Moments Of Degree Distribution Research Articles

Impact of noise and damage on collective dynamics of scale-free neuronal networks

We study the role of scale-free structure and noise in collective dynamics of neuronal networks. For this purpose, we simulate and study analytically a cortical circuit model with stochastic neurons. We compare collective neuronal activity of networks with different topologies: classical random graphs and scale-free networks. We show that, in scale-free networks with divergent second moment of degree distribution, an influence of noise on neuronal activity is strongly enhanced in comparison with networks with a finite second moment. A very small noise level can stimulate spontaneous activity of a finite fraction of neurons and sustained network oscillations. We demonstrate tolerance of collective dynamics of the scale-free networks to random damage in a broad range of the number of randomly removed excitatory and inhibitory neurons. A random removal of neurons leads to gradual decrease of frequency of network oscillations similar to the slowing of the alpha rhythm in Alzheimer's disease. However, the networks are vulnerable to targeted attacks. A removal of a few excitatory or inhibitory hubs can impair sustained network oscillations.

A general geometric growth model for pseudofractal scale-free web

We propose a general geometric growth model for pseudofractal scale-free web (PSW), which is controlled by two tunable parameters. We derive exactly the main characteristics of the networks: degree distribution, second moment of degree distribution, degree correlations, distribution of clustering coefficient, as well as the diameter, which are partially determined by the parameters. Analytical results show that the resulting networks are disassortative and follow power-law degree distributions with a more general degree exponent tuned from 2 to 1 + ln 3 ln 2 ; the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order; the diameter grows logarithmically with the number of network nodes. All these reveal that the networks described by our model have small-world effect and scale-free topology.

Diffusive capture process on complex networks

We study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of N. We find that the lifetime {T} of a lamb scales as {T} approximately N and the survival probability S(N-->infinity, t) becomes finite on scale-free networks with degree exponent gamma > 3. However, S(N, t) for gamma < 3 has a long-living tail on tree-structured scale-free networks and decays exponentially on looped scale-free networks. This suggests that the second moment of degree distribution {k2} is the relevant factor for the dynamical properties in the diffusive capture process. We numerically find that the normalized number of capture events at a node with degree k, n(k), decreases as n(k) approximately k(-sigma). When gamma < 3, n(k) still increases anomalously for k approximately kmax, where kmax is the maximum value of k of given networks with size N. We analytically show that n(k) satisfies the relation n(k) approximately {k2}P(k) for any degree distribution P(k) and the total number of capture events Ntot is proportional to {k2}, which causes the gamma -dependent behavior of S(N, t) and {T}.