Fig 1.
Geometric comparison and properties of hyperbolic space.
(A) Comparison of geometric proximity between Euclidean and hyperbolic spaces. In Euclidean space, points A′ and B′ located on different branches of the diagram appear closer to each other, whereas in hyperbolic space, points A and B on different branches exhibit distances that more accurately reflect their hierarchical relationship. (B) Characteristics of hyperbolic space and its tangent plane. The diagram illustrates two lines, e1 and e2, passing through point c that never intersect, demonstrating the deviation from the fifth postulate of Euclidean geometry.
Table 1.
Comparison of MOHGCAA with related models.
Table 2.
The notation used in this work.
Fig 2.
Depictions of three hyperbolic graphical models: the Lorentz model, the Kälin model, and the Poincaré ball model.
Fig 3.
The operation maps data from Euclidean space to hyperbolic space, whereas the
operation maps data from hyperbolic space to its Euclidean tangent plane.
Fig 4.
Upon initialization of the data phase in hyperbolic space, it is projected into the tangent plane of its o-points via the function, and subsequent to the aggregation manipulate, it is implicitly re-mapped into the new hyperbolic space using the
function.
Fig 5.
The overall framework of this study: First, the data representation and its corresponding adjacency matrix are obtained.
Next, is mapped into hyperbolic space via the function, producing its hyperbolic representation. Then, the
function is applied to project onto its tangent space, yielding the representation
. On this basis, a multi-order graph convolution network is employed to derive the multi-order representation
. Subsequently, an attention-based network is used to generate the hyperbolic multi-order graph representation
. Finally, the convolution attention representation
is mapped into a new hyperbolic space through the
function, resulting in the hyperbolic representation
.
Fig 6.
The framework of the multi-order unsupervised hyperbolic graph convolution and aggregated attention for social event detection (MOUHGCAASED) model.
represents the obtained node feature,
signifies the augmented node feature of
,
and
indicate the hyperbolic feature following hyperbolic multi-order aggregation,
and
denote the Euclidean spatial feature, and
and
are the ultimate feature representations.
Fig 7.
The framework of the multi-order hyperbolic graph convolution and aggregated model for social event detection (MOHGCAASED) model.
represents the node characteristics,
signifies hyperbolic features subsequent to hyperbolic multi aggregation,
indicates Euclidean space features following the logarithmic transformation, and
constitutes the ultimate feature representation.
Table 3.
The statistics of datasets.
Table 4.
Parameters of MOHGCAA in unsupervised settings.
Table 5.
Performance comparison of different models on various datasets.
Fig 8.
Different multi-orders and dimension of MOHGCAA in unsupervised settings.
Table 6.
Parameters of MOHGCAA in supervised settings.
Table 7.
Supervised performance on the Twitter dataset.
Fig 9.
Different multi-orders and dimension of MOHGCAA in supervised settings.
Fig 10.
Comparison of Euclidean and hyperbolic spaces in unsupervised and supervised settings.
(a) Micro-F1 scores for Euclidean and hyperbolic spaces in the unsupervised scenario. (b) Macro-F1 scores for Euclidean and hyperbolic spaces in the unsupervised scenario. (c) Various metric scores for Euclidean and hyperbolic spaces in the supervised scenario.
Fig 11.
Comparison in different hyperbolic spaces in unsupervised and supervised settings.
(a) Micro-F1 scores in the unsupervised scenario. (b) Macro-F1 scores in the unsupervised scenario. (c) Various metric scores in the supervised scenario.