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# Graph traversal#

To generate graphs of impact like supply chain or Sankey diagrams, we need to traverse the graph of the supply chain. The GraphTraversal class does this in a relatively intelligent way, assessing each inventory activity only once regardless of how many times it is used, and prioritizing activities based on their LCA score. It is usually possible to create a reduced graph of the supply chain, with only the most relevant pathways and flows included, in a few seconds.

## Illustration of graph traversal#

It’s easiest to understand how graph traversal is implemented with a simple example. Take this system:

• This system has four nodes, which are LCI processes, also called transforming activities. Each node has one reference product, and a set of zero or more technosphere inputs. By convention, node A produces one unit of product A.

• This system has four edges which define the inputs of each node. An edge has a start, an end, and an amount.

• We consider solving this system for a functional unit of one unit of A.

As we traverse this supply chain, we will keep different data for the nodes and the edges. For nodes, we are interested in the following:

• amount: The total amount of this node needed to produce the functional unit.

• cum: The cumulative LCA impact score attributable to the needed amount of this node, including its specific supply chain.

• ind: The individual LCA impact score directly attributable to one unit of this node, i.e. the score from the direct emissions and resource consumption of this node.

For edges, we want to know:

• to: The row index of the node consuming the product.

• from: The row index of the node producing the product.

• amount: The total amount of product from needed for the amount of to needed.

• exc_amount: The amount of from needed for one unit of to, i.e. the value given in the technosphere matrix.

• impact: The total LCA impact score embodied in this edge, i.e. the individual score of from times amount.

Our functional unit is one unit of A. Before starting any calculations, we need to set up our data structures. First, we have an empty list of edges. We also have a heap, a list which is automatically sorted, and keeps track of the nodes we need to examine. nodes are identified by their row index in the technosphere matrix. Finally, we have a dictionary of nodes, which looks up nodes by their row indices.

nodes, edges, heap = {}, [], []


We create a special node, the functional unit, and insert it into the nodes dictionary:

nodes[-1] = {
'amount': 1,
'cum': total_lca_score,
'ind': 1e-6 * total_lca_score
}


The cumulative LCA impact score is obviously the total LCA score; we set the individual LCA score to some small but non-zero value so that it isn’t deleted in graph simplification later on.

We next start building our list of edges. We start with all the inputs to the functional unit, which in this case is only one unit of A. Note that the functional unit can have multiple inputs.

for node_id, amount in functional_unit:
edges.append({
"to": -1,  # Special id of functional unit
"from": node_id,
"amount": amount,
"exc_amount": amount,
"impact": LCA(node_id, amount).score,  # Evaluate LCA impact score for node_id/amount
})


Finally, we push each node to the heap:

for node_id, amount in functional_unit:
heappush(heap, (abs(1 / LCA(node_id, amount).score), node_id))


This is not so easy to understand at first glance. What is 1 / LCA(node_id, amount).score? Why the absolute value? What is this heappush thing?

We want one divided by the LCA impact score for node A because our heap is sorted in ascending order, and we want the highest score to be first.

We take the absolute value because we are interested in the magnitude of node scores in deciding which node to process next, not the sign of the score - leaving out the absolute value would put all negative scores at the top of the heap (which is sorted in ascending order).

heappush is just a call to push something on to the heap, which is our automatically sorted list of nodes to examine.

After this first iteration, we have the following nodes and edges in our graph traversal:

nodes = {-1: {'amount': 1, 'cum': some number, 'ind': some small number}}
edges = [{
'to': -1,
'from': 0,  # Assuming A is 0
'amount': 1,
'exc_amount': 1,
'impact': some number
}]
heap = [(some number, 0)]


After this, it is rather simple: pull off the next node from the heap, add it to the list of nodes, construct its edges, and add its inputs to the heap. Iterate until no new nodes are found.

Because the heap is automatically sorted, at each iteration we will take the node with the highest impact that hasn’t yet been assessed.

There are two more things to keep in mind:

• We use a cutoff criteria to stop traversing the supply chain - any node whose cumulative LCA impact score is too small is not added to the heap.

• We only visit each node once. The is functionality in bw2analyzer to “unroll” the supply chain so that afterwards each process can occur more than once.