In order to quantify the abundance of simple cycles in food webs, one should know the maximum possible number of simple cycles. The maximum number of simple cycles will be associated with a completely connected food web, that is, a food web whose adjacency matrix contains just 1s.
Table 1 Number of simple cycles of length k (column) in a completely connected food web formed by n species (rows)
6 6 15 40 90 144 120 415
7 7 21 70 210 504 840 720 2 372
8 8 28 112 420 1344 3360 5760 5040 16072
9 9 36 168 756 3024 10080 25 920 45 360 40320 125 673
10 10 45 240 1260 6048 25 200 86400 226800 403200 362880 1 112 083
In order to count the maximum number of simple cycles, we start from the ones with maximum length (in graph theory they are called Hamiltonian cycles). In a completely connected food web composed of n species, the number of simple cycles of level (i.e., length) n is (n — 1)!. This simple formula can be explained combina-torically using permutations: we can see a cycle oflevel n as a permutation ofthe n labels ofthe nodes: for example, ABCD will represent the cycle A ! B ! C ! D ! A. Now, the number of permutations of n elements is n!. We note, however, that every cycle gives rise to n possible sequences (e.g., ABCD, BCDA, CDAB, and DABC represent the same cycle of length 4). Therefore, the total number of simple cycles of maximum length is n !/n = (n — 1)!.
This is an enormous number, as soon as n becomes large. For example, in a 100 species food web, we can find almost 10155 simple cycles of level n.
Now that we know the total number of simple cycles of level n in a completely connected food web, we can easily derive the number of simple cycles of level (n — 1). For each subgraph containing (n — 1) species we will have (n — 1)!/(n — 1) = (n — 2)! simple cycles of length (n — 1). The number of possible subgraphs containing (n — 1) species is given by the binomial coefficient
Therefore, the total number of simple cycles of level (n — 1) in a completely connected food web composed of n species is n(n — 2)!.
Similarly, we can define the total number of simple cycles of length k in a completely connected food web of n species as
Table 1 represents the number of cycles of level k (column) for a completely connected food web of n species (rows).
The total number of cycles is therefore given by the following formula:
The first 10 values are represented in Table 1. Note that this sequence is defined, in combinatorics, as 'logarithmic numbers'.
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