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This is obtained by extending an algorithm designed for the adjacency matrix, due to Jacobs and Trevisan (2011).
The memory usage scales as O ( N b 2 ), accounting for the adjacency matrix of the buffered graph.
Write (A(G)) for the adjacency matrix of G and let (D(G)) be the diagonal matrix of the degrees of G.
We use the notation (mathbf{Y}_{{r}} = ( r_{ij})) and (mathbf{Y}_{{c}} = ( c_{ij})) for the adjacency matrices which describe the airport and commuting ground-truth flows, respectively.
Remark 4.1 For the adjacency spectral radius of a unicyclic graph, it is well known that ρ ( G ) < ρ ( C 3 n − 3 ) for G ≇ C 3 n − 3. It is interesting that for the skew-spectral radius of an oriented unicyclic graph, we also have ϱ ( G σ ) < ϱ ( D 3 n − 3 ) ( n > 5 ) for G ≇ C 3 n − 3.
This is not true for the adjacency network, and shows that the statistically validated networks are able to identify the many preferential links inside communities and the few preferential links bridging different communities of organisms.
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(1998) developed a molecular distance-edge between carbon atoms vector (MDE) based on two fundamental structural variables: one for distance between atoms in the molecular graph, and one for edges of the adjacency in the graph.
Given a DCJ operation ρ, let λ0 and λ1 be, respectively, the sum of the indel-potentials for the components of the adjacency graph before and after ρ, and let Δ λ= λ1− λ1−
Given a DCJ operation ρ, let σ0 and σ1 be, respectively, the sum of the substitution-potentials for the components of the adjacency graph before and after ρ, and let Δ σ= σ1− σ1−
To make use of the prior information (in the original leaf networks), RefineML uses a belief coefficient k b for each entry of the adjacency matrices of these networks, which represents how much we trust the prediction by the base network inference algorithm.
Insets show the spectral graphs of the adjacency matrices for each period.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com