A non-backtracking random walk on a graph is a random walk where, at each step, it is not allowed to return back to the node that has just been left. Non-backtracking random walks can model physical diffusion phenomena in a more realistic way than traditional random walks. However, the interest in these stochastic processes has grown only in recent years and, for this reason, there are still various open questions. In this work, we show how to compute the average number of steps a non-backtracking walker takes to reach a specific node starting from a given node. This problem can be reduced to solving a linear system that, under suitable conditions, has a unique solution. Finally, we compute the average number of steps required to perform a round trip from a given node and we show that mean return times for non-backtracking random walks coincide with their analogue for random walks in all finite, undirected graphs in which both stochastic processes are well-defined.

Random graph models are a recurring tool‐of‐the‐trade for studying network structural properties and benchmarking community detection and other network algorithms. Moreover, they serve as test‐bed generators for studying diffusion and routing processes on networks. In this paper, we illustrate how to generate large random graphs having a power‐law degree distribution using the Chung–Lu model. In particular, we are concerned with the fulfillment of a fundamental hypothesis that must be placed on the model parameters, without which the generated graphs lose all the theoretical properties of the model, notably, the controllability of the expected node degrees and the absence of correlations between the degrees of two nodes joined by an edge. We provide explicit formulas for the model parameters to generate random graphs that have several desirable properties, including a power‐law degree distribution with any exponent larger than 2, a prescribed asymptotic behavior of the largest and average expected degrees, and the presence of a giant component.

Being able to produce synthetic networks by means of generative random graph models and scalable algorithms is a recurring tool-of-the-trade in network analysis, as it provides a well-founded basis for the statistical analysis of various properties in real-world networks. In this paper, we illustrate how to generate large random graphs having a power-law degree distribution by means of the Chung-Lu model. In particular, we are concerned with the fulfillment of a fundamental hypothesis that must be placed on the model parameters, without which the generated graphs loose all the theoretical properties of the model, notably, the controllability of the expected node degrees and the absence of correlations between the degrees of two nodes joined by an edge. We provide explicit formulas for the model parameters in order to generate random graphs which fulfill a number of requirements on the behavior of the smallest, largest, and average expected degrees and have several desirable properties, including a power-law degree distribution with any prescribed exponent larger than 2, the presence of a giant component and no potentially isolated nodes.

A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is not allowed to revisit a node in one step. Second-order random walks can model physical diffusion phenomena in a more realistic way than traditional random walks and have been very successfully used in various network mining and machine learning settings. However, numerous questions are still open for this type of stochastic processes. In this work, we extend well-known results concerning mean hitting and return times of standard random walks to the second-order case. In particular, we provide simple formulas that allow us to compute these numbers by solving suitable systems of linear equations. Moreover, by introducing the ‘pullback’ first-order stochastic process of a second-order random walk, we provide second-order versions of the renowned Kac’s and Random Target Lemmas.