Node-node distance distribution for growing networks (original) (raw)

Remarks on Distance Distributions for Graphs Modeling Computer Networks

Electronic Notes in Discrete Mathematics, 2002

The Wiener polynomial of a graph G is a generating function for the distance distribution dd(G) = (D 1 , D 2 , . . . , D t ), where D i is the number of unordered pairs of distinct vertices at distance i from one another and t is the diameter of G. We use the Wiener polynomial and several related generating functions to obtain generating functions for distance distributions of unweighted and weighted graphs that model certain large classes of computer networks. These provide a straightforward means of computing distance and timing statistics when designing new networks or enlarging existing networks.

Distance Distributions in Finite Uniformly Random Networks: Theory and Applications

IEEE Transactions on Vehicular Technology, 2000

In wireless networks, the knowledge of nodal distances is essential for several areas such as system configuration, performance analysis and protocol design. In order to evaluate distance distributions in random networks, the underlying nodal arrangement is almost universally taken to be an infinite Poisson point process. While this assumption is valid in some cases, there are also certain impracticalities to this model. For example, practical networks are non-stationary, and the number of nodes in disjoint areas are not independent. This paper considers a more realistic network model where a finite number of nodes are uniformly randomly distributed in a general d-dimensional ball of radius R and characterizes the distribution of Euclidean distances in the system. The key result is that the probability density function of the distance from the center of the network to its n th nearest neighbor follows a generalized beta distribution. This finding is applied to study network characteristics such as energy consumption, interference, outage and connectivity.

Degree Distributions of Growing Networks

Physical Review Letters, 2001

The in-degree and out-degree distributions of a growing network model are determined. The indegree is the number of incoming links to a given node (and vice versa for out-degree). The network is built by (i) creation of new nodes which each immediately attach to a pre-existing node, and (ii) creation of new links between pre-existing nodes. This process naturally generates correlated in-and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained.

Distance distribution in random graphs and application to networks exploration

2007

We consider the problem of determining the proportion of edges that are discovered in an Erdos-Renyi 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 new 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.

Typical distances in a geometric model for complex networks

Internet Mathematics

We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al. [Krioukov et al., 2010] as a model for complex networks, N vertices are drawn randomly within a bounded subset of the hyperbolic plane and any two of them are joined if they are within a threshold hyperbolic distance. With appropriately chosen parameters, the random graph is sparse and exhibits power law degree distribution as well as local clustering. In this paper we show a further property: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in N , i.e., the graph is an ultra-small world. More precisely, we show that the distance rescaled by log log N converges in probability to a certain constant that depends on the exponent of the power law. The same constant emerges in an analogous setting with the well-known Chung-Lu model for which the degree distribution has a power law tail.

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 < ∞.

Average Distance in Growing Trees

International Journal of Modern Physics C, 2003

Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barab\'asi--Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to m=1m=1m=1 nodes. Average node-node distance ddd is calculated numerically in evolving trees as dependent on the number of nodes NNN. The results for NNN not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance ddd for large NNN can be approximated by d=2ln(N)+c1d=2\ln(N)+c_1d=2ln(N)+c1 for the exponential trees, and d=ln(N)+c2d=\ln(N)+c_2d=ln(N)+c2 for the scale-free trees, where the cic_ici are constant. We derive also iterative equations for ddd and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.

Matrix Representation of Evolving Networks

Acta Physica Polonica B, 2005

We present the distance matrix evolution for different types of networks: exponential, scale-free and classical random ones. Statistical properties of these matrices are discussed as well as topological features of the networks. Numerical data on the degree and distance distributions are compared with theoretical predictions.

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.