6,742 research outputs found
On the physical relevance of random walks: an example of random walks on a randomly oriented lattice
Random walks on general graphs play an important role in the understanding of
the general theory of stochastic processes. Beyond their fundamental interest
in probability theory, they arise also as simple models of physical systems. A
brief survey of the physical relevance of the notion of random walk on both
undirected and directed graphs is given followed by the exposition of some
recent results on random walks on randomly oriented lattices.
It is worth noticing that general undirected graphs are associated with (not
necessarily Abelian) groups while directed graphs are associated with (not
necessarily Abelian) -algebras. Since quantum mechanics is naturally
formulated in terms of -algebras, the study of random walks on directed
lattices has been motivated lately by the development of the new field of
quantum information and communication
Thermodynamic formalism for dissipative quantum walks
We consider the dynamical properties of dissipative continuous-time quantum
walks on directed graphs. Using a large-deviation approach we construct a
thermodynamic formalism allowing us to define a dynamical order parameter, and
to identify transitions between dynamical regimes. For a particular class of
dissipative quantum walks we propose a quantum generalization of the the
classical PageRank vector, used to rank the importance of nodes in a directed
graph. We also provide an example where one can characterize the dynamical
transition from an effective classical random walk to a dissipative quantum
walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page
An edge-based matching kernel through discrete-time quantum walks
In this paper, we propose a new edge-based matching kernel for graphs by using discrete-time quantum walks. To this end, we commence by transforming a graph into a directed line graph. The reasons of using the line graph structure are twofold. First, for a graph, its directed line graph is a dual representation and each vertex of the line graph represents a corresponding edge in the original graph. Second, we show that the discrete-time quantum walk can be seen as a walk on the line graph and the state space of the walk is the vertex set of the line graph, i.e., the state space of the walk is the edges of the original graph. As a result, the directed line graph provides an elegant way of developing new edge-based matching kernel based on discrete-time quantum walks. For a pair of graphs, we compute the h-layer depth-based representation for each vertex of their directed line graphs by computing entropic signatures (computed from discrete-time quantum walks on the line graphs) on the family of K-layer expansion subgraphs rooted at the vertex, i.e., we compute the depth-based representations for edges of the original graphs through their directed line graphs. Based on the new representations, we define an edge-based matching method for the pair of graphs by aligning the h-layer depth-based representations computed through the directed line graphs. The new edge-based matching kernel is thus computed by counting the number of matched vertices identified by the matching method on the directed line graphs. Experiments on standard graph datasets demonstrate the effectiveness of our new kernel
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