## Number of Roles and Development Capacity

Number of nodes, flow organization, and transfer intensities are strongly affected by the level of network detail. Therefore, indices such as TST, AMI, A, and C depend on arbitrary choices taken when an ecosystem network is built. For instance, if we are interested in decomposition activities, we will emphasize resolution of microbial and nonliving nodes, whereas to evaluate the susceptibility to environmental conditions (water and nutrient availability, solar radiation, temperature, wind intensity, salinity levels, etc.), each compartment will group species showing the same behavior.

To define development capacity variations as a function of the network framework, the connectivity (x), calculated as flows per node, and the number of roles (p) are introduced:

with n = number of nodes and f= total number of flows.

It appears evident how connectivity and number of flows are directly affected by network topology and, in what follows, are proposed as an alternative way to define whole-system indices.

Although the concept ofrole has been fundamental to develop ecological niche idea and food web research, it has never been formally defined. Trophic position (TP; computed as the sum of the fractions of trophic activity that each species performs at different trophic levels), trophic niche (the 'ecological function' carried out by species in a given ecosystem), ecological guild (two species belong to the same ecological guild when they exploit the same class ofenvironmental resources and in a similar way), and trophospecies (recently interpreted as a set of species with similar diet or predators) describe, in a slightly different way, the trophic role in an ecosystem, but none is completely satisfactory in wholly capturing relation with respect to food and enemies.

An effective definition can be obtained adopting concepts from social network analysis, where the role is seen as a specialized function joining structurally equivalent nodes. In the framework ofecological flow networks, this means that species belonging to the same role take input from one source and show outflows to a single destination. Then, in a recursive definition, two species (or group of species) are regular equivalent, exhibiting the same role, when eaten by and feeding on equivalent species.

Nevertheless, when role is calculated as a function of connectivity and number of flows, eqn [38] remains a different concept with respect to regular equivalence.

A |
B |
A |
B | |||

I) |
C |
P = 1 |
I |
C | ||

Figure 4 (a) The fully connected topology with four compartments (n = 4) and 16 flows (f = 16). In this case, an average number of four flows per node (x = 4) was calculated, with only one role (p = 1). (b) The more articulated network with n = 4 nodes, f = 4 flows, connectivity x = 1 flow per node, number of roles p = n = 4. Rather than considering the two approaches in contrast with one another, it is interesting to stress the potential for their integration. The relationship between regular equivalence and number of roles is clearest when the network contains no cycles. In a linear chain with n nodes we identify n roles, while in a fully connected topology of n compartments, there is only one role (see Figure 4). The example of Figure 4 represents two hypothetical unweighted networks, not really suitable to describe the real world. In what follows, a method to extend the role calculation when dealing with real ecosystems (where flows have unequal size, with sometimes extraordinary differences) is suggested. In weighted networks, the effective connectivity is estimated accounting for the weights of flows and the portion of TST processed by each node. Applying a weighted geometric mean, connectivity (x) and total system flows ( f) become n+2 n+2 x=niL and the derived number of roles (p) is |

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