19,785 research outputs found
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
Clustering and the hyperbolic geometry of complex networks
Clustering is a fundamental property of complex networks and it is the
mathematical expression of a ubiquitous phenomenon that arises in various types
of self-organized networks such as biological networks, computer networks or
social networks. In this paper, we consider what is called the global
clustering coefficient of random graphs on the hyperbolic plane. This model of
random graphs was proposed recently by Krioukov et al. as a mathematical model
of complex networks, under the fundamental assumption that hyperbolic geometry
underlies the structure of these networks. We give a rigorous analysis of
clustering and characterize the global clustering coefficient in terms of the
parameters of the model. We show how the global clustering coefficient can be
tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur
Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces
We show that complex (scale-free) network topologies naturally emerge from
hyperbolic metric spaces. Hyperbolic geometry facilitates maximally efficient
greedy forwarding in these networks. Greedy forwarding is topology-oblivious.
Nevertheless, greedy packets find their destinations with 100% probability
following almost optimal shortest paths. This remarkable efficiency sustains
even in highly dynamic networks. Our findings suggest that forwarding
information through complex networks, such as the Internet, is possible without
the overhead of existing routing protocols, and may also find practical
applications in overlay networks for tasks such as application-level routing,
information sharing, and data distribution
From Graph Theory to Network Science: The Natural Emergence of Hyperbolicity (Tutorial)
Network science is driven by the question which properties large real-world networks have and how we can exploit them algorithmically. In the past few years, hyperbolic graphs have emerged as a very promising model for scale-free networks. The connection between hyperbolic geometry and complex networks gives insights in both directions:
(1) Hyperbolic geometry forms the basis of a natural and explanatory model for real-world networks. Hyperbolic random graphs are obtained by choosing random points in the hyperbolic plane and connecting pairs of points that are geometrically close. The resulting networks share many structural properties for example with online social networks like Facebook or Twitter. They are thus well suited for algorithmic analyses in a more realistic setting.
(2) Starting with a real-world network, hyperbolic geometry is well-suited for metric embeddings. The vertices of a network can be mapped to points in this geometry, such that geometric distances are similar to graph distances. Such embeddings have a variety of algorithmic applications ranging from approximations based on efficient geometric algorithms to greedy routing solely using hyperbolic coordinates for navigation decisions
Discrete-time Temporal Network Embedding via Implicit Hierarchical Learning in Hyperbolic Space
Representation learning over temporal networks has drawn considerable
attention in recent years. Efforts are mainly focused on modeling structural
dependencies and temporal evolving regularities in Euclidean space which,
however, underestimates the inherent complex and hierarchical properties in
many real-world temporal networks, leading to sub-optimal embeddings. To
explore these properties of a complex temporal network, we propose a hyperbolic
temporal graph network (HTGN) that fully takes advantage of the exponential
capacity and hierarchical awareness of hyperbolic geometry. More specially,
HTGN maps the temporal graph into hyperbolic space, and incorporates hyperbolic
graph neural network and hyperbolic gated recurrent neural network, to capture
the evolving behaviors and implicitly preserve hierarchical information
simultaneously. Furthermore, in the hyperbolic space, we propose two important
modules that enable HTGN to successfully model temporal networks: (1)
hyperbolic temporal contextual self-attention (HTA) module to attend to
historical states and (2) hyperbolic temporal consistency (HTC) module to
ensure stability and generalization. Experimental results on multiple
real-world datasets demonstrate the superiority of HTGN for temporal graph
embedding, as it consistently outperforms competing methods by significant
margins in various temporal link prediction tasks. Specifically, HTGN achieves
AUC improvement up to 9.98% for link prediction and 11.4% for new link
prediction. Moreover, the ablation study further validates the representational
ability of hyperbolic geometry and the effectiveness of the proposed HTA and
HTC modules.Comment: KDD202
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