No abstract

No abstract

a b s t r a c tStandard random walks on finite graphs select the vertex visited next to the adjacent vertices at random with the same probability. Despite not using any global topological information, they guarantee O(n 3 ) hitting and cover times for any graph, where n is the order of the graph. Motivated by network protocol applications, this paper investigates the impact of local topological information on designing ''better'' random walks. We first show that (a) for any transition probability matrix, the hitting (and hence the cover) time of a path graph is Ω(n 2 ). We next investigate for any graph G = (V , E) a transition probability matrix P = (p(u, v)) u,v∈V defined bywhere N(u) and deg(u) are respectively the set of adjacent vertices of u and the u's degree. Random walks obeying this transition probability matrix are shown to guarantee the following: For any graph, (b) the hitting time is O(n 2 ), and (c) the cover time is O(n 2 log n).Facts (a) and (b) show that the degree information on the adjacent vertices is powerful enough for random walks to achieve the optimum hitting time.

No abstract

We investigate the cosmological applications of scalar-tensor theories that arise effectively from the Lorentz fiber bundle of a Finsler-like geometry. We first show that the involved nonlinear connection induces a new scalar degree of freedom and eventually a scalar-tensor theory. Using both a holonomic and a non-holonomic basis, we show the appearance of an effective dark energy sector, which additionally acquires an explicit interaction with the matter sector, arising purely from the internal structure of the theory. Applying the theory at late times we find that we can obtain the thermal history of the universe, namely the sequence of matter and radiation epochs, and moreover the effective dark-energy equation-of-state parameter can be quintessence-like, phantom-like, or experience the phantom-divide crossing during cosmological evolution. Furthermore, applying the scenario at early times we see that one can acquire an exponential de Sitter solution, as well as obtain an inflationary realization with the desired scale-factor evolution. These features arise purely from the intrinsic geometrical structure of Finsler-like geometry, and reveal the capabilities of the construction.PACS numbers: 95.36.+x, 04.50.Kd