The connectivity probability of edge evolving network driven by compound Poisson process

Abstract

In this paper, we propose a kind continuous edge evolving network, edges in the network grow by a linear function of time t and sudden deletion described by compound Poisson process with parameter $\lambda t$. Under some mild assumption, we prove that the mean expectation of the first connectivity time show exponential form and the probability of the connectivity also has asymptotic exponential form.

Introduction

Many systems in real world can usefully be represented as networks. Examples can be found in the Internet [1], www [2], [3], social networks [4], citation networks [5], [6], food webs [7], biochemical networks [8], [9]. Each of these networks consists of a set of nodes or vertices representing, for instance, computers or routers on the Internet or people in social network, connected together by links or edges, representing data connections between computers, friendships between people, and so on.

In the investigation of various properties of complex networks, one feature that has been emphasized is the network connectivity which can determine many fundamental properties of networks. This feature is especially important in network control, communication, estimation, wireless sensor and genes's interaction and regulation [10], [11], [12], [13]. This feature was studied in different models and aspects. It was initiated by Erdös and Rényi [14], in the ER model, the number of vertices is fixed, while edges connect one vertex to another randomly with certain probability, Penrose [15] introduced random geometrical graph, edges are connected when two nodes are in a range of distance and studied the connectivity probability, Chung and Lu [16] introduced random weighted graph and gave connectivity probability, based on the work of [17], [18] studied configuration network and connectivity probability for random graphs with given degree distribution, Berkowitz [19] proposed fracture network and gave an analysis of its connectivity using percolation theory, Beveridge [20] proposed random cubic sum graph and obtained thresholds for connectivity which coincides with the disappearance of the last isolated vertex, Caro et al. [21] studied the connected domination set of spanning trees with many leaves and gave an upper bound of connected domination number. In communication network, Wu [22] studied the connectivity in mobile linear network. In Vehicular and Ad Hoc network, for more works in theory and application, we can see [23], [24], [25].

In network, one of the important indexes which describe connectivity is edge, as for the works on the mechanism of edge formation, most previous works concerned on discrete time case, few works were done to the continuous case which has tight connection to actual networks. In considering the change of network edges, most existed evolution models considered the increase or decrease of edges with nodes at the same time, but, in real network, such as Internet network, communication network, interpersonal relationship network, and sexual relationship network, the number of nodes keeps all the time, while the relation between nodes changes with the time. This relationship leads to the number of edge shows a linear growth trend due to the inertia, on the other hand, due to the impact of some unexpected events, the number of edges shows sudden increase or decrease trend. Motivated by previous reasons, we introduce an edge evolution model based on the theory of risk [26], edges in the network change according to the following rule: at first, there exist u edges in the network; the edge grows c per unit time; in the time interval $[0,t]$, some sudden deleting edge behaviors will happen, the times of the sudden deletion obeys Poisson distribution with parameter $\lambda t$, every time, for example, the ith time, Z_i edges are deleted or added, $Z_i,i=1,2,\dots$ are independent random variables and have identity distribution with random variable Z which has values in integer. Let X_t denote the number of edges in the network at time t, we have

$$X_t=\mathit{ct}-\sum _{n=1}^{N_t}{Z}_n+u$$

Comparing to the classical risk model, constraint conditions for coefficients are relaxed, we allow variables to take negative values, which can reflect the sudden changes in the link of the network. Based on our proposed model, we concern the connectivity probability and the first connectivity time in the proposed network. The whole paper are arranged as follows, in Section 2, we discuss the first connectivity time of the network; in Section 3, we give the connectivity probability of the network.