Role of fractal dimension in random walks on scale-free networks

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called $(x,y)$-flowers; the other is random, which is a combination of $(1,3)$-flower and $(2,4)$-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the $(x,y)$-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.Comment: Definitive version published in European Physical Journal

Mean first-passage time for random walks on undirected networks

In this paper, by using two different techniques we derive an explicit formula for the mean first-passage time (MFPT) between any pair of nodes on a general undirected network, which is expressed in terms of eigenvalues and eigenvectors of an associated matrix similar to the transition matrix. We then apply the formula to derive a lower bound for the MFPT to arrive at a given node with the starting point chosen from the stationary distribution over the set of nodes. We show that for a correlated scale-free network of size $N$ with a degree distribution $P(d)\sim d^{-\gamma}$, the scaling of the lower bound is $N^{1-1/\gamma}$. Also, we provide a simple derivation for an eigentime identity. Our work leads to a comprehensive understanding of recent results about random walks on complex networks, especially on scale-free networks.Comment: 7 pages, no figures; definitive version published in European Physical Journal

Astronomical Distance Determination in the Space Age: Secondary Distance Indicators

The formal division of the distance indicators into primary and secondary leads to difficulties in description of methods which can actually be used in two ways: with, and without the support of the other methods for scaling. Thus instead of concentrating on the scaling requirement we concentrate on all methods of distance determination to extragalactic sources which are designated, at least formally, to use for individual sources. Among those, the Supernovae Ia is clearly the leader due to its enormous success in determination of the expansion rate of the Universe. However, new methods are rapidly developing, and there is also a progress in more traditional methods. We give a general overview of the methods but we mostly concentrate on the most recent developments in each field, and future expectations. © 2018, The Author(s)

1st Postgraduate Seminar on Agriculture and Forestry 2021

We present velocity-resolved reverberation results for five active galactic nuclei. We recovered velocity-delay maps using the maximum-entropy method for four objects: Mrk 335, Mrk 1501, 3C120, and PG2130+099. For the fifth, Mrk 6, we were only able to measure mean time delays in different velocity bins of the Hbeta emission line. The four velocity-delay maps show unique dynamical signatures for each object. For 3C120, the Balmer lines show kinematic signatures consistent with both an inclined disk and infalling gas, but the HeII 4686 emission line is suggestive only of inflow. The Balmer lines in Mrk 335, Mrk 1501, and PG 2130+099 show signs of infalling gas, but the HeII emission in Mrk 335 is consistent with an inclined disk. We also see tentative evidence of combined virial motion and infalling gas from the velocity-binned analysis of Mrk 6. The maps for 3C120 and Mrk 335 are two of the most clearly defined velocity-delay maps to date. These maps constitute a large increase in the number of objects for which we have resolved velocity-delay maps and provide evidence supporting the reliability of reverberation-based black hole mass measurements.Comment: 23 pages, 17 Figures. Accepted for publication in ApJ. For a brief video explaining the key results of this paper, see http://www.youtube.com/watch?v=8xAflzp-Yl