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In the previous chapters, we explored the effects of resetting on networks considering one and two nodes. In this chapter, we will describe a generalization of random walks with resetting to an arbitrary number of nodes $\mathcal{M}$. In…
Diffusion processes are instrumental to describe the movement of a continuous quantity in a generic network of interacting agents. Here, we present a probabilistic framework for diffusion in networks and propose to classify agent…
Dynamical systems having many coexisting attractors present interesting properties from both fundamental theoretical and modelling points of view. When such dynamics is under bounded random perturbations, the basins of attraction are no…
In this minireview we present the main results regarding the transport properties of stochastic movement with relocations to known positions. To do so, we formulate the problem in a general manner to see several cases extensively studied…
An abstract network approach is proposed for the description of the dynamics in reactive processes. The phase space of the variables (concentrations in reactive systems) is partitioned into a finite number of segments, which constitute the…
Power-law-shaped avalanche-size distributions are widely used to probe for critical behavior in many different systems, particularly in neural networks. The definition of avalanche is ambiguous. Usually, theoretical avalanches are defined…
Quantum repeater networks distribute entanglement over lossy links while many users share a limited pool of entangled pairs. Most existing routing schemes either always use a single best path or rely on global optimizations that are hard to…
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…
A control scheme to reduce the size of avalanches of the Bak-Tang-Wiesenfeld model on complex networks is proposed. Three network types are considered: those proposed by Erd\H{o}s-Renyi, Goh-Kahng-Kim, and a real network representing the…
We prove that for a broad class of permutation-equivariant learning rules (including SGD, Adam, and others), the training process induces a bi-Lipschitz mapping between neurons and strongly constrains the topology of the neuron distribution…
We study the avalanche dynamics in the data packet transport on scale-free networks through a simple model. In the model, each vertex is assigned a capacity proportional to the load with a proportionality constant $1+a$. When the system is…
Complex behavioral contagion in collective evasion of mobile animal groups can be predicted by reconstructing quantitative interaction networks. Based on the assumption of time-scale separation between a fast contagion process and a slower…
Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized…
We consider a discrete-time random walk where the random increment at time step $t$ depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition…
It is well known that the distribution of simple random walks on $\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on…
We consider a random walk among a Poisson cloud of moving traps on ${\mathbb Z}^d$, where the walk is killed at a rate proportional to the number of traps occupying the same position. In dimension $d=1$, we have previously shown that under…
This paper mainly discusses the diffusion on complex networks with time-varying couplings. We propose a model to describe the adaptive diffusion process of local topological and dynamical information, and find that the Barabasi-Albert…
Living neuronal networks in dissociated neuronal cultures are widely known for their ability to generate highly robust spatiotemporal activity patterns in various experimental conditions. These include neuronal avalanches satisfying the…
We discuss basic features of emergent complexity in dynamical systems far from equilibrium by focusing on the network structure of their state space. We start by measuring the distributions of avalanche and transient times in Random Boolean…
We study a simple model for a neuron function in a collective brain system. The neural network is composed of uncorrelated random scale-free network for eliminating the degree correlation of dynamical processes. The interaction of neurons…