Many times the nodes of a complex network, whether deliberately or not, are aggregated for technical, ethical, legal limitations or privacy reasons. A common example is the geographic position: one may uncover communities in a network of places, or of individuals identified with their typical geographical position, and then aggregate these places into larger entities, such as municipalities, thus obtaining another network. The communities found in the networks obtained at various levels of aggregation may exhibit various degrees of similarity, from full alignment to perfect independence. This is akin to the problem of ecological and atomic fallacies in statistics, or to the Modified Areal Unit Problem in geography. We identify the class of community detection algorithms most suitable to cope with node aggregation, and develop an index for aggregability, capturing to which extent the aggregation preserves the community structure. We illustrate its relevance on real-world examples (mobile phone and Twitter reply-to networks). Our main message is that any node-partitioning analysis performed on aggregated networks should be interpreted with caution, as the outcome may be strongly influenced by the level of the aggregation.
We present a new online algorithm for detecting overlapping communities. The main ingredients are a modification of an online k-means algorithm and a new approach to modelling overlap in communities. An evaluation on large benchmark graphs shows that the quality of discovered communities compares favorably to several methods in the recent literature, while the running time is significantly improved.
The theory of community structure is a powerful tool for real networks, which can simplify their topological and functional analysis considerably. However, since community detection methods have random factors and real social networks obtained from complex systems always contain error edges, evaluating the robustness of community structure is an urgent and important task. In this letter, we employ the critical threshold of resolution parameter in Hamiltonian function, $γ_C$, to measure the robustness of a network. According to spectral theory, a rigorous proof shows that the index we proposed …
Many networks display community structure which identifies groups of nodes within which connections are denser than between them. Detecting and characterizing such community structure, which is known as community detection, is one of the fundamental issues in the study of network systems. It has received a considerable attention in the last years. Numerous techniques have been developed for both efficient and effective community detection. Among them, the most efficient algorithm is the label propagation algorithm whose computational complexity is O(|E|). Although it is linear in the number of…
MIT Lincoln Laboratory（麻省理工学院林肯实验室）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
Learning the right graph representation from noisy, multi-source data has garnered significant interest in recent years. A central tenet of this problem is relational learning. Here the objective is to incorporate the partial information each data source gives us in a way that captures the true underlying relationships. To address this challenge, we present a general, boosting-inspired framework for combining weak evidence of entity associations into a robust similarity metric. Building on previous work, we explore the extent to which different local quality measurements yield graph representa…
Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjac…
A particular challenge in the area of social media analysis is how to find communities within a larger network of social interactions. Here a community may be a group of microblogging users who post content on a coherent topic, or who are associated with a specific event or news story. Twitter provides the ability to curate users into lists, corresponding to meaningful topics or themes. Here we describe an approach for crowdsourcing the list building efforts of many different Twitter users, in order to identify topical communities. This approach involves the use of ensemble community finding t…
Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the tradeoffs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with $N$ nodes and $L$ layers, which a…
Network science is a powerful tool for analyzing complex systems in fields ranging from sociology to engineering to biology. This paper is focused on generative models of large-scale bipartite graphs, also known as two-way graphs or two-mode networks. We propose two generative models that can be easily tuned to reproduce the characteristics of real-world networks, not just qualitatively, but quantitatively. The characteristics we consider are the degree distributions and the metamorphosis coefficient. The metamorphosis coefficient, a bipartite analogue of the clustering coefficient, is the pro…
Many complex systems can be represented as networks and separating a network into communities could simplify the functional analysis considerably. Recently, many approaches have been proposed for finding communities, but none of them can evaluate the communities found are significant or trivial definitely. In this paper, we propose an index to evaluate the significance of communities in networks. The index is based on comparing the similarity between the original community structure in network and the community structure of the network after perturbed, and is defined by integrating all the sim…
We introduce a novel algorithm of community detection that maintains dynamically a community structure of a large network that evolves with time. The algorithm maximizes the modularity index thanks to the construction of a randomized hierarchical clustering based on a Monte Carlo Markov Chain (MCMC) method. Interestingly, it could be seen as a dynamization of Louvain algorithm (see Blondel et Al, 2008) where the aggregation step is replaced by the hierarchical instrumental probability.
The increasing penetration of variable renewable energy (VRE) has brought significant challenges for power systems planning and operation. These highly variable sources are typically distributed in the grid; therefore, a detailed representation of transmission bottlenecks is fundamental to approximate the impact of the transmission network on the dispatch with VRE resources. The fine grain temporal scale of short term and day-ahead dispatch, taking into account the network constraints, also mandatory for mid-term planning studies, combined with the high variability of the VRE has brought the n…
We study the community structure of the multi-network of commodity-specific trade relations among world countries over the 1992-2003 period. We compare structures across commodities and time by means of the normalized mutual information index (NMI). We also compare them with exogenous community structures induced by geographical distances and regional trade agreements. We find that commodity-specific community structures are very heterogeneous and much more fragmented than that characterizing the aggregate ITN. This shows that the aggregate properties of the ITN may result (and be very differe…
Understanding social dynamics that govern human phenomena, such as communications and social relationships is a major problem in current computational social sciences. In particular, given the unprecedented success of online social networks (OSNs), in this paper we are concerned with the analysis of aggregation patterns and social dynamics occurring among users of the largest OSN as the date: Facebook. In detail, we discuss the mesoscopic features of the community structure of this network, considering the perspective of the communities, which has not yet been studied on such a large scale. To…
We show that the recently introduced label propagation method for detecting communities in complex networks is equivalent to find the local minima of a simple Potts model. Applying to empirical data, the number of such local minima was found to be very high, much larger than the number of nodes in the graph. The aggregation method for combining information from more local minima shows a tendency to fragment the communities into very small pieces.
The international trade network is a complex system where multiple trade blocs with varying sizes coexist and overlap with each other. However, the resulting structures of community detection in trade networks are often inconsistent and fails to capture the complex landscape of international trade. To address these problems, we propose a multiresolution framework that aggregates all the configuration information from a range of resolutions. This allows us to consider trade communities of different sizes and illuminate the underlying hierarchical structure of trade networks and its constituting…
Detecting and analyzing dense groups or communities from social and information networks has attracted immense attention over last one decade due to its enormous applicability in different domains. Community detection is an ill-defined problem, as the nature of the communities is not known in advance. The problem has turned out to be even complicated due to the fact that communities emerge in the network in various forms - disjoint, overlapping, hierarchical etc. Various heuristics have been proposed depending upon the application in hand. All these heuristics have been materialized in the for…
Complex systems show the capacity to aggregate information and to display coordinated activity. In the case of social systems the interaction of different individuals leads to the emergence of norms, trends in political positions, opinions, cultural traits, and even scientific progress. Examples of collective behavior can be observed in activities like the Wikipedia and Linux, where individuals aggregate their knowledge for the benefit of the community, and citizen science, where the potential of collectives to solve complex problems is exploited. Here, we conducted an online experiment to inv…
Community structure exists in many real-world networks and has been reported being related to several functional properties of the networks. The conventional approach was partitioning nodes into communities, while some recent studies start partitioning links instead of nodes to find overlapping communities of nodes efficiently. We extended the map equation method, which was originally developed for node communities, to find link communities in networks. This method is tested on various kinds of networks and compared with the metadata of the networks, and the results show that our method can id…
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, wh…
The discovery of community structure is a common challenge in the analysis of network data. Many methods have been proposed for finding community structure, but few have been proposed for determining whether the structure found is statistically significant or whether, conversely, it could have arisen purely as a result of chance. In this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure. We propose a suitable method for perturbing networks and a measure of the resulting change in comm…
Many systems exhibit complex temporal dynamics due to the presence of different processes taking place simultaneously. An important task in such systems is to extract a simplified view of their time-dependent network of interactions. Community detection in temporal networks usually relies on aggregation over time windows or consider sequences of different stationary epochs. For dynamics-based methods, attempts to generalize static-network methodologies also face the fundamental difficulty that a stationary state of the dynamics does not always exist. Here, we derive a method based on a dynamic…
We study the Twitter conversation following the August 2017 `Unite the Right' rally in Charlottesville, Virginia, using tools from network analysis and data science. We use media followership on Twitter and principal component analysis (PCA) to compute a `Left'/`Right' media score on a one-dimensional axis to characterize nodes. We then use these scores, in concert with retweet relationships, to examine the structure of a retweet network of approximately 300,000 accounts that communicated with the #Charlottesville hashtag. The retweet network is sharply polarized, with an assortativity coeffic…
