Trapping in complex networks

Abstract

We investigate the trapping problem in Erdos-Renyi (ER) and Scale-Free (SF) networks. We calculate the evolution of the particle density ρ(t) of random walkers in the presence of one or multiple traps with concentration c. We show using theory and simulations that in ER networks, while for short times ρ(t) ∝ exp(−Act), for longer times ρ(t) exhibits a more complex behavior, with explicit dependence on both the number of traps and the size of the network. In SF networks we reveal the significant impact of the trap's location: ρ(t) is drastically different when a trap is placed on a random node compared to the case of the trap being on the node with the maximum connectivity. For the latter case we findIntroduction. -The properties of random walk greatly vary depending on the dimension and the structure of the medium in which it is confined [1][2][3][4], where a particularly interesting medium for the study of the random walk is complex networks [5][6][7][8][9]. Networks describe systems from various fields, such as communication (e.g. the Internet), the social sciences, transportation, biology, and others. Many of these networks are scale-free (SF) [10][11][12][13]. This class of networks is defined by a broad degree distribution, such as a power law P (k) ∝ k −γ (k ≥ m), where γ is a parameter which controls the broadness of the distribution.Trapping is a random walk problem in which traps are placed in random locations, absorbing all walkers that visit them. This problem was shown to yield different results over different geometries, dimensions and time regimes [2,3,[14][15][16][17]. The main property of interest during such a process is the survival probability ρ(t), which denotes the probability that a particle survives after t steps. The problem was studied in regular lattices and in fractal spaces [2,[14][15][16][17][18][19] and recently, in small-world networks [6].In this Letter we study the problem of trapping in networks. This is a model for the propagation of information in certain communication networks. This follows since in some cases data packets traverse the network in a random fashion (for example, in wireless sensor networks [20], ad-hoc networks [21] and peer-to-peer networks [22]). A malfunctioning node in which information is lost (e.g., a router which cannot transmit data due to some failure) acts just like a trap in the model. This model can also be applied to loss of information in messages over communication systems, e.g. in the case of e-mail messages, where a malfunctioning e-mail server acts as a node absorbing, but not transmitting, all e-mail

Close or connected: Distance and connectivity effects on transport in networks

“…The impact of connectivity on transport properties has been put forward in [8,[23][24][25], where it was found in different examples that transport towards a target node can be favored by a high connectivity of the target, and different functional forms of the dependence of the MFPT on the target connectivity were proposed. On the other hand, the dependence of the MFPT on geometric properties, such as the volume of the network and the source to target distance, has been obtained recently in [26][27][28][29], where it was shown that the starting position of the random walker plays a crucial role in the target search problem.…”

Random walks on deterministic scale-free networks: Exact results

“…Due to the strong variability of the topology in the samples, the quenched properties could not coincide with the behavior found in the typical cases, as described by the probability distribution. Therefore, the study of quenched samples and deterministic topologies is a very interesting task [10,11,12,13,14,15].…”

Exact eigenvalue spectrum of a class of fractal scale-free networks

“…A fundamental issue in network science is to understand diverse dynamical processes defined on different networks [4][5][6]. A paradigmatic dynamical process that has attracted increasing research interest is random walks [7][8][9][10][11][12][13][14][15][16][17], which have found a broad range of applications in various areas of science and engineering [18][19][20][21].…”