Hyperbolic Networks Research Articles

Intra-community link formation and modularity in ultracold growing hyperbolic networks

Hyperbolic network models offer a straightforward yet powerful method for understanding the small-world, scale-free, highly clustered, and modular characteristics typical of complex systems that are often called as real-world networks. These models involve randomly positioning nodes in a hyperbolic space and connecting them based on a probability that diminishes with distance. In this study, we examine the community structure within networks created by the Popularity Similarity Optimization model, a fundamental hyperbolic model, when the temperature parameter (responsible for tuning the clustering coefficient) is set to the limiting value of zero. We focus on link formation within communities and show their close relation with non-linear preferential attachment processes. Based on this, we derive analytical expressions that significantly improve previous estimates of the expected modularity for partitions formed by equally sized angular sectors in the 2d hyperbolic space. Our formulas can now predict average modularity, confirmed by numerical simulations, with high accuracy over a broader range of model parameters and community sizes relative to the whole network. These results advance our understanding of module formation in hyperbolic networks, highlighting the surprising emergence of communities despite the lack of explicit community formation steps in the model definition.

Anomalous and Chern topological waves in hyperbolic networks

Hyperbolic lattices are a new type of synthetic materials based on regular tessellations in non-Euclidean spaces with constant negative curvature. While so far, there has been several theoretical investigations of hyperbolic topological media, experimental work has been limited to time-reversal invariant systems made of coupled discrete resonances, leaving the more interesting case of robust, unidirectional edge wave transport completely unobserved. Here, we report a non-reciprocal hyperbolic network that exhibits both Chern and anomalous chiral edge modes, and implement it on a planar microwave platform. We experimentally evidence the unidirectional character of the topological edge modes by direct field mapping. We demonstrate the topological origin of these hyperbolic chiral edge modes by an explicit topological invariant measurement, performed from external probes. Our work extends the reach of topological wave physics by allowing for backscattering-immune transport in materials with synthetic non-Euclidean behavior.

Long‐range quantum energy teleportation and distribution on a hyperbolic quantum network

AbstractTeleporting energy to remote locations is new challenge for quantum information science and technology. Developing a method for transferring local energy in laboratory systems to remote locations will enable non‐trivial energy flows in quantum networks. From the perspective of quantum information engineering, we propose a method for distributing local energy to a large number of remote nodes using hyperbolic geometry. Hyperbolic networks are suitable for energy allocation in large quantum networks since the number of nodes grows exponentially. To realise long‐range quantum energy teleportation (QET), we propose a hybrid method of quantum state telepotation and QET. By transmitting local quantum information through quantum teleportation and performing conditional operations on that information, QET can theoretically be realized independent of geographical distance. The method we present will provide new insights into new applications of future large‐scale quantum networks and potential applications of quantum physics to information engineering.

Greedy routing optimisation in hyperbolic networks

Finding the optimal embedding of networks into low-dimensional hyperbolic spaces is a challenge that received considerable interest in recent years, with several different approaches proposed in the literature. In general, these methods take advantage of the exponentially growing volume of the hyperbolic space as a function of the radius from the origin, allowing a (roughly) uniform spatial distribution of the nodes even for scale-free small-world networks, where the connection probability between pairs decays with hyperbolic distance. One of the motivations behind hyperbolic embedding is that optimal placement of the nodes in a hyperbolic space is widely thought to enable efficient navigation on top of the network. According to that, one of the measures that can be used to quantify the quality of different embeddings is given by the fraction of successful greedy paths following a simple navigation protocol based on the hyperbolic coordinates. In the present work, we develop an optimisation scheme for this score in the native disk representation of the hyperbolic space. This optimisation algorithm can be either used as an embedding method alone, or it can be applied to improve this score for embeddings obtained from other methods. According to our tests on synthetic and real networks, the proposed optimisation can considerably enhance the success rate of greedy paths in several cases, improving the given embedding from the point of view of navigability.

Unexpected paths to cooperation on tied hyperbolic networks

Hyperbolic networks have high clustering, short average path lengths, and community structure, which are all properties that are commonly associated with social networks. As such, these networks constitute the perfect playing ground for probing factors that may affect public cooperation in realistic scenarios. And although much is already known about the evolution of cooperation on networks, we here consider the public goods game on tied hyperbolic networks, such that payoffs in one network layer influence the payoffs in the other and vice versa. We also consider random, assortative, and disassortative mixing in the networks to account for varying connections between players over time. While our research confirms the overall positive impact of interdependent payoffs, we also find that mixing on the network where cooperation thrives may strongly promote the cooperation in the other network, while destroying it completely in the former. We show that this is related to the mapping of lower payoffs from one network to the other, where cooperators in one network benefit from the failure of cooperators in the other network. Namely, as soon as the multiplication factor for the public goods is high enough to nullify the negative effects of mixing and cooperators thus recover, the positive effect on cooperation in the other network vanishes. We determine optimal conditions for this phenomenon in terms of the frequency of mixing and the strength of ties between the payoffs on both networks, and we discuss the implications of our research for enhanced cooperation in coupled populations, in particular in the light of mutual success not always being desirable for cooperation to thrive.

Maximally modular structure of growing hyperbolic networks

Hyperbolic network models provide a particularly successful approach to explain many peculiar features of real complex networks including, for instance, the small-world and scale-free properties, or the relatively high clustering coefficient. Here we show that for the popularity-similarity optimisation (PSO) model from this family, the generated networks become also extremely modular in the thermodynamic limit, despite lacking any explicitly built-in community formation mechanism in the model definition. In particular, our analytical calculations indicate that the modularity in PSO networks can get arbitrarily close to its maximal value of 1 as the network size is increased. We also derive the convergence rate, which turns out to be dependent on the popularity fading parameter controlling the degree decay exponent of the generated networks.

Geometrical congruence, greedy navigability and myopic transfer in complex networks and brain connectomes

We introduce in network geometry a measure of geometrical congruence (GC) to evaluate the extent a network topology follows an underlying geometry. This requires finding all topological shortest-paths for each nonadjacent node pair in the network: a nontrivial computational task. Hence, we propose an optimized algorithm that reduces 26 years of worst scenario computation to one week parallel computing. Analysing artificial networks with patent geometry we discover that, different from current belief, hyperbolic networks do not show in general high GC and efficient greedy navigability (GN) with respect to the geodesics. The myopic transfer which rules GN works best only when degree-distribution power-law exponent is strictly close to two. Analysing real networks—whose geometry is often latent—GC overcomes GN as marker to differentiate phenotypical states in macroscale structural-MRI brain connectomes, suggesting connectomes might have a latent neurobiological geometry accounting for more information than the visible tridimensional Euclidean.

Observation of novel topological states in hyperbolic lattices

The discovery of novel topological states has served as a major branch in physics and material sciences. To date, most of the established topological states have been employed in Euclidean systems. Recently, the experimental realization of the hyperbolic lattice, which is the regular tessellation in non-Euclidean space with a constant negative curvature, has attracted much attention. Here, we demonstrate both in theory and experiment that exotic topological states can exist in engineered hyperbolic lattices with unique properties compared to their Euclidean counterparts. Based on the extended Haldane model, the boundary-dominated first-order Chern edge state with a nontrivial real-space Chern number is achieved. Furthermore, we show that the fractal-like midgap higher-order zero modes appear in deformed hyperbolic lattices, and the number of zero modes increases exponentially with the lattice size. These novel topological states are observed in designed hyperbolic circuit networks by measuring site-resolved impedance responses and dynamics of voltage packets. Our findings suggest a useful platform to study topological phases beyond Euclidean space, and may have potential applications in the field of high-efficient topological devices, such as topological lasers, with enhanced edge responses.

Meta hyperbolic networks for zero-shot learning

Zero-Shot Learning (ZSL) aims at generalizing the classification experience from seen classes to unseen classes with auxiliary side information, among which word vectors of class names and class attributes are popular ones. While word vectors technique is more practical for large datasets, its performance is usually slightly worse than that of attributes since it requires no human annotations. In this paper, we focus on the utilization of word vectors in ZSL by exploiting their hierarchical knowledge for large dataset and propose Meta Hyperbolic Networks (MHN). Specifically, we present Poincaré Graph Convolutional Networks (P-GCN), which transforms the word vectors into a Poincaré ball, and then encodes them with Poincaré Graph Convolutional layers. During training, the image classifiers from Convolutional Neural Networks (CNN) and P-GCN weights are aligned to ensure an accurate mapping between them. Moreover, we further develop a short-term memory episode learning on it to relieve the model’s inherent bias towards seen classes. Extensive experiments on the popular ImageNet dataset show the competitive performance for both ZSL and generalized ZSL.

Generalised popularity-similarity optimisation model for growing hyperbolic networks beyond two dimensions

Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks. One of the most widely known models of this type is the popularity-similarity optimisation (PSO) model, working in the native disk representation of the two-dimensional hyperbolic space and generating networks with small-world property, scale-free degree distribution, high clustering and strong community structure at the same time. With the motivation of better understanding hyperbolic random graphs, we hereby introduce the dPSO model, a generalisation of the PSO model to any arbitrary integer dimension d>2. The analysis of the obtained networks shows that their major structural properties can be affected by the dimension of the underlying hyperbolic space in a non-trivial way. Our extended framework is not only interesting from a theoretical point of view but can also serve as a starting point for the generalisation of already existing two-dimensional hyperbolic embedding techniques.

The inherent community structure of hyperbolic networks

A remarkable approach for grasping the relevant statistical features of real networks with the help of random graphs is offered by hyperbolic models, centred around the idea of placing nodes in a low-dimensional hyperbolic space, and connecting node pairs with a probability depending on the hyperbolic distance. It is widely appreciated that these models can generate random graphs that are small-world, highly clustered and scale-free at the same time; thus, reproducing the most fundamental common features of real networks. In the present work, we focus on a less well-known property of the popularity-similarity optimisation model and the {mathbb {S}}^1/{mathbb {H}}^2 model from this model family, namely that the networks generated by these approaches also contain communities for a wide range of the parameters, which was certainly not an intention at the design of the models. We extracted the communities from the studied networks using well-established community finding methods such as Louvain, Infomap and label propagation. The observed high modularity values indicate that the community structure can become very pronounced under certain conditions. In addition, the modules found by the different algorithms show good consistency, implying that these are indeed relevant and apparent structural units. Since the appearance of communities is rather common in networks representing real systems as well, this feature of hyperbolic models makes them even more suitable for describing real networks than thought before.

이종 그래프상의 비유클리디안 데이터 분석을 위한 쌍곡 그래프 변형 인공 신경망

합성곱 기반인 합성곱 인공 신경망(CNNs)은 이미지 분류, 이미지 생성, 시계열 분석 등에 다양하게 쓰이고 있다. 하지만 일반적인 유클리디안 공간과는 달리 그래프와 같은 비유클리디안 공간에서는 합성곱을 바로 적용할 수 없다. 이를 극복하기 위해 다양한 기법으로 합성곱을 그래프 상으로 확장하였으며, 다양한 그래프 인공 신경망(GNNs)이 제안되어 왔다. 하지만 기존의 그래프 인공 신경망 연구는 간선의 타입이 하나인 동종 그래프 분석에 국한되어 있는데 반해, 현실의 데이터는 간선의 타입이 많은 이종그래프 데이터인 경우가 많기 때문에 이를 기존의 그래프 인공 신경망으로 해결하려 하면 큰 왜곡이 생기게 된다. 본 연구는 계층적 구조를 가진 이종 그래프 데이터를 효과적으로 다루기 위하여 그래프 변형 네트워크(GTNs) 모델과 쌍곡 그래프 합성곱 네트워크(HGCNs) 모델을 통합하여 새로운 모델인 쌍곡 그래프 변형 네트워크(HGTNs)를 제안한다.

Reliable Learning with PDE-Based CNNs and DenseNets for Detecting COVID-19, Pneumonia, and Tuberculosis from Chest X-Ray Images

It has recently been shown that the interpretation by partial differential equations (PDEs) of a class of convolutional neural networks (CNNs) supports definition of architectures such as parabolic and hyperbolic networks. These networks have provable properties regarding the stability against the perturbations of the input features. Aiming for robustness, we tackle the problem of detecting changes in chest X-ray images that may be suggestive of COVID-19 with parabolic and hyperbolic CNNs and with domain-specific transfer learning. To this end, we compile public data on patients diagnosed with COVID-19, pneumonia, and tuberculosis, along with normal chest X-ray images. The negative impact of the small number of COVID-19 images is reduced by applying transfer learning in several ways. For the parabolic and hyperbolic networks, we pretrain the networks on normal and pneumonia images and further use the obtained weights as the initializers for the networks to discriminate between COVID-19, pneumonia, tuberculosis, and normal aspects. For DenseNets, we apply transfer learning twice. First, the ImageNet pretrained weights are used to train on the CheXpert dataset, which includes 14 common radiological observations (e.g., lung opacity, cardiomegaly, fracture, support devices). Then, the weights are used to initialize the network which detects COVID-19 and the three other classes. The resulting networks are compared in terms of how well they adapt to the small number of COVID-19 images. According to our quantitative and qualitative analysis, the resulting networks are more reliable compared to those obtained by direct training on the targeted dataset.

Public goods games on random hyperbolic graphs with mixing

Understanding the evolution of cooperation in structured populations remains one of the fundamental challenges of the 21st century, with far-reaching implications for the wellbeing of modern human societies. Studies over the past two decades showed that the structure of the network of contacts plays a crucial role in determining whether cooperation prevails or not. An important step to more realistic networks was made with the shift from regular grids and lattices to complex social networks at the turn of the century. Real networks exhibit a high mean local clustering coefficient, short average path lengths, and community structure. Recent studies have revealed that random geometric graphs in hyperbolic spaces exhibit properties that are frequently found in real networks. We here study the public goods game on random geometric graphs in hyperbolic spaces, and we consider assortative and disassortative mixing with different frequencies. We show that in hyperbolic spaces heterogeneous networks promote the evolution of public cooperation in comparison to the more homogeneous networks. We also confirm that assortative and disassortative mixing on random hyperbolic networks both impair the evolutionary success of cooperators, regardless of the network architecture. The differences between the two mixing protocols are most expressed at low mixing frequencies, whilst at high mixing frequencies the two almost converge.

Comparison between a bone-embedded and two marker-based coordinate systems of the hindfoot

Coexistence of cooperation and defection is a common phenomenon in nature. The topic of why cooperation occurs among selfish individuals has attracted much attention in recent years. Generally, in the snowdrift game, when a defector exploits a cooperator, the defector would get the maximum benefit without paying any cost. However, the elapse of time is also a kind of loss. Even if the defector chooses to stay in the car waiting for cooperator to shovel snow, the defector also has to bear a loss of time. Under such circumstances, the time cost also should be considered in the total cost of shovelling snow. We focus on how time cost impacts the evolution of cooperation in the snowdrift game. More importantly, the experiments were implemented on hyperbolic networks which are closer to real world networks. Simulated results show that the larger proportion of time cost is, the more cooperation could be promoted.

Effects of time cost on the evolution of cooperation in snowdrift game

Abstract Coexistence of cooperation and defection is a common phenomenon in nature. The topic of why cooperation occurs among selfish individuals has attracted much attention in recent years. Generally, in the snowdrift game, when a defector exploits a cooperator, the defector would get the maximum benefit without paying any cost. However, the elapse of time is also a kind of loss. Even if the defector chooses to stay in the car waiting for cooperator to shovel snow, the defector also has to bear a loss of time. Under such circumstances, the time cost also should be considered in the total cost of shovelling snow. We focus on how time cost impacts the evolution of cooperation in the snowdrift game. More importantly, the experiments were implemented on hyperbolic networks which are closer to real world networks. Simulated results show that the larger proportion of time cost is, the more cooperation could be promoted.

Self-organised criticality and emergent hyperbolic networks: blueprint for complexity in social dynamics

Online social dynamics based on human endeavours exhibit prominent complexity in the emergence of new features embodied in the appearance of collective social values. The vast amount of empirical data collected at various websites provides a unique opportunity for the quantitative study of the underlying social dynamics in full analogy with complex systems in the physics laboratory. Here, we briefly describe the extent of these analogies and indicate the methods from other science disciplines that the physics theory can incorporate in order to provide an adequate description of human entities and principles of their self-organisation. We demonstrate the approach on two examples using empirical data regarding knowledge-creation processes in online chats and questions-and-answers. Specifically, we describe the self-organised criticality as the acting mechanism in the social knowledge-sharing dynamics and demonstrate the emergence of the hyperbolic geometry of the co-evolving networks that underlie these stochastic processes.

Scalable Algorithms for Large and Dynamic Networks: Reducing Big Data for Small Computations

In this paper we summarize recent research regarding a novel characterization of large-scale real-life informational networks which can be leveraged to speed computations for network analytics purposes by orders of magnitude. First, using publicly available data, we show that informational networks not only satisfy well-known principles such as the small-world property and variants of the power law degree distribution, but that they also exhibit the geometric property of large-scale negative curvature, also referred to as hyperbolicity. We then provide examples of large-scale physical networks that universally lack this property, thus showing that hyperbolicity is not an ever-present feature of real-life networks in general. We document how hyperbolicity leads to unusually high centrality in informational networks. We then describe an approximation of hyperbolic networks that leverages the observed property of high centrality. We provide evidence that the fidelity of the proposed approximation is not only high for applications such as distance approximation, but that it can speed computation by a factor of 1000X or more. Finally, we discuss two applications of our proposed linear-time distance approximation for informational networks: one for personalized ranking and the other for clustering. These and many more algorithms yet to be developed take full advantage of our proposed tree-approximation of hyperbolic networks and further demonstrate its power and utility.

From explosive to infinite-order transitions on a hyperbolic network.

We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous ("explosive") transition in ordinary percolation. To this end, we solve the q-state Potts model in the analytic continuation for noninteger q with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in q shows that the discontinuous transition is generic in the regime q<2 that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, q=2. For q>2 the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless type.

Scaling of clusters near discontinuous percolation transitions in hyperbolic networks.

We investigate the onset of the discontinuous percolation transition in small-world hyperbolic networks by studying the systems-size scaling of the typical largest cluster approaching the transition, p ↗ p(c). To this end, we determine the average size of the largest cluster 〈s(max)〉 ∼ N(Ψ(p)) in the thermodynamic limit using real-space renormalization of cluster-generating functions for bond and site percolation in several models of hyperbolic networks that provide exact results. We determine that all our models conform to the recently predicted behavior regarding the growth of the largest cluster, which found diverging, albeit subextensive, clusters spanning the system with finite probability well below p(c) and at most quadratic corrections to unity in Ψ(p) for p ↗ p(c). Our study suggests a large universality in the cluster formation on small-world hyperbolic networks and the potential for an alternative mechanism in the cluster formation dynamics at the onset of discontinuous percolation transitions.