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Based on the complex network theory, robustness analysis of condition monitoring wireless sensor network under uncertain interference is present. In the evolution of the topology of sensor networks, the density weighted algebraic connectivity is taken into account, and the phenomenon of removing and repairing the link and node in the network is discussed. Numerical simulation is conducted to explore algebraic connectivity characteristics and network robustness performance. It is found that nodes density has the effect on algebraic connectivity distribution in the random graph model; high density nodes carry more connections, use more throughputs, and may be more unreliable. Moreover, the results show that, when network should be more error tolerant or robust by repairing nodes or adding new nodes, the network should be better clustered in median and high scale wireless sensor networks and be meshing topology in small scale networks.

Currently, wireless sensor networks have been deployed for condition monitoring application. In industrial harsh environment, there are many kinds of uncertain interference, for example, energy dependence, dynamic topological update, and varying large number of nodes, and these make WSN a type of complex system.

Under uncertain industrial environment, robustness is an important property. Robustness is often defined as invariance degree of state, behavior, and function or the adaptation/flexibility degree under interference of perturbations. Robust analysis of wireless sensor networks is intractable and challenging.

There are three models of complex network [

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With the fundamentals of these, the paper focuses on the research of topology choice and repairing control based on density weighted algebraic connectivity; when the vertex and links are not always constant, they can change with time.

The contributions of the paper are

The paper is arranged as follows: in Section

A network is represented as an undirected graph

Erdős and Rényi define a random graph as

In ER model, there is a critical probability

The critical probability at which almost every graph contains a subgraph with

The useful threshold probabilities for WSN at which different subgraphs appear in a random graph are shown in Figure

The expectation value of the number of nodes with degree

A general conclusion is that, for most values of

The clustering coefficient of a random graph is formula

Threshold probabilities at which different subgraphs appear in a random graph.

The Laplacian matrix of a graph

The asymptotic behavior of the algebraic connectivity in the Erdős-Rényi random graph

Paper [

At large graph size

And the larger the algebraic connectivity is, the better the graph’s robustness of node and link failures is.

Figure

Simplest topology of normal condition monitoring WSN.

Star topology of WSN

Single hop mesh topology of WSN

In Figure

In time domain, this omitted network graph can be looked as a random graph. (For the data transmit, the connected link is at probability

Here the average degree of a vertex in this network is

If

Modern complexity topology of normal condition monitoring WSN.

The omitted network graph can also be looked to as a random graph. And the network made from route nodes and sink nodes has similar property with mesh network.

The eigenvalues of

Algebraic connectivity of wireless star, cluster-tree, or mesh network has below properties:

For an ideal full-function mesh network, algebraic connectivity is proportional with density of network.

For an ideal star or cluster-tree network, the end nodes have similar algebraic connectivity; algebraic connectivity of the route nodes and AP (or coordinator) is proportional with density of network.

For complex WSN, algebraic connectivity has a weight; the simplest example of the weight is density of network.

To discuss the robustness of WSN, here two connectivity metrics of

The robustness of network graph has a relationship with algebraic connectivity; the algebraic connectivity of a graph is increased with the node and the link connectivity. In another way, robustness has a relationship with error tolerance of network. Two types of node removal were considered. The first type was that all the nodes were randomly removed. The second type was that the most highly connected nodes were of more reliability nodes; other was the same as type 1.

In graph

If

The sum of

If

Propose the density (weighted) stands for more easy to produce congest, and to be more unreliability. Here density weighted influence function set of network is defined. It has a relationship with throughput, energy cost, and so on.

An ideal random network means that there are no determining factors that can infect network communication, for example, (nodes) density, (effective) communication distances. Then the mean or least square mean of connectivity is optimum estimation value, and meanwhile the network may be microstable (less intermoving) or reliable.

There are some different kinds of model of time-varying topologies:

When wireless sensor networks are working under uncertain interference, nodes and links may lose effect momentarily or permanently. And the topology may be different with random topologies.

When the links are added or removed unpredictably from the set at any time, the graph can be looked to as the realization of a random process. WSN are normally exposed to random communication failures caused by uncertain interference, and these communication impairments cause abrupt changes in the connectivity of the network, which are described by means of a random graph,

When considering formula (

Then the algebraic connectivity

The state of the network can be simplified as linear system:

A discrete implementation of the expression of (

Different from weighted algebraic connectivity,

More generally, the linear control systems are described by the following state equation:

In wireless condition monitoring network, supposing whether nodes can be connected or not only relies on its effective communication distance. As shown in Figure

Different topology and its communication links in a certain area (left is random mesh; right is star).

The connectivity is defined as the successful connection possibility when random deploy nodes in this area 100 times. When numbers of nodes are above than or equal to 10, the networks have a reliability of connectivity (as shown in Figure

The relationship between density and connectivity.

This means, in wireless condition monitoring network, an effective distance of random network should be larger than half of monitoring area. Considering its physical communication ability, the edge of monitoring cell is defined as twice times of its effective distance.

The relationship between algebraic connectivity and connectivity.

The relationship between throughput of network and density.

As discussed in

The relationship between random destruction number of node and connectivity.

There are two kinds of data; one kind is distributed data. It combined lots of nodes to get a useful data, as shown in formula

The another kind is standalone data, when can be described as a function, like

An example delay of wireless HART network is

Suppose that loss of effective nodes is relay nodes, and then delay time value is changed in normality distribution way; its means is zero. Its standard deviation has a relationship with connectivity.

As a result, if delay time is shorter than sample time, delays will do less interference with LTI system. And if the packet loss rate was less than 15% (use true time simulate using CSMA/CA, TDMA), spring damper system should work normally with little performance degrade (as shown in Figure

The relationship between loss packet rate and step response of spring damper.

If the system is not connected, should repair it. This robust control problem is to find control scheme to obtain system robustness. That includes two factors: the first is topology selection; the second is repairing scheme.

If the connected degree distribution was Poisson distribution, the Packet loss rate under uncertain interface should follow the Poisson distribution.

Then this becomes a time-space transmitting processing problem.

In every time slot different density (data transmitting rate), the most possibility of losing packet is shown in Figures

The time and space domain losing packet rate at lower space density.

The time and space domain losing packet rate at higher space density.

But center nodes are more important and fragile than other nodes in these topologies. If they are to be novel stronger than other nodes, in median and large networks, this topology may have large possibility to be more robust than mesh topology in median and large scale networks. And, in small scale network, mesh topology may have large possibility to be more robust than other topology.

The cluster-tree, star-mesh topologies are easier than only mesh network to repair the network and may have large possibility to be more robust than mesh topology in median and large scale networks. In small scale network, mesh topology may have large possibility to be more robust than other topology.

If random destruction of node was larger than its algebraic connectivity, the connection of network should be broken out.

The properties of density weighted algebraic connectivity are two factors: one is that the useful value of algebraic connectivity is in certain density; the other is an effective network in which the density of network should not be large.

For support and contributions to discuss the authors would like to thank the rest of MCC (Mobile Computer Center) of UESTC (University of Electronic Science and Technology of China), Canada partner of ISTP, and others.