## Structure of Cycles in Ecological Networks Strongly Connected Components

Two nodes A and B are said to belong to the same strongly connected component (SCC) if they are reachable from each other, that is to say if we can find a path going from A to B and a path coming back from B to A.

If A and B belong to the same SCC, then they are connected by cycles. A graph can be divided into its SCCs, considering every node that is not involved in cycles as an SCC by itself.Figure 3a represents the Baltic Sea ecosystem. One can individuate 6 SCCs: 4 of them are composed by a single node, while 2 of them comprise more than 1 node (Figure 3b).

Figure 3 Schematic representation of Baltic Sea ecosystem (a). The boxes define different strongly connected components. Condensing each box into a single node yields an acyclic graph (b). This graph can be sorted so that all arrows point in the same direction, showing the underlying straight flow between compartments (c).

Figure 3 Schematic representation of Baltic Sea ecosystem (a). The boxes define different strongly connected components. Condensing each box into a single node yields an acyclic graph (b). This graph can be sorted so that all arrows point in the same direction, showing the underlying straight flow between compartments (c).

If we compact every SCC into a single node, we produce an acyclic graph (Figure 3 c). Further analysis shows how one component contains just pelagic species and the other one just benthic. Acyclic graphs can be ordered so that all edges point in the same direction (from bottom to top in Figure 3 c) using a procedure known as 'topological sort' (or partial ordering). Acyclic graphs are therefore intrinsically hierarchical. In this case, the flows find a sink in the benthic compartment, while the pelagic compartment acts as a bridge between the primary producer 1 and the benthic compartment. The same structure was found for other aquatic networks as well. Note that this feature depends drastically on the presence/absence of resuspension of nutrients. If this is negligible, then the network presents several SCCs. When remineralization is strong, however, the process joins the benthic and pelagic components, thus forming a giant SCC.

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