Approaching a Problem as a Network

Introduction

The purpose of the "network" view is to look holistically at an environmental problem. In the "systems" view, we broke down the problem into sub-models and expressed those using five universal tools. In the "network" view we want to learn to describe the behavior of the whole system and to be able to predict that behavior from characteristics of the network of processes that are taking place. The description of behavior will require a new and very specific vocabulary.

The "network" view is very useful for systems that have a medium number of objects that interact in very specific ways. We will be using the network view to understand the behavior of food webs; are they stable, do they bounce back after a stress event, and how important are the specificity of the linkages that have developed.

In systems with a small number of objects, the network view can easily be made to be congruent to the "systems" view. We will examine a food web (with only a few organisms) from both a "systems" and a "network" view. Even though we can force congruency in these simple network/systems, the goal is to learn to approach more complex networks. A holistic network approach can be very different, and provide very different insight into the problem than a dynamic systems approach. The network view looks at the web of relationships and the systems view tries to reduce all objects, flows and controls into a stringent format.

The network diagram

The network diagram looks very similar to the systems diagram we used before. There are nodes and connections between the nodes. For example, we might construct a network diagram for a simple 5 species food web (Figure 1).

Figure 1. A node and arrow network diagram for a food web with five participants. "A" and "B" can pass energy or materials on to "D". "B" and "C" can pass energy or materials on to "E". The focus in the network view is on the interactions, in this case there are four unique interactions, AD, DB, BE, and EC. The nodes are where these interactions connect. In this diagram all of the connection strengths are the same. For simplicity subsequent figures will show only the lines rather than the arrows, but you should focus your attention on the links, not the nodes.

In this foodweb, "D" and "E" are the predators and "A", "B" and "C" are the prey. There is also some competition, "D" and "E" compete for "B".

In this network the changes in any one component will have immediate effects and subsequent compensatory responses. For example if the amount of "A" is diminished, there could be an immediate negative effect on "D" which could be compensated if "D" switches to consuming more of "B". The decrease in "B" would effect "E" and that would ripple over to effect "C". Thus a change in one species could effect the entire network, with all of the other species helping the system adjust to the initial perturbation.

Description of network structure

Network structure and function are related. The structure of the network is also called the "trophic" structure. The first level of the description is the network diagram, the nodes and arrows as shown in Figure 1. Two important characteristics of this network linkage are the connectance and the linkage density.The connectance is the proportion of the number of links to the total links possible. The total number of links possible can be easily calculated from the number of nodes as:

total_possible_links = n*(n-1)/2

For example a network with 5 nodes has a maximum of 10 links. The link density is simply the average number of links per node.

Description of network behavior

We are going to focus on attempting to describe the stability of a food web or other network. Stability could broadly be considered the ability of the network to return to its starting condition after a perturbation. Assuming that the foodweb is in a healthy state to start with, having the appropriate number of connections, it will return to that state after an amount of time.

The ability to tolerate these perturbations is called the "resilience" but it has two different interpretations in the current literature. Some authors use the term "resilience" to indicate the amount of time the network takes to return to its original state whereas others use the term "resilience" to indicate the maximum magnitude of a perturbing stress for which the network will recover. The general sense of resilience is that it indicates the ability of the network to handle stress.

Figure 2. A common metaphor for the resilience of a system. Figure A is the stable state. Figure B shows how far you need to move the ball and yet have it still roll back to its its original state. Figure C shows that the system was pushed to far and moved to another stable state, different from the original.

At any one time all of the nodes and connections make up the current "state" of the network. This state can be constant or it can be variable. The initial examples we will explore assume that there is a preferred stable and constant state. It is possible to have stable variable states as well, network states in which the concentration or activity of each species is varying about some value. These states of the network can be described as oscillating or being periodic. There are also states of a network in which the variation is unpredictable and may swing widely. Although this is a stable, it might make more sense to call this a persistent condition of the network for clarity. These three types of behaviors can be relate to a network characteristics called an "attractor". A "point attractor" is a simple constant stable state, where all the values and activities of the network converge on specific values. A "periodic attractor" is a stable or persistent state of the network in which the values or activities of each species cycles through a predictable set of values. A "chaotic attractor" is the state of the network in which the values or activities all stay within a range, and the network is stable, but the value of any value or activity of a single species is unpredictable.

Visualization of a food web network response to a single perturbation

The following food web diagram (Figure 3a) is used to describe the linkages in network that is assumed to be in a stable configuration. Imagine that the links are springs and that the tension of the links is equal. If one of the nodes is pulled a little out of its current position (Figure 3b), there will be an immediate effect on all the springs that are attached to that node and a subsequent, compensatory effect of the entire network to re-establish equal tension (Figure 3c). In this visual/mechanical metaphor for a network, the position of each node in XY space represents how a species deals with its environment. A shift of position of a node should be interpreted as a required change by a species to acclimate to new environmental stresses or conditions. In this metaphor, it is also necessary to envision that the nodes don't move instantaneously, but rather slowly drift toward a new position.

Figure 3. A network that starts in a stable state in which all the links have equal tension (a) until one node is disturbed and the link is stretched (b), followed by compensation by the entire network (c). During the period of compensation, some links are stretched a little and others may actually be compressed (such as the link between H1 and G2).

If the perturbed node is also allowed to respond, the entire network should return to the same geometry as it started with. If the perturbed node is held in a position for a period of time, the rest of the network may readjust itself to the same geometry but shifted over a bit.

This visualization of network behavior is supposed to give you a feel for how a change in any one of the nodes will lead to a compensation by the entire network. This view seems to be a cause and effect type system and you can imagine that it could also be represented by a systems diagram. The visualization of a shifting set of nodes and rearrangement of the links however can be applied to more variable systems that act more like real food web networks.

Visualization of the behavior in a network with variable nodes

In the previous diagrams, the position of the node in XY space represented both the environmental condition that the species was dealing with and how it dealt with it. For example, the shift to the left of G1 could represent how a species of grass dealt with a particularly dry spell of weather. What we need to visualize now is what the network behavior would be if the nodes were constantly varying on their own (or being driven by environmental conditions) and what a network of constantly moving nodes and stretching/condensing links would look like. This will be represented below in a series of figures that show how the oscillation in just one node, "G1", would propagate oscillations to other nodes in the network. The oscillation in G1 could be caused by a daily or tidal environmental forcing function for example. In a real food web network, we should expect that several of the species might be responding to environmental conditions and that the network behavior could be described more as a set of dancing nodes than a simple response to a perturbation.

Figure 4. Propagation of an oscillation from G1 to other nodes in the network. Each subsequent diagram shows how the oscillation from the previous diagram might propagate next. As the nodes are further away from G1, the response can be considerably attenuated.

An important part of this analysis is the number steps that it would take to have the original perturbation propagate through the entire network. In the above example, the next two steps after the perturbation are shown and it would only take one more step to effect all of the nodes. The level of connectivity determines the number of steps.

Summary of the network view of food webs

A food web network has the following characteristics that can be used to understand and describe its behavior:

1. Each link between two species represent specific activities such as predatory prey interactions.

2. Each node should only have several links. More links represent generalist species and fewer links represent specialist species.

3. The food web should have an intermediate level of connectedness.

4. A single perturbation will cause an immediate reaction and then several levels of response from the full network, depending on the connectedness.

5. Continued variability in the environment and the response of individual species can result in a highly complex variation in all of the species all the time. Even though there is continuous variability, this can be a stable state of the network.

6. Individual perturbations or environmental fluctuations can cause changes in the network that are temporary, with the food web returning to a stable state. If individual or environmental perturbations are too large, the food web network could flip to an entirely different stable state. The amount of perturbation that it takes to just reach the border for a network transition is called the resilience.

7. Healthy natural networks should have a high threshold of resilience.