The HyperKron Graph Model for Higher-Order Features

Eikmeier, Nicole; Ramani, Arjun S.; Gleich, David F.

doi:10.1109/icdm.2018.00115

Cited by 12 publications

(8 citation statements)

References 30 publications

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“…Then, once the matrix attains dimension n × n, it is used to generate a graph on n nodes in which edge (i, j) exists with probability p (f ) ij . The hypergraph generalization, called HyperKron [255], works essentially in the same way: one starts with a small k dimensional tensor P (0) and obtains a large final tensor of dimension k and n × n × .. × n. One can then use the tensor to generate a random hypergraph, with hyperedges of size k. The model has found application in generating large realistic graphs and hypergraph quickly.…”

Section: Hypergraphs Modelsmentioning

confidence: 99%

Networks beyond pairwise interactions: structure and dynamics

Battiston,

Cencetti,

Iacopini

et al. 2020

Preprint

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Section: Hypergraphs Modelsmentioning

confidence: 99%

Networks beyond pairwise interactions: structure and dynamics

Battiston,

Cencetti,

Iacopini

et al. 2020

Preprint

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“…Nevertheless, recent studies have shown that networks evolve through higherorder interactions, i.e., much of the structure in evolving networks involves interactions between more than just two nodes [10]. Moreover, random graph models constructed from distributions of triangles have shown to be good fits for realworld data [21], providing additional evidence that triadic relationships are important to the assembly of networks.…”

Section: Background and Related Workmentioning

confidence: 99%

Pairwise link prediction

Nassar

Benson

Gleich

2019

Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

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Link prediction is a common problem in network science that transects many disciplines. The goal is to forecast the appearance of new links or to find links missing in the network. Typical methods for link prediction use the topology of the network to predict the most likely future or missing connections between a pair of nodes. However, network evolution is often mediated by higher-order structures involving more than pairs of nodes; for example, cliques on three nodes (also called triangles) are key to the structure of social networks, but the standard link prediction framework does not directly predict these structures. To address this gap, we propose a new link prediction task called "pairwise link prediction" that directly targets the prediction of new triangles, where one is tasked with finding which nodes are most likely to form a triangle with a given edge. We develop two PageRank-based methods for our pairwise link prediction problem and make natural extensions to existing link prediction methods. Our experiments on a variety of networks show that diffusion based methods are less sensitive to the type of graphs used and more consistent in their results. We also show how our pairwise link prediction framework can be used to get better predictions within the context of standard link prediction evaluation.

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“…Then add an edge connecting u to v. PA is meant to model the power-law behavior that is often seen in real-world networks [Faloutsos et al, 1999;Huberman, 2001;Medina et al, 2000], that is a few vertices tend to have very large degree while most vertices have fairly low degree. A set of values x 1 , x 2 , .…”

Section: P R E F E R E N T I a L At Ta C H M E N Tmentioning

confidence: 99%

“…For example the triad formation model described in Section 2.3 [Holme and Kim, 2002], and the family of PA models [Ostroumova et al, 2013] discussed in Section 1. Another model, HyperKron, places a distribution over hyperedges and inserts motifs instead of edges [Eikmeier et al, 2018] and is specifically shown to have higher order clustering.…”

Section: H I G H E R O R D E R F E At U R E S I N G R a P H Smentioning

confidence: 99%

Triangle Preferential Attachment Has Power-law Degrees and Eigenvalues; Eigenvalues Are More Stable to Network Sampling

Eikmeier,

Gleich

2019

Preprint

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Preferential attachment models are a common class of graph models which have been used to explain why power-law distributions appear in the degree sequences of real network data. One of the things they lack, however, is higher-order network clustering, including non-trivial clustering coefficients. In this paper we present a specific Triangle Generalized Preferential Attachment Model (TGPA) that, by construction, has nontrivial clustering. We further prove that this model has a power-law in both the degree distribution and eigenvalue spectra. We use this model to investigate a recent finding that power-laws are more reliably observed in the eigenvalue spectra of real-world networks than in their degree distribution. One conjectured explanation for this is that the spectra of the graph is more robust to various sampling strategies that would have been employed to collect the real-world data compared with the degree distribution. Consequently, we generate random TGPA models that provably have a power-law in both, and sample subgraphs via forest fire, depth-first, and random edge models. We find that the samples show a power-law in the spectra even when only 30% of the network is seen. Whereas there is a large chance that the degrees will not show a power-law. Our TGPA model shows this behavior much more clearly than a standard preferential attachment model. This provides one possible explanation for why power-laws may be seen frequently in the spectra of real world data.

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The HyperKron Graph Model for Higher-Order Features

Cited by 12 publications

References 30 publications

Networks beyond pairwise interactions: structure and dynamics

Networks beyond pairwise interactions: structure and dynamics

Pairwise link prediction

Triangle Preferential Attachment Has Power-law Degrees and Eigenvalues; Eigenvalues Are More Stable to Network Sampling

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