The names of Grünwald and Letnikov are associated with discrete convolutions of mesh h, multiplied by h−α. When h tends to zero, the result tends to a Marchaud’s derivative (of the order of α) of the function to which the convolution is applied. The weights wkα of such discrete...

We studied random walks on two-dimensional patterns formed by the sequence of configurations of complex elementary cellular automata (CA) with random initial configurations. The walkers are allowed to jump between nearest neighbours or next nearest neighbours 1 sites. On patterns of rules 22,...

We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0α⩽2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the...

We model the time series of the S&P500 index by a combined process, the AR+GARCH process, where AR denotes the autoregressive process which we use to account for the short-range correlations in the index changes and GARCH denotes the generalized autoregressive conditional heteroskedastic process...

We show how effectively the diffusive capture processes (DCP) on complex networks can be applied to information search in the networks. Numerical simulations show that our method generates only 2% of traffic compared with the most popular flooding-based query-packet-forwarding (FB) algorithm. We...

The one-dimensional random walk (RW), where steps up and down are performed according to the occurrence of special primes, is defined. Some quantities characterizing RW are investigated. The mean fluctuation function F(l) displays perfect power-law dependence F(l)∼l1/2 indicating that the...

In complex networks a common task is to identify the most important or “central” nodes. There are several definitions, often called centrality measures, which often lead to different results. Here, we introduce fingerprints of networks, which we define as correlation plots of local and...

We propose a model of time evolving networks in which a kind of transport between vertices generates new edges in the graph. We call the model “Network formed by traces of random walks”, because the transports are represented abstractly by random walks. Our numerical calculations yield...

We discuss various features of the trajectories of spider monkeys looking for food in a tropical forest, as observed recently in an extensive in situ study. Some of the features observed can be interpreted as the result of social interactions. In addition, a simple model of deterministic walk in...

For arbitrary random walks in any d-dimensional space, a 1/d expansion of the most probable size ratio, i.e., squared radius of gyration s2 divided by 〈s2〉 of open random walks, has been developed, which, at O(1/d3), yields a very good approximation to the exact value for chains (d ⩾ 2)...