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i, due to the assumption of sparseness. As we shall see in Sec. , such an assumption does not hold for some real systems.
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i, due to the assumption of sparseness. As we shall see in Sec., such an assumption does not hold for some real systems.
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We observe that we defined an ordering which is just the opposite of the one defined in. The reason for this stems from the interesting role played by order theory to understand the particular properties of DAGs. In this way, in our ordering, a maximal will display a number smaller than any of its neighbors, and the opposite happens in the case of minimal, leading this definition of order to being more intuitive for the reader. It is clear, however, that any choice is equivalent, provided that the construction is internally consistent.
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We observe that we defined an ordering which is just the opposite of the one defined in. The reason for this stems from the interesting role played by order theory to understand the particular properties of DAGs. In this way, in our ordering, a maximal will display a number smaller than any of its neighbors, and the opposite happens in the case of minimal, leading this definition of order to being more intuitive for the reader. It is clear, however, that any choice is equivalent, provided that the construction is internally consistent.
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ij =1 position in A and m and n are the numbers obtained in the randomization method. For the sake of simplicity we avoid the use of this double notation saying that node position is invariant in the matrix.
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i j = 1 position in A and m and n are the numbers obtained in the randomization method. For the sake of simplicity we avoid the use of this double notation saying that node position is invariant in the matrix.
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note
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A measure quantifying how random or how deterministic is a structure in relation to the space allowed by the topological invariants was required. The degree-degree joint entropy of a graph holds this property. Other valuable measures, such as assortativity or mutual information, have been pointed out. Assortativity measures degree-degree correlations and degree-degree mutual information quantifies the predictability of neighbors' degrees from the sole knowledge of the degree of a given node in relation to the available degree richness of the system. The former case strictly looks for linear relationships, and it is supposed to be a more appropriate measure for normally distributed data. Furthermore, both approaches naturally require certain degree-degree variance within the graph. For instance, a large feed-forward single chain of nodes has a strong degree-degree determinism that none of these two measurements would capture. The reason is that most of the degree-degree pairs would be (2,2) for undirected and (1,1) for any directed degree analyses. In this sense, degree-degree joint entropy provides a suitable measure of the relation or determinism of degree-degree relations with neither parametric assumptions nor degree-degree variance requisites. Accordingly, we used the concept of degree-degree relations instead of degree-degree correlations.
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