Based on the complex network theory, a new topological evolving model is
proposed. In the evolution of the topology of sensor networks, the
energy-aware mechanism is taken into account, and the phenomenon of
change of the link and node in the network is discussed. Theoretical
analysis and numerical simulation are conducted to explore the
topology characteristics and network performance with different node
energy distribution. We find that node energy distribution has the
weak effect on the degree distribution

Recently, complex networks have attracted considerable intention to investigate various real-world dynamic networks, such as scientific collaboration, the Internet, worldwide web, social networks, biological networks, transportation networks, e-mail networks, software engineering, and ad hoc networks; see [

The motivation for considering dynamic networks comes, in part, from the recent interest in designing wireless sensor networks as a prime example. Sensor networks have recently received increasing interests due to their extensive application in areas such as information collection, environmental monitoring, industrial automation, health tracking, and military surveillance [

Well-known examples of such dynamical network models are proposed including preferential attachment and its variants [

Several recently proposed models have addressed the link and node deletion process for dynamical sensor network. Kong and Roychowdhury [

Motivated by the above analysis, in this paper, we aim to investigate the topological evolving model for wireless sensor network, which is combined energy-aware mechanism with both addition and removal of link and node based on complex network theory. To the best of the authors’ knowledge, the proposed mechanism has not yet been addressed for WSNs. The main contributions of this paper are summarized as follows. (1) a new evolution model is proposed to describe dynamical sensor network. (2) a combination of two important mechanisms of energy preferential attachment for link and node addition and energy antipreferential attachment for link and deletion contributes to investigating the complexity of WSNs. (3) Degree distribution

The rest of this paper is organized as follows. In Section

In this section, we present the following model to capture the particular features of such WSNs evolving networks. In the initial state, the network has a small number

At each time step, a new node is added to the system. And

In real wireless sensor network, the node which has more connetivities will carry more traffic load and consume its energy more quickly. For the balance the energy consumption, we assume the more energy a node has, the strong ability it will have of being connected to the new coming nodes. Therefore,

At each time step, with probability

Then node

Topological characterization is of great importance for network structure in reality. To have a better understanding of the complex dynamics in the considered model and of the influence of

The degree distribution

By the mean-field theory, let

It is easy to know that the first term in (

From the mean-field sense, we have

Supposing that sensor networks which undergo a large number of time steps

It is obvious that, at every time step

In this case, there are only link and node additions without link and node deletions in the evolving process as in [

With the initial condition

The probability that a node has a connectivity which satisfy

Assuming that we add the node to the network at equal time intervals in evolving process for WSNs, the probability density at the time

The probability density function of the degree of a node with energy

The overall probability density function is

In this case, links and nodes in the evolving network model are not monotonously growing. Instead, links and nodes can be added in some occasion and removed in other case. We rewrite (

With the initial condition that node

We can get from (

With the same about the probability density at the time

The probability density function of the degree of a node with energy

To obtain the overall probability density function

In this paper, we consider three kinds of node energy distribution

In Figure

The degree distribution

In Figures

The degree distribution

The degree distribution

To clearly understand the influence of

The relation between node degree and node energy obtained by simulations for two kinds of node energy distribution

Connectivity correlation is also quantified by reporting the numerical value of the slope of

The average degree of neighboring node

We investigate the effect of node energy distribution on network’s cluster coefficient, which quantifies the extent to which nodes adjacent to a given node are linked [

In Figure

The clustering coefficient obtained by simulations for three kinds of node energy distribution

In WSN, the sensor nodes forward the data by multihops. The average path length

Since

The average shortest path length obtained by simulations for three kinds of node energy distribution

The network efficiency obtained by simulations for three kinds of node energy distribution

In this paper, we have addressed a novel topology evolution model for wireless sensor networks. A notion of energy-aware mechanism combined with additions and removals of link and node has been first defined to characterize the evolution model of WSNs. Subsequently, by using mean-field approach, numerical calculation shows the network evolving into the scale-free state with a horse-head-like initial section. Finally, experimental simulations have been employed to demonstrate the effectiveness of the results derived in this paper. Node energy distribution has a weak effect on the degree distribution

This model articulates the topology dynamics of the real WSNs and provides some useful guidelines for constructing WSNs.

This work was partially supported by the NSF of China under Grants no. 60773094 and no. 50803016 and Shanghai Shuguang Program under Grant no. 07SG32.