In the past 20 years network science has proven its strength in modeling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Nevertheless, 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 multibody interactions. Here we propose and study a class of random walks defined on such higher-order structures and grounded on a microscopic physical model where multibody proximity is associated with highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterization of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalized 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 full control of the high-order structures and on real-world networks where higher-order interactions are at play. As the first application of the method, we compare the behavior 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 the 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 unraveling the effect of higher-order interactions on diffusive processes in higher-order networks, shedding light on mechanisms at the heart of biased information spreading in complex networked systems.

中文翻译：

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随机走在超图上

在过去的20年中，网络科学已经证明了其在将许多现实世界中的交互系统建模为成对代理（即成对边缘的节点）方面的实力。但是，在许多相关情况下，交互不是成对的，而是一次涉及更大的节点集。因此，这些系统在超图框架中得到了更好的描述，其超边有效地说明了多体交互作用。在这里，我们提出并研究一类在这种高阶结构上定义并基于微观物理模型的随机游动，其中多体邻近性与属于同一超边缘的代理之间的高度可能交换相关。我们提供了该过程的分析特性，得出了步行者固定分布的一般解决方案。动力学最终由广义随机游走拉普拉斯算子驱动，该算子在所有超边均具有大小2时简化为标准随机游走拉普拉斯算子，因此意在描述成对耦合。我们将在完全控制高阶结构的综合模型以及在高阶交互作用中的真实世界中说明我们的结果。作为该方法的首次应用，我们将超图上的随机游走者的行为与相应投影网络上的传统随机游走者的行为进行了比较，得出了协作网络中节点排名的有趣结论。作为第二个应用程序，我们展示了从超图上的随机游走所获得的信息如何成功用于涉及具有多个特征的对象的分类任务，每个都由超边缘表示。综上所述，我们的工作有助于揭示高阶交互作用对高阶网络中扩散过程的影响，从而阐明复杂网络系统中偏向信息传播核心的机制。