Distance distribution in random graphs and application to networks exploration (original) (raw)
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Distance distribution in random graphs and application to network exploration
Physical Review E, 2007
We consider the problem of determining the proportion of edges that are discovered in an Erdős-Rényi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of determining the proportion of edges connecting nodes that are at identical distance from the source node. The evolution of this quantity with the probability of existence of the edges exhibits intriguing oscillatory behavior. In order to perform our analysis, we introduce a different way of computing the distribution of distances between nodes. Our method outperforms previous similar analyses and leads to estimates that coincide remarkably well with numerical simulations. It allows us to characterize the phase transitions appearing when the connectivity probability varies.
The Optimal Path in an Erdos-Renyi Random Graph
We study the optimal distance opt in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path ("cost") is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in Erdős-Rényi (ER) random graphs, opt scales as N 1/3 , where N is the number of nodes in the graph. Thus, opt increases dramatically compared to the known small world result for the minimum distance min , which scales as log N . We also find the functional form for the probability distribution P ( opt ) of optimal paths. In addition we show how the problem of strong disorder on a random graph can be mapped onto a percolation problem on a Cayley tree and using this mapping, obtain the probability distribution of the maximal weight on the optimal path.
First-passage properties of the Erdos–Renyi random graph
Journal of Physics A: Mathematical and General, 2005
We study the mean time for a random walk to traverse between two arbitrary sites of the Erdős-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behavior can be understood by simple heuristic arguments.
The distribution of path lengths of self avoiding walks on Erdős–Rényi networks
Journal of Physics A: Mathematical and Theoretical, 2016
We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a deadend node from which they cannot proceed. Focusing on Erdős-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, p t (k), with an average degree, k t , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, n T (ℓ), increases dramatically as a function of ℓ. We also obtain analytical results for the path-length distribution, P (ℓ), of the SAW paths which are actually pursued, starting from a random initial node. It turns out that P (ℓ) follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.
Scaling of Optimal-Path-Lengths Distribution in Complex Networks
We study the distribution of optimal path lengths in random graphs with random weights associated with each link ͑"disorder"͒. With each link i we associate a weight i = exp͑ar i ͒, where r i is a random number taken from a uniform distribution between 0 and 1, and the parameter a controls the strength of the disorder. We suggest, in an analogy with the average length of the optimal path, that the distribution of optimal path lengths has a universal form that is controlled by the expression ͑1/ p c ͒͑ᐉ ϱ / a͒, where ᐉ ϱ is the optimal path length in strong disorder ͑a → ϱ͒ and p c is the percolation threshold. This relation is supported by numerical simulations for Erdős-Rényi and scale-free graphs. We explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network.
Journal of Physics A: Mathematical and Theoretical, 2016
Rare event statistics for random walks on complex networks are investigated using the large deviations formalism. Within this formalism, rare events are realized as typical events in a suitably deformed path-ensemble, and their statistics can be studied in terms of spectral properties of a deformed Markov transition matrix. We observe two different types of phase transition in such systems: (i) rare events which are singled out for sufficiently large values of the deformation parameter may correspond to localized modes of the deformed transition matrix; (ii) "mode-switching transitions" may occur as the deformation parameter is varied. Details depend on the nature of the observable for which the rare event statistics is studied, as well as on the underlying graph ensemble. In the present letter we report on the statistics of the average degree of the nodes visited along a random walk trajectory in Erdős-Rényi networks. Large deviations rate functions and localization properties are studied numerically. For observables of the type considered here, we also derive an analytical approximation for the Legendre transform of the large-deviations rate function, which is valid in the large connectivity limit. It is found to agree well with simulations.
2006
Erdös-Renyi Random Graphs The first random graph model was introduced by Erdös and Rényi in the late 1950's. To define the model, we begin with the set of vertices V = {1, 2,. .. n}. For 1 ≤ x < y ≤ n let η x,y be independent = 1 with probability p = λ/n and 0 otherwise. Let η y ,x = η x,y. If η x,y = 1 there is an edge from x to y. A large Erdös-Renyi random graph has a degree distribution that is Poisson with mean λ. However in many technological and social networks, the degree distribution p k follows a power law: p k ∼ Ck −α. Rick Durrett (Cornell) Random Graph Dynamics 2 / Figure: Sweden sex partners follow power law Rick Durrett (Cornell) Random Graph Dynamics 3 / 109 Fixed Degree Distributions Molloy and Reed (1995) were the first to construct graphs with specified degree distributions. We will use the approach of Newman, Strogatz, and Watts (2001, 2002) to define the model. Let d 1 ,. .. d n be independent and have P(d i = k) = p k. Since we want d i to be the degree of vertex i, we condition on E n = {d 1 + • • • + d n is even}. If the probability P(E 1) ∈ (0, 1) then P(E n) → 1/2 as n → ∞ so the conditioning will have little effect on the finite dimensional distributions. Rick Durrett (Cornell) Random Graph Dynamics 4 / Attach d i half-edges to vertex i and then pair the half-edges at random. This can produce parallel edges or self-loops, but if Ed 2 i < ∞ then with probability bounded away from 0 we get an ordinary graph. Rick Durrett (Cornell) Random Graph Dynamics 6 / Contact Process Consider the contact process on a power-law random graph. In this model infected individuals become healthy at rate 1 (and are again susceptible to the disease) susceptible individuals become infected at a rate λ times the number of infected neighbors. Pastor-Satorras and Vespigniani (2001a, 2001b, 2002) have made an extensive study of this model using mean-field methods (See Section 4.8.). Rick Durrett (Cornell) Random Graph Dynamics 7 / 109 Contact Process Conjectures Let λ c be the critical value for prolonged persistence. If λ > λ c there will be a quasi-stationary distribution with density ρ(λ) ∼ C (λ − λ c) β If α ≤ 3 then λ c = 0. If 3 < α < 4, λ c > 0 but the critical exponent β > 1. If α > 4 then λ c > 0 and β = 1. Problem. Berger, Borgs, Chayes, Saberi (2005) prove persistence for time exp(cn 1/2) for any λ > 0 when α = 3. Does it last for exp(cn)? Rick Durrett (Cornell) Random Graph Dynamics 8 / Voter models Vertex voter model. Each vertex x changes at rate 1. Pick a neighbor at random and set ξ(x) = ξ(y). Genealogical process jumps at rate 1. Stationary distribution π(x) = cd(x). Edge voter model. Each edge becomes active at rate 1. Flip a coin to give it an orientation (x, y) then set ξ t (x) = ξ t (y). Genealogical process of a site is a random walk that jumps to a randomly chosen neighbor at rate d(x). Stationary distribution is uniform. Rick Durrett (Cornell) Random Graph Dynamics 9 / 109 Rick Durrett (Cornell) Random Graph Dynamics 11 / 109 Rick Durrett (Cornell) Random Graph Dynamics 17 / 109 Rick Durrett (Cornell) Random Graph Dynamics 23 / 109 Rick Durrett (Cornell) Random Graph Dynamics 29 / 109 Rick Durrett (Cornell) Random Graph Dynamics 65 / 109 Rick Durrett (Cornell) Random Graph Dynamics 83 / 109 Rick Durrett (Cornell) Random Graph Dynamics 89 / 109 Rick Durrett (Cornell) Random Graph Dynamics 95 / 109
Connectivity of Growing Random Networks
Physical Review Letters, 2000
A solution for the time-and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites which link to earlier sites with a probability A k which depends on the number of pre-existing links k to that site. For homogeneous connection kernels, A k ∼ k γ , different behaviors arise for γ < 1, γ > 1, and γ = 1. For γ < 1, the number of sites with k links, N k , varies as stretched exponential. For γ > 1, a single site connects to nearly all other sites. In the borderline case A k ∼ k, the power law N k ∼ k −ν is found, where the exponent ν can be tuned to any value in the range 2 < ν < ∞.
Analytical results for the distribution of shortest path lengths in random networks
EPL (Europhysics Letters), 2015
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Node-node distance distribution for growing networks
2003
We present the simulation of the time evolution of the distance matrix. The result is the node-node distance distribution for various kinds of networks. For the exponential trees, analytical formulas are derived for the moments of the distance distribution.