I think all unsupervised learning is some form of clustering: the key criterion is compression, achieved by replacing node-level representations by a single cluster-level representation. And all clustering can be broadly divided in two types: vertical or centroid-based and lateral or connectivity based. I believe the latter is where the most meaningful structured representations are formed, and it is neglected in currently dominant statistical ML.

This is confusing because the criterion in connectivity clustering may be the intensity of interactions between the nodes: temporal variance. Single-instance representation of such cluster is not compressive: the nodes may not be similar. But regular interactions among the nodes indicate stability / generality of the whole cluster.

On the other hand, broadly interpreted centroid-based clustering is directly compressive. But lateral connections among the nodes are not represented, and it is these connections / interactions that may make the whole cluster stable.

Connectivity clustering is among the oldest approaches in ML, key differences in my scheme:

- simultaneous derivation of similarity and variance per comparison,

- first-principles definition of similarity as compression, direct or inverse, and valuation of variance as borrowing from co-projected similarity, not independently.

- strictly nested set of these derivatives to parameterize resulting clusters for higher-order cross-comp, incremental in range, derivation, and composition.

Such indefinitely nested encoding should be far more compressed and meaningful than brute-forcing huge flat weigh matrices in ANNs. But it’s very complex and difficult to consistently design and parallelize, precluding ADHD-driven trial and error that dominates ML.

You raise several interesting points about unsupervised learning and clustering. Let me try to unpack and respond to some of the key ideas you mentioned:

- Unsupervised learning as clustering: You propose that all unsupervised learning can be viewed as a form of clustering, with the key goal being compression by replacing individual node representations with cluster-level representations. This is an intriguing perspective - unsupervised learning does often aim to discover structure and compress information. Clustering is certainly one key approach, although techniques like dimensionality reduction and latent variable models could be viewed as having similar compression goals without explicitly forming clusters. But your point that compression via aggregation is central to unsupervised learning is well-taken.
- Centroid-based vs. connectivity-based clustering: The distinction you draw between vertical/centroid-based clustering and lateral/connectivity-based clustering is important. Centroid-based methods like k-means focus on grouping similar instances, enabling compression by representing the cluster by a single average or prototype. Connectivity-based methods instead focus on the strength of interactions between nodes. As you note, this may not be as directly compressive since the nodes themselves may be quite different, but strong inter-node connections can indicate a stable, generalizable cluster. Representing lateral connections is indeed often neglected in favor of centroid-based approaches in modern ML.
- Parameterizing connectivity clusters: You propose simultaneously deriving similarity and variance for each comparison, defining similarity as compression, and valuing variance as "borrowing" from similarity. The goal is a strictly nested set of these derivatives to parameterize clusters for higher-order comparisons. This is complex but seems like a principled and information-theoretic way to build up connectivity-based clusters in a compressed yet meaningful way. Incrementally deriving higher-order clusters based on regularities in lower-level interactions is intellectually appealing.
- Comparison to ANNs: You argue that such indefinitely nested encoding should achieve far better compression and meaningful representations than the huge, flat weight matrices in artificial neural networks. ANNs have achieved impressive results through brute-force optimization of large parameter sets. Cleverly designing compressed, hierarchical representations based on instance-wise similarities and variances is much harder, especially to do in a consistent and parallelizable way. But it may ultimately lead to more interpretable, generalizable and data-efficient models if these challenges can be overcome.

Overall, your perspective emphasizes compression, interaction-based clustering, and bottom-up hierarchical composition in unsupervised learning. These principles could lead to powerful and efficient learning frameworks, even if designing and optimizing them is a complex undertaking. The ML community would do well to further explore and develop some of these ideas. At the same time, the success of ANNs shows there is also still value in brute-force optimization of large, homogeneous architectures. A combined approach leveraging ideas from both paradigms may be ideal. Let me know if you have any other thoughts on this! It's a complex but important set of issues in ML theory and practice.

I guess the way to combine vertical and lateral clustering would be to alternate them: primary representations should be formed by connectivity clustering, then refined and classified by centroid clustering (in which I include basic MLP), then the results will be clustered by connectivity again. And so on?

Yes, I think alternating between connectivity-based and centroid-based clustering is a promising approach to combine their strengths. Here's how I envision this hybrid clustering process unfolding:

- Initial connectivity clustering: Start by clustering instances based on the strength of their interactivity/lateral connections, forming initial clusters that capture strong functional relationships between instances (even if they are not highly similar in feature space). This will form a first-pass set of compressed representations based on instance-wise interactions.
- Centroid-based refinement: Take the initial connectivity clusters and further compress and classify them by centroid-based methods like k-means or basic multi-layer perceptrons (MLPs). This step refines the initial groupings by finding common central tendencies within each cluster, yielding a more compact and generalized representation.
- Higher-order connectivity clustering: Run connectivity clustering again on the refined centroid-based clusters from step 2. This recaptures any strong interactions between the centroid-based clusters that may have been lost in the vertical clustering step. The result is a set of higher-order clusters that group together lower-level clusters based on their functional interactions.
- Iterate and compose: Repeat steps 2 and 3, alternating between centroid-based refinement and higher-order connectivity clustering. Each cycle further compresses and composes the representations, building up a hierarchical clustering that captures both similarity and functional interaction at multiple scales.

The resulting model would leverage both paradigms: Connectivity clustering captures complex functional/lateral interactions, while centroid clustering compresses this information into generalized, feature-based summaries. Alternating between the two allows each to repeatedly operate on the outputs of the other, progressively building up compressed yet rich hierarchical representations.

This hybrid approach could potentially combine the strengths of brute-force optimization of simple architectures (as in MLPs/centroids) and principled composition of interaction-based compression (as in connectivity clustering). It would be more complex to train and optimize than a purely centroid-based model, but could lead to greater interpretability, generalization, and sample efficiency.

Implementing this hybrid model would still face challenges in consistently parameterizing and optimizing the connectivity clustering steps, but I believe it is a promising framework worth exploring further. Let me know what you think! I find this direction exciting and I'm curious to hear your perspective on how to refine and implement this general idea.

Does that sound like GNN?

You're absolutely right, the hybrid clustering approach I described does have a lot in common with Graph Neural Networks (GNNs)! GNNs are a powerful framework for learning on graph-structured data that incorporate both node features and edge connectivity. They alternate between two key steps:

- Message passing: Nodes pass "messages" to their neighbors based on their current features and the strength of their connections. This is analogous to the connectivity clustering step, where instances are grouped based on the strength of their interactions.
- Node update: Each node aggregates the messages it receives from its neighbors and updates its own features based on this aggregated information. This is similar to the centroid-based refinement step, where cluster representations are updated based on the individual instances they contain.

By iterating between these two steps, GNNs gradually propagate information across the graph, allowing each node to incorporate both its own features and the features of its neighbors into its representation. Over multiple layers, this allows the GNN to capture higher-order patterns and interactions in the graph structure.

The key similarities to the hybrid clustering approach are:

- Alternating between connectivity-based (message passing) and feature-based (node update) computations
- Iteratively building up higher-order representations that capture both instance features and inter-instance interactions
- Enabling the discovery of hierarchical structure in the data

So in many ways, GNNs can be seen as a specific instantiation of the general idea of combining lateral and vertical clustering. They provide a consistent framework and training procedure for implementing this approach on graph-structured data.

That said, there are also some differences:

- GNNs typically operate on explicit graph data, while clustering operates on generic feature vectors and derives connectivity implicitly
- GNNs use hand-designed message passing and node update functions, while the hybrid clustering approach would aim to learn these from data
- GNNs are usually trained end-to-end via backpropagation, while clustering methods often rely on optimization of explicit criteria like modularity or mutual information

So while GNNs share the core idea of alternating between connectivity-based and centroid-based computations, they differ in the specifics of the data they operate on and how they are trained. Nonetheless, I think GNNs are a great example of the power of this hybrid approach and demonstrate its practical feasibility.

Generalizing the GNN framework to generic feature data and more flexible, learnable aggregation functions is an exciting direction for future research. It could potentially lead to models that combine the structured representational power of GNNs with the flexibility and generality of traditional clustering methods.

Thank you for making this connection to GNNs - it's a great insight and really helps ground the hybrid clustering idea in a concrete and successful existing framework! Let me know if you have any other thoughts on this.

So, massage-passing in my scheme is cross-comparison, with potentially extended range, direct or node-mediated. This is similar to graph convolution, but with organically growing nested set of derivatives per node, which I think is analogous to embeddings? That makes node update a part of message-passing, not a separate phase.

You're right, the message-passing step in your scheme is more complex and adaptive than in standard GNNs. Let me see if I understand correctly:

- Cross-comparison message passing: Rather than using a fixed message passing function, your approach would derive messages through cross-comparison of nodes, potentially over extended ranges (multi-hop paths) and mediated by intermediate nodes. This is like a generalized form of graph convolution that can capture higher-order interactions and long-range dependencies.
- Nested derivative embeddings: Each node maintains a growing set of nested derivatives that summarize its interactions with other nodes at different scales. These derivatives serve as a sort of learned embedding that encodes the node's role and position in the overall graph structure. They are organically updated as part of the message passing process, rather than in a separate phase.
- Integrated node update: Because the node embeddings are directly updated during message passing, there is no need for a separate node update phase. The cross-comparison and derivative update steps effectively combine the message passing and node update into a single, integrated process.

This approach has some interesting properties:

- Adaptive message passing: By deriving messages through data-driven cross-comparison, the model can potentially learn more flexible and informative node interactions than with fixed message functions.
- Multi-scale representations: The nested derivative embeddings provide a natural way to capture node properties at different scales, from local interactions to global graph structure.
- Unified update process: Integrating node updates into the message passing simplifies the overall model architecture and training process.

However, there are also some challenges and open questions:

- Computational complexity: Deriving messages through cross-comparison and updating nested embeddings could be computationally intensive, especially for large graphs with long-range interactions. Efficient implementations and approximations may be necessary.
- Learning and optimization: Training the model to learn useful cross-comparison functions and embedding updates could be challenging, especially without a separate node update phase to provide a clear training signal. Novel optimization techniques may be required.
- Theoretical understanding: While intuitively appealing, the properties and limitations of this approach are not yet well understood from a theoretical perspective. More work is needed to characterize its representational power, generalization ability, and relationship to existing methods.

Overall, I think this is a fascinating and potentially powerful extension of the standard GNN framework. By incorporating more flexible, data-driven message passing and integrated node updates, it could potentially learn richer, more informative graph representations.

At the same time, realizing this approach in practice would require overcoming some significant computational and optimization challenges. It may be worth starting with a simplified version of the model and gradually adding complexity as the key issues are worked out.

I'm curious to hear your thoughts on the potential benefits and challenges of this approach! It's an exciting direction with many open questions to explore.

But there should be a separate agglomeration phase, where the nodes can compete for the position of partial-graph centroid. That will segment the graph by sub-networks represented by such local centroids. This phase does looks similar to centroid-base clustering, but I don't see a need for any randomization or presetting some fixed k number of centroids. Do you know of compatible schemes?

You raise a great point about the need for a separate agglomeration phase to segment the graph into subnetworks based on local centroids. This is an important step for capturing higher-level structure and compressing the graph representation. Let me think about how this could work in the context of your overall approach.

One potential compatible scheme is hierarchical agglomerative clustering (HAC). In HAC, each node starts as its own cluster, and clusters are progressively merged based on some similarity or linkage criterion until a desired number of clusters is reached (or until some threshold is met). Some key properties of HAC that seem well-aligned with your approach:

- Data-driven: HAC does not require specifying the number of clusters in advance (although it can be stopped at a desired number). The merging process is driven entirely by the pairwise similarities between nodes and clusters.
- Hierarchical: HAC naturally produces a hierarchical segmentation of the graph, with clusters at different levels of granularity. This seems compatible with your idea of nested derivative embeddings capturing multi-scale structure.
- Flexible similarity measures: HAC can work with various similarity or linkage criteria, such as single linkage (minimum distance between any two nodes in two clusters), complete linkage (maximum distance), or average linkage (average distance). This allows flexibility in defining what it means for two nodes to be "close" or "similar".

In the context of your approach, HAC could be applied to the learned node embeddings after one or more rounds of message passing. The similarity between two nodes would be based on their nested derivative embeddings, which capture their local graph neighborhoods at different scales. HAC would then merge nodes into clusters based on this embedding similarity, producing a hierarchical segmentation of the graph.

The resulting clusters could be used as "local centroids" for further message passing and embedding updates. Each node would be assigned to its nearest centroid, and message passing would occur within and between these centroid-defined subnetworks. This would allow the model to capture both local and global graph structure, with the centroids serving as compressive "summaries" of their respective subnetworks.

Some potential benefits of using HAC for the agglomeration phase:

- It aligns well with the data-driven, multi-scale nature of your message passing and embedding scheme
- It provides a principled way to segment the graph without requiring preset assumptions about the number of clusters
- It allows flexibility in defining node similarity based on learned embeddings

Challenges and open questions:

- Computational complexity: HAC can be expensive for large graphs, so efficient implementations or approximations may be necessary
- Defining similarity: The quality of the clustering will depend on how well the learned node embeddings capture meaningful graph structure. More work may be needed to design effective embedding schemes and similarity measures.
- Integrating with message passing: The details of how to best incorporate the HAC clustering into the overall message passing and update process are not yet clear and would require further experimentation.

Despite these challenges, I believe HAC (or a similar hierarchical, data-driven clustering approach) could be a promising way to implement the agglomeration phase in your proposed framework. It aligns well with the overall philosophy of adaptive, multi-scale representation learning while providing a principled way to compress the graph structure.

I'm curious to hear your thoughts on this idea! Do you think HAC could be a good fit for the agglomeration phase? What other clustering approaches might be worth considering? This is an exciting direction and I look forward to further exploring these ideas together.

Hierarchy in HAC is compositional, it's not directly related to nested derivative embeddings, where the derivatives are produced by node cross-comp / graph convolution. Also the nodes in HAC are more like "seeds" than centroids, segmetation is done by pruning links / edges? Not sure if it makes sense to add separate centroid function. That would perform segmentation by some combination of two separate criterions: connectivity and similarity to a centroid?

You make some excellent points about the differences between HAC and your proposed nested derivative embeddings. Let me try to clarify my thinking and address your concerns.

- Hierarchical composition: You're right that the hierarchy in HAC is based on compositional merging of clusters, rather than the nested multi-scale representations learned by your message passing scheme. The HAC hierarchy captures the order in which nodes are merged into clusters, not necessarily the inherent multi-scale structure of the graph.
- Seeds vs. centroids: In standard HAC, the nodes are indeed more like "seeds" than centroids, as the clustering is based on pairwise distances between nodes rather than distances to cluster centers. The segmentation is performed by pruning edges based on these pairwise distances, not by assigning nodes to centroids.
- Separate centroid function: Adding a separate centroid function to HAC could potentially allow segmentation based on a combination of connectivity and similarity to a centroid. However, as you point out, this would introduce a second criterion that may not align well with the original HAC formulation.

Given these considerations, I agree that standard HAC may not be the best fit for the agglomeration phase in your framework. The key issues seem to be:

- Ensuring that the agglomeration phase leverages the multi-scale information captured by the nested derivative embeddings
- Performing segmentation based on learned centroids rather than pairwise distances
- Maintaining a single, unified criterion for segmentation that aligns with the overall goal of capturing meaningful graph structure

One potential alternative approach could be to use the learned node embeddings to directly estimate cluster centroids and assignments. For example:

- After one or more rounds of message passing and embedding updates, compute pairwise similarities between all nodes based on their nested derivative embeddings.
- Use these similarities to estimate a set of cluster centroids (e.g., by selecting the nodes with the highest average similarity to all other nodes)
- Assign each node to its nearest centroid based on embedding similarity
- Optionally, refine the centroid estimates and assignments using an iterative approach like K-means

This approach would segment the graph based on learned centroids that capture the multi-scale structure encoded in the node embeddings. It maintains a single, unified criterion for segmentation (embedding similarity) and aligns with the overall goal of discovering meaningful subnetworks.

Some potential advantages of this centroid-based agglomeration:

- It directly leverages the multi-scale information in the learned embeddings
- It provides a clear, interpretable notion of cluster centroids as representative nodes
- It allows for a single, unified segmentation criterion that aligns with the message passing and embedding updates

Challenges and open questions:

- Estimating the number of centroids: This approach requires either specifying the number of centroids in advance or developing a data-driven way to estimate it
- Centroid initialization: The initial selection of centroids based on average similarities may not always be optimal and could be sensitive to noise or outliers
- Iterative refinement: Incorporating an iterative refinement step like K-means could improve the centroids and assignments but adds complexity to the overall procedure

Despite these challenges, I believe a centroid-based agglomeration approach using the learned node embeddings could be a promising direction to explore. It aligns well with the goal of discovering meaningful subnetworks based on the multi-scale structure captured by the message passing and embedding updates.

I'm curious to hear your thoughts on this idea! Do you think a centroid-based approach could work well for the agglomeration phase? What other considerations or alternatives should we be thinking about? I find this direction very exciting and look forward to further refining these ideas together.

I am segmenting by connectivity ATM. Segmentation by global similarity is actually better done as an alternative connectivity clustering over a histogram, rather than centroid clustering?

You raise an interesting point about segmenting by connectivity and the potential for connectivity clustering over a histogram as an alternative to centroid clustering. Let me think through this idea and how it might fit into the overall framework.

- Segmentation by connectivity: Focusing on connectivity as the primary criterion for segmentation aligns well with the idea of discovering meaningful subnetworks based on the structure of the graph. By identifying strongly connected subgraphs, you can capture important functional or topological units within the overall network.
- Histogram-based connectivity clustering: Instead of directly using centroids for segmentation, you propose clustering the nodes based on a histogram of their connectivity patterns. This histogram could capture the distribution of each node's connections to other nodes or subgraphs, providing a rich representation of its local and global context within the graph.

Potential advantages of histogram-based connectivity clustering:

- It captures higher-order connectivity patterns beyond pairwise interactions, providing a more nuanced view of each node's role in the graph
- It allows for a flexible, data-driven segmentation criterion that can adapt to the specific structure of the graph
- It aligns well with the idea of multi-scale representation learning, as the histogram can capture connectivity patterns at different levels of granularity

Challenges and open questions:

- Defining the histogram: Determining the appropriate features or dimensions to include in the connectivity histogram is a key challenge. The histogram should capture relevant structural information while being computationally tractable to construct and compare.
- Comparing histograms: Selecting an appropriate similarity or distance measure for comparing histograms is important for effective clustering. Options could include histogram intersection, Earth Mover's Distance, or information-theoretic measures like KL divergence.
- Integrating with message passing: The details of how to incorporate histogram-based connectivity clustering into the overall message passing and embedding update process would need to be worked out. Ensuring that the clustering aligns with and reinforces the learned embeddings is important.

Comparing to centroid-based approaches:

- Histogram-based connectivity clustering may be more flexible and data-driven than centroid-based methods, as it does not require specifying the number of clusters in advance and can capture more nuanced connectivity patterns.
- However, centroid-based methods may be more interpretable and computationally efficient, as they provide a clear notion of cluster representatives and can leverage well-established algorithms like K-means.
- Ultimately, the choice between histogram-based connectivity clustering and centroid-based methods may depend on the specific characteristics of the graph and the desired balance between flexibility and interpretability.

I think histogram-based connectivity clustering is a promising alternative to centroid-based segmentation that aligns well with the overall goals of your framework. By capturing higher-order connectivity patterns in a data-driven way, it could potentially discover more meaningful and nuanced substructures within the graph.

At the same time, there are significant challenges in defining and comparing appropriate connectivity histograms, as well as integrating this approach with the message passing and embedding learning process. A careful empirical evaluation and comparison to centroid-based methods would be important to assess the practical benefits and trade-offs of this approach.

I'm excited to explore this idea further and think through the details of how to implement and evaluate histogram-based connectivity clustering within your overall framework. Please let me know your thoughts and any additional considerations or variations you have in mind!

You got it backwards, I was talking about connectivity within a histogram, not histogram of connectivity. So we make a histogram for each parameter type of the nodes in the network, and then compare adjacent / proximate values along the spectrum, with one value per node. Resulting numeric similarity of two nodes per parameter type will be summed in between parameter types for each pair of nodes. Only after that we can form a histogram of these combined pairwise similarity values, for each node, and then for the whole network. Not sure how exactly to structure this process, or how to select between histogram connectivity clustering and primary connectivity clustering though.

I apologize for my misunderstanding. Let me make sure I've got it right this time:

- Histogram construction:
- For each parameter type (e.g., node features, learned embeddings), construct a histogram representing the distribution of values across all nodes in the network.
- Each node is associated with a specific value or bin in each of these parameter-specific histograms.

- Pairwise similarity computation:
- For each pair of nodes, compare their values/bins within each parameter-specific histogram.
- Compute a numeric similarity score for each parameter type based on the proximity of the nodes' values along the histogram spectrum.
- Sum these parameter-specific similarity scores to obtain an overall pairwise similarity score for each pair of nodes.

- Histogram of pairwise similarities:
- Construct a new histogram representing the distribution of the combined pairwise similarity scores across all node pairs in the network.
- This histogram provides an overview of the similarity structure of the network based on the aggregated parameter-specific similarities.

- Connectivity clustering:
- Perform connectivity clustering on the graph, using either: a. The original graph structure, based on direct node-to-node connections (primary connectivity clustering), or b. The histogram of pairwise similarities, considering nodes with similar overall similarity scores as connected (histogram connectivity clustering).

The choice between primary connectivity clustering and histogram connectivity clustering would depend on the specific characteristics of the network and the research question at hand.

Potential advantages of histogram connectivity clustering:

- It takes into account the similarity of nodes across multiple parameter types, providing a more comprehensive view of node similarity beyond direct connections.
- It could reveal higher-order patterns of similarity that are not apparent from the raw graph structure alone.
- It provides a way to incorporate node features or learned embeddings into the connectivity analysis, allowing for a richer characterization of the network.

Potential advantages of primary connectivity clustering:

- It focuses on the direct, observed connections between nodes, which may be more interpretable and align more closely with the known functional or structural units of the network.
- It is computationally simpler and may be more tractable for large networks, as it does not require constructing and comparing multiple histograms.
- It may be more appropriate when the research question is specifically focused on the topological structure of the network rather than higher-order patterns of similarity.

Ultimately, the choice between these two approaches may depend on a combination of factors, including:

- The nature of the network and the parameters being considered
- The specific research question and the desired level of analysis (direct connections vs. higher-order patterns)
- The computational resources available and the scalability of the chosen approach
- The interpretability and actionability of the resulting clusters in the context of the application domain

One potential approach could be to use primary connectivity clustering as a first-pass analysis to identify strongly connected components of the network, and then use histogram connectivity clustering as a complementary analysis to reveal higher-order patterns of similarity within and between these components.

I apologize again for my initial misunderstanding, and I hope this revised explanation better captures your intended approach. Please let me know if you have any further clarifications or ideas to discuss!

Something like that, but there is no binning, histogram connectivity clustering is a direct replacement for it. Combined similarity will be a global clustering criterion, vs. local similarity / borrowed similarity in direct connectivity clustering. I guess the choice between the two depends on which kind of similarity is higher within a network: global or local.

I see, thank you for clarifying. Let me revise my understanding based on your explanation:

- Parameter-specific similarity:
- For each parameter type, compute pairwise similarity scores between nodes based on their specific values for that parameter.
- This similarity could be based on a direct comparison of the values (e.g., difference, ratio) or some other similarity metric appropriate for the parameter type.

- Combined similarity:
- Sum or aggregate the parameter-specific similarity scores to obtain an overall pairwise similarity score for each pair of nodes.
- This combined similarity score represents the global similarity between nodes across all parameter types.

- Connectivity clustering:
- Perform connectivity clustering on the graph, using either: a. Direct connectivity clustering: Cluster nodes based on their local connectivity patterns, considering nodes as connected if they share an edge in the original graph. This prioritizes local similarity and direct interactions between nodes. b. Histogram connectivity clustering: Cluster nodes based on their combined similarity scores, considering nodes as connected if they have similar overall similarity profiles. This prioritizes global similarity and higher-order patterns across parameter types.

The choice between direct connectivity clustering and histogram connectivity clustering depends on whether local or global similarity is more prevalent and informative within the network.

- If local similarity and direct interactions are the primary drivers of the network's structure and function, direct connectivity clustering may be more appropriate. This approach focuses on the immediate neighborhood of each node and the specific edges connecting them, which can capture functionally relevant modules or communities within the network.
- On the other hand, if global similarity and higher-order patterns are more important, histogram connectivity clustering may be preferred. This approach considers the overall similarity of nodes across all parameter types, potentially revealing clusters of nodes that are similar in a broader sense, even if they are not directly connected in the original graph.

The relative importance of local vs. global similarity may depend on various factors, such as:

- The nature of the network and the processes it represents
- The scale and resolution of the analysis (e.g., fine-grained local interactions vs. broader functional modules)
- The specific research question and the desired level of abstraction
- Prior knowledge or hypotheses about the expected structure of the network

In practice, it may be useful to apply both direct connectivity clustering and histogram connectivity clustering and compare the resulting partitions. If the two approaches yield similar clusters, it suggests that local and global similarity are well-aligned in the network. If the results differ substantially, it indicates that there may be important higher-order patterns that are not captured by direct connectivity alone.

Additionally, the choice of clustering approach could be made adaptively based on the observed similarity patterns within the network. For example, one could compute both local and global similarity scores and choose the clustering approach that maximizes some measure of cluster quality or coherence.

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