Community identification of network components enables us to understand the mesoscale clustering structure of networks. A number of algorithms have been developed to determine the most likely community structures in networks. Such a probabilistic or stochastic nature of this problem can naturally involve the ambiguity in resultant community structures. More specifically, stochastic algorithms can result in different community structures for each realization in principle. In this study, instead of trying to "solve" this community degeneracy problem, we turn the tables by taking the degeneracy as a chance to quantify how strong companionship each node has with other nodes. For that purpose, we define the concept of companionship inconsistency that indicates how inconsistently a node is identified as a member of a community regarding the other nodes. Analyzing model and real networks, we show that companionship inconsistency discloses unique characteristics of nodes, thus we suggest it as a new type of node centrality. In social networks, for example, companionship inconsistency can classify outsider nodes without firm community membership and promiscuous nodes with multiple connections to several communities. In infrastructure networks such as power grids, it can diagnose how the connection structure is evenly balanced in terms of power transmission. Companionship inconsistency, therefore, abstracts individual nodes' intrinsic property on its relationship to a higher-order organization of the network.

Power grids sustain modern society by supplying electricity and thus their stability is a crucial factor for our civilization. The dynamic stability of a power grid is usually quantified by the probability of its nodes' recovery to phase synchronization of the alternating current it carries, in response to external perturbation. Intuitively, the stability of nodes in power grids is supposed to become more robust as the coupling strength between the nodes increases. However, we find a counterintuitive range of coupling strength values where the synchronization stability suddenly droops as the coupling strength increases, on a number of simple graph structures. Since power grids are designed to fulfill both local and long-range power demands, such simple graph structures or graphlets for local power transmission are indeed relevant in reality. We show that the observed nonmonotonic behavior is a consequence of transitions in multistability, which are related to changes in stability of the unsynchronized states. Therefore, our findings suggest that a comprehensive understanding of changes in multistability are necessary to prevent the unexpected catastrophic instability in the building blocks of power grids.

The calculation of nestedness has become a routine analysis in the study of ecological networks, as it is commonly associated with community resilience, robustness and species persistence. While meaningful in species distributional patterns, for an interaction matrix to be nested, specialist species must interact with ordered subsets of subsequently more generalized speciesnot just with a lower number of species. However, after reviewing 419 papers on mutualistic networks published since nestedness was introduced for the study of species interactions in 2003, we have found that only two theoretical studies considered explicitly ordered subsets. Instead, most studies interpret nestedness as a core of densely connected generalist species, surrounded by a periphery of specialist species attached to this corea so-called core-periphery structure . Such a topological feature is generally perceived as a core-periphery structure in network science. Here, we argue that the concept of core-periphery may be more relevant for studies on mutualistic networks than the concept of nestedness, as ecologists are usually not interested in exploring in detail the ordered subsets that characterize nestedness but instead use nestedness to describe a topology with a core of densely linked generalist species surrounded by a sparsely linked periphery of specialists. To illustrate our arguments and the quantification of core-periphery structures, we calculate core-periphery and nestedness in a large publicly available dataset of mutualistic networks. We also describe the calculation of core-periphery structures, its relationship with nestedness, and provide the code inside the R package econetwork for its calculation in mutualistic networks. We hope that our review will help ecologists to move beyond nestedness towards a more explicit representation of the structure of ecological networks.

The gravity model (GM) analogous to Newton's law of universal gravitation has successfully described the flow between different spatial regions, such as human migration, traffic flows, international economic trades, etc. This simple but powerful approach relies only on the 'mass' factor represented by the scale of the regions and the 'geometrical' factor represented by the geographical distance. However, when the population has a subpopulation structure distinguished by different attributes, the estimation of the flow solely from the coarse-grained geographical factors in the GM causes the loss of differential geographical information for each attribute. To exploit the full information contained in the geographical information of subpopulation structure, we generalize the GM for population flow by explicitly harnessing the subpopulation properties characterized by both attributes and geography. As a concrete example, we examine the marriage patterns between the bride and the groom clans of Korea in the past. By exploiting more refined geographical and clan information, our generalized GM properly describes the real data, a part of which could not be explained by the conventional GM. Therefore, we would like to emphasize the necessity of using our generalized version of the GM, when the information on such nongeographical subpopulation structures is available.

We perform an in-depth analysis on the inequality in 863 Wikimedia projects. We take the complete editing history of 267 304 095 Wikimedia items until 2016, which not only covers every language edition of Wikipedia, but also embraces the complete versions of Wiktionary, Wikisource, Wikivoyage, etc. Our findings of common growth pattern described by the interrelations between four characteristic growth yardsticks suggest a universal law of communal data formation. In this encyclopaedic data set, we observe the interplay between the number of edits and the degree of inequality. In particular, the rapid increase in the Gini coefficient suggests that this entrenched inequality stems from the nature of such open-editing communal data sets, namely the abiogenesis of the supereditors' oligopoly. We show that these supereditor groups were created at the early stages of these open-editing media and are still active. Furthermore, our model considers both short-term and long-term memories to successfully elucidate the underlying mechanism of the establishment of oligarchy in Wikipedia. Our results anticipate a noticeable prospect of such communal databases in the future: the disparity will not be resolved spontaneously.