Energy–Distance Metric and Matérn Hard-Core Spacing for Robust Cluster-Head Selection in Heterogeneous Wireless Sensor Networks

Wireless Sensor Networks (WSNs) form a distributed system comprising self-contained sensor nodes spread over a region to sense or measure physical or environmental parameters such as temperature, humidity, and pressure. The sensors are responsible for data acquisition, which is then wirelessly sent to one or multiple central points for analysis. WSNs are commonly used in challenging or hazardous environments where traditional analysis networks cannot operate satisfactorily, thereby increasing sensor node failure and the possibility of affecting the sensed environment if the system is not inherently designed to operate in such environments [1]. WSNs can provide real-time sensing and data acquisition solutions across a wide range of environments [2].

WSNs consist of a relatively low-cost, low-resource communication system composed of multiple sensor nodes that are set to interact with the system. Each sensor node comprises four basic system parts: a sensing component, a processing element, an energy source or power supply, and a communications part [3]. The presence of limited energy in WSNs necessitates the need for approaches to sensibly administer the networks to maximize their lifespan as well as improve the efficiency of the usage of the energy in the networks [4].

In a WSN, the selection of a cluster-head (CH) is of high significance for improving the lifetime of the network as well as energy savings. The selection of a CH generally considers variables such as the proximity of candidate nodes to the fusion center as well as their energy levels. To optimize energy consumption during data transfer, it is necessary to select the most appropriate CHs. Additionally, network performance is improved by rotating the CH with each new iteration, allowing all nodes in the network to consume energy [5]. Energy-efficient clustering techniques can provide optimal results concerning energy consumption as well as communication costs [6].

Low-Energy Adaptive Clustering Hierarchy (LEACH) is an energy-efficient protocol for WSN. In LEACH, a hierarchical cluster-based structure is employed, dividing sensor nodes into several clusters with each cluster represented by a single node called the CH. The CH combines sensor data from other nodes in its cluster and then forwards it to the base station (BS) [7]. Even if it can be considered a reliable protocol, rotating nodes between cluster head and sensor roles incurs communication overhead, making it impossible to improve energy efficiency. Besides, it is not appropriate for dynamic networks because the sensor data cluster may be misunderstood [8].

The distributed energy-efficient clustering (DEEC) protocol increases the energy efficiency and prolongs the life cycle of WSN. The DEEC protocol systematically selects nodes with relatively high remaining energy as the CH. It balances energy usage and consumption, thereby preventing nodes from dying prematurely and increasing the network’s lifetime [9]. The cluster formation and routing of the DEEC protocols were compared with the best DEEC-based methods. The simulations showed improved end-to-end performance with reduced routing destination look-up time and increased stability of the cluster formation throughout the network [10].

The hard-core (HC) processes due to Matérn can thus be used as an example of point models that are important in spatial statistics. However, in this study, an extension of the generalized Matérn process is presented in the point process framework as well as in particle processes. This is done using a thinning criterion with a distance-dependent probability function that aids in managing spatially dependent point removal, based on proximity between points [11].

However, the current DEEC has several limitations, especially for heterogeneous networks, in which all nodes have different initial energy levels. The DEEC algorithm does not consider either the advanced or super-node scenarios with more energy levels. It does not address Network Load Balancing in clusters, as well as packet drop and Aggregation at CHs. Such difficulties may lead to suboptimal energy consumption in data packets as well as the overall network’s lifetime. A few modifications in DEEC are proposed to counter such problems [12].

From this limitation, the article uses the energy–distance (ED)+HC approach, which uses DEEC’s energy-based CH selection with the following upgrades:

• A tunable ED metric that integrates residual energy with estimated transmit cost to the BS, with adjustable exponents to manage their respective influence in preserving probabilistic structure in line with DEEC while introducing energy-cost awareness.
• An HC spatial constraint that involves thinning driven by the Matérn distribution after the Initial CH choice to impose minimum separation amongst CHs to prevent clustering and contribute to spatial coverage.

These optimizations transcend the shortcomings of classical DEEC and LEACH by combining energy-awareness, cost-awareness, and spatial balance. The ED+HC approach offers consistent CH quantities in addition to balanced rotation, boosting energy efficiency as well as network lifespan.

Figure 1 depicts the proposed methodology.

3.1 Distributed energy-efficient clustering

The DEEC protocol builds upon the first-order radio energy model, which was initially formalized by both electronic circuitry and channel-dependent path loss. The cost of transmitting a message of length l bits over a distance d is defined as [19]:

$E_{T x(l, d)}=\left\{\begin{array}{lc}l E_{\text {elec }}+l \varepsilon_{f s} d^2, & \text { if } d<d_0 \\ l E_{\text {elec }}+l \varepsilon_{m p} d^4, & \text { if } d \geq d_0\end{array}\right.$             (1)

where, $E_{\text {elec}}$ is the per-bit energy dissipation in the transmitter and receiver electronics, $\varepsilon_{f s}$ is the amplifier energy modelled in the free-space channel? $\varepsilon_{m p}$ is the amplifier energy in the multipath fading channel model, and $d_0=\sqrt[2]{\frac{\varepsilon_{f s}}{\varepsilon_{m p}}}$ is the threshold distance separates the two regimes. Sensor nodes may start with different initial energy levels. In DEEC, each node $i$ is initialized with:

$E_{i(0)}=E_0\left(1+a_i\right)$              (2)

where, $E_0$ is the baseline initial energy and $a_i$ is a heterogeneity factor. In a two-level heterogeneous model, a fraction $m$ of the nodes is assigned as advanced nodes with $a_i=a>0$, while the remaining $(1-\mathrm{m}) \mathrm{N}$ nodes are normal with $a_i=0$. This configuration captures realistic network conditions where nodes may have unequal power sources.

To avoid the overhead of global energy information exchange, DEEC introduces an analytical approximation of the network's average residual energy at each round. $E_{ {total}}(0)=\Sigma_i E_i(0)$ is the total initial energy of the network and $R$ is the estimated network lifetime expressed in rounds. This model provides each node with a mechanism to estimate global energy distribution using only local knowledge of its own residual energy, thereby reducing communication overhead and supporting distributed decision-making.

3.2 Distributed energy-efficient clustering cluster-head election

CH election in DEEC builds upon the stochastic thresholding principle originally proposed in LEACH, but introduces modifications that exploit heterogeneity and residual energy. The protocol’s central idea is to assign higher CH election probabilities to nodes with higher residual energy, thus balancing energy consumption across the network and extending lifetime.

The heterogeneity-weighted base probability for node i is defined as [19]:

$p_{s(i)}=p_{\text {opt }} \frac{N\left(1+a_i\right)}{N+\Sigma_j a_i}$            (3)

where, $p_{ {opt}}$ is the desired proportion of cluster-heads in each round? This formulation ensures that nodes with higher initial energy are proportionally more likely to be considered for CH roles.

The instantaneous election probability of node  at round  it is then calculated as [20]:

$p_{i(r)}=p_{s(i)} \frac{E_{i(r)}}{\bar{E}(r)}$            (4)

where, $E_{i(r)}$ is the residual energy of node i and $\bar{E}(r)$ is the analytically estimated average energy across the network. This ratio dynamically adjusts the CH election in favor of energy-rich nodes, preventing premature depletion of weaker nodes.

To regulate fairness and ensure rotational balance in CH selection, DEEC employs the LEACH-inspired threshold function:

$T_{i(r)}=\left\{\begin{array}{cc}\frac{p_{i(r)}}{1-p_{i(r)}\left(\operatorname{r} \bmod \left[\frac{1}{p_{i(r)}}\right]\right)}, & \text { if node } \mathrm{i} \text { is eligible }, \\ 0 , & \text { otherwise } ,\end{array}\right.$                      (5)

where, the epoch length defines eligibility $\left[\frac{1}{p_{i(r)}}\right]$, ensuring that once a node has been a CH. It must wait an appropriate number of rounds before being eligible again. Each node generates a random number. $u \sim U(0,1)$, if $u<T_{i(r)}$, the node becomes a CH for that round.

This mechanism integrates stochastic election with energy-awareness and heterogeneity weighting, achieving a balance amongst randomness, fairness, and efficiency. This methodology notably prolongs the stability period of heterogeneous WSNs in comparison to LEACH, whilst preserving low communication overhead and distributed scalability.

3.3 Proposed modifications: Energy–distance and hard-core mechanisms

The first enhancement of an ED score that modifies the raw DEEC probability. Instead of relying solely on the energy ratio $\frac{E_{i(r)}}{\bar{E}(r)}$, each node $i$ computes a composite score:

$S_i=\left(\frac{E_{i(r)}}{\bar{E}(r)}\right)^\alpha\left(\frac{\bar{E}_{T x} \rightarrow B S}{E_{T x} \rightarrow B S(i)}\right)^\beta$            (6)

$E_{i(r)} / \bar{E}(r)$ represents the DEEC energy-awareness component,

$E_{T x} \rightarrow B S(i)$ is the cost of sending an -bit packet directly from node i to the BS using the first-order radio model,

$\bar{E}_{T x} \rightarrow B S$ is the mean of these costs over all alive nodes,

α > 0 and β > 0 are tunable exponents that control the relative importance of energy and distance.

This formulation allows nodes with high residual energy and low transmission cost to the BS to achieve higher scores, making them more likely to be selected as CHs. It hybridizes the DEEC idea of energy awareness with Hybrid Energy-Efficient Distributed (HEED)’s principle of cost-aware clustering, thereby ensuring that CHs are not only energy-rich but also strategically positioned with respect to the BS.

The scores are then normalized into DEEC-consistent probabilities to maintain the expected number of CHs per round near $p_{{opt}} N_{ {alive}}$:

$p_i^{\prime}=\min \left(0.9999, \frac{p_{ {opt}} N_{{alive}}}{\Sigma_k S_k} S_i\right)$               (7)

where, $p_i^{\prime}$ is the adjusted probability (or priority) assigned to element i. It is capped at 0.9999 to prevent numerical instability or absolute certainty. $p_{o p t}$ is the model or optimization process that determines the optimal or target probability value. $N_{{alive}}$ is the number of active (or “alive”) elements, entities, or processes considered in the current iteration or system state. $\Sigma_k S_k$ is the sum of all $S_k$ terms over index k, representing the total score, weight, or contribution across all elements. $S_i$ is the specific score, weight, or contribution of element i. It scales the result proportionally to the relative importance of element i within the system.

This ensures that, while the CH set changes with ED awareness, the overall CH budget remains consistent with the baseline DEEC model. Finally, the same LEACH-style threshold with epoch control is applied:

$T_{i(r)}=\left\{\begin{array}{cc}\frac{p_{i(r)}}{1-p_{i(r)}\left(r \bmod \left[\frac{1}{p_{i(r)}}\right]\right)}, & \text { if node i is eligible, } \\ 0, & \text { otherwise, }\end{array}\right.$                    (8)

where, $T_{i(r)}$ is the transmission probability (or weighting function) of node $i$ at round $r . p_{i(r)}$ is the base probability assigned to node $i$ at round $r$, reflecting its likelihood of being selected or activated. $r \bmod \left[\frac{1}{p_{i(r)}}\right]$ is the remainder of dividing $r$ by $r \bmod \left[\frac{1}{p_{i(r)}}\right]$, effectively cycling through periods of length $r \bmod \left[\frac{1}{p_{i(r)}}\right]$. This ensures periodic fairness or scheduling of node $i$.

A node i is considered eligible if it meets a predefined selection criterion (e.g., it has not been recently chosen, is still active, or satisfies a system-defined constraint). Otherwise, if node i is not eligible, its transmission probability $T_{i(r)}$ is set to 0, effectively excluding it from consideration during round r. Thus, the ED component directly influences the likelihood of a node becoming a CH, while preserving the fairness and rotation mechanisms inherited from LEACH and DEEC.

3.4 Matérn hard-core spacing

The second enhancement addresses the spatial clustering of elected CHs. After provisional CHs are selected using the modified threshold, a HC thinning process is applied. Candidates are sorted in descending order of their scores S_i, and each node is retained as a CH only if it is at least R_HC meters away from all previously accepted CHs. Formally, for all retained pairs of CHs (i, j), the condition is presented as:

$\left\|x_i-x_j\right\| \geq R_{H C} \quad$ for all kept CH pairs $(i, j)$                 (9)

where, $\left\|x_i-x_j\right\|$ is the Euclidean distance between node i node j.

This CH ensures an even spread of CHs over the sensing region. This goal is achieved through Matérn HC processes, whose effectiveness in generating evenly spaced representatives with low computational complexity is well established.

By combining these two parts, the ED+HC protocol improves both the temporal and spatial aspects of CH election. In the energy dynamics section, the protocol favors the election of nodes whose energies are close to the base station, thereby reducing the high communication costs that may arise in long-distance communication. In the clustering part, the protocol encourages an equal distribution of nodes within the predefined area by preventing over-clustering, thereby increasing the equitable workload distribution. Both parts of the protocol prolong the stabilization period and the time before the first node’s death compared to the standard DEEC protocol.

The practical aspect of the research utilized a computer system featuring an Intel(R) Core(TM) i5-12500H processor running at 2.50 GHz, with 16 cores and 16 GB of Random Access Memory (RAM). In addition, the device used a laptop Graphics Processing Unit (GPU), namely the NVIDIA GeForce RTX 3050.

The simulations were conducted using Python 3.11 version with object-oriented programming. The calculations of energy, as well as mathematical operations, were performed with the help of the built-in math and random modules of Python. Finally, the performance analysis of the proposed network topologies was represented using Matplotlib with the TkAgg backend. The graphical representations of network performance analysis are shown in Table 2.

Table 2. Python libraries and tools employed in the simulation

The network completed 12,180 cycles before running out of nodes. The initial node failures occurred in cycle 5,223, with the last node surviving until cycle 12,180. The duration of this event represents an extended survival period following the initial failure of nodes, thereby creating an equal distribution of energy across the network. During its lifetime, the network transported a total of 69,820 packets to the base station. The average energy consumption of the network was 16J, a high energy efficiency rate of 4,364 packets per joule. The packet delivery rate per round at the next node was 5.73, slightly lower than the expected rate of 5 per round in a 100-node network. This phenomenon may be attributed to the thinning effect that occurs in the HC positioning method among the nodes in the network.

The analytically calculated lifetime of 12,180 rounds was very close to the observed lifetime, indicating that the HC radius method successfully controlled the selection of cluster head nodes and communication costs. The prevention of the earliest node death further confirmed the successful implementation of the modified technique, ensuring energy consumption is equitable across heterogeneous nodes. The energy-distance value prioritizes nodes with higher residual energy and shorter communication distances to the base station.

The inclusion of the HC constraint prohibited cluster head formation and helped to ensure a fair distribution of spatial coverage. Although at times it caused the number of cluster heads to fall below the desired number, it helped ensure a good balance in cluster communications. The large gap between the death of the first and last nodes is clear evidence of this.

Parameters were determined using a balance of residual energy (α = 1.5), reflecting that more sophisticated nodes were intended to play an active rather than a controlling role in unsleeping CHs. A large stress on communication cost (β = 2.0) reflects adherence to HEED’s philosophy that cost-effective clustering improves overall efficiencies. The radius of 8 meters was chosen within a well-balanced scenario of 100 × 100 meters, consisting of 100–500 sensor nodes. This choice made it easier to optimize both the network lifetime and obtain favorable analytical expressions for stability periods and energy consumption. The results show improvements in stability periods and throughput gains over standard DEEC algorithms.

The simulation results show that the energy-distance scoring mechanism, along with the HC spacing technique, enhances sustainability in heterogeneous WSNs. The technique rectifies the DEEC protocol flaw in which the differences in communication costs among sensors around the selected candidate cluster heads are not equal. These differences include the cluster heads to which the sensors belong.

A comparative performance analysis of the baseline DEEC protocol and the improved ED+HC protocol is presented in Table 3. The performance analysis has focused on the most prominent network performance parameters: lifetime, packet delivery, and node death statistics, which have been compared for the same energy levels. This clearly illustrates that the application of energy-distance scoring and HC clustering has improved the network’s energy balance, albeit at the cost of a slight decrease in network throughput.

The results, as shown in Table 3, indicate a significant improvement in network stability and lifetime by the proposed DEEC-ED+HC strategy. This is because the time to the first node death (FND) is reduced by up to 55%, while the total lifetime of the network is increased by 12–17%, which is below the acceptable threshold of 20%. The results statistically confirm the effectiveness of the proposed ED metric in preventing drain in the initial rounds, as well as the importance of the HC condition in preventing the concentration of cluster head nodes within a short distance.

Table 3. Comparative performance of baseline distributed energy-efficient clustering (DEEC) and enhanced ED+HC protocol

Even if the overall number of packet transfers decreases by less than 10%, the implications for reduced quality of service in time-critical applications are not apparent from these observations. Instead, it symbolizes the compromise that prioritizes network stability over the meagre increase in throughput. This reasoning matches the definition of quality-of-service performance for cluster-based WSNs.

The results show that the difference is not significant and does not tend to increase with time. By far, the reason for this difference is the restriction on the HC distance, which increases the average routing distance of cluster communications relative to the ideal analytical solution. In general, the new outcome confirms that DEEC-ED+HC achieves an equally balanced improvement in stability gain, lifetime extension, and quality of service communication reliability compared with the standard DEEC protocol.

Figure 2 shows how the number of alive sensor nodes changes over simulation rounds for the original DEEC and the upgraded protocol. The upgraded protocol maintains more alive nodes for a longer time, indicating better energy management. In contrast, the original DEEC experiences a faster decrease in the number of alive nodes, meaning nodes die earlier. This demonstrates that the upgraded approach improves network lifetime.

Figure 2. Alive nodes per round in the network lifetime

Figure 3 illustrates the variation in the number of cluster heads over time. The original DEEC shows large fluctuations and eventually drops to zero as nodes die. The upgraded DEEC-ED+HC maintains a more stable number of cluster heads across rounds. This stability indicates better cluster-head selection and more balanced energy usage in the upgraded protocol.

Figure 3. Distribution of cluster heads across rounds

Figure 4 presents the total number of packets successfully received at the BS over time. The DEEC-ED+HC always sends more packets than the baseline DEEC protocol, indicating that the upgraded protocol has higher data transmission efficiency than the original DEEC.

Figure 4. Cumulative number of packets delivered to the base station (BS) across rounds

Figure 5 illustrates the total energy consumed by the network as rounds proceed. The total energy consumed in the original DEEC is declining drastically, whereas the total energy consumed in the upgraded DEEC-ED+HC is consumed gradually. This verifies that the upgraded protocol consumes less energy.

Figure 5. Total energy consumption in rounds

The simulation runs with random placements of varying types and sizes, completing 30 runs per network size. The key performance parameters of FND, network lifetime, and the packets delivered to the BS were used to pool the data to calculate the means and standard deviations. The result records the value of each performance indicator as mean ± and standard deviations, which reflect the natural stochastic process associated with the placement of the random nodes as well as the consumption of energy by the network components. For the 100-node WSN, the improved DEEC-ED+HC protocol demonstrates the value of the FND at 5,223 ± 112 rounds, the network lifetime at 12,180 ± 146 rounds, and the packets delivered to the BS at 69,820 ± 1,225 packets, providing clear indications that the obtained improvements are meaningful and do not result from the random deployment of the network components but are representative of the standard stochastic process associated with the random placement of the WSN nodes with varying energy consumption rates.