A community detection algorithm is considered to have a resolution limit if the scale of the smallest modules that can be resolved depends on the size of the analyzed subnetwork. The resolution limit is known to prevent some community detection algorithms from accurately identifying the modular structure of a network. In fact, any global objective function for measuring the quality of a two-level assignment of nodes into modules must have some sort of resolution limit or an external resolution parameter. However, it is yet unknown how the resolution limit affects the so-called map equation, which is known to be an efficient objective function for community detection. We derive an analytical estimate and conclude that the resolution limit of the map equation is set by the total number of links between modules instead of the total number of links in the full network as for modularity. This mechanism makes the resolution limit much less restrictive for the map equation than for modularity, and in practice orders of magnitudes smaller. Furthermore, we argue that the effect of the resolution limit often results from shoehorning multi-level modular structures into two-level descriptions. As we show, the hierarchical map equation effectively eliminates the resolution limit for networks with nested multi-level modular structures.
We consider the problem of estimating community memberships of nodes in a network, where every node is associated with a vector determining its degree of membership in each community. Existing provably consistent algorithms often require strong assumptions about the population, are computationally expensive, and only provide an overall error bound for the whole community membership matrix. This paper provides uniform rates of convergence for the inferred community membership vector of each node in a network generated from the Mixed Membership Stochastic Blockmodel (MMSB); to our knowledge, thi…
The maximization of generalized modularity performs well on networks in which the members of all communities are statistically indistinguishable from each other. However, there is no theory bounding the maximization performance in more realistic networks where edges are heterogeneously distributed within and between communities. Using the random graph properties, we establish asymptotic theoretical bounds on the resolution parameter for which the generalized modularity maximization performs well. From this new perspective on random graph model, we find the resolution limit of modularity maximi…
Community structure is one of the most important features of complex networks. Modularity-based methods for community detection typically rely on heuristic algorithms to optimize a specific community quality function. Such methods are limited by two major defects: (1) the resolution limit problem, which prohibits communities of heterogeneous sizes being simultaneously detected, and (2) divergent outputs of the heuristic algorithm, which make it difficult to differentiate relevant and irrelevant results. In this paper, we propose an improved method for community detection based on a scalable co…
This paper introduces a spatiotemporal analysis framework for estimating hourly changing population distribution in urban areas using geo-tagged tweets (the messages containing users' physical locations), land use data, and dasymetric maps. We collected geo-tagged social media (tweets) within the County of San Diego during one year (2015) by using Twitter's Streaming Application Programming Interfaces (APIs). A semi-manual Twitter content verification procedure for data cleaning was applied first to separate tweets created by humans and non-human users (bots). The next step is to calculate the…
Modularity maximization using greedy algorithms continues to be a popular approach toward community detection in graphs, even after various better forming algorithms have been proposed. Apart from its clear mechanism and ease of implementation, this approach is persistently popular because, presumably, its risk of algorithmic failure is not well understood. This Rapid Communication provides insight into this issue by estimating the algorithmic performance limit of modularity maximization. This is achieved by counting the number of metastable states under a local update rule. Our results offer …
Various attempts have been made in recent years to solve the Resolution Limit (RL) problem in community detection by considering variants of the modularity metric in the detection algorithms. These metrics purportedly largely mitigate the RL problem and are preferable to modularity in many realistic scenarios. However, they are not generally suitable for analyzing weighted networks or for detecting hierarchical community structure. Resolution limit problems can be complicated, though, and in particular it can be unclear when it should be considered as problem. In this paper, we introduce a met…
Multilayer networks capture pairwise relationships between the components of complex systems across multiple modes or scales of interactions. An important meso-scale feature of these networks is measured though their community structure, which defines groups of strongly connected nodes that exist within and across network layers. Because interlayer edges can describe relationships between different modalities, scales, or time points, it is essential to understand how communities change and evolve across layers. A popular method for detecting communities in multilayer networks consists of maxim…
Mining community structures from the complex network is an important problem across a variety of fields. Many existing community detection methods detect communities through optimizing a community evaluation function. However, most of these functions even have high values on random graphs and may fail to detect small communities in the large-scale network (the so-called resolution limit problem). In this paper, we introduce two novel node-centric community evaluation functions by connecting correlation analysis with community detection. We will further show that the correlation analysis can pr…
The concept of community detection has long been used as a key device for handling the mesoscale structures in networks. Suitably conducted community detection reveals various embedded informative substructures of network topology. However, regarding the practical usage of community detection, it has always been a tricky problem to assign a reasonable community resolution for networks of interest. Because of the absence of the unanimously accepted criterion, most of the previous studies utilized rather ad hoc heuristics to decide the community resolution. In this work, we harness the concept o…
Heuristic and exact modularity optimization with size-constrained communities
Kellogg School of Management, Northwestern University（西北大学凯洛格管理学院）；Department of Mechanical and Industrial Engineering, University of Toronto（多伦多大学机械与工业工程系）；Centre for Science and Technology Studies (CWTS), Leiden University（莱顿大学科学技术研究中心）；Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University Bloomington（印第安纳大学布鲁明顿分校卢迪信息、计算与工程学院复杂网络与系统研究中心）
Modularity based community detection encompasses a number of widely used, efficient heuristics for identification of structure in networks. Recently, a belief propagation approach to modularity optimization provided a useful guide for identifying non-trivial structure in single-layer networks in a way that other optimization heuristics have not. In this paper, we extend modularity belief propagation to multilayer networks. As part of this development, we also directly incorporate a resolution parameter. We show that adjusting the resolution parameter affects the convergence properties of the a…
Bipartite networks composed of dichotomous node sets are ubiquitous in nature and society. Partly for simplicity's sake, many studies have focused on their projection onto their unipartite versions where one only needs to care about a single type of node. When it comes to mesoscale structures such as communities, however, properly incorporating a priori structural restrictions such as bipartivity is ever more important. In this paper, as a case study, we take the community structure of bipartite networks in various scales to examine the amount of information of bipartivity encoded in the commu…
Alternative approach to community detection in networks
社区检测的问题是相关的许多学科的科学和模块化优化是广泛接受的方法，用于此目的。它最近已被证明，这种方法提出了一个分辨率限制，它是不可能的，以检测社区的大小小于一个阈值，这取决于网络的大小。此外，这种做法产生的社区可能不符合通常的社区定性定义，即，社区中的节点之间的连接比社区外的节点之间的连接更多。本文介绍了一种新的复杂网络社区发现方法。我们基于Radicchi等人（Proc.Nat.Acad. Sci. USA 101，2658-2663（2004）），并且我们表明这种局部定义避免了在模块化优化方法中发现的分辨率限制问题。
Optimizing statistical measures for community structure is one of the most popular strategies for community detection, but many of them lack the flexibility of resolution and thus are incompatible with multi-scale communities of networks. Here, we further studied a statistical measure of interest for community detection, asymptotic surprise, an asymptotic approximation of surprise. We discussed the critical behaviors of asymptotic surprise in phase transition of community partition theoretically. Then, according to the theoretical analysis, a multi-resolution method based on asymptotic surpris…
Global warming imposes us to reflect on the way we carry research, embarking on the obligation to minimize the environmental impact of our research programs, with the reduction of our travel footprint being one of the easiest actions to implement, thanks to the advance of digital technology. The X-ray Integral Field Unit (X-IFU), the cryogenic spectrometer of the Athena space X-ray observatory of the European Space Agency will be developed by a large international consortium. The travel footprint associated with the development of the X-IFU is to be minimized. For that purpose, a travel footpr…
University of Colorado（科罗拉多大学）；BioFrontiers Institute（生物前沿研究所）
In bipartite networks, community structures are restricted to being disassortative, in that nodes of one type are grouped according to common patterns of connection with nodes of the other type. This makes the stochastic block model (SBM), a highly flexible generative model for networks with block structure, an intuitive choice for bipartite community detection. However, typical formulations of the SBM do not make use of the special structure of bipartite networks. Here we introduce a Bayesian nonparametric formulation of the SBM and a corresponding algorithm to efficiently find communities in…
