We propose a mathematical model to describe the athermal fluctuations of thin sheets driven by the type of random driving that might be experienced prior to weak crumpling. The model is obtained by merging the Föppl-von Kármán equations from elasticity theory with techniques from out-of-equilibrium statistical physics to obtain a nonlinear strongly coupled φ 4 -Langevin field equation with spatially varying kernel. With the aid of the self-consistent expansion (SCE), this equation is analytically solved for the structure factor of a fluctuating sheet. In contrast to previous research which has suggested that the structure factor follows an anomalous power-law, we find that the structure factor in fact obeys a logarithmically corrected rational function. Numerical simulations of our model confirm the accuracy of our analytical solution.
We present analytical results for the distribution of shortest path lengths (DSPL) in a network growth model which evolves by node duplication (ND). The model captures essential properties of the structure and growth dynamics of social networks, acquaintance networks, and scientific citation networks, where duplication mechanisms play a major role. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network, forming a link to the mother node, and with probability p to each one of its neighbors. The degree distribution of the resulting network turns out to follow a power-law distribution, thus the ND network is a scale-free network. To calculate the DSPL we derive a master equation for the time evolution of the probability P_{t}(L=ℓ), ℓ=1,2,⋯, where L is the distance between a pair of nodes and t is the time. Finding an exact analytical solution of the master equation, we obtain a closed form expression for P_{t}(L=ℓ). The mean distance 〈L〉_{t} and the diameter Δ_{t} are found to scale like lnt, namely, the ND network is a small-world network. The variance of the DSPL is also found to scale like lnt. Interestingly, the mean distance and the diameter exhibit properties of a small-world network, rather than the ultrasmall-world network behavior observed in other scale-free networks, in which 〈L〉_{t}∼lnlnt.
We present exact analytical results for the degree distribution in a directed network model that grows by node duplication (ND). Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p . In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. The corresponding undirected ND model was studied before and was found to exhibit a power-law degree distribution. We obtain analytical results for the in-degree distribution P t (K in = k), and for the out-degree distribution P t (K out = k), of the corded DND network at time t. It is found that the in-degrees follow a shifted powerlaw distribution, so the network is asymptotically scale free. In contrast, the out-degree distribution is a narrow distribution, that converges to a Poisson distribution in the limit of p 1 and to a Gaussian distribution in the limit of p 1. Such distinction between a broad in-degree distribution and a narrow out-degree distribution is common in empirical networks such as scientific citation networks. Using these distributions we calculate the mean degree K in t = K out t , which converges to 1/(1 − p) in the large network limit, for the whole range of 0 < p < 1.
We present exact analytical results for the distribution of shortest path lengths (DSPL) in a directed network model that grows by node duplication. Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p. In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. In this network not all pairs of nodes are connected by directed paths, in spite of the fact that the corresponding undirected network consists of a single connected component. More specifically, in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths. To calculate the DSPL between those pairs of nodes that are connected by directed paths we derive a master equation for the time evolution of the probability Pt(L = ), = 1, 2, . . . , where is the length of the shortest directed path. Solving the master equation, we obtain a closed form expression for Pt(L = ). It is found that the DSPL at time t consists of a convolution of the initial DSPL P0(L = ), with a Poisson distribution and a sum of Poisson distributions. The mean distance Et[L|L < ∞] between pairs of nodes which are connected by directed paths is found to depend logarithmically on the network size Nt. However, since in the large network limit the fraction of pairs of nodes that are connected by directed paths is diminishingly small, the corded DND network is not a small-world network, unlike the corresponding undirected network.
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