Analytical results for the distribution of cover times of random walks on random regular graphs

We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of N nodes of degree c (c ≥ 3). Starting from a random initial node at time t = 1, at each time step t ≥ 2 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time TCis the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution Pt(S = s) of the number of distinct nodes s visited by an RW up to time t and solve it analytically. Inserting s = N we obtain the cumulative distribution of cover times, namely the probability P (TC ≤ t) = Pt(S = N) that up to time t an RW will visit all the N nodes in the network. Taking the large network limit, we show that P (TC ≤ t) converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times P (TPC,k = t), which is the probability that at time t an RW will complete visiting k distinct nodes. We also calculate the distribution of random cover (RC) times P (TRC,k = t), which is the probability that at time t an RW will complete visiting all the nodes in a subgraph of k randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

The power of two choices for random walks

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

New archaeomagnetic secular variation data from Central Europe, I: Directions

Summary Archaeomagnetic directions of one hundred and forty-one archaeological structures have been studied from 21 sites in Austria, 31 sites in Germany and one site in Switzerland. Characteristic remanent magnetisation directions obtained from alternating field and thermal demagnetisations provided 82 and 78 new or updated (12 and 10 per cent) directions of Austria and Germany, respectively. Nine of the directions are not reliable for certain reasons (e.g. displacement) while three of the features are not well dated. Apart from this some updated age information for the published databases is provided. Rock magnetic experiments revealed magnetite as main magnetic carrier of the remanences. The new data agree well with existing secular variation reference curves. The extended data set covers now the past 3500 years and a lot of progress were made to cover times BC with data. Here enhanced secular variation is observed manifested in declinations with values up to 70°. The new data will allow for recalculation of archaeomagnetic calibration curves for Central Europe from mid Bronze Age until today.

A Spectral Characterization for Concentration of the Cover Time

We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time.