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Detecting hyperbolic geometry in networks: why triangles are not enough
Physics and Society · Submitted 3 Jun 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces.
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The abstract, sentence by sentence
In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the past decade, geometric network models have received vast attention in the literature. | background 1.00 | 0.03 | 0.01 | asserted 1.00 |
| s1 | These models formalize the natural idea that similar vertices are likely to connect. | background 1.00 | 0.05 | 0.02 | asserted 0.93 |
| s2 | Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. | background 1.00 | 0.07 | 0.02 | asserted 0.90 |
| s3 | Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. | background 1.00 | 0.09 | 0.03 | asserted 0.80 |
| s4 | Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. | gap 0.63 | 0.10 | 0.04 | asserted 0.66 |
| s5 | Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. | background 0.97 | 0.05 | 0.03 | asserted 1.00 |
| s6 | In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. | result 0.41 | 0.92 | 0.88 | asserted 1.00 |
| s7 | We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. | contribution 0.63 | 0.94 | 0.10 | asserted 0.93 |
| s8 | We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks. | result 0.94 | 0.97 | 0.56 | asserted 1.00 |
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Original abstract
In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks.
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In its original formulation, Bringmann et al. [ref] defined the GIRG as an entire class of models, where the connection probability [ref] is only required to hold asymptotically.
Average Distance in a General Class of Scale-Free Networks with Underlying Geometry · Models · 0.95
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