## 1 Introduction  
Due to the benefits of low cost, rapid deployment, self-organization capability and cooperative data processing, wireless sensor networks (WSNs) have been proposed as a practical solution for a wide range of applications [1], such as battlefield surveillance, habitat monitoring, intelligent agriculture, home automation, _etc._ A typical WSN is composed of a large number of sensor nodes responsible for sensing data and a sink node responsible for collecting and processing data. Since the energy supply for each sensor node is usually extremely limited, energy efficiency is the primary challenge of WSNs. Previous research works have indicated that a clustered structure [2,3] and multi-hop routing [4] are essential for better energy efficiency in large scale WSNs.  
In WSNs, the data traffic follows a many-to-one communication pattern. Nodes closer to the sink tend to carry heavier traffic loads, which will deplete their energy faster [5-9]. [PERSON] _et al._[8] argued that by the time nodes closest to the sink deplete their energy, nodes farther away from the sink may still have up to 93% of their initial energy available, assuming all nodes have the same fixed transmission range and nodes are uniformly distributed in the network. In the literature, this problem is referred as the _energy hole_ problem and a number of studies have been conducted to mitigate its impact on the lifetime of a WSN, including energy-efficient clustering schemes [5,7,10], analysis of the energy hole problem [11,12], nonuniform node distribution strategies [13,14], and utilization of mobile nodes [15].  
In our previous work, a highly scalable network architecture, named Progressive Multi-hop Rotational Clustered (PMRC) structure [16], was proposed for the construction of large scale WSNs. Like other multi-hop structures, PMRC also suffers from the energy hole problem. This issue was alleviated to some extent by the Overlapping Layers (OL) scheme [17], which tends to balance the relay load at the cluster heads for all layers by overlapping the neighbor layers following a desired overlap range. However, due to the fixed layer boundary and range overlap, the network lifetime of OL-enabled WSN is still limited by those nodes which have exactly one candidate cluster head. To overcome this limit, the Minimum Overlapping Layers (MOL) scheme [18] was proposed to gradually increase the required minimum overlap between neighbor layers during network lifetime. As the traffic follows a many-to-one pattern, the network lifetime of MOL-enabled PMRC-based WSNs is still limited by the number of sensor nodes in the initial first layer. Unfortunately, the existing schemes for the energy hole problem in WSNs may not be suitable for MOL-enabled PMRC-based WSNs in which the layer boundary changes dynamically during network lifetime.  
In this paper, we propose a load-similar node distribution strategy to address the energy hole problem of PMRC-based WSNs. First, a load analysis in the continuous space of the network is performed, which reflects the gradual change of the layer boundary in the MOL scheme. Then, based on the analysis, a load-similar node distribution strategy is proposed. Simulation results confirm the superiority of the proposed load-similar node distribution over the non-uniform node distribution [14] and uniform node distribution strategies for MOL-enabled PMRC-based WSNs.  
The rest of this paper is organized as follows. Related work is reviewed in Section 2. The preliminaries, including the PMRC structure, the MOL scheme, and media access control are briefly described in Section 3. In Section 4, the traffic load in PMRC-based WSNs is analyzed and theload-similar node distribution strategy is described. Simulation results are presented in Section 5 and the paper is concluded in Section 6.