Random walks and flights over connected graphs and complex networks (original) (raw)

Random walks on graphs: ideas, techniques and results

Journal of Physics A: Mathematical and General, 2005

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects. Contents 1 Introduction 2 Mathematical description of graphs 3 The random walk problem 4 The generating functions 5 Random walks on finite graphs 6 Infinite graphs 7 Random walks on infinite graphs 8 Recurrence and transience: the type problem 9 The local spectral dimension 10 Averages on infinite graphs 11 The type problem on the average 1 12 The average spectral dimension 21 13 A survey of analytical results on specific networks 23 13.1 Renormalization techniques. .

Random walk with memory on complex networks

2020

We study random walks on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs of nodes, for a random walk with a memory of one step. We have analyzed one particular model of random walk, where the transition probabilities depend on the number of paths to the second neighbors. The numerical experiments on paradigmatic complex networks verify the validity of the theoretical expressions, and also indicate that the flattening of the stationary occupation probability accompanies a nearly optimal random search.

Exploring Complex Graphs by Random Walks

AIP Conference Proceedings, 2003

We present an algorithm [1] to grow a graph with scale-free structure of in-and out-links and variable wiring diagram in the class of the worldwide Web. We then explore the graph by intentional random walks using local next-near-neighbor search algorithm to navigate through the graph. The topological properties such as betweenness are determined by an ensemble of independent walkers and efficiency of the search is compared on three different graph topologies. In addition we simulate interacting random walks which are created by given rate and navigated in parallel, representing transport with queueing of information packets on the graph.

Navigation by anomalous random walks on complex networks

Scientific Reports, 2016

Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.

Random walks on complex networks with inhomogeneous impact

Physical Review E, 2005

In many complex systems, for the activity fi of the constituents or nodes i a power-law relationship was discovered between the standard deviation σi and the average strength of the activity: σi ∝ fi α ; universal values α = 1/2 or 1 were found, however, with exceptions. With the help of an impact variable we introduce a random walk model where the activity is the product of the number of visitors at a node and their impact. If the impact depends strongly on the node connectivity and the properties of the carrying network are broadly distributed (like in a scale free network) we find both analytically and numerically non-universal α values. The exponent always crosses over to the universal value of 1 if the external drive dominates.

Static and dynamic properties of selected stochastic processes on complex networks

This thesis is concerned with the properties of a number of selected processes taking place on complex networks and the way they are affected by structure and evolution of the networks. What is meant here by 'complex networks' is the graph-theoretical representations and models of various empirical networks (e.g., the Internet network) which contain both random and deterministic structures, and are characterised among others by the small-world phenomenon, power-law vertex degree distributions, or modular and hierarchical structure. The mathematical models of the processes taking place on these networks include percolation and random walks we utilise.The results presented in the thesis are based on five thematically coherent papers. The subject of the first paper is calculating thresholds for epidemic outbreaks on dynamic networks, where the disease spread is modelled by percolation. In the paper, known analytical solutions for the epidemic thresholds were extended to a class...

Dynamics of Nonlinear Random Walks on Complex Networks

Journal of Nonlinear Science

In this paper we study the dynamics of nonlinear random walks. While typical random walks on networks consist of standard Markov chains whose static transition probabilities dictate the flow of random walkers through the network, nonlinear random walks consist of nonlinear Markov chains whose transition probabilities change in time depending on the current state of the system. This framework allows us to model more complex flows through networks that may depend on the current system state. For instance, under humanitarian or capitalistic direction, resource flow between institutions may be diverted preferentially to poorer or wealthier institutions, respectively. Importantly, the nonlinearity in this framework gives rise to richer dynamical behavior than occurs in linear random walks. Here we study these dynamics that arise in weakly and strongly nonlinear regimes in a family of nonlinear random walks where random walkers are biased either towards (positive bias) or away from (negative bias) nodes that currently have more random walkers. In the weakly nonlinear regime we prove the existence and uniqueness of a stable stationary state fixed point provided that the network structure is primitive that is analogous to the stationary distribution of a typical (linear) random walk. We also present an asymptotic analysis that allows us to approximate the stationary state fixed point in the weakly nonlinear regime. We then turn our attention to the strongly nonlinear regime. For negative bias we characterize a period-doubling bifurcation where the stationary state fixed point loses stability and gives rise to a periodic orbit below a critical value. For positive bias we investigate the emergence of multistability of several stable stationary state fixed points.

All-time dynamics of continuous-time random walks on complex networks

The Journal of Chemical Physics, 2013

The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena. However, theoretical analysis has been done mostly for systems with a simple geometry. Here we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks. Specific calculations are performed for models on lattices with branches and for models on coupled parallel-chain lattices. Exact expressions for velocities and dispersions are obtained. Generalized fluctuations theorems for CTRW models on complex networks are discussed.