Matrix-weighted networks for modeling multidimensional dynamics

Networks are powerful tools for modeling interactions in complex systems. While traditional networks use scalar edge weights, many real-world systems involve multidimensional interactions. For example, in social networks, individuals often have multiple interconnected opinions that can affect different opinions of other individuals, which can be better characterized by matrices. We propose a novel, general framework for modeling such multidimensional interacting dynamics: matrix-weighted networks (MWNs). We present the mathematical foundations of MWNs and examine consensus dynamics and random walks within this context. Our results reveal that the coherence of MWNs gives rise to non-trivial steady states that generalize the notions of communities and structural balance in traditional networks.

Residual2Vec: Debiasing graph embedding with random graphs

Graph embedding maps a graph into a convenient vector-space representation for graph analysis and machine learning applications. Many graph embedding methods hinge on a sampling of context nodes based on random walks. However, random walks can be a biased sampler due to the structural properties of graphs. Most notably, random walks are biased by the degree of each node, where a node is sampled proportionally to its degree. The implication of such biases has not been clear, particularly in the context of graph representation learning. Here, we investigate the impact of the random walks' bias on graph embedding and propose residual2vec, a general graph embedding method that can debias various structural biases in graphs by using random graphs. We demonstrate that this debiasing not only improves link prediction and clustering performance but also allows us to explicitly model salient structural properties in graph embedding.

Unsupervised embedding of trajectories captures the latent structure of mobility

Human mobility and migration drive major societal phenomena such as the growth and evolution of cities, epidemics, economies, and innovation. Historically, human mobility has been strongly constrained by physical separation -- geographic distance. However, geographic distance is becoming less relevant in the increasingly-globalized world in which physical barriers are shrinking while linguistic, cultural, and historical relationships are becoming more important. As understanding mobility is becoming critical for contemporary society, finding frameworks that can capture this complexity is of paramount importance. Here, using three distinct human trajectory datasets, we demonstrate that a neural embedding model can encode nuanced relationships between locations into a vector-space, providing an effective measure of distance that reflects the multi-faceted structure of human mobility. Focusing on the case of scientific mobility, we show that embeddings of scientific organizations uncover cultural and linguistic relations, and even academic prestige, at multiple levels of granularity. Furthermore, the embedding vectors reveal universal relationships between organizational characteristics and their place in the global landscape of scientific mobility. The ability to learn scalable, dense, and meaningful representations of mobility directly from the data can open up a new avenue of studying mobility across domains.