TY - JOUR
T1 - Detecting hyperbolic geometry in networks
T2 - Why triangles are not enough
AU - Michielan, Riccardo
AU - Litvak, Nelly
AU - Stegehuis, Clara
N1 - Publisher Copyright:
© 2022 American Physical Society.
PY - 2022/11
Y1 - 2022/11
N2 - 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 scale 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 differerent statistic, weighted triangles, which weighs triangles based on their evidence for geometry. We show analytically, as well as on synthetic and real-world data, that weighted triangles are a powerful statistic to detect hyperbolic geometry in networks.
AB - 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 scale 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 differerent statistic, weighted triangles, which weighs triangles based on their evidence for geometry. We show analytically, as well as on synthetic and real-world data, that weighted triangles are a powerful statistic to detect hyperbolic geometry in networks.
KW - 2023 OA procedure
UR - http://www.scopus.com/inward/record.url?scp=85142144952&partnerID=8YFLogxK
U2 - 10.1103/PhysRevE.106.054303
DO - 10.1103/PhysRevE.106.054303
M3 - Article
C2 - 36559472
AN - SCOPUS:85142144952
VL - 106
JO - Physical review E: covering statistical, nonlinear, biological, and soft matter physics
JF - Physical review E: covering statistical, nonlinear, biological, and soft matter physics
SN - 2470-0045
IS - 5
M1 - 054303
ER -