Scalable detection of statistically significant communities and hierarchies,using message passing for modularity

Modularity is a popular measure of community structure. However, maximizing the modularity can lead to many competing partitions, with almost the same modularity, that are poorly correlated with each other. It can also produce illusory ‘‘communities’’ in random graphs where none exist. We address this problem by using the modularity as a Hamiltonian at finite temperature and using an efficient belief propagation algorithm to obtain the consensus of many partitions with high modularity, rather than looking for a single partition that maximizes it. We show analytically and numerically that the proposed algorithm works all of the way down to the detectability transition in networks generated by the stochastic block model. It also performs well on real-world networks, revealing large communities in some networks where previous work has claimed no communities exist. Finally we show that by applying our algorithm recursively, subdividing communities until no statistically significant subcommunities can be found, we can detect hierarchical structure in real-world networks more efficiently than previous methods.Community detection, or node clustering, is a key problem in network science, computer science, sociology, and biology. It aims to partition the nodes in a network into groups such that there are many edges connecting nodes within the same group and comparatively few edges connecting nodes in different groups.Many methods have been proposed for this problem. These include spectral clustering, where we classify nodes according to the eigenvectors of a linear operator such as the adjacency matrix, the random walk matrix, the graph Laplacian, or other linear operators (1–3); statistical inference, where we fit the network with a generative model such as the stochastic block model (4–7); and a wide variety of other methods, e.g., refs. 8–10. See ref. 11 for a review.We focus here on a popular measure of the quality of a partition, the modularity (e.g., refs. 8 and 12–14). We think of a partition {t} into q groups as a function t:V → {1, …, q}, where t_i is the group to which node i belongs. The modularity of a partition {t} of a network with n nodes and m edges is defined asQ({t})=1m(∑〈ij〉∈ℰδtitj−∑〈ij〉didj2mδtitj).[1]Here ? is the set of edges, d_i is the degree of node i, and δ is the Kronecker delta function. The modularity is proportional to the number of edges connecting nodes in the same community minus the expected number of such edges if the graph were random conditioned on its degree distribution, that is, the expectation in a null model where i and j are connected with probability proportional to d_id_j.However, maximizing over all possible partitions often gives a large modularity even in random graphs with no community structure (15–18). Thus, maximizing the modularity can lead to overfitting, where the “optimal” partition simply reflects random noise. Even in real-world networks, the modularity often exhibits a large amount of degeneracy, with multiple local optima that are poorly correlated with each other and are not robust to small perturbations (19).Thus, we need to add some notion of statistical significance to our algorithms. One approach is hypothesis testing, comparing various measures of community structure to the distribution we would see in a null model such as Erdős–Rényi (ER) graphs (20–22). However, even when communities really exist, the modularity of the true partition is often no higher than that of random graphs. In Fig. 1, we show partitions of two networks with the same size and degree distribution: an ER graph (Left) and a graph generated by the stochastic block model (Right), in the detectable regime where it is easy to find a partition correlated with the true one (5, 6). The true partition of the network in Fig. 1, Right has a smaller modularity than the partition found for the random graph in Fig. 1, Left. We can find a partition with higher modularity (and lower accuracy) in Fig. 1, Right, using, e.g., simulated annealing, but then the modularities we obtain for the two networks are similar. Thus, the usual approach of null distributions and P values for hypothesis testing does not appear to work.Open in a separate windowFig. 1.The adjacency matrices of two networks, partitioned to show possible community structure. Each blue point is an edge. (Left) The network is an ER graph, with no real community structure; however, a search by simulated annealing finds a partition with modularity 0.391. (Right) The network has true communities and is generated by the stochastic block model, but the true partition has modularity of just 0.333. Thus, illusory communities in random graphs can have higher modularity than true communities in structured graphs. Both networks have size n=5,000 and a Poisson degree distribution with mean c = 3; the network at Right has c_out/c_in = 0.2, in the easily detectable regime of the stochastic block model.We propose to solve this problem with the tools of statistical physics. As in ref. 16, we treat the modularity as the Hamiltonian of a spin system. We define the energy of a partition {t} as E({t}) = ?mQ({t}) and introduce a Gibbs distribution as a function of inverse temperature β, P({t}) ∝ e^?βE({t}). Rather than maximizing the modularity by searching for the ground state of this system, we focus on its Gibbs distribution at a finite temperature, looking for many high-modularity partitions rather than a single one. In analogy with previous work on the stochastic block model (5, 6), we define a partition {t^} by computing the marginals of the Gibbs distribution and assigning each node to its most likely community. Specifically, if ψti is the marginal probability that i belongs to group t, then t^i=argmaxtψti, breaking ties randomly if more than one t achieves the maximum. We call {t^} the retrieval partition and call its modularity Q({t^}) the retrieval modularity. We claim that {t^} is a far better measure of significant community structure than the maximum-modularity partition. In the language of statistics, the maximum marginal prediction is better than the maximum a posteriori prediction (e.g., ref. 23). More informally, the consensus of many good solutions is better than the ‘‘best’’ single one (24, 25).We give an efficient belief propagation (BP) algorithm to approximate these marginals, which is derived from the cavity method of statistical physics. This algorithm is highly scalable; each iteration takes linear time on sparse networks if the number of groups is fixed, and it converges rapidly in most cases. It is optimal in the sense that for synthetic graphs generated by the stochastic block model, it works all of the way down to the detectability transition. It provides a principled way to choose the number of communities, unlike other algorithms that tend to overfit. Finally, by applying this algorithm recursively, subdividing communities until no statistically significant subcommunities exist, we can uncover hierarchical structure.We validate our approach with experiments on real and synthetic networks. In particular, we find significant large communities in some large networks where previous work claimed there were none. We also compare our algorithm with several others, finding that it obtains more accurate results, both in terms of determining the number of communities and in terms of matching their ground-truth structure.     

HPLC-CAD法测定元胡止痛胶囊中生物碱和香豆素

目的建立高效液相色谱法联合二极管阵列检测器(HPLC-DAD)法同时测定元胡止痛胶囊中延胡索乙素、盐酸黄连碱、去氢延胡索甲素、延胡索甲素、欧前胡素和异欧前胡素。方法采用HPLC-CAD法,Thermo Hypersil BDS C18色谱柱(250 mm×4.6 mm,5μm);流动相:乙腈–10 mmol/L甲酸铵(甲酸调pH 4.0),梯度洗脱;体积流量0.8 mL/min;CAD雾化器温度为35℃;柱温35℃;进样量为20μL。结果延胡索乙素、盐酸黄连碱、去氢延胡索甲素、延胡索甲素、欧前胡素和异欧前胡素分别在0.041～1.020、0.013～0.320、0.068～1.700、0.041～1.020、0.101～2.520、0.039～0.980μg与峰面积线性关系良好;平均回收率分别为98.41%、98.89%、98.32%、99.35%、98.05%、98.67%,RSD值分别为0.9%、1.4%、0.7%、1.1%、0.7%、0.9%。结论该方法准确、简便、灵敏度高,可用于元胡止痛胶囊中生物碱和香豆素类多成分的质量控制研究。     

新型冠状病毒肺炎130例CT分布特点及征象研究

目的探讨2019新型冠状病毒肺炎(COVID-19)影像学表现。方法根据纳入标准和排除标准收集2020年1月20日至2月5日来自全国多家医院确诊COVID-19病例130例,按分布特点进行分型,分析其影像学特征。结果(1)分布:单侧14例(10.7%),双侧116例(89.3%);胸膜下型(102例78.4%),小叶核心型99例(76.1%),弥漫型8例(6.1%);(2)数目:单发病灶9例(6.9%),多发病灶113例(86.9%),弥漫8例(6.1%);(3)密度:仅为磨玻璃影(GGO)70例(53.8%),GGO与实变影兼有60例(46.2%);(3)伴随征象:血管增粗100例(76.9%),胸膜平行征98例(75.3%),"细网格征"100例(76.9%),"晕征"13例(10%),"反晕征"6例(4.6%),3例胸腔积液(2.3%),2例肺气囊(1.5%)。未见空洞。35患者行CT复查,21例(60%)好转,14例(40%)加重。结论COVID-19影像学特点主要以胸膜下及小叶核心分布为主,两者均可融合成片,重症者发展为双肺弥漫;最有价值的特征是"胸膜平行征";恢复期表现为边缘收缩的实变影,支气管扩张,胸膜下线或纤维条索影。     

Analysis of density and epitopes of D antigen on the surface of erythrocytes from DEL phenotypic individuals carrying the RHD1227A allele

Background

The characteristics of the D antigen are important as they influence the immunogenicity of D variant cells. Several studies on antigenic sites have been reported in normal D positive, weak D and partial D cases, including a comprehensive analysis of DEL types in Caucasians. The aim of this study was to assess D antigen density and epitopes on the erythrocyte surface of Asian type DEL phenotypic individuals carrying the RHD1227A allele in the Chinese population.

Materials and methods

A total of 154 DEL phenotypic individuals carrying the RHD1227A allele were identified through adsorption and elution tests and polymerase chain reaction analysis with sequence-specific primers in the Chinese population. D antigen density on the erythrocyte surface of these individuals was detected using a flow cytometric method. An erythrocyte sample with known D antigen density was used as a standard. Blood samples from D-negative and D-positive individuals were used as controls. In addition, D antigen epitopes on the erythrocyte surface of DEL individuals carrying the RHD1227A allele were investigated with 18 monoclonal anti-D antibodies specific for different D antigen epitopes.

Results

The means of the median fluorescence intensity of D antigen on the erythrocyte membrane surface of D-negative, D-positive and DEL individuals were 2.14±0.25, 193.61±11.43 and 2.45±0.82, respectively. The DEL samples were estimated to have approximately 22 D antigens per cell. The samples from all 154 DEL individuals reacted positively with 18 monoclonal anti-D antibodies specific for different D antigen epitopes.

Discussion

In this study, D antigen density on the erythrocyte surface of DEL individuals carrying the RHD1227A allele was extremely low, there being only very few antigenic molecules per cell, but the D antigen epitopes were grossly complete.