Modeling Complex Network Interactions: Streams, Roads, and Ecosystems

Modeling Complex Network Interactions: Streams, Roads, and Ecosystems Luquillo Experimental Forest, Puerto Rico Andrew Pike Dept. of Earth and Environmental Science University of Pennsylvania Philadelphia, PA

Biocomplexity Flow Diagram Hydrology Vegetation Geology Stream Network Societal Needs Habitat Predator – Prey Interactions Aquatic Ecosystems Human Ecosystems Recreation Migration Economic Needs Road Network Physical Landscape Road Funding Population Centers

Watersheds of the LEF Rio Mameyes Rio Espiritu Santo Rio Sabana Rio Fajardo Rio Blanco

Study Area Slope Rainfall Geology Land Use

Field Methods Cross Sectional Survey n = 105 Survey Sites Pebble Counts Salt Addition Total Station Laser Theodolite

GIS Model Digital Elevation Model (DEM) Topographic Map Slope Curvature Rainfall

GIS Model Topographic Map Flow Direction Flow Accumulation

GIS Model 50ha 10ha 6ha

GIS Model USGS Map ArcGIS Generated Superimposed

Physical Template • Elevation • Slope • Rainfall • Drainage Area • Mean Annual Discharge • Geopotential Energy Flux • Stream Power • Flow Frequency • Land Use • Substrate • Est. Width, Depth, Velocity • - Est. Grain Size

Network Topology What can topology tell us about the form and function of a network? Stream Network Road Network Food Web Network

2 1 4 5 6 3 Network Topology andGraph Theory 100 101 102 105 106 103 104

2 1 4 5 6 Undirected 3 Directed 2 1 4 5 6 3 Network Adjacency A(1) = 14 <ai>= 2.33 A(2) = 7 <ai>= 1.17

2 1 4 5 6 Undirected 3 Directed 2 1 4 5 6 3 Network Distance D(1) = 52 <di>= 8.66 D(2) = 24 <di>= 4.00

2 1 4 5 6 3 _A_ D A/D Indicator = _ai_ di Complexity Indicator B = SUM _A_ V2 _2E_ V2 = Conn = Complexity Indices Undirected Conn = .39 A/D Indicator = .27 Complexity Indicator B = 1.81 Directed 2 1 4 5 6 3 Conn = .19 A/D Indicator = .29 Complexity Indicator B = 2.04

Complexity Indices

2 1 4 5 6 Undirected 3 Cluster Coefficient and Length _First Neighbor Connections_ Total Possible Connections ci = _2Ei_ ai(ai–1) = c1 = 2/3 c2= 1/1 c3= 1/1 c4 = 2/6 c5 = 0/1 c6 = 0 D(1) = 52 L(p) = 52/36 = 1.44 C(p)= Avg(ci) = .50

Watts-Strogatz Model Watts and Strogatz (1998)

Scale-Free Networks • For Scale-Free Networks: • Networks expand continuously by the addition of new vertices • New vertices attach preferentially to sites that are already well-connected • (Barabasi and Albert 1999) In the random network, the five nodes with the most links (in red) are connected to only 27% of all nodes (green). In the scale-free network, the five most connected nodes (red) are connected to 60% of all nodes (green). Scale Free Network = “Hub and Spoke”

Scale-Free Networks How would an agent route itself through each network? Bifurcation Scale-Free

Food Webs from Waide and Reagan 1996

Food Web Topology from Dunne et al. 2002

Summary • Topology yields insight into form and function of a network • More complex models needed to find network-network interactions • Questions?

Network Motifs Milo et al. 2004

Modeling Complex Network Interactions: Streams, Roads, and Ecosystems