Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.Comment: In Press, Accepted Manuscript, Physics Reports 201

Representation Learning for Texts and Graphs: A Unified Perspective on Efficiency, Multimodality, and Adaptability

[...] This thesis is situated between natural language processing and graph representation learning and investigates selected connections. First, we introduce matrix embeddings as an efficient text representation sensitive to word order. [...] Experiments with ten linguistic probing tasks, 11 supervised, and five unsupervised downstream tasks reveal that vector and matrix embeddings have complementary strengths and that a jointly trained hybrid model outperforms both. Second, a popular pretrained language model, BERT, is distilled into matrix embeddings. [...] The results on the GLUE benchmark show that these models are competitive with other recent contextualized language models while being more efficient in time and space. Third, we compare three model types for text classification: bag-of-words, sequence-, and graph-based models. Experiments on five datasets show that, surprisingly, a wide multilayer perceptron on top of a bag-of-words representation is competitive with recent graph-based approaches, questioning the necessity of graphs synthesized from the text. [...] Fourth, we investigate the connection between text and graph data in document-based recommender systems for citations and subject labels. Experiments on six datasets show that the title as side information improves the performance of autoencoder models. [...] We find that the meaning of item co-occurrence is crucial for the choice of input modalities and an appropriate model. Fifth, we introduce a generic framework for lifelong learning on evolving graphs in which new nodes, edges, and classes appear over time. [...] The results show that by reusing previous parameters in incremental training, it is possible to employ smaller history sizes with only a slight decrease in accuracy compared to training with complete history. Moreover, weighting the binary cross-entropy loss function is crucial to mitigate the problem of class imbalance when detecting newly emerging classes. [...