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A weight-reshuffled graph can be obtained by collecting all the link weights in the original network and by randomly reassigning them to the links in the original network topology. Equivalently, half, e.g., upper right of the original adjacency matrix is randomly shuffled and then is mapped to the lower-left part of the adjaceny matrix, whose symmetry is reserved in this weight-reshuffling process
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A weight-reshuffled graph can be obtained by collecting all the link weights in the original network and by randomly reassigning them to the links in the original network topology. Equivalently, half, e.g., upper right of the original adjacency matrix is randomly shuffled and then is mapped to the lower-left part of the adjaceny matrix, whose symmetry is reserved in this weight-reshuffling process.
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With N=150, we are able to compare the link weight correlation around a node via ( si + sj ) /2 (N-1 ) versus wij in networks measured before and after neurosurgery, because almost the same link weight distribution is followed
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With N = 150, we are able to compare the link weight correlation around a node via (s i + s j) / 2 (N - 1) versus w i j in networks measured before and after neurosurgery, because almost the same link weight distribution is followed.
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Instead of making scatter plots of all [ ( si + sj ) /2 (N-1 ) , wij ] pairs which heavily overlap, we divide the link weight range [0,1] into 100 bins, and over each bin we calculate the average ( si + sj ) /2 (N-1 ) corresponding to those wij that belong to the same bin
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Instead of making scatter plots of all [(s i + s j) / 2 (N - 1), w i j] pairs which heavily overlap, we divide the link weight range [0,1] into 100 bins, and over each bin we calculate the average (s i + s j) / 2 (N - 1) corresponding to those w i j that belong to the same bin.
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Weighted network analysis could be noisy if the weakest link weights may not represent real coupling, but statistical fluctuations in the coupling measure
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Weighted network analysis could be noisy if the weakest link weights may not represent real coupling, but statistical fluctuations in the coupling measure.
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min (MaST )
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The construction of a MaST shows that, if node i is the last node to join the MaST by including link (i, k), w i k = w min (MaST) and w i k must be the largest link weight among all links connected to node i, or else node i can join the MaST by another link incident to it, but with higher link weight. Any other subgraph G s (N) spanning all nodes must contain at least one link connected to node i, whose weight is smaller or equal to w i k = w min (MaST) and which can be larger than w min (G s (N)). Hence, w min (G s (N)) ≤ w min (MaST).
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Thus, E [D] /N=20% of the original links with the highest link weights are included, resulting in a small number of networks with only few disconnected nodes. These disconnected nodes, although small in number, are all taken into account in the unweighted analysis, because such disconnectivity also reveals information about the weighted networks
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Thus, E [D] / N = 20 % of the original links with the highest link weights are included, resulting in a small number of networks with only few disconnected nodes. These disconnected nodes, although small in number, are all taken into account in the unweighted analysis, because such disconnectivity also reveals information about the weighted networks.
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