A New Measure of Modularity in Hypergraphs: Theoretical Insights and Implications for Effective Clustering

Kumar, Tarun; Vaidyanathan, Sankaran; Ananthapadmanabhan, Harini; Parthasarathy, S.; Ravindran, Balaraman

doi:10.1007/978-3-030-36687-2_24

Cited by 14 publications

(23 citation statements)

References 21 publications

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“…We began by applying a higher-order generalization of the standard community detection approach using a hypergraph modularity maximization algorithm [33]. this algorithm partitions collections of (potentially overlapping) sets of nodes called hyperedges into communities that have a high degree of internal integration and a lower A-B.…”

Section: B Characterizing Higher-order Brain Structuresmentioning

confidence: 99%

Partial entropy decomposition reveals higher-order structures in human brain activity

Varley¹,

Pope²,

Puxeddu³

et al. 2023

Preprint

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The standard approach to modeling the human brain as a complex system is with a network, where the basic unit of interaction is a pairwise link between two brain regions. While powerful, this approach is limited by the inability to assess higher-order interactions involving three or more elements directly. In this work, we present a method for capturing higher-order dependencies in discrete data based on partial entropy decomposition (PED). Our approach decomposes the joint entropy of the whole system into a set of strictly non-negative partial entropy atoms that describe the redundant, unique, and synergistic interactions that compose the system's structure. We begin by showing how the PED can provide insights into the mathematical structure of both the FC network itself, as well as established measures of higher-order dependency such as the O-information. When applied to resting state fMRI data, we find robust evidence of higher-order synergies that are largely invisible to standard functional connectivity analyses. This synergistic structure is symmetrical across hemispheres, largely conserved across individual subjects, and is distinct from structural features based on redundancy that have previously dominated FC analyses. Our approach can also be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies and synergies change over the course of a recording. We find that different ensembles of regions can transiently change from being redundancy-dominated to synergy-dominated, and that the temporal pattern is structured in time. These results provide strong evidence that there exists a large space of unexplored structures in human brain data that have been largely missed by a focus on bivariate network connectivity models. This synergistic "shadow structures" is dynamic in time and, likely will illuminate new and interesting links between brain and behavior. Beyond brain-specific application, the PED provides a very general approach for understanding higher-order structures in a variety of complex systems.

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Section: B Characterizing Higher-order Brain Structuresmentioning

confidence: 99%

Partial entropy decomposition reveals higher-order structures in human brain activity

Varley¹,

Pope²,

Puxeddu³

et al. 2023

Preprint

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“…There are some recent attempts to deal with hypergraphs in the context of graph clustering. Kumar et al [142,143] still reduce the problem to graphs but use the original hypergraphs to iteratively adjust weights to encourage some hyperedges to be included in some cluster but discourage other ones. Moreover, in [129] a number of extensions of the classic null model for graphs are proposed that can potentially be used by true hypergraph algorithms.…”

Section: Potential Research Direction # 10 (Hypergraph Modularity Fun...mentioning

confidence: 99%

Survey of Generative Methods for Social Media Analysis

Matwin¹,

Milios²,

Prałat³

et al. 2021

Preprint

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“…D v ∈ R n×n , D e ∈ R m×m and W ∈ R m×m are the diagonal matrices containing node degrees, hyperedge degrees and hyperedge weights respectively. Then the adjacency matrix of hypergraph G is defined as [13]: A = HW H T − D v Node Degree Preserving Reduction [21]: During clique reduction, the node degree of a vertex v is over counted by a factor of (δ (e) − 1) for each hyperedge containing v. To preserve the node degree, we can scale down each w(e) by a factor of (δ (e) − 1). This results in the following adjacency matrix,…”

Section: Hypergraphsmentioning

confidence: 99%

“…δ (e)−1 (this is equal to A ndp (x, y) [21]). Figure 1 illustrates the direct transfer of resource between nodes in a toy hypergraph.…”

Section: Hypergraph Resource Allocation (Hra)mentioning

confidence: 99%

HPRA: Hyperedge Prediction using Resource Allocation

Kumar,

Darwin,

Parthasarathy

et al. 2020

Preprint

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Many real-world systems involve higher-order interactions and thus demand complex models such as hypergraphs. For instance, a research article could have multiple collaborating authors, and therefore the co-authorship network is best represented as a hypergraph. In this work, we focus on the problem of hyperedge prediction. This problem has immense applications in multiple domains, such as predicting new collaborations in social networks, discovering new chemical reactions in metabolic networks, etc. Despite having significant importance, the problem of hyperedge prediction hasn't received adequate attention, mainly because of its inherent complexity. In a graph with n nodes the number of potential edges is O(n 2 ), whereas in a hypergraph, the number of potential hyperedges is O(2 n ). To avoid searching through the huge space of hyperedges, current methods restrain the original problem in the following two ways. One class of algorithms assume the hypergraphs to be k-uniform where each hyperedge can have exactly k nodes. However, many real-world systems are not confined only to have interactions involving k components. Thus, these algorithms are not suitable for many real-world applications. The second class of algorithms requires a candidate set of hyperedges from which the potential hyperedges are chosen. In the absence of domain knowledge, the candidate set can have O(2 n ) possible hyperedges, which makes this problem intractable. More often than not, domain knowledge is not readily available, making these methods limited in applicability. We propose HPRA -Hyperedge Prediction using Resource Allocation, the first of its kind algorithm, which overcomes these issues and predicts hyperedges of any cardinality without using any candidate hyperedge set. HPRA is a similarity-based * Both authors contributed equally to this research. method working on the principles of the resource allocation process. In addition to recovering missing hyperedges, we demonstrate that HPRA can predict future hyperedges in a wide range of hypergraphs. Our extensive set of experiments shows that HPRA achieves statistically significant improvements over state-of-the-art methods.

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A New Measure of Modularity in Hypergraphs: Theoretical Insights and Implications for Effective Clustering

Cited by 14 publications

References 21 publications

Partial entropy decomposition reveals higher-order structures in human brain activity

Partial entropy decomposition reveals higher-order structures in human brain activity

Survey of Generative Methods for Social Media Analysis

HPRA: Hyperedge Prediction using Resource Allocation

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