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Wireless sensor networks (WSN) are becoming increasingly promising in practice. As the predeployment design and optimization are usually unpractical in random deployment scenarios, the global optimum of the WSN’s performance is achievable only if the topology dependent self-organizing process acquires the overview of the WSN, in which the boundary is the most important. The idea of this paper comes from the fact that contours only break on the geometrical boundary and the WSN are discrete sampling systems of real environments. By simulating a diffusion process in discrete form, the end point of semi-contours suggests the boundary nodes of a WSN. The simulation cases show the algorithm is well worked in WSN with average degree higher than 10. The boundary recognition could be very valuable for other algorithms dedicated to optimize the overall performance of WSN.

There are some areas where we have interests in what is happening, but environments are hostile for a man or too costly to sending a man for the duty. Wireless sensor networks (WSN), which are usually at low cost and self-organized, are appropriate for those tasks [

Some existed algorithms assume that the sensor fields are convex in shape [

Our viewpoint is to regard the WSN as a discrete sampling of the geometric environment. This is inspired by the fact that the WSN are used for providing intense monitoring of the environment. So, the boundaries of the sensor field usually represent the physical boundary of the underlying environments, such as walls of buildings and changes of topography. More importantly, newly appeared boundaries, which means a majority of local sensors are off duty due to destruction or power deficient, could be an indicator of emergency. For example, a wild fire in forest damages all sensors in fire line and also creates new boundary in the sensor field. An inner boundary is also an important indicator of the unhealthiness of the network, such as insufficient connectivity and coverage, revealing the locations where additional sensor nodes are required.

Furthermore, bottleneck recognition [

It is always easy to find the boundary when an overview is offered. For example, in Figure

Sensor nodes deployed in geometric areas.

There are some distributed algorithms trying to recognize the boundary in the literature. They can be classified into three categories by their basic ideas: geometric-based algorithms, statistical algorithms, and topological-based algorithms.

The geometric-based algorithms assume that a node of WSN realizes the exact locations of itself and the nodes in its neighborhood. Fang proposed the algorithm based on the fact that a data packet can only get stuck in a node at boundary in a geographical forwarding [

The information of nodes’ location definitely benefits the boundary recognition. However, the boundary recognition is also needed in the WSN which do not have the ability of locating. So statistical algorithms and topological based algorithms are developed for such WSN.

Statistical algorithm assumes that the nodes are uniformly randomly deployed in sensor field. Fekete proposed an algorithm with such assumption [

Topological based algorithms assume that a node knows only which other nodes are connected directly [

This paper proposes a distributed algorithm for recognizing the boundary of WSN, using only direct connection information. We do not assume that any location information, distance information, or angular information is collected.

This paper is based on the following assumptions:

the nodes in WSN are provided with limited computation ability, energy, and memory;

the communication range of a node is much greater than sensing range; so, the average degree is reasonable if the sensing field is well covered;

the nodes are uniformly randomly deployed in the sensing field;

the nodes are deployed in a closed area;

the sensing data are not required; that is, the algorithm does not require any positioning information about the nodes.

The basic idea of this paper comes from an intense observation of a gas diffusion process in a closed space. We are motivated by the fact that some features of concentration field suggest the boundary of a closed space and then realize that the boundary of WSN can be recognized by simulating similar process.

Consider the following scenario. Bounded space

Fick’s first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative):
^{−2}·s^{−1}]. ^{2}·s^{−1}]. ^{−4}].

Equation (^{−1}] and ^{2}]. The direction is to the outward normal of the element.

Equation (

If the surface

Inside object

The net amount of substance that flows into

In the scenario we are observing, there are a group of sources

Thus, the net gain of gas

Meanwhile, the net gain of gas

By the law of mass conservation, (

Considering the case introduced above, there is not any gas

And, as the space

For the differential equation (

In previous works, the boundary of the space

Given time

There is at least one point

Let

If so, for all

Therefore, for all

Consequently, iso-contour never breaks in

The result is encouraging. However, we should notice that “

In a map with contour, it is possible that contour is a tangent curve to the boundary of the map at somewhere, as the arrow points to in Figure

A map with contour [

Iso-contours when source is at the center of a circle.

Although the worst case happens in small probability, we should and can avoid it by generating another type of inactive gas

A space

Subspace

Similar to (

And the change of concentration is due to diffusion in adjacent subspace and the source effect:

So,

Let

The concentration of

In a WSN application, a lot of sensor nodes are deployed in a sensing area. Our viewpoint is to regard the WSN as a discrete sampling of the environment. Every sensor node is a sample of local area. So, we virtually start a simulation of multigas diffusion process.

Assuming that

At the very beginning, for all

At time

After repeating several times, the diffusion process spreads virtual gas everywhere in the sensor field. Figure

A concentration map and semi-iso-contours. The virtual sources are marked as star. The hotter colour represents higher concentration value, while the cooler colour represents lower concentration value.

The end points of semi-iso-contours roughly show the boundary of the sensor field in Figure

Semi-iso-contours and their end points of the other 9 types of virtual gas.

Reading Figure

Final result of boundary recognition.

Our approach for boundary recognition consists of 3 steps as follows:

simulating the process of diffusion;

drawing semi-iso-contours;

determining whether to be an end point or not.

The 1st step repeats multiple times of communication in neighborhood and calculation. In each repeat, every node should communicate with all its 1-hop neighbors and update

The 2nd step requires a comparison in neighborhood. That is

The 3rd step requires a count over

So, the overall complexity in time is

In all three steps, the nodes record current concentration value of adjacent nodes and itself. All historical data are discarded. So the overall complexity in memory is

A recent paper proposes a topological based algorithm [

The algorithm discussed above is applied in different sensor fields. The results are shown in Figure

Boundary recognition in multiple cases. (a) 3023 nodes with average degree 13.1; (b) 2094 nodes with average degree 12.6; (c) 2381 nodes with average degree 13.0; (d) 2115 nodes with average degree 12.5; (e) 2024 nodes with average degree 13.4; (f) 1311 nodes with average degree 13.2; (g) 6811 nodes with average degree 13.3.

In all these cases, the nodes that are recognized as boundary nodes generally cover the geometrical boundary of the sensor fields. A few inner nodes, which are at least 1-hop range away from actual geometrical boundary, are faultily identified. Table

Statistics of faulty recognition.

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Number of nodes | 3023 | 2094 | 2381 | 2115 | 2024 | 1311 | 6811 |

Inner nodes | 2468 | 1402 | 1866 | 1448 | 1738 | 1065 | 5618 |

Faulty recognition | 9 | 25 | 2 | 18 | 13 | 9 | 2 |

Faulty rate (%) | 0.36 | 1.78 | 0.08 | 1.24 | 0.7 | 0.85 | 0.04 |

It is predictable that the result of boundary recognition is better if the average degree is higher, because when

Boundary recognition in sparse WSN. (a) 10.9 in average degree and (b) 8.9 in average degree.

Faulty rate versus average degree.

When faulty rate increases up to 5% or higher (Figure

A comparison between algorithms. (a–d) Funke’s algorithm with average degree at 5 in (a), 10 in (b), 18 in (c), and 39 in (d). The black dots are identified as boundary nodes, while gray ones are inner nodes. (e)–(f) Our algorithm with average degree at 10.

In this paper, a distributed algorithm for boundary recognition in WSN is proposed. The idea comes from the facts that iso-contours only break on the geometrical boundary and the WSN is a discrete sampling system of real environment. Then, we virtually start a diffusion process to create concentration gradient field in WSN, and finally the nodes that are often identified as end points of semi-iso-contours are regarded as boundary nodes. The simulation results show that the algorithm works well for the WSN with average degree over 10. Further, as diffusion in 3D space is well studied, our algorithm is potentially to be improved to recognize boundary of a 3D WSN, which is also a hot research topic [

The authors declare that there is no conflict of interests regarding the publication of this paper.

Research for this work was supported by the Fundamental Research Funds for the Central Universities (JUSRP1027 and JUSRP51407B).