Latent Geometry for Complementarity-Driven Networks

Networks of interdisciplinary teams, biological interactions as well as food webs are examples of networks that are shaped by complementarity principles: connections in these networks are preferentially established between nodes with complementary properties. We propose a geometric framework for complementarity-driven networks. In doing so we first argue that traditional geometric representations, e.g., embeddings of networks into latent metric spaces, are not applicable to complementarity-driven networks due to the contradiction between the triangle inequality in latent metric spaces and the non-transitivity of complementarity. We then propose the cross-geometric representation for these complementarity-driven networks and demonstrate that this representation (i) follows naturally from the complementarity rule, (ii) is consistent with the metric property of the latent space, (iii) reproduces structural properties of real complementarity-driven networks, if the latent space is the hyperbolic disk, and (iv) allows for prediction of missing links in complementarity-driven networks with accuracy surpassing existing similarity-based methods. The proposed framework challenges social network analysis intuition and tools that are routinely applied to complementarity-driven networks and offers new avenues towards descriptive and prescriptive analysis of systems in science of science and biomedicine

Hyperbolic matrix factorization improves prediction of drug-target associations

Past research in computational systems biology has focused more on the development and applications of advanced statistical and numerical optimization techniques and much less on understanding the geometry of the biological space. By representing biological entities as points in a low dimensional Euclidean space, state-of-the-art methods for drug-target interaction (DTI) prediction implicitly assume the flat geometry of the biological space. In contrast, recent theoretical studies suggest that biological systems exhibit tree-like topology with a high degree of clustering. As a consequence, embedding a biological system in a flat space leads to distortion of distances between biological objects. Here, we present a novel matrix factorization methodology for drug-target interaction prediction that uses hyperbolic space as the latent biological space. When benchmarked against classical, Euclidean methods, hyperbolic matrix factorization exhibits superior accuracy while lowering embedding dimension by an order of magnitude. We see this as additional evidence that the hyperbolic geometry underpins large biological networks

based on resting state fMRI

학위논문(박사) -- 서울대학교대학원 : 융합과학기술대학원 분자의학 및 바이오제약학과, 2021.8. 위원석.대부분의 실세계 네트워크에서 네트워크의 구성에 있어서 기하학이 중요한 역할을 하며, 최근 연구에서 구조적 뇌 네트워크는 쌍곡기하적 특성을 가지고 있음이 밝혀졌다. 뇌의 구조와 기능은 밀접한 연관을 지니고 있으므로, 기능적 뇌 네트워크 역시 쌍곡기하적 특성을 지니고 있음을 추정할 수 있다. 이번 연구에서, 우리는 휴식기 뇌 자기공명영상(rs-fMRI)을 통해 추출한 기능적 뇌 커넥톰(connectome)을 분석하여 이 가설을 증명하고자 하였으며, 이를 쌍곡공간에 임베드(embed)함으로써 기능적 뇌 네트워크의 특성을 새로이 조사하고자 하였다. 네트워크의 꼭지점은 274개의 미리 정의된 관심영역(ROI) 혹은 6mm 크기의 복셀(voxel)의 두 가지 스케일로 정의되었으며, 꼭지점 사이의 연결성은 자기공명 영상에서 각 영역의 시간에 따른 BOLD 신호의 상관관계를 측정하고 일정 문턱값(threshold)을 적용함으로서 결정되었다. 먼저 쌍곡기하 네트워크의 특징인 스케일-프리(scale-free)를 만족함을 확인하기 위해, 네트워크의 차수(degree) 분포의 급수성(power-law)을 평가하였다. 차수의 확률분포곡선은 로그-로그 스케일의 그래프에서 우하향하는 직선 모양의 분포를 보였으며, 이는 즉 차수 분포가 차수의 음의 급수함수에 의해 나타내어짐을 의미한다. 이어서 기능적 뇌 네트워크에 가장 적합한 기저 기하를 확인하기 위하여, 그래프를 유클리드, 쌍곡, 구면적 틍성을 가진 다양체들에 임베드하여 임베딩의 충실성 척도(fidelity measure)들을 비교하였다. 임베드 대상이 된 적 다양체들 중, 10차원 및 2차원 쌍곡공간의 평균 뒤틀림(distortion)이 동일 차원의 유클리드 다양체와 비교하여 더 낮았다. 이어, 네트워크를 구체화 및 시각화하고 그 특징을 확인하기 위하여, 네트워크를 이차원의 쌍곡 원판에 1/ℍ2 기하학적 모델에 따라 임베드하였다. 이 이차원의 극좌표 형태의 모델에서 반경 및 각 차원의 좌표는 각각 꼭지점의 연결 인기도 및 유사도를 나타낸다. ROI 수준의 분석에서는 특별히 높은 인기도를 갖는 영역은 관찰되지 않아 임베드된 원판의 중심부에 빈 공간으로 나타났다. 한편 같은 해부학적 엽(lobe)에 속한 영역들은 비슷한 각도 영역 내에 밀집되었으며, 반대측 동일 엽에 속한 영역들 역시 그 각좌표의 분포가 구분되지 않았다. 이는 기능적 뇌 네트워크의 해부학적 연관성과 반대측 동일 엽 간의 기능적 연관성을 나타내는 것으로 볼 수 있다. 또한, 복셀 수준의 분석에서는 소뇌에 속한 복셀들 중 다수가 넓은 각좌표 영역에 흩뿌려진 현상이 나타났으며, 이는 개개 복셀의 기능적 이질성을 시사한다. 또한, 전 영역에 걸쳐 매우 유사한 각좌표를 가진 방사형의 막대 모양의 점의 집합이 관찰되었으며, 높은 기능적 유사성을 가진 복셀들로 볼 수 있다. 복셀 수준의 네트워크에서 뇌의 독립성분 분석(ICA) 의 결과로 나온 성분 네트워크들을 플로팅한 결과, 각 네트워크 성분이 높은 밀집도를 보여 두 방법론 간 결과의 유사성을 확인할 수 있었다. 자폐스펙트럼장애의 ABIDE II 오픈 데이터셋을 이용하여 1/ℍ2 모델에 근거하여, 대조군 환자 그룹과 질병군 환자 개인의 네트워크를 비교하는 분석을 시행한 결과, 질병군에서 다양한 패턴을 보였으나, 그 중 자폐증 진단을 받은 환자에서 피질-선조체 경로의 이상이, 아스퍼거증후군 진단을 받은 환자에서 후위관자고랑 (posterior superior temporal sulcus) 을 포함하는 경로의 이상을 발견할 수 있었다. 분석의 재현성을 확인하기 위하여 같은 네트워크를 대상으로 임베딩 과정을 반복 시행하였을 때, 네트워크 말단의 일부 꼭지점을 제외하면 높은 재현성을 보였다. 영상의 시계열(time series) 내 일관성을 확인하기 위하여 영상을 시간 구간에 따라 분리하여 분석하였을 때, 4구간으로 나눈 시계열 영상에서는 유사한 결과를 얻었으나 30초 길이의 30구간으로 나뉘었을 때는 일관적인 결과가 관찰되지 않았다. 이 연구는 뇌 기능적 네트워크에 대한 분석 중 최초로 기하학적 관점에서 진행된 것이며, 이러한 새로운 관점 및 질병군 대상에서 뇌 네트워크의 이상을 찾기 위한 새로운 방법론을 제시한다는 의의가 있다.For most of the real-world networks, geometry plays an important role in organizing the network, and recent works have revealed that the geometry in the structural brain network is most likely to be hyperbolic. Therefore, it can be assumed that the geometry of the functional brain network would also be hyperbolic. In this study, we analyzed the functional connectomes from functional magnetic resonance imaging (fMRI) to prove this hypothesis and investigate the characteristics of the network by embedding it into the hyperbolic space, by utilizing human connectome project (HCP) dataset for healthy young adults and Autism Brain Imaging Data Exchange II (ABIDE II) dataset for diseased autism subject and control group. Nodes of the network were defined at two different scales: by 274 predefined ROIs and 6mm-sized voxels. The adjacency between the nodes was determined by computing the correlation of the time-series of the BOLD signal of brain regions and binarized by adopting threshold value. First, we aimed to find out whether the network was scale-free by investigating the degree distribution of the functional brain network. The probability distribution function (PDF) versus degree was plotted as a straight line at a log-log scale graph versus the degree of nodes. This indicates that degree distribution is roughly proportional to a power function of degree, or scale-free. To clarify the most fitting underlying geometry of the network, we then embedded the graph into manifolds of Euclidean, hyperbolic, or spherical spaces and compared the fidelity measures of embeddings. The embedding to the hyperbolic spaces yielded a better fidelity measure compared to other manifolds. To get a discrete and visible map and investigate the characteristics of the network, we embedded the network in a two-dimensional hyperbolic disc by the 1/ℍ2 model. The radial and angular dimensions in the embedding is interpreted as popularity and similarity dimensions, respectively. The ROI-wise analysis revealed that no nodes with particularly high popularity were found, which was revealed by a vacant area in the center of the disk. Nodes in the same lobe were more likely to be clustered in narrow similarity dimensions, and the nodes from the homotopic lobes were also functionally clustered. The results indicate the anatomic relevance of the functional brain network and the strong functional coherence of the homotopic area of the cerebral cortex. The voxel-wise analysis revealed additional features. A large number of voxels from the cerebellum were scattered in the whole angular position, which might reflect the functional heterogeneity of the cerebellum in the sub-ROI level. Additionally, multiple rod-shaped substructures of radial direction were found, which indicates sets of voxels with functional similarity. When compared with independent component analysis (ICA)-driven results, each large-scale component of the brain acquired by ICA showed a consistent pattern of embedding between the subjects. To find the abnormality of the network in the diseased patient, we utilized the autistic spectrum disorder (ASD) dataset. The two groups of ASD and the control group were found to be comparable in means of the quality of embedding. We calculated the hyperbolic distance between all edges of the network and searched for the alteration of the distance of the individual brain network. Among the variable results among the networks of ASD group subjects, the alteration of the cortico-striatal pathway in an autism patient and posterior superior temporal sulcus (pSTS) in an Asperger’s syndrome patient were present, respectively. The two different anatomically-scaled layers of the network showed a certain degree of correspondence in terms of degree-degree correlation and spreading pattern of network. But anatomically parcellated ROI did not guarantee the functional similarity between the voxels composing it. Finally, to investigate the reproducibility of the embedding process, we repeatedly performed the embedding process and computed the variance of distance matrices. The result was stable except for end-positioned non-popular nodes. Furthermore, to investigate consistency along time-series of fMRI, we compared network yielded by segments of the time series. The segmented networks showed similar results when divided into four frames, but the result lost consistency when divided into 30 frames of 30 seconds each. This study is the first to investigate the characteristics of the functional brain network on the basis of hyperbolic geometry. We suggest a new method applicable for assessing the network alteration in subjects with a neuropsychiatric disease, and these approaches grant us a new understanding in analyzing the functional brain network with a geometric perspective.1. Introduction 1 1.1. Human brain networks 1 1.1.1. Geometry of human brain networks 2 1.2. Scale-free network 3 1.2.1. Definition of a scale-free network 4 1.3. Embedding of the network in hyperbolic space 5 1.3.1. Hyperbolic spaces and Poincaré disk 5 1.3.2. Geometric model of 1/ℍ2 9 1.4. The aim of the present study 10 2. Methods 12 2.1. Subjects and image acquisition 12 2.1.1. Human connectome project (HCP) dataset 12 2.1.2. Autism Brain Imaging Data Exchange II (ABIDE II) dataset 12 2.2. Preprocessing for resting-state fMRI 15 2.3. Resting-state networks and functional connectivity analysis 16 2.3.1. Analyzing degree distribution 18 2.4. Assessing underlying geometry 18 2.4.1. The three component spaces 18 2.4.2. Embedding into spaces 20 2.5. Embedding of the network in the 1/ℍ2 model 22 2.6. Comparison with ICA-driven method 23 2.7. Assessing the quality of embedding 23 2.8. Abnormality detection in the diseased subject 24 2.9. Assessing variability of analysis 27 3. Results 29 3.1. Global characteristics of the network 29 3.1.1. The degree distribution 31 3.1.2. Determining the threshold value of network 34 3.2. Graph embedding into spaces 36 3.3. 1/ℍ2 model analysis 39 3.4. Quality of the embedding 58 3.5. Alteration of the network in the diseased subject 61 3.6. Variability of results 63 3.6.1. Reproducibility of Mercator 63 3.6.2. Time variance of results 67 4. Discussion 70 4.1. Composition of the network 70 4.2. Scale-freeness of brain network 71 4.3. The underlying geometry of brain network 73 4.4. Hyperbolic plane representation 75 4.4.1. Voxelwise approach 78 4.4.2. Compatibility with ICA 80 4.5. Alteration of the network in ASD subjects 81 4.6. Variability and reproducibility of methods 83 4.7. Further applications 85 5. Conclusion 87 References 89 국문 초록 106박

Latent geometry of bipartite networks

Despite the abundance of bipartite networked systems, their organizing principles are less studied, compared to unipartite networks. Bipartite networks are often analyzed after projecting them onto one of the two sets of nodes. As a result of the projection, nodes of the same set are linked together if they have at least one neighbor in common in the bipartite network. Even though these projections allow one to study bipartite networks using tools developed for unipartite networks, one-mode projections lead to significant loss of information and artificial inflation of the projected network with fully connected subgraphs. Here we pursue a different approach for analyzing bipartite systems that is based on the observation that such systems have a latent metric structure: network nodes are points in a latent metric space, while connections are more likely to form between nodes separated by shorter distances. This approach has been developed for unipartite networks, and relatively little is known about its applicability to bipartite systems. Here, we fully analyze a simple latent-geometric model of bipartite networks, and show that this model explains the peculiar structural properties of many real bipartite systems, including the distributions of common neighbors and bipartite clustering. We also analyze the geometric information loss in one-mode projections in this model, and propose an efficient method to infer the latent pairwise distances between nodes. Uncovering the latent geometry underlying real bipartite networks can find applications in diverse domains, ranging from constructing efficient recommender systems to understanding cell metabolism