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We find it convenient even on a undirected graph to consider the ends of the edges to be distinguishable—each edge has a unique A-end and B-end, which are marked in some way. We can think of one of the ends as having a dot or other identifying feature on it. This makes the counting of edges simpler: the matrix element (Formula presented) is defined as the probability that a randomly chosen edge is connected to a vertex of type i at its A end and type j at its B end. Thus every edge, whether it joins unlike vertices or like ones, appears only once in the matrix—no edge appears both above and below the diagonal. It is possible to construct a theory in which the ends of undirected edges are indistinguishable, but in this case each edge that joins unlike vertices appears twice in the matrix, both above and below the diagonal, and edges joining like vertices appear only once. This necessitates the introduction of an extra factor of 2 into the off-diagonal terms. This approach is adopted for example in Ref. 43
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We find it convenient even on a undirected graph to consider the ends of the edges to be distinguishable—each edge has a unique A-end and B-end, which are marked in some way. We can think of one of the ends as having a dot or other identifying feature on it. This makes the counting of edges simpler: the matrix element (Formula presented) is defined as the probability that a randomly chosen edge is connected to a vertex of type i at its A end and type j at its B end. Thus every edge, whether it joins unlike vertices or like ones, appears only once in the matrix—no edge appears both above and below the diagonal. It is possible to construct a theory in which the ends of undirected edges are indistinguishable, but in this case each edge that joins unlike vertices appears twice in the matrix, both above and below the diagonal, and edges joining like vertices appear only once. This necessitates the introduction of an extra factor of 2 into the off-diagonal terms. This approach is adopted for example in Ref. 43.
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One can also calculate a value for r by simply ignoring the directed nature of the edges in a directed network. This approach, which we adopted in Ref. 22, will in general give a different figure from that given by Eq. (25). While Eq. (25) will normally give a more meaningful result for a directed network, there may be cases in which ignoring direction is the correct thing to do. For example, in a food web one might only be interested in which species have tropic relations with with others, and not in which direction that relation lies in terms of energy or carbon flow
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One can also calculate a value for r by simply ignoring the directed nature of the edges in a directed network. This approach, which we adopted in Ref. 22, will in general give a different figure from that given by Eq. (25). While Eq. (25) will normally give a more meaningful result for a directed network, there may be cases in which ignoring direction is the correct thing to do. For example, in a food web one might only be interested in which species have tropic relations with with others, and not in which direction that relation lies in terms of energy or carbon flow.
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Strictly these probabilities are only correct in a “canonical ensemble” of graphs in which the degree distribution is fixed rather than the degree sequence. This ensemble and the fixed-degree-sequence one studied here, however, become equivalent in the limit of large graph size; the error introduced here by substituting one for the other is of the order of (Formula presented) and is small compared with other sources of error in our simulations
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The degree is not recalculated after each removal. Removal is in the order of vertices’ starting degree in the network before any deletion has taken place.
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