Random walks on hypergraphs (1911.06523v2)

Abstract: In the last twenty years network science has proven its strength in modelling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Yet, in many relevant cases, interactions are not pairwise but involve larger sets of nodes, at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multi-body interactions. We hereby propose a new class of random walks defined on such higher-order structures, and grounded on a microscopic physical model where multi-body proximity is associated to highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterisation of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalised random walk Laplace operator that reduces to the standard random walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have a full control of the high-order structures, and real-world networks where higher-order interactions are at play. As a first application of the method, we compare the behaviour of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As a second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unveiling the effect of higher-order interactions on diffusive processes in higher-order networks, shading light on mechanisms at the hearth of biased information spreading in complex networked systems.

• The paper introduces a novel random walk framework on hypergraphs that accounts for multi-body interactions through hyperedge-weighted transitions.
• It adapts transition probabilities based on hyperedge sizes, simulating a bias that mirrors natural information spread in complex systems.
• The authors validate their model using synthetic data and real co-authorship networks, demonstrating superior node ranking over traditional methods.

Insightful Overview of "Random Walks on Hypergraphs"

The paper "Random Walks on Hypergraphs" by Carletti et al. provides a comprehensive exploration of random walks within the framework of hypergraphs, addressing the limitations of traditional graph theory in modeling multi-body interactions. While networks conventionally capture pairwise relationships among entities, many real-world systems involve complex interactions that encompass more than two entities simultaneously. Hypergraphs offer a potent extension of graph theory by utilizing hyperedges to account for such multi-nodal interactions. The authors introduce an innovative class of random walks that fully leverages the innate higher-order structures of hypergraphs, presenting both theoretical insights and practical applications.

Theoretical Advancements and Analytical Foundation

The authors establish a new class of random walks formulated on hypergraphs through a physically grounded microscopic model. A salient feature of their framework is the adaptation of the transition probability, which considers the densities of hyperedges: at every step, a walker chooses an adjacent hyperedge based on a probability proportional to its size and jumps to a random node within this hyperedge. The large hyperedges effectively retain walkers, simulating a bias mirroring natural processes like group information spread. Mathematically, this dynamic is dictated by a generalized Laplacian operator. The conventional Laplacian is a special case within this framework when each hyperedge involves exactly two nodes.

Synthetic and Real-world Applications

Carletti et al. thoroughly investigate the properties of random walks on hypergraphs using both synthetic models and real-world datasets. In synthetic models, they show how hypergraphs with varying hyperedge sizes influence the walkers' stationary distribution, revealing significant rankings differences compared to traditional pairwise network projections. The influence of hyperedges becomes particularly pronounced in models that transition from linear arrangements to complete connectivity, demonstrating the distinct behaviors predicted by their hypergraph-based model against standard graph structures.

Real-world applicability is showcased through empirical analyses on co-authorship networks extracted from the arXiv platform. Here, each paper's co-authors comprise a hyperedge, inherently capturing the collaborative nature of academic research. The authors illustrate how traditional network-based approaches mistakenly prioritize nodes solely based on individual connections, whereas the hypergraph-based approach provides a more nuanced nodal ranking reflecting substantive collaborations.

Implications and Future Prospects

The research posits significant implications for understanding networked systems characterized by multi-body interactions, as seen in social networks, biological systems, and collaborative structures. Practically, it offers refined algorithms for tasks such as node ranking and classification, bolstered by the nuanced treatment of information flows implicit in hypergraph dynamics. Theoretically, extending random walk theories to hypergraphs paves the way for further exploration into nonlinear dynamics and temporally-evolving hypergraphs.

Future inquiries are likely to explore nonlinear transition rates and more intricate biases in hypergraph-based random walks. This prospect opens avenues for enhanced data clustering, machine learning techniques, and even novel studies on diffusion processes within multi-dimensional interactions.

In sum, "Random Walks on Hypergraphs" establishes a foundational framework for exploring higher-order network interactions, presenting both a meaningful advancement in theoretical network science and practical improvements for analyzing complex systems. The integration of hypergraph structures stands to significantly enhance our capacity to model, understand, and utilize the intricate webs of multi-entity relationships that shape the contemporary data landscape.