The Flood Complex: Large-Scale Persistent Homology on Millions of Points

We thank the reviewer for the very positive feedback and the time spent. We address all weaknesses (W) and questions (Q) point by point, enumerated in the order they appeared in the review.

W1 Choice of $L$. Although the paper treat it by refering to fpsample, there are no guarantees...

▶ Thank you for pointing this out. We will remark that when using FPS, the Haussdorf distance between the landmarks $L$ and the point cloud $X$ decreases monotonically #landmarks increases. In fact, the landmarks, if selected this way, are an $\epsilon$-net of $X$, with $\epsilon$ the min. distance between the last selected landmark to any other landmark.

W2 Choice of $L$. The accuracies seem to reflect that this choice is not too impactful, but the paper do not provide an analysis on this choice. It would be interesting to see what ...

▶ Our approx. guarantee (Cor. 4) wrt the full Alpha complex depends on the Hausdorff distance between $L$ and $X$. For an empirical study, we kindly refer to our response to Sf3t. In terms of theoretical aspects, we do not have a full answer yet. Still, if a sequence $(X_1,\dots,X_n)$ is drawn iid from a probability measure on a totally bounded subset of any metric space $(𝒳, d)$, then the minimal distance $d_n=\min_{i<n}d(X_i,X_n)$ satisfies $ℙ[d_n>\epsilon]\le𝒩(\epsilon/2)/(2n)$, where $𝒩$ denotes the covering radius of that subset. This result (and others) is proven in [Kulkarni et al. "Rates of convergence of nearest neighbor estimation under arbitrary sampling", In: IEEE Trans. on Inf. Theory '95]. In our setting, this means that if $X$ is an iid sample from a measure on a bounded subset of Euclidean space (and $L$ is considered as the first #landmarks points), then $ℙ[d_H(X,L)>\epsilon]\le 𝒩(\epsilon/2)/(2|L|)$, where $𝒩(r)$ scales as $r^{-d}$ where $d$ is the intrinsic dimension, i.e., $d=2$ for a surface (mesh) and $d=3$ for a volume. Stronger guarantees are possible if the point cloud geometry is taken into account.

W3 ...issues with fpsample, so I couldn't run it...

▶ We are sorry to hear you had issues; fpsample worked on all of our test systems. If you still want to try out our code, you could use torch_cluster's FPS implementation as a replacement (or select $L$ randomly).

W4 ...the code for the experiments is not given.

▶ You are correct, only a few examples were provided with our code. Unfortunately, we are not allowed to upload (or link to) additional material for the rebuttal. However, we will release the full code with all necessary parts for reproducibility after the reviewing phase.

W5 This is not fully on the GPU since a Delaunay complex is still needed in the end. This can be problematic in higher...

▶ We agree. The Flood complex is designed for low-dim. point clouds as an efficient alternative to the Alpha complex. We want to add, however, that there is currently no viable option to compute large-scale PH in high dimensions. E.g., computing PH (up to dim. 8) of only 100 uniformly distributed points in the 8-dim. unit cube already takes ~1min when using an Alpha or a sparse Rips complex with sparsity parameter 1 (theoretical guarantees require <1) on our hardware.

W6 It seems to me that this filtration is differentiable w.r.t. the initial point cloud, ... Would you be more specific in this point...

▶ The (approx.) filtration function is given by $f(σ)=\max_{p\in P_σ}\min_{x\in X}d(p,x)$, which is differentiable unless there are multiple minimizers (similar to max-pool. layers or L1 regularization). In practice, the Triton kernel prevents differentiability, as we did not implement the derivative function (yet). This will be done in the final release version.

W7 Experiments. I was expecting to see bootstrap methods (...) or witness / sparse filtrations to be in the main body of the paper...

▶ You likely have seen it; we compare to averaged persistence vectorizations in our appendix (Table 2) and can move them to the main part. We also compare to sparse Rips in Fig. 2. Regarding witness complexes, these methods shine in higher dimensions (as does the Rips complex) but are neither competitive to the Alpha nor to the Flood complex in the low-dim. regime. To the best of our knowledge, there are no sparse versions of the Alpha complex.

W8 Experiments 2. I feel that they are too few datasets...

▶ For our comparison to the Alpha complex, we consider two settings: one (Alpha-full) where the available information is the same, i.e., Alpha on $X$, and one (Alpha-†) where the available runtime budget is equal. However, in case of the former, runtime of the Alpha complex is prohibitive. Still, for the rebuttal we ran classification experiments using Alpha-full on swisscheese (93% ± 2%, 70s per point cloud) and MCB (60% ± 3%, 200s). Using more points does not improve accuracy, likely because of oversmoothing effects during vectorization. Moreover, the latter becomes slow and difficult to tune for large point clouds, e.g., vectorizing all Alpha-full diagrams from MCB takes 30min per split.

To address your datasets concern, we generated a 2nd synthetic dataset using the porespy library [Gostick J et al. "PoreSpy: A Python Toolkit for Quantitative Analysis of Porous Media Images". JOSS, 2019. ]. Point clouds in this dataset (abbr. as rocks) are extracted from boolean voxel grids of shape (256,256,256) that were constructed from two different generators (blobs and fractal noise), yielding point clouds of varying size up to 16M points. We evaluate (1) classification accuracy wrt the data generating method (i.e., 2 classes) and (2) regression accuracy wrt the surface area computed from the voxel grid. Please see our results in response to Sf3t.

W9 Scope. I feel that the main contribution of this paper is in the field of computational geometry. I don't know how pertinent this work is to the NeurIPS audience.

▶ We understand your point, but emphasize that our approach represents a critical advancement for the ML community, especially in domains where understanding data topology at scale is crucial. From our perspective, Flood PH facilitates TDA at modern data scales and thereby offers a way to integrate global geometric reasoning (via PH) into ML pipelines, which has been largely underutilized due to computational constraints. Further, from a practical perspective, our work aligns well with what is prominently mentioned in a recent ICML position paper [Papamarkou et al. "Position: Topological Deep Learning is the New Frontier for Relational Learning". In: ICML 2024] in terms of availability of efficient TDA methods and software for ML problems.

W10 Theorem 3. It would be very nice if this was true for any Euclidean space

▶ Yes, this would be great. So far, we only have a proof in 2D where things can easily be illustrated, but we conjecture the theorem to hold in higher dimensions as well. Most of the building blocks for the general result are already there. The only extension missing is of Lemma 10, i.e., that "a simplex $𝜏\in\mathrm{Del}(X)$ is a free face of $\mathrm{Flood}_{f\_{\mathrm{Flood}(𝜏)}}$ iff $|𝜏|\cap\mathrm{int}|𝜏|^\dagger=\emptyset$."

▶ Q1 When Alpha is computable (which it is given enough time budget), one reason we identified is the inability of existing vectorization strategies to deal with massive persistence diagrams. Essentially, we observed over-smoothing effects (e.g., due to a large number of summations in the method we selected, but this similarly holds for other strategies) that confound the vectorization quality and thus negatively impact downstream classification performance. The rather small persistence diagrams of Flood PH are advantageous in this regard.

▶ Q2 In 2D, the Flood complex is stable wrt $L$, which follows directly from Col 4. Admittedly, the multiplicative constant of 6 is quite large and a more direct proof (also in higher dimensions) would be more satisfying. The difficulty when studying stability wrt $L$ is that the resulting filtrations are defined on two different simplicial complexes, i.e., on $\mathrm{Del}(L)$ and $\mathrm{Del}(L')$. We tried using a common subdivision, but were unable to link the difference in filtration values to the Hausdorff distance between the landmark sets.
In the special case that there is a bijection $L'\ni l_i'=l_i+\epsilon_i$ such that $\{l_i,\dots,l_k\}\in\mathrm{Del}(L)$ iff $\{l'_i,\dots,l'_k\}\in\mathrm{Del}(L')$ it can be shown that $\mathrm{Flood}_r(X,L)\subset\mathrm{Flood}\_{r+\epsilon}(X,L')$ and vice versa, and therefore $d_B(\mathrm{dgm}(\mathrm{Alpha}(X)),\mathrm{dgm}(\mathrm{Flood}(X,L))<\epsilon$ with $\epsilon=\max_i\epsilon_i\le d_H(L,L')$.

▶ Q3 Extending the Flood filtration to other complexes such as Rips would be great. Unfortunately however, there is no natural inclusion $\mathrm{Del}(L) \hookrightarrow \mathrm{Alpha}(X)_r \subset\mathrm{Del}(X)$, unless no point of $X$ is inside a circumsphere of a simplex in $\mathrm{Del}(L)$, which can be considered an unlikely boundary case. (On the contrary, we would like $L$ to sample $X$ "well".)

▶ Q4 You are correct regarding the uniform sampling. However, for efficiency reasons, it is not viable to optimize the point selection for each simplex separately based on its geometry. However, it is possible to select an evenly-spaced grid on the standard 3-simplex with $m$ points per edge (obtained from $\{(i,j,k,l)\in ℕ^4\colon i+j+k+l=m\}$) and use that as the barycentric coordinates of the simplex points. This approach has the benefit that filtration values of the faces can directly be computed from the 3-simplices by taking the max. over a subset of simplex points. We already used this deterministic approach to isolate the effect of landmark selection for the sensitivity analysis wrt the landmarks (see response to S3ft) and will support it in the upcoming code release.

Thank you for your fast response. We are happy to hear that we could already address most points.

Choice of L.

When using FPS, we can obtain similar asymptotic results. Indeed, the Hausdorff distance between $L$ and $X$ decreases monotonically with $|L|$, and with order $d_H(X,L)\in O(D\cdot |L|^{-1/d})$, where $D$ is the diameter of the point cloud and $d$ its intrinsic dimension, i.e., $2$ for a surface (mesh) and $3$ for a volume.

To be more precise, [González, Clustering to minimize the maximum intercluster distance, Theorem 2.2] shows that FPS yields an $\epsilon$-net of the point cloud, meaning that any two landmarks are at distance at least $\epsilon$ (packing) and that $d_H(X,L)\leq\epsilon$ (covering), for some $\epsilon >0$. Moreover, we have that (e.g., [Feldmann & Filtser, Highway Dimension: a Metric View, Lemma 6 ]), given a $\epsilon$-packing $L$ of a set with diameter $D$, its size satisfies $|L|\leq 2^{d \log(2D/\epsilon)}$. We therefore have that $\epsilon\leq 2D\cdot k^{-1/d}$.

In fact, Hausdorff distance is the criterion in FPS for selecting the next landmark, and therefore can in principle be used as a stopping criterion instead of pre-defining the number of landmarks $|L|$.

We will add this discussion and the one for random sampling in the final version of the manuscript, following your suggestion.

Scope

Thank you for acknowledging that our approach is "necessary for advencements of TDA in ML" and that it "fits in the TDA + ML scope". We hope that our work will foster the currently limited synergy between ML and topology, also in real-world applications.

Please also note that we have added in our last reply to Sf3t some preliminary (given the limited time we had) results with two stronger neural baselines, PointMLP and PVT. In line with the original experiments, Flood PH succeeds on the structurally more challenging cheese, rocks and (real-world) corals datasets, while all tested neural architectures struggle.

Dear Reviewer,

Many thanks for engaging with the authors in this discussion thread. Your contribution is making the review process of Neurips better.

I just wonder if you could also follow up with the authors' latest response to your concerns? Many thanks in advance.

Best,

Ac

Thank you for your overall positive feedback and recognizing the creativity and the relevance of our work.

Weaknesses: In fact, I am completely unfamiliar with this direction. So, I don't know if it has any disadvantages.

▶ We understand that the paper is somewhat technical in parts. Yet, your summary very well captures the essence of our work, i.e., computing PH for low-dimensional Euclidean point cloud data at scale. We agree that this is, at the moment, one of the key limitations for ML practitioners when seeking to distill topological information out of data. Our approach is a first step towards mitigating this limitation in the low-dimensional regime.

Further, thank you for giving your best effort. We appreciate your perspective and, aside from technicalities, we would be grateful if you would point us to any potentially unclear points so that we can make our work more accessible to a broader audience (outside the topological and geometric deep learning communities).

We thank the reviewer for the constructive/positive feedback and the time spent. We address all weaknesses (W) and questions (Q) point by point enumerated as in the review and will adjust the paper accordingly.

W1 Limited consideration of Baselines:

▶ Thank you, we will acknowledge both references. In fact, [1] and [2] are conceptually closer to the Rips than to the Alpha complex. [1] presents a method to compute a distance matrix between equivalence classes of points on which a Rips complex is built. [2] proposes a method to select landmarks for a lazy witness complex. Due to the underlying simplicial complex (which is much larger than an Alpha complex) in both works, the PH computation takes very long even on a few points.

To mitigate this issue, we use edge collapses as in [Boissonnat & Pritam. "Edge Collapse and Persistence of Flag Complexes", SoCG '20], but still the number of landmarks / equivalence classes that can be used is limited, and we face additional problems, briefly described below.

In case of [1], the selection strategy requires building a Delaunay triangulation on the entire point cloud, i.e., the very step we try to avoid with the Flood complex. To achieve a reasonable runtime, we first run FPS to obtain 2k points as input to their selection process. Balanced classification accuracies are ~50% on swisscheese and corals, and 45% on MCB.

In case of [2], when constructing lazy witness complexes (in Gudhi), only computing filtration values already takes long. E.g., for 100 landmarks on 10k witnesses from the swisscheese data, complex construction takes ~20s., and for 1M witnesses ~20min. However, in the former setting, we again obtain only 50% acc. on swisscheese.

W2 Missing sensitivity analysis wrt choice of Landmarks.

We evaluate the sensitivity wrt the landmarks selection on the data from Sec. 5.1 in terms of the bottleneck distance $d_B$ to Alpha PH of the entire point cloud $X$. We compare FPS to (uniform) random selection (Rand). To isolate the effect of landmark selection, we compute the flooding radii based on a evenly spaced grid (instead of random, see response to b9wn) of points on each simplex. For reference, we also list results for Alpha† (on 75k points).

Bottleneck distances (the metric of interest) of Flood PH (FPS) are significantly smaller than for Flood PH (Rand). The same applies to Alpha†, notably despite Hausdorff distances of Alpha† are larger.

W3 Tradeoff between #landmarks and Approximation quality

▶ Approx. quality is upper bounded by the Hausdorff distance $d_H$ between $L$ and $X$, see Cor. 4. The latter decreases monotonically with #landmarks selected by FPS. For an empirical evaluation of #landmarks vs bottleneck distance in $ℍ_2$ on a swisscheese point cloud, we refer to Fig. 4 column (d). Further, we computed $d_B$ between diagrams of $\text{Flood}(X,L_{\text{FPS}})$ and $\text{Alpha}(X)$ on the two point clouds from Sec. 5.1. using $10^k$ Landmarks for $k\in \{2.75,2.875,...,3.5\}$.

Virus

Coral

W4a Tradeoff between #Landmarks and Acc. of Flood complex across various datasets in Table 1.

▶ Below, we list classification acc / mse as a function of the #landmarks. As b9wN requested another dataset, the table includes two new columns for rocks, i.e., one binary classification task (cls) and a surface area regression task (reg). For a description of the dataset, please see our comments to b9wN.

W4b In Table 1, the improvement in Acc. (Due to Flood Complex) is negligible compared to Alpha+.

▶ Our Table 1 results show a small but consistent improvement over Alpha†, including on the new (requested by b9wN) rocks data (see table above). On closer inspection (via a paired t-test, as we evaluate on the same splits), the difference in mean between Flood PH and Alpha† PH is statistically significant (at 5%) on swisscheese, MCB and rocks, while on corals and Modelnet10 it is not.

Notably, for fair comparison, the #points used for Alpha† PH hinges on the runtime of Flood PH. In case of the latter, small implementation tweaks such as (1) allowing for larger batch sizes during Triton kernel computation (by transposing the computation grid), (2) implementing a Triton kernel for masking (see Sec. 4.2), and (3) adjusting block sizes for the GPU arch., yielded substantial runtime improvements (cf. Q1). Under these circumstances, the #points available to Alpha† decreases drastically and then all results clearly favor Flood PH (e.g., Alpha† on corals reduces to 0.63).

W4c Similarly, in Figure 3, it seems the accuracy of alpha(X) with 100k landmarks is almost the same as Flood complex with 500k landmarks...

▶ While it is true that the accuracy of Alpha(X) on 100k points is close to that of Flood(X,L), the latter is much faster (2 vs. 11s). One might expect better performance of Alpha PH with more points, but this is not necessarily the case. In fact, existing vectorization methods (including the one we use) suffer from oversmoothing effects as the number of points in the diagrams increases, resulting in less discriminative vectorizations and a drop in downstream performance. On swisscheese Alpha PH on all of $X$ yields 93% ± 2% (at 70s/point cloud) and on MCB 60% ± 3% (at 200s).

W4d In Figure 4, explain why Alpha is slower than the Flood complex?

▶ Flood complex construction (1s) is faster than Alpha complex construction (90s) for both reasons you mention, and even a pure CPU-based implementation (which we offer in our code) is much faster (17s). The complex construction uses only a subset of the points, and the rest of the data is used only for the (computationally) cheap flooding process. Our GPU implementation of the latter then yields a notable speedup over the CPU-only variant.

Dim. 2 column in Fig. 4(b) is missing.

▶ Thank you for pointing this out; we will add the values to Fig. 4(b). For the Flood complex, runtime is 0.2s. For the Alpha complex, it is 1.3s (0.8s for construction and 0.5s for persistence).

W5 Better clarification on the role of GPU in Flood complex construction...

▶ For landmark selection using FPS and the Delaunay triangulation of the landmarks, CPU-based implementations are efficient when using dedicated data structures such as $k$-d trees.

In terms of runtime, the bottlenecks are (1) the pre-selection of relevant points per simplex and (2) the distance computations, see Sec. 4.2. Both steps are speed up when done on the GPU. That being said, we support a full ($k$-d tree based) CPU version (not submitted) to compute filtration values. For a typical case of 1M points and 2k landmarks in 3D, runtimes are around 17s, compared to 1s on GPU.

▶ Q1 That is correct. PH computation time is negligible (usually, <0.1s), as PH is computed from a filtration of a comparably small complex, i.e., the Delaunay complex on the landmarks.

Could you provide a more detailed breakdown of the total time in a table?

▶ We report relative runtimes (in %) of landmark selection (FPS), complex construction, and PH computation on the vertices (ranging from 28k to 5.3M) of the coral meshes and on the swisscheese dataset (always 500k points) when using 1k and 2k landmarks. Avg runtimes are bout 1.3s (1k) and 1.8s (2k) on coral vertices, and 0.7s and 1.2s on cheese.

When #landmarks is fixed, the share of time spent on complex construction increases with the point cloud size. When increasing #landmarks, the runtime of complex construction and PH computation increases more than that of FPS sampling.

▶ Q2 The idea of a flooding process on a triangulation of a subset of a point cloud can be extended to different metrics in $R^n$, but guarantees (if there are any) are harder to show without the Voronoi-Delaunay duality. For (non-Euclidean) graphs, extending the Flood complex is difficult, as the flooding idea relies on distances between $X$ and points in the convex hull of vertices, and therefore on an ambient space.

▶ Q3 As pointed out on l198 of the paper, an upper bound is given by iterating, for each of the $m=O(|L|^{d/2})$ simplices, over the random points in $P_σ$ and the points in $X$, for a total of $O(m|P_σ||X|)$. With masking, one has $O(m|X|)$ distance computations to build the balls (see l216), and then for each simplex $O(|P_σ||\tilde X|)$, where $\tilde X\subset X$ is the set of points inside the masking ball. We will clarify this!

A small remark:

We also remark that, in most practical scenarios, the selection of the number of landmarks will be determined by computational constraints, possibly based on some validation data

Such reasoning is problematic. One may also argue the following:

"If I know that I can get $\epsilon$ guarantee in bottleneck distance by choosing, say, 10 landmarks (and if that is sufficent for my task), why would I waste computational resources by computing Flood complex with 1000 landmarks just because I can?"

I thank the authors for their efforts during the rebuttal and discussion phase. Since I do not have any more concerns left, I have raised my score to Accept.

I encourage the authors to incorporate the theoretical and empirical results, discussions, and missing related works into the final version. Good luck!

We thank you for the valuable feedback and for recognizing the importance of our work. Below, we address all weaknesses (W) and questions (Q) point by point, enumerated as in the review. We specifically clarify the availability of the appendix as part of the supplementary material ZIP file.

Summary While I would not classify the contribution of the work to be under "AI", it does make persistent homology a viable tool to then be used to study problems in AI.

▶ We understand your point, but respectfully disagree with the assertion that our work lies outside the (broad) domain of AI. In fact, we address (and remove) a key barrier to the adoption of TDA methods in modern AI/ML research, namely the inability to compute persistent homology at scale. By facilitating PH computation on millions of points, our approach allows the extraction of stable, interpretable features at practical scales for tasks where data comes with a complex geometric structure. The latter has direct implications for domains such as robotics and 3D vision, but also generative modeling (e.g., to ensure global structural fidelity) or biomedical AI (e.g., molecule analysis). From our perspective, Flood PH advances the computational and algorithmic foundations necessary to apply a theoretically rich tool (PH) in precisely these settings.

W1 A limitation of the work that I see is that it still relies on considering smaller subsets of the data, which is a natural/classical approach and takes away from the originality but nevertheless, the contribution of the work is definitely very important.

▶ We want to clarify that our method consists of two building blocks: (1) the Delaunay complex on a subset of the data, i.e., the landmarks, and (2) a filtration on it that depends on the entire set of points. Contrary to simply considering only a subset of the point cloud (which is typically done in practice), in our construction, all points contribute to the filtration values. This is beneficial as it retains information that would otherwise be lost. On the swisscheese data (Appendix, Table 2), for example, we very clearly see that Flood PH allows to distinguish the two classes, while (even) repeated subsampling and computing Alpha PH fails to extract class-specific discriminative information due to insufficient subsample size.

W2 Additionally, proposing another complex is not very original, but again, nevertheless, a solid way to workaround the problem.

▶ We understand the criticism. However, we observed that the existing complexes, even when paired with efficient/optimized PH computation (such as the ones provided by Gudhi or Ripser variants), are unable to compute PH beyond dimension 1 for large-scale problems, or for datasets with thousands of point cloud instances, as common in ML tasks. Given the unfavorable computational complexity for computing PH on large (in terms of #simplices) filtered simplicial complexes, the only viable solution is to introduce a new complex that is (1) small enough (achieved via the landmarks) to precisely control computational complexity of subsequent PH computation, and (2) amenable to construction on dedicated GPU-computing hardware. The proposed Flood complex realizes both of these points.

W3 However, the biggest weakness I see and most problematic aspect is that it appears that the appendix was not submitted and so I could not check details (and would have liked to), which I see as quite a serious oversight. This is the main reason for my borderline assessment of the paper.

▶ The appendix (as well as the code) is available in the supplementary material (i.e., a ZIP file in our case), downloadable at the top of this page. Therein, we provide proofs for all theoretical statements, as well as additional details and experimental results.

Q1 Some comparisons are carried out wrt the alpha and witness complexes, for example, which I suppose provides a satisfactory (but somewhat underwhelming) "guarantee" of the performance of the method,

▶ We primarily present experiments with the Alpha complex (including subsampling experiments in the Appendix). From a computational perspective, Alpha PH is the only viable competitor given the point cloud sizes in our experiments. Even publicly-available Witness complex implementations (e.g., in Gudhi) cannot be computed at that scale unless the number of landmarks is extremely low (around 100); see tables in our reply to reviewer Sf3t.

Q2 ... an important outstanding question ... is the correctness of the algorithm. In the experiments, persistent homology was computed on massive datasets where it was to date not possible: how can we be sure that the resulting barcodes are indeed the correct barcodes? If correctness cannot be proven, then some theoretical results on convergence of the method to the "true" barcode is a really important missing theoretical result. Are there details on this in the missing appendix? If so, they should be moved to the main body of the paper.

▶ In Secs. 3.2 and 4.1, we present several approximation guarantees for our method. Specifically, we show (Theorem 3) in $ℝ^2$, that the Flood complex is topologically (homotopy) equivalent to the Alpha complex, if the landmark set $L$ is the entire point cloud $X$. Moreover, if $L\subset X$, then the (bottleneck) distance between the persistent homology of the Alpha and Flood complex is bounded by the Hausdorff distance between $X$ and $L$, see Cor. 4. We hope that our clarification on the availability of the appendix (and all proofs therein) as part of the supplementary material ZIP file clarifies this issue.

Limitations were discussed briefly in the main body of the paper, but given the potential impact of the work, I would like to see a more honest, sober analysis and ablation study of the contribution.

▶ In Appendix A.1., we specifically address limitations. We will (1) move the key elements of this section (e.g., homotopy equivalence only proven in $ℝ^2$) to the main part of the manuscript and (2) extend it by consolidating our sensitivity analysis - requested by multiple reviewers - to provide the reader with a clear picture of how the Flood complex behaves with varying parameters.