Several networks occurring in real life have modular structures that are arranged in an hierarchical fashion. In this paper, we have proposed a model for such networks, using a stochastic generation method. Using this model we show that, the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks, as had previously been suggested on the basis of a deterministically constructed model. We also look at dynamics on such networks, in particular, the stability of equilibria of network dynamics and of synchronized activity in the network. For both of these, we find that, increasing modularity or the number of hierarchical levels tends to increase the probability of instability. As both hierarchy and modularity are seen in natural systems, which necessarily have to be robust against environmental fluctuations, we conclude that additional constraints are necessary for the emergence of hierarchical structure, similar to the occurrence of modularity through multi-constraint optimization as shown by us previously.
Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that modular hierarchical topologies may provide an optimal structure for certain dynamical processes. We revisit a modular hierarchical network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erdős-R…
Organizational network analysis (ONA) is a method for studying interactions within formal organizations. The utility of ONA has grown substantially over the years as means to analyze the relationships developed within and between teams, departments, and other organizational units. The mapping and quantifying of these relationships have been shown to provide insight into the exchange of information and resources, the building of social capital, and the spread of culture within and between organizations. However, the ethical concerns regarding personally identifiable information (PII) that exist…
Around 80% of global trade by volume is transported by sea, and thus the maritime transportation system is fundamental to the world economy. To better exploit new international shipping routes, we need to understand the current ones and their complex systems association with international trade. We investigate the structure of the global liner shipping network (GLSN), finding it is an economic small-world network with a trade-off between high transportation efficiency and low wiring cost. To enhance understanding of this trade-off, we examine the modular segregation of the GLSN; we study provi…
Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effe…
In network science, a group of nodes connected with each other at higher probability than with those outside the group is referred to as a community. From the perspective that individual communities are associated with functional modules constituting complex systems described by networks, discovering communities is primarily important for understanding overall functions of these systems. Much effort has been devoted to developing methods to detect communities in networks since the early days of network science. Nevertheless, the method to reveal key characteristics of communities in real-world…
Detecting Hierarchical Clusters and Estimating their Modularity Directly from Dendrograms
Institute of Mathematics and Statistics - DCC University of São Paulo（圣保罗大学数学与统计学院）；São Carlos Institute of Physics - DFCM University of São Paulo（圣保罗大学圣卡洛斯物理学院）
Functional networks provide a topological description of activity patterns in the brain, as they stem from the propagation of neural activity on the underlying anatomical or structural network of synaptic connections. This latter is well known to be organized in hierarchical and modular way. While it is assumed that structural networks shape their functional counterparts, it is also hypothesized that alterations of brain dynamics come with transformations of functional connectivity. In this computational study, we introduce a novel methodology to monitor the persistence and breakdown of hierar…
Hierarchical modularity in human brain functional networks
复杂系统具有层次模块化组织的思想起源于20世纪60年代初，最近吸引了大规模现实网络定量研究的新支持。在这里，我们调查的层次模块（或“模块中的模块”）分解的人脑功能网络，测量功能性磁共振成像（fMRI）在18个健康志愿者无任务或休息条件下。我们使用定制的模板来提取具有超过1800个区域节点的网络，并应用快速算法来识别多个层次的嵌套模块结构。我们使用互信息，0 < I < 1，来估计不同主题中网络的社区结构的相似性，并确定最具代表性的个体网络。结果表明，人脑功能网络具有层次模块化组织，受试者之间具有相当程度的相似性，I=0.63。最大的5个模块在最高层次的内侧枕，外侧枕，中央，顶额和额颞系统;枕模块表现出较少的子模块组织比模块包括多模态关联皮层的区域。连接器节点和枢纽，模块间的连接的关键作用，也集中在关联皮质区。我们的结论是，方法可用于分层模块化分解的大量高分辨率的脑功能网络使用计算方便的算法。这可能使未来的调查西蒙的原始假设，层次或近分解的物理符号系统是一个关键的设计功能，为他们的快速适应不断变化的环境条件。
Groups of firms often achieve a competitive advantage through the formation of geo-industrial clusters. Although many exemplary clusters, such as Hollywood or Silicon Valley, have been frequently studied, systematic approaches to identify and analyze the hierarchical structure of the geo-industrial clusters at the global scale are rare. In this work, we use LinkedIn's employment histories of more than 500 million users over 25 years to construct a labor flow network of over 4 million firms across the world and apply a recursive network community detection algorithm to reveal the hierarchical s…
Hierarchical organization of H. Eugene Stanley scientific collaboration community in weighted network representation
通过映射当代社会互动的最先进的元素，世界科学合作网络发展了一个极其复杂的组织。这种异质性的选择特征进行了研究，并确定了重点放在科学合作社区的H。尤金·斯坦利-目前世界上最多产的学者之一。基于截至2016年3月28日的Web of Science记录，构建了几种网络变体。结果发现，斯坦利#1网络-这在类比的Erdos的# -发展了一个很大程度上一致的层次结构和斯坦利自己遵守规则的相同层次结构。然而，这仅在加权网络表示中看到。当这样的加权网络的发展，现有的相关模型表明，权重的传播得到刺激的乘法突发在相邻的节点，这导致它们之间的互连的平衡增长。虽然不是斯坦利独有的，但这种行为并不是一种规则。在这里研究的其他杰出学者的网络更经常发展成一种星状形式，中心枢纽构成了离群点。这项研究的补充频谱分析的归一化拉普拉斯矩阵来自相应的网络的加权变体，除其他外，它指出了这样一个程序的效率，用于确定组件社区和它们之间的关系在复杂的加权网络。
Often exhibiting hierarchical and overlapping structures, communities or modular groups are fundamental and complex in network science. One of the most exploited tools to detect the mesoscopic structure is synchronization. Several phenomena including convergence rate and local convergence under constraints are studied to uncover the existence and features of communities. Here, employing a background of opinion dynamics, we study the divergence of agents' opinions, i.e. the state differences of nodes, in the progress of reaching global consensus and then reveal its connections with hierarchy an…
Knowledge silos emerge when structural properties of organizational interaction networks limit the diffusion of information. These structural barriers are known to take many forms at different scales - hubs in otherwise sparse organisations, large dense teams, or global core-periphery structure - but we lack an understanding of how these different structures interact. Here we bridge the gap between the mathematical literature on localization of spreading dynamics and the more applied literature on knowledge silos in organizational interaction networks. To do so, we introduce a new model that c…
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…
Identifying and explaining the structure of complex networks at different scales has become an important problem across disciplines. At the mesoscale, modular architecture has attracted most of the attention. At the macroscale, other arrangements --e.g. nestedness or core-periphery-- have been studied in parallel, but to a much lesser extent. However, empirical evidence increasingly suggests that characterizing a network with a unique pattern typology may be too simplistic, since a system can integrate properties from distinct organizations at different scales. Here, we explore the relationshi…
As new instances of nested organization --beyond ecological networks-- are discovered, scholars are debating around the co-existence of two apparently incompatible macroscale architectures: nestedness and modularity. The discussion is far from being solved, mainly for two reasons. First, nestedness and modularity appear to emerge from two contradictory dynamics, cooperation and competition. Second, existing methods to assess the presence of nestedness and modularity are flawed when it comes to the evaluation of concurrently nested and modular structures. In this work, we tackle the latter prob…
