Evolution equation for a model of surface relaxation in complex networks

Rocca, C. E. La; Braunstein, Lidia A.; Macri, P. A.

doi:10.1103/physreve.77.046120

Cited by 19 publications

(42 citation statements)

References 38 publications

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“…When the process reaches the steady state [20] of the evolution with this relaxation (R), we construct the gradient network and measure the pressure congestion J R after the relaxational dynamic is performed. In accordance with previous observations [20,22], the steady state is reached extremely quickly: the saturation time actually does not scale with the system size N but it approaches an N -independent constant [23,24].…”

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confidence: 69%

Using relaxational dynamics to reduce network congestion

et al. 2008

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We study the effects of relaxational dynamics on congestion pressure in scale free networks by analyzing the properties of the corresponding gradient networks [1]. Using the Family model [2] from surface-growth physics as single-step load-balancing dynamics, we show that the congestion pressure considerably drops on scale-free networks when compared with the same dynamics on random graphs. This is due to a structural transition of the corresponding gradient network clusters, which self-organize such as to reduce the congestion pressure. This reduction is enhanced when lowering the value of the connectivity exponent λ towards 2.

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confidence: 69%

Using relaxational dynamics to reduce network congestion

et al. 2008

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“…In Fig. 3 we plot W S as function of N in log-log scale for different values of p. We can see that for p = 0 there is a pure power law with exponent 0.186 (see Fig 1) and for p = 1 the scaling behavior of W s with N corresponds to a logarithm as expected for λ < 3 [24,25]. For values 0 < p < 1 we find that there is a crossover between both regimes.…”

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confidence: 78%

“…al. [25] who derived the evolution equation of this model on complex networks. However, up to our knowledge, neither the BD model nor the competition between different processes on complex networks was ever reported.…”

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confidence: 99%

Competition between surface relaxation and ballistic deposition models in scale-free networks

Rocca¹,

Macri²,

Braunstein³

2013

EPL

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In this paper we study the scaling behavior of the fluctuations in the steady state WS with the system size N for a surface growth process given by the competition between the surface relaxation (SRM) and the ballistic deposition (BD) models on degree uncorrelated scale-free (SF) networks, characterized by a degree distribution P (k) ∼ k −λ , where k is the degree of a node. It is known that the fluctuations of the SRM model above the critical dimension (dc = 2) scale logarithmically with N on Euclidean lattices. However, Pastore y Piontti et al. (Phys. Rev. E, 76 (2007) 046117) found that the fluctuations of the SRM model in SF networks scale logarithmically with N for λ < 3 and as a constant for λ 3. In this letter we found that for a pure ballistic deposition model on SF networks WS scales as a power law with an exponent that depends on λ. On the other hand when both processes are in competition, we find that there is a continuous crossover between a SRM behavior and a power law behavior due to the BD model that depends on the occurrence probability of each process and the system size. Interestingly, we find that a relaxation process contaminated by any small contribution of ballistic deposition will behave, for increasing system sizes, as a pure ballistic one. Our findings could be relevant when surface relaxation mechanisms are used to synchronize processes that evolve on top of complex networks.

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“…More recently, research has focused on dynamical processes taking place on the underlying network [4][5][6][7][8][9][10]. Particularly, many mathematical and numerical models have been elaborated to study the problem of synchronization [11][12][13][14][15][16], a phenomenon present in the behavior of many collective systems. In these processes the state of the system evolves to a p-1 synchronized state, where the coupled units adjust their dynamics with one another.…”

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confidence: 99%

“…Solving the discrete EW equation numerically for finite size systems, in [12] the authors found that W s decreases with N , which is not representative of any growth model. With a different approach, La Rocca et al [14] developed a Langevin stochastic equation that describes the evolution of the interface, and solved it up to second order by numerical integration for finite system sizes, recovering Eq. (4).…”

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confidence: 99%

Synchronization in scale-free networks: The role of finite-size effects

Torres¹,

Muro²,

Rocca³

et al. 2015

EPL

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-Synchronization problems in complex networks are very often studied by researchers due to its many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, Scale Free networks with degree distribution P (k) ∼ k −λ , are widely used in research since they are ubiquitous in nature and other real systems. In this paper we focus on the surface relaxation growth model in Scale Free networks with 2.5 < λ < 3, and study the scaling behavior of the fluctuations, in the steady state, with the system size N . We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of N = N * that depends on λ: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above N * , the fluctuations decrease with λ, which means that the synchronization of the system improves as λ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size N and the exponent of the degree distribution.Since a great variety of systems can be represented by complex networks, over the last decades many researchers have studied both the topology and processes that evolve on top of these networks. Systems such as neural networks, the Internet and airlines networks [1][2][3] can be described by a set of nodes connected by links that represent a relationship between them, such as an electric impulse, friendship or air traffic. Many of these real networks were found to be characterized by a Scale Free (SF) topology, given by a degree distributionwhere k is the degree of the nodes and m ≤ k ≤ k max , where m and k max are the minimum and maximum degree respectively, and λ represents the broadness of the distribution. On most real systems, such as the World Wide Web or metabolic networks, it was found that 2 < λ < 3 [1, 2]. More recently, research has focused on dynamical processes taking place on the underlying network [4][5][6][7][8][9][10]. Particularly, many mathematical and numerical models have been elaborated to study the problem of synchronization [11][12][13][14][15][16], a phenomenon present in the behavior of many collective systems. In these processes the state of the system evolves to a p-1

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Evolution equation for a model of surface relaxation in complex networks

Cited by 19 publications

References 38 publications

Using relaxational dynamics to reduce network congestion

Using relaxational dynamics to reduce network congestion

Competition between surface relaxation and ballistic deposition models in scale-free networks

Synchronization in scale-free networks: The role of finite-size effects

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