## Removing Cycles in Ecological Networks

We have stated above that it is possible to enumerate all the cycles in a food web. In an ecological network, however, each cycle will also possess a 'weight', given by the amount of flow passing through the cycle.

Some network analysis applications (e.g., the so-called 'Lindeman spine') require an acyclic network as an input. The removal of the cycles therefore becomes an important topic for network analysis.

The current procedure requires the removal of cycles according to their 'nexus'. Two cycles are in the same nexus if they share the same weak arc, defined as the smallest flow in the cycle. Cycles are then removed dividing the flow constituting the weak arc among all the cycles sharing the same nexus. The resulting amounts are then subtracted from each edge of the cycles. This process results in the removal of the weak arc. The procedure is then repeated until the resulting network is acyclic.

A nice by-product of the procedure is the creation of a network composed of all the cycles in the original network. This is usually referred to as 'aggregated cycles' network in ecological literature. This network will receive no input, produce no output, and will be balanced (i.e., incoming flows equal outgoing ones) for all nodes. If the resulting aggregated cycle network is composed of several subgraphs, each subgraph is a strongly connected component.

Note that while some applications require acyclic networks, most of them are actually based on the fact that empirical networks contain millions of cycles. As explained in the next section, in fact, cycles are among the most important features of ecosystems.

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