DedeBac
commented
1 month ago First of all, thanks for your insightful comments. I will try to reply point by point to the issues you raised.
Regarding the notion of "affinity weights" and "topological weights", there is actually no strict definition of such concepts. It's a terminology we introduced to distinguish between (1) the weights associated to the input higher-order data (if the input Hypergraph is unweighted one considers the weights to be 0 and 1), and (2) the weights associated with the simplicial version of the data (it can be seen as a topological version of the input). The advantage of using the topological weights is that they allow for a convenient normalization of the Hodge Laplacian, which permits to explore topological properties of a simplicial complex.
Regarding the issue about the edge/triangle weights in collaboration networks, I need to double check but at a first glance it is possible that the wrong version of the files was uploaded by mistake. Thanks for bringing this up, I will check this asap. However, it is correct to compute the weights of edges using Eq. 63, and those of the simplices of maximum order (in this case, triangles) with Eq. 65. If you do this, it is verified that the corresponding TOPOLOGICAL weights (not the bare weights) have the property that the sum of the weights incident to a node is the total number of articles written by the node.
I agree that it would be useful to add an extra example. While I work on updating the files, I will also consider providing an example.
Da: Matt Piekenbrock @.> Inviato: venerdì 5 luglio 2024 09:01 A: DedeBac/WeightedSimplicialComplexes @.> Cc: Subscribed @.***> Oggetto: [DedeBac/WeightedSimplicialComplexes] Can't decipher the notation (Issue #1)
I'm having trouble following the salient parts of the paper that don't involve the Heat kernel.
For example, here are four statement made at some point in the paper:
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"We assume to have as input data some bare affinity weights ω(α) ≥ 0 associated to each simplex of a simplicial complex."
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"The aim is to define the correct definition of the bare affinity weights ω(α) such that the topological weight (given by Eq.(3)) associated to each node of the simplicial complex is equal to the number of papers written by the corresponding author."
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"Our goal is to define a proper measure of the strength of the collaboration ω(α) associated to the simplices of K_d, so that the topological weights defined recursively by Eq. (3) attribute to each node a topological weight given by the number of the paper they have co-authored."
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We encode the higher-order collaboration data into a weighted simplicial complex of dimension d = 2 where the affinity weights are choosen according to Eq.(65) for triangles (with n = d) and Eq.(63) nodes and links (with n < d).
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It is easily verified from case (c) that the bare affinity weight of a link can be obtained by subtracting the topological weight of the link by the sum of the topological weights of the triangles incident to it.
It's hard to get a handle on what precisely are these notions of "bare affinity weight(s)" and "topological weights" and what properties they should have.
For example, I was able to reproduce both the edge and triangle weights you produce in your example data set using (65), though it's confusing because: a. You say in (4) above that the edges and nodes would use eq. (63), but the data you supply uses (65) b. The sum of the edge weights for a given node is not equal to the number of papers that author participated in
It's also a little unclear how the given input formats relate to hypergraphs. Given a hyper graph, to produce an input like e.g. the edges_matrix_with_Mn_ConnComp, is the idea something like the following:
- Start with a set of hyperedges $H$ (i.e. subsets)
- Extract all the unique $d$-simplices for some $d$
- For each $d$-simplex $\sigma \in K_d$, record both the number of hyperedges containing $\sigma$ and their sizes
It would be nice if there was a smaller example to demonstrate the weight calculation, bigger than the one if Fig 3 (but still very small), or some code to produce the weighted simplices.
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I'm having trouble following the salient parts of the paper that don't involve the Heat kernel.
For example, here are four statement made at some point in the paper:
"We assume to have as input data some bare affinity weights ω(α) ≥ 0 associated to each simplex of a simplicial complex."
"The aim is to define the correct definition of the bare affinity weights ω(α) such that the topological weight (given by Eq.(3)) associated to each node of the simplicial complex is equal to the number of papers written by the corresponding author."
"Our goal is to define a proper measure of the strength of the collaboration ω(α) associated to the simplices of K_d, so that the topological weights defined recursively by Eq. (3) attribute to each node a topological weight given by the number of the paper they have co-authored."
We encode the higher-order collaboration data into a weighted simplicial complex of dimension d = 2 where the affinity weights are choosen according to Eq.(65) for triangles (with n = d) and Eq.(63) nodes and links (with n < d).
It is easily verified from case (c) that the bare affinity weight of a link can be obtained by subtracting the topological weight of the link by the sum of the topological weights of the triangles incident to it.
It's hard to get a handle on what precisely are these notions of "bare affinity weight(s)" and "topological weights" and what properties they should have.
For example, I was able to reproduce both the edge and triangle weights you produce in your example data set using (65), though it's confusing because: a. You say in (4) above that the edges and nodes would use eq. (63), but the data you supply uses (65) b. The sum of the edge weights for a given node is not equal to the number of papers that author participated in
It's also a little unclear how the given input formats relate to hypergraphs. Given a hyper graph, to produce an input like e.g. the edges_matrix_with_Mn_ConnComp, is the idea something like the following:
It would be nice if there was a smaller example to demonstrate the weight calculation, bigger than the one if Fig 3 (but still very small), or some code to produce the weighted simplices.