-image: url("https://live.staticflickr.com/65535/51198451539_f09703e3d8_c_d.jpg") background-position: center --- # Networks as a Metaphor Much of what we do in Landscape (and Popualation) Genetics can be considered as a set of network problems. Both genetic and ecological distances are commonly represented as distance matrices, which mathematically is also an adjacency matrix. - Between Sites connectivity `\(\to\)` gene flow. - Between Site separation `\(\to\)` spatial/ecological separation. - Source/Sink dynamics --- class: center, middle  --- # Graph Components - Nodes .pull-left[ Definable sample locaions defined by sites, ponds, habitats, etc. - Nodes, Vertices, Points - .redinline[**AT**] site data - Spatially Relevant ] .pull-right[  ] --- # Graph Components - Edges .pull-left[ Connections between nodes. - Edges, arcs, links - .orangeinline[**BETWEEN**] site data - Spatially, ecologically, or topographically relevant. ] .pull-right[  ] --- # Graph Types  --- class: center, middle  --- # Graph Jargon  --- # Graph Visualization A `layout` is a configuration of nodes in either 2- or 3-dimensional space. - Spatial - Based upon internal structure. --- class: center, middle  ## This does exist --- # Basic Network Analysis in R .pull-left[  ] .pull-right[ ```r library( igraph ) A <- matrix( 0, nrow=5,ncol=5) A[1,2] <- A[2,1] <- A[1,3] <- A[2,3] <- 1 A[3,4] <- A[4,5] <- 1 rownames(A) <- colnames(A) <- c("X","Y","Z","W","U") A ``` ``` ## X Y Z W U ## X 0 1 1 0 0 ## Y 1 0 1 0 0 ## Z 0 0 0 1 0 ## W 0 0 0 0 1 ## U 0 0 0 0 0 ``` ] --- # Basic Network Analysis in R .pull-left[  ] .pull-right[ ```r G <- graph.adjacency( A, mode="directed" ) G ``` ``` ## IGRAPH 9e448f9 DN-- 5 6 -- ## + attr: name (v/c) ## + edges from 9e448f9 (vertex names): ## [1] X->Y X->Z Y->X Y->Z Z->W W->U ``` ```r vertex_attr( G ) ``` ``` ## $name ## [1] "X" "Y" "Z" "W" "U" ``` ```r edge_attr( G ) ``` ``` ## list() ``` ] --- ```r plot(G) ``` <img src="slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" /> --- class: center, middle # .red[Rule Based Networks] --- # Genetic Distance - A metric that can be used to describe pair-wise structure of populations. - Configured as a network, the topological shape of the network is a snapshot of structure. - Require specific rules for construction. --- # *Posidonia oceanica* - clonal seagrass .pull-left[  ] .pull-right[  ] --- # *Posidonia oceanica* - clonal seagrass  --- # Threshold Connectivities .pull-left[ Edges are In/Out based upon geographic separation between sample locations. ] .pull-right[  ] --- # Topological Rules - Graph Metrics Perhaps these networks are defined by graph-construction algorithms, many of which have been sucessful in describing other network structures (internet, epidemoloigcal disease networks, etc.) In these models, the topological connections define the network connection rules.  --- # Challenges to Rules-Based Networks Rules based networks require `someone` to decide on the rules. - Male intuition? - "Body of Knowledge"... -- Making .redinline[arbitrary] decisions about modifying your data based upon your own *intuition* or *body of knowledge* is suspect. - No matter what kind of *expert* you think you are in some system (with fancy papers to show off), your intuition is just a **guess**, nothing more. - Use **data**, **math**, and **experimentation** to verify the input into your models. -- To what extent doe the .redinline[mechanisms that influence realized connectivity for a species] sampled among different locations operate in a way that is aligned with a fancy graph-construction algorithm thought up by some mathematician? --- class: center, middle # .red[Population Graphs] ### .red[Statistical Based Topologies] --- # Model-Based Graph Construction - The raw data itself as valuable information about the *spatial distribution of genetic variation*. - The demographic processes that connect populations (or localles) is **exactly** the process that creates the genetic structure we observe. - Initiated by wanting to find a *post hoc* analysis on genetic structure statitics. --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle, center  --- # Non-Uniqueness of Structure A structure statistic such as `\(F_{ST}\)` is a very powerful tool for understanding connectivity. However, there are some topological concerns related to its non-reversability. &nbsp;  --- class: middle, center  --- .pull-left[ # *Lophocereus schottii* Shape of genetic connectivity.  ] .pull-right[  ] --- # Conditional Graph Distance As Surrogate  --- # Visualizing Topologies ```r library( popgraph ) data( lopho ) lopho ``` ``` ## IGRAPH 6af7beb UNW- 21 52 -- ## + attr: name (v/c), size (v/n), color (v/c), Region (v/c), weight (e/n) ## + edges from 6af7beb (vertex names): ## [1] BaC--LaV BaC--Lig BaC--PtC BaC--PtP BaC--SnE BaC--SnI ## [7] BaC--StR Ctv--PtP Ctv--SLG Ctv--SnF Ctv--SenBas LaV--Lig ## [13] LaV--PtC LaV--SnE LaV--SnF LaV--TsS Lig--PtC Lig--SnI ## [19] Lig--StR Lig--TsS PtC--SnE PtC--StR PtC--TsS PtC--SenBas ## [25] PtP--SnF PtP--SnI PtP--SenBas SLG--SnF SLG--SnI SnE--StR ## [31] SnE--TsS SnF--SnI SnI--StR StR--TsS StR--SenBas CP --Seri ## [37] CP --SG CP --SN CP --TS LF --PL LF --SG LF --SI ## [43] PL --SenBas PL --SG PL --SI PL --SN ## + ... omitted several edges ``` --- ```r plot( lopho ) ``` <img src="slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" /> .footnote[ The default layout is based upon the *Fruchterman Reinghold* algorithm.] --- # Customizing Layouts - Topological Layouts .pull-left[ Projection onto 2-dimensional space based upon *spring* model of connectivity. ```r l <- igraph::layout.kamada.kawai( lopho ) head(l) ``` ``` ## [,1] [,2] ## [1,] -1.0095891 1.1397088 ## [2,] 0.1515580 0.8346170 ## [3,] -1.2728255 2.1898410 ## [4,] -1.4344552 1.8100753 ## [5,] -2.4123245 0.2252931 ## [6,] -0.1723895 0.8333827 ``` ] .pull-right[ ```r plot( lopho, layout = l ) ``` <img src="slides_files/figure-html/unnamed-chunk-7-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Customizing Network Plots .pull-left[ ```r plot( lopho, edge.color = "black", vertex.label.color = "darkred", vertex.color = "gray", vertex.label.dist = 2 ) ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Interactive Visualization .pull-left[ ```r library( networkD3 ) lopho.d3 <- igraph_to_networkD3( lopho, group=V(lopho)$Region) forceNetwork( Links = lopho.d3$links, Nodes = lopho.d3$nodes, Source = "source", Target = 'target', NodeID = 'name', Group = 'group') ``` ] .pull-right[ <div id="htmlwidget-a03f357a8b31136be5d9" style="width:504px;height:504px;" class="forceNetwork html-widget"></div> <script type="application/json" data-for="htmlwidget-a03f357a8b31136be5d9">{"x":{"links":{"source":[0,0,2,13,2,3,0,0,1,1,4,14,5,10,12,13,16,12,14,17,13,14,1,12,14,16,18,4,0,2,1,5,6,2,0,3,8,5,6,0,3,9,7,4,18,19,12,10,4,3,2,7],"target":[2,3,3,14,4,4,4,5,5,15,15,15,15,15,16,17,17,17,17,18,18,18,6,19,19,19,19,7,7,7,8,8,8,8,9,9,9,9,9,10,10,10,10,10,20,20,20,11,11,11,11,11],"colour":["#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666","#666"]},"nodes":{"name":["BaC","Ctv","LaV","Lig","PtC","PtP","SLG","SnE","SnF","SnI","StR","TsS","CP","LF","PL","SenBas","Seri","SG","SI","SN","TS"],"group":["Baja","Baja","Baja","Baja","Baja","Baja","Baja","Baja","Baja","Baja","Baja","Baja","Sonora","Sonora","Sonora","Sonora","Sonora","Sonora","Sonora","Sonora","Sonora"]},"options":{"NodeID":"name","Group":"group","colourScale":"d3.scaleOrdinal(d3.schemeCategory20);","fontSize":7,"fontFamily":"serif","clickTextSize":17.5,"linkDistance":50,"linkWidth":"function(d) { return Math.sqrt(d.value); }","charge":-30,"opacity":0.6,"zoom":false,"legend":false,"arrows":false,"nodesize":false,"radiusCalculation":" Math.sqrt(d.nodesize)+6","bounded":false,"opacityNoHover":0,"clickAction":null}},"evals":[],"jsHooks":[]}</script> ] --- # Visualizing Topologies - Spatially ```r data( baja ) lopho <- decorate_graph(lopho, baja) lopho ``` ``` ## IGRAPH 6af7beb UNW- 21 52 -- ## + attr: name (v/c), size (v/n), color (v/c), Region (v/c), Latitude ## | (v/n), Longitude (v/n), weight (e/n) ## + edges from 6af7beb (vertex names): ## [1] BaC--LaV BaC--Lig BaC--PtC BaC--PtP BaC--SnE BaC--SnI ## [7] BaC--StR Ctv--PtP Ctv--SLG Ctv--SnF Ctv--SenBas LaV--Lig ## [13] LaV--PtC LaV--SnE LaV--SnF LaV--TsS Lig--PtC Lig--SnI ## [19] Lig--StR Lig--TsS PtC--SnE PtC--StR PtC--TsS PtC--SenBas ## [25] PtP--SnF PtP--SnI PtP--SenBas SLG--SnF SLG--SnI SnE--StR ## [31] SnE--TsS SnF--SnI SnI--StR StR--TsS StR--SenBas CP --Seri ## [37] CP --SG CP --SN CP --TS LF --PL LF --SG LF --SI ## + ... omitted several edges ``` --- # Spatial Locations .pull-left[ ```r library( ggplot2 ) p <- ggplot() p <- p + geom_edgeset( aes(x=Longitude,y=Latitude), lopho) p ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Spatial Locations .pull-left[ ```r p <- p + geom_nodeset( aes(x=Longitude,y=Latitude), lopho) p <- p + coord_map() + xlab("Longitude") + ylab("Latitude") p ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Overlaying onto Other Features .pull-left[ ```r map_data("world", "mexico") -> map ggplot() + geom_polygon( aes(long,lat, group=group), data=map, fill="gray") + geom_edgeset( aes(x=Longitude,y=Latitude), lopho, color="gray40") + geom_nodeset( aes(x=Longitude,y=Latitude, color=Region, size=size, alpha=0.7), lopho) + coord_sf( xlim=c(-116,-109), ylim=c(22,35), crs = 4326) ``` ] .pull-right[ <img src="slides_files/figure-html/unnamed-chunk-18-1.png" width="504" style="display: block; margin: auto;" /> ] --- # Leaflet Overlays .pull-left[ We can convert the node and edge sets into `sf` objects and then overlay them using the leaflet framework. ```r nodes <- to_sf(lopho, what="nodes" ) edges <- to_sf(lopho, what="edges" ) ``` ] .pull-right[ ```r head( nodes ) ``` ``` ## Simple feature collection with 6 features and 4 fields ## Geometry type: POINT ## Dimension: XY ## Bounding box: xmin: -114.72 ymin: 24.04 xmax: -109.99 ymax: 29.73 ## Geodetic CRS: WGS 84 ## name size color Region geometry ## BaC BaC 12.158810 #FDAE61 1 POINT (-111.79 26.59) ## Ctv Ctv 3.880886 #FDAE61 1 POINT (-114.72 29.73) ## LaV LaV 3.533757 #FDAE61 1 POINT (-109.99 24.04) ## Lig Lig 4.731355 #FDAE61 1 POINT (-111.27 25.73) ## PtC PtC 4.684652 #FDAE61 1 POINT (-111.15 24.19) ## PtP PtP 10.925375 #FDAE61 1 POINT (-113.9 29.03) ``` ```r head( edges ) ``` ``` ## Simple feature collection with 6 features and 2 fields ## Geometry type: LINESTRING ## Dimension: XY ## Bounding box: xmin: -113.9 ymin: 24.04 xmax: -109.99 ymax: 29.03 ## Geodetic CRS: WGS 84 ## # A tibble: 6 × 3 ## EdgeGroup Weight geometry ## <chr> <dbl> <LINESTRING [°]> ## 1 BaC-LaV 9.05 (-109.99 24.04, -111.79 26.59) ## 2 BaC-Lig 9.72 (-111.27 25.73, -111.79 26.59) ## 3 BaC-PtC 12.4 (-111.15 24.19, -111.79 26.59) ## 4 BaC-PtP 6.54 (-113.9 29.03, -111.79 26.59) ## 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One way to test this while at the same time preserve any local habitat heterogeneity is to permute the edges. `\(H_O: Edges \; Random \; WRT \; Habitat\)` If this is true, then we should be able to permute edges across the landscape and `raster::extract()` the habitat types traversed. The set of observed ones should be just a random sample of the randomized ones **if and only if** the null hypothesis is correct. See the function `?randomize_graph()` on how you can randomize the edges in a network. --- # Ecological Edge Crossing  .footnote[Dyer *et al.* (2012) [doi](https://doi.org/10.1007/s10980-011-9696-x) [pdf](https://drive.google.com/open?id=0B0T81CzLjtfPOXlQcXdLczM2c2c)] --- # Ecological Edge Crossing  --- class: center, middle # .redinline[Popgraph Topolgical Features] --- # Topological Properties There are several different statistics measuring properties of nodes and edges. Here are some examples: - *Degree:* The number of edges connected to a node. - *Closeness:* Standardized distance from a node to other nodes (firestation). - *Betweenness:* Standardized distance from everywhere else to the node/edge (library). --- # Node-Specific Topological Features ```r degree( lopho ) ``` ``` ## BaC Ctv LaV Lig PtC PtP SLG SnE SnF SnI StR ## 7 4 6 6 7 5 3 5 5 6 7 ## TsS CP LF PL SenBas Seri SG SI SN TS ## 5 4 3 5 5 3 5 5 5 3 ``` ```r closeness( lopho ) ``` ``` ## BaC Ctv LaV Lig PtC PtP ## 0.003006606 0.003443042 0.002396129 0.002421674 0.002416570 0.003532856 ## SLG SnE SnF SnI StR TsS ## 0.003276693 0.002471779 0.003226610 0.003134548 0.003220898 0.002202304 ## CP LF PL SenBas Seri SG ## 0.002466197 0.002956398 0.003267921 0.003844318 0.002513050 0.002864214 ## SI SN TS ## 0.002918707 0.002702668 0.002430486 ``` ```r betweenness( lopho ) ``` ``` ## BaC Ctv LaV Lig PtC PtP SLG SnE SnF SnI StR ## 9 31 4 0 0 25 3 0 16 8 30 ## TsS CP LF PL SenBas Seri SG SI SN TS ## 0 2 0 92 98 1 32 15 0 0 ``` ```r eigen_centrality( lopho )$vector ``` ``` ## BaC Ctv LaV Lig PtC PtP ## 0.6687298120 0.0567830271 0.7013285708 0.8081182427 1.0000000000 0.1514600463 ## SLG SnE SnF SnI StR TsS ## 0.0347729211 0.6113445775 0.1620895869 0.3248050072 0.7041867362 0.7112031591 ## CP LF PL SenBas Seri SG ## 0.0007395873 0.0028570764 0.0598602506 0.3783327302 0.0005974582 0.0040620964 ## SI SN TS ## 0.0038824940 0.0054114197 0.0007822407 ``` --- class: center, middle ## .red[Draw it Dummy!] ###### .red[ &dash; Message from *Historical* Dyer to *Current* Dyer to Draw Graph Distance IBD] --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle, center # .red[Topological Congruence] --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle, center  --- class: middle background-image: url("https://live.staticflickr.com/65535/50367566131_85c1285e2f_o_d.png") background-position: right background-size: auto .pull-left[  ] .pull-right[ # 🙋🏻 Questions? If you have any questions for about<br/> the content presented herein<br/> now is your time. If you think of something later though, <br/>feel free to [ask me via email](mailto://rjdyer@vcu.edu) and I'll<br/> get back to you as soon as possible. ]