  Random walks on graphs are a fundamental concept in graph theory and play a
crucial role in solving a wide range of theoretical and applied problems in
discrete math, probability, theoretical computer science, network science, and
machine learning. The connection between Markov chains on graphs and their
geometric and topological structures is the main reason why such a wide range
of theoretical and practical applications exist. Graph connectedness ensures
irreducibility of a Markov chain. The convergence rate to the stationary
distribution is determined by the spectrum of the graph Laplacian which is
associated with lower bounds on graph curvature. Furthermore, walks on graphs
are used to infer structural properties of underlying manifolds in data
analysis and manifold learning. However, an important question remains: can
similar connections be established between Markov chains on simplicial
complexes and the topology, geometry, and spectral properties of complexes?
Additionally, can we gain topological, geometric, or analytic information about
a manifold by defining appropriate Markov chains on its triangulations? These
questions are not only theoretically important but answers to them provide
powerful tools for the analysis of complex networks that go beyond the analysis
of pairwise interactions. In this paper, we provide an integrated overview of
the existing results on random walks on simplicial complexes, using the novel
perspective of signed graphs. This perspective sheds light on previously
unknown aspects such as irreducibility conditions. We show that while up-walks
on higher dimensional simplexes can never be irreducible, the down walks become
irreducible if and only if the complex is orientable. We believe that this new
integrated perspective can be extended beyond discrete structures and enables
exploration of classical problems for triangulable manifolds.
