## 1 Introduction
To achieve unlimited network lifetime when sensor nodes have the only energy source harvested from environment, the network wide energy neutral state should be guaranteed which requires each node to consume less energy than the amount of energy harvested during a certain period of time [27]. In [28], a protocol named Distributed Energy Harvesting Aware Routing (DEHAR) is proposed to find energy optimized routers in EH-WSNs. This protocol finds and maintains an energy efficient route from every source node to BS, taking into account the current energy status of the network. Sensor nodes with too little energy are excluded from these optimized routes so that they can regain the energy level through energy harvesting. Simulations show that perpetual network operation is possible for this protocol. In addition, in [18], an Energy Neutral Routing (ENR) protocol is proposed to guarantee the network wide energy neutral state in EH-WSNs. Admission controls are carried out locally at every node based on its energy budgets to admit or reject traffic routing requests, which can prevent each node from consuming more energy than harvested. In recent years, cluster-based routing for EH-WSNs has aroused wide concern. In addition, many protocols have been proposed to solve the clustering problem in single-hop EH-WSNs [20; 25; 29; 30]. In [29], EH-Sensors are introduced to serve as the dedicated relay nodes for cluster heads (CHs), for the purposed of prolonging the network lifetime. In addition, in [20; 25; 30], sensor nodes have the only energy source harvested from the environment. To achieve perpetual network operation, sensor nodes' energy harvesting ability and available energy are usually considered in these protocols when selecting CHs. To the best of our knowledge, there are fewer works about clustering in multi-hop EH-WSNs. Because of the characteristics of wireless channel, single-hop communication method may result in low energy consumption efficiency [31]. In addition, sensor nodes usually have limited communication abilities in real scenario. Thus, it is better to employ the multi-hop inter-cluster communication method so that CHs in the network can transmit data to BS cooperatively. In this paper, we concentrate on the problem of how to construct the optimal multi-hop clustering architecture in EH-WSNs, for the purpose of achieving perpetual network operation and maximizing network information gathering. Our main contributions are listed as follows:  
* First, we propose a Multi-hop Energy Neutral Clustering (MENC) algorithm that ensures the network-wide energy neutral operation. MENC clusters the network with goal of providing perpetual network operation by controlling the network data transmission rate and selecting the qualified CHs based on the available energy of sensor nodes. * Second, we construct an easy-to-deploy network architecture in which the sensor field is divided into several ring-based units, and CHs in different units compose a multi-hop routing backbone. Based on this network architecture, we analyze the energy consumption of sensor nodes for intra- and inter-cluster data transmission and then give the energy neutrality constraints. Under these constraints, every sensor node can work in an energy neutral state. * Third, by balancing the average energy consumption of nodes in different units, we conclude a constraint formula of the number of clusters between neighbor units. Under this constraint, we can avoid packet loss problem which is caused by the limited data forwarding ability of the CHs in the unit closer to BS. * At last, under the energy neutrality constraints, we optimize the parameters appeared in our proposed protocol through convex optimization, including the optimal number of units, minimum network data transmission cycle and number of clusters in the first unit, for the purpose of maximizing network information gathering. The rest of this paper is organized as follows: Section 2 summarizes the existing works related to our clustering scheme.