Over the last decade, random hyperbolic graphs have proved successful in providing geometric explanations for many key properties of real-world networks, including strong clustering, high navigability, and heterogeneous degree distributions. These properties are ubiquitous in systems as varied as the internet, transportation, brain or epidemic networks, which are thus unified under the hyperbolic network interpretation on a surface of constant negative curvature. Although a few studies have shown that hyperbolic models can generate community structures, another salient feature observed in real networks, we argue that the current models are overlooking the choice of the latent space dimensionality that is required to adequately represent clustered networked data. We show that there is an important qualitative difference between the lowest-dimensional model and its higher-dimensional counterparts with respect to how similarity between nodes restricts connection probabilities. Since more dimensions also increase the number of nearest neighbors for angular clusters representing communities, considering only one more dimension allows us to generate more realistic and diverse community structures.
Technische Universität Dresden（德累斯顿工业大学）；Tsinghua University（清华大学）
Analysis of 'big data' characterized by high-dimensionality such as word vectors and complex networks requires often their representation in a geometrical space by embedding. Recent developments in machine learning and network geometry have pointed out the hyperbolic space as a useful framework for the representation of this data derived by real complex physical systems. In the hyperbolic space, the radial coordinate of the nodes characterizes their hierarchy, whereas the angular distance between them represents their similarity. Several studies have highlighted the relationship between the an…
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 popul…
The hidden metric space behind complex network topologies is a fervid topic in current network science and the hyperbolic space is one of the most studied, because it seems associated to the structural organization of many real complex systems. The Popularity-Similarity-Optimization (PSO) model simulates how random geometric graphs grow in the hyperbolic space, reproducing strong clustering and scale-free degree distribution, however it misses to reproduce an important feature of real complex networks, which is the community organization. The Geometrical-Preferential-Attachment (GPA) model was…
We propose that hypergraphs can be used to model social networks with overlapping communities. The nodes of the hypergraphs represent the communities. The hyperlinks of the hypergraphs denote the individuals who may participate in multiple communities. The hypergraphs are not easy to analyze, however, the line graphs of hypergraphs are simple graphs or weighted graphs, so that the network theory can be applied. We define the overlapping depth $k$ of an individual by the number of communities that overlap in that individual, and we prove that the minimum adjacency eigenvalue of the correspondin…
The location check-ins of users through various location-based services such as Foursquare, Twitter, and Facebook Places, etc., generate large traces of geo-tagged events. These event-traces often manifest in hidden (possibly overlapping) communities of users with similar interests. Inferring these implicit communities is crucial for forming user profiles for improvements in recommendation and prediction tasks. Given only time-stamped geo-tagged traces of users, can we find out these implicit communities, and characteristics of the underlying influence network? Can we use this network to impro…
Cliques are frequently used to model communities: a community is a set of nodes where each pair is equally likely to be connected. But studying real-world communities reveals that they have more structure than that. In particular, the nodes can be ordered in such a way that (almost) all edges in the community lie below a hyperbola. In this paper we present three new models for communities that capture this phenomenon. Our models explain the structure of the communities differently, but we also prove that they are identical in their expressive power. Our models fit to real-world data much bette…
As a fundamental structure in real-world networks, in addition to graph topology, communities can also be reflected by abundant node attributes. In attributed community detection, probabilistic generative models (PGMs) have become the mainstream method due to their principled characterization and competitive performances. Here, we propose a novel PGM without imposing any distributional assumptions on attributes, which is superior to the existing PGMs that require attributes to be categorical or Gaussian distributed. Based on the block model of graph structure, our model incorporates the attrib…
Human proximity networks are temporal networks representing the close-range proximity among humans in a physical space. They have been extensively studied in the past 15 years as they are critical for understanding the spreading of diseases and information among humans. Here we address the problem of mapping human proximity networks into hyperbolic spaces. Each snapshot of these networks is often very sparse, consisting of a small number of interacting (i.e., non-zero degree) nodes. Yet, we show that the time-aggregated representation of such systems over sufficiently large periods can be mean…
Large Question-and-Answer (Q&A) platforms support diverse knowledge curation on the Web. While researchers have studied user behavior on the platforms in a variety of contexts, there is relatively little insight into important by-products of user behavior that also encode knowledge. Here, we analyze and model the macroscopic structure of tags applied by users to annotate and catalog questions, using a collection of 168 Stack Exchange websites. We find striking similarity in tagging structure across these Stack Exchange communities, even though each community evolves independently (albeit under…
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explor…
Speech sounds of the languages all over the world show remarkable patterns of cooccurrence. In this work, we attempt to automatically capture the patterns of cooccurrence of the consonants across languages and at the same time figure out the nature of the force leading to the emergence of such patterns. For this purpose we define a weighted network where the consonants are the nodes and an edge between two nodes (read consonants) signify their co-occurrence likelihood over the consonant inventories. Through this network we identify communities of consonants that essentially reflect their patte…
New network models of complex systems use layers, state nodes, or hyperedges to capture higher-order interactions and dynamics. Simplifying how the higher-order networks change over time or depending on the network model would be easy with alluvial diagrams, which visualize community splits and merges between networks. However, alluvial diagrams were developed for networks with regular nodes assigned to non-overlapping flat communities. How should they be defined for nodes in layers, state nodes, or hyperedges? How can they depict multilevel, overlapping communities? Here we generalize alluvia…
Understanding overlapping community structures is crucial for network analysis and prediction. AGM (Affiliation Graph Model) is one of the favorite models for explaining the densely overlapped community structures. In this paper, we thoroughly re-investigate the assumptions made by the AGM model on real datasets. We find that the AGM model is not sufficient to explain several empirical behaviors observed in popular real-world networks. To our surprise, all our experimental results can be explained by a parameter-free hypothesis, leading to more straightforward modeling than AGM which has many …
Determining the functional structure of biological networks is a central goal of systems biology. One approach is to analyze gene expression data to infer a network of gene interactions on the basis of their correlated responses to environmental and genetic perturbations. The inferred network can then be analyzed to identify functional communities. However, commonly used algorithms can yield unreliable results due to experimental noise, algorithmic stochasticity, and the influence of arbitrarily chosen parameter values. Furthermore, the results obtained typically provide only a simplistic view…
We propose a generative model to detect globally optimal community structures in networks by utilizing random walks. Sophisticated parameter optimization algorithms are developed based on the Markov chain Monte Carlo methods to overcome limitations of the EM algorithm, which has been used in previous works but is sometimes trapped in local optima depending on initial conditions. We apply the algorithms to synthetic and real-world networks to examine their performance in terms of precision and robustness of detected communities. It is found that the Gibbs samplers outperform the previous approa…
Interactions between users in cyberspace may lead to phenomena different from those observed in common social networks. Here we analyse large data sets about users and Blogs which they write and comment, mapped onto a bipartite graph. In such enlarged Blog space we trace user activity over time, which results in robust temporal patterns of user--Blog behavior and the emergence of communities. With the spectral methods applied to the projection on weighted user network we detect clusters of users related to their common interests and habits. Our results suggest that different mechanisms may pla…
Algorithms to find communities in networks rely just on structural information and search for cohesive subsets of nodes. On the other hand, most scholars implicitly or explicitly assume that structural communities represent groups of nodes with similar (non-topological) properties or functions. This hypothesis could not be verified, so far, because of the lack of network datasets with information on the classification of the nodes. We show that traditional community detection methods fail to find the metadata groups in many large networks. Our results show that there is a marked separation bet…
Recent years have shown a promising progress in understanding geometric underpinnings behind the structure, function, and dynamics of many complex networks in nature and society. However these promises cannot be readily fulfilled and lead to important practical applications, without a simple, reliable, and fast network mapping method to infer the latent geometric coordinates of nodes in a real network. Here we present HyperMap, a simple method to map a given real network to its hyperbolic space. The method utilizes a recent geometric theory of complex networks modeled as random geometric graph…
Embedding a network in hyperbolic space can reveal interesting features for the network structure, especially in terms of self-similar characteristics. The hidden metric space, which can be thought of as the underlying structure of the network, is able to preserve some interesting features generally observed in real-world networks such as heterogeneity in the degree distribution, high clustering coefficient, and small-world effect. Moreover, the angular distribution of the nodes in the hyperbolic plane reveals a community structure of the embedded network. It is worth noting that, while a larg…
Hypergraphs, encoding structured interactions among any number of system units, have recently proven a successful tool to describe many real-world biological and social networks. Here we propose a framework based on statistical inference to characterize the structural organization of hypergraphs. The method allows to infer missing hyperedges of any size in a principled way, and to jointly detect overlapping communities in presence of higher-order interactions. Furthermore, our model has an efficient numerical implementation, and it runs faster than dyadic algorithms on pairwise records project…
We discuss two views on extending existing methods for complex network modeling which we dub the communities first and the networks first view, respectively. Inspired by the networks first view that we attribute to White, Boorman, and Breiger (1976)[1], we formulate the multiple-networks stochastic blockmodel (MNSBM), which seeks to separate the observed network into subnetworks of different types and where the problem of inferring structure in each subnetwork becomes easier. We show how this model is specified in a generative Bayesian framework where parameters can be inferred efficiently usi…
Many methods have been proposed to detect communities, not only in plain, but also in attributed, directed or even dynamic complex networks. In its simplest form, a community structure takes the form of a partition of the node set. From the modeling point of view, to be of some utility, this partition must then be characterized relatively to the properties of the studied system. However, if most of the existing works focus on defining methods for the detection of communities, only very few try to tackle this interpretation problem. Moreover, the existing approaches are limited either in the ty…
Topic lifecycle analysis on Twitter, a branch of study that investigates Twitter topics from their birth through lifecycle to death, has gained immense mainstream research popularity. In the literature, topics are often treated as one of (a) hashtags (independent from other hashtags), (b) a burst of keywords in a short time span or (c) a latent concept space captured by advanced text analysis methodologies, such as Latent Dirichlet Allocation (LDA). The first two approaches are not capable of recognizing topics where different users use different hashtags to express the same concept (semantica…
Many methods have been proposed to detect communities, not only in plain, but also in attributed, directed or even dynamic complex networks. From the modeling point of view, to be of some utility, the community structure must be characterized relatively to the properties of the studied system. However, most of the existing works focus on the detection of communities, and only very few try to tackle this interpretation problem. Moreover, the existing approaches are limited either by the type of data they handle, or by the nature of the results they output. In this work, we see the interpretatio…
Most existing approaches for community detection require complete information of the graph in a specific scale, which is impractical for many social networks. We propose a novel algorithm that does not embrace the universal approach but instead of trying to focus on local social ties and modeling multi-scales of social interactions occurring in those networks. Our method for the first time optimizes the topological entropy of a network and uncovers communities through a novel dynamic system converging to a local minimum by simply updating the membership vector with very low computational compl…
Most random graph models are locally tree-like - do not contain short cycles - rendering them unfit for modeling networks with a community structure. We introduce the hierarchical configuration model (HCM), a generalization of the configuration model that includes community structures, while properties such as the size of the giant component, and the size of the giant percolating cluster under bond percolation can still be derived analytically. Viewing real-world networks as realizations of the HCM, we observe two previously unobserved power-law relations: between the number of edges inside a …
The past analyses of datasets of social networks have enabled us to make empirical findings of a number of aspects of human society, which are commonly featured as stylized facts of social networks, such as broad distributions of network quantities, existence of communities, assortative mixing, and intensity-topology correlations. Since the understanding of the structure of these complex social networks is far from complete, for deeper insight into human society more comprehensive datasets and modeling of the stylized facts are needed. Although the existing dynamical and static models can gene…
Information spreading in online social communities has attracted tremendous attention due to its utmost practical values in applications. Despite that several individual-level diffusion data have been investigated, we still lack the detailed understanding of the spreading pattern of information. Here, by comparing information flows and social links in a blog community, we find that the diffusion processes are induced by three different spreading mechanisms: social spreading, self-promotion and broadcast. Although numerous previous studies have employed epidemic spreading models to simulate inf…
Background: We study mechanisms underlying the collective emotional behavior of Bloggers by using the agent-based modeling and the parameters inferred from the related empirical data. Methodology/Principal Findings: A bipartite network of emotional agents and posts evolves through the addition of agents and their actions on posts. The emotion state of an agent,quantified by the arousal and the valence, fluctuates in time due to events on the connected posts, and in the moments of agent's action it is transferred to a selected post. We claim that the indirect communication of the emotion in the…
Understanding community structure of social media is critical due to its broad applications such as friend recommendations, user modeling and content personalizations. Existing research uses structural metrics such as modularity and conductance and functional metrics such as ground truth to measure the qualify of the communities discovered by various community detection algorithms, while overlooking a natural and important dimension, community size. Recently, anthropologist Dunbar suggests that the size of stable community in social media should be limited to 150, referred to as Dunbar's numbe…
Predicting risk profiles of individuals in networks (e.g.~susceptibility to a particular disease, or likelihood of smoking) is challenging for a variety of reasons. For one, `local' features (such as an individual's demographic information) may lack sufficient information to make informative predictions; this is especially problematic when predicting `risk,' as the relevant features may be precisely those that an individual is disinclined to reveal in a survey. Secondly, even if such features are available, they still may miss crucial information, as `risk' may be a function not just of an ind…
Dynamic community detection provides a coherent description of network clusters over time, allowing one to track the growth and death of communities as the network evolves. However, modularity maximization, a popular method for performing multilayer community detection, requires the specification of an appropriate null model as well as resolution and interlayer coupling parameters. Importantly, the ability of the algorithm to accurately detect community evolution is dependent on the choice of these parameters. In functional temporal networks, where evolving communities reflect changing functio…
