Disaster Recovery in Wireless Networks:A Homology-Based Algorithm

Disaster Recovery in Wireless Networks:
A Homology-Based Algorithm

A. Vergne, I. Flint, L. Decreusefond and P. Martins Institut Telecom, TELECOM ParisTech, LTCI
Paris, France
Email: avergne, flint, decreuse, martins@telecom-paristech.fr

In this paper, we present an algorithm for the recovery of wireless networks after a disaster. Considering a damaged wireless network, presenting coverage holes or/and many disconnected components, we propose a disaster recovery algorithm which repairs the network. It provides the list of locations where to put new nodes in order to patch the coverage holes and mend the disconnected components. In order to do this we first consider the simplicial complex representation of the network, then the algorithm adds supplementary vertices in excessive number, and afterwards runs a reduction algorithm in order to reach an optimal result. One of the novelty of this work resides in the proposed method for the addition of vertices. We use a determinantal point process: the Ginibre point process which has inherent repulsion between vertices, and has never been simulated before for wireless networks representation. We compare both the determinantal point process addition method with other vertices addition methods, and the whole disaster recovery algorithm to the greedy algorithm for the set cover problem.

I Introduction

Wireless networks are present everywhere, must it be sensor networks or cellular networks. Fields where wireless sensor networks can be used range from battlefield surveillance to target enumeration in agriculture and include environmental monitoring. In most applications, the topology of the network, such as its connectivity and its coverage, is a critical factor. Cellular networks are used for radio communication, where coverage is also a critical factor. Indeed the covered area is often the main characteristic of a cellular network. However such networks are not necessary built with redundancy and can be sensitive to disasters.

In case of a disaster, a wireless network can be seriously damaged: some of its nodes can be completely destroyed. Coverage holes can appear resulting in no signal for communication or no monitoring at all of a whole area, connectivity can be lost between nodes. Paradoxically, reliable and efficient communication and/or monitoring are especially needed in such situations. Therefore solutions for damage recovery for the coverage of wireless networks are much needed. Extensive research on the coverage problem in wireless networks exists: we can cite location-based [7] and range-based [14] methods. However, connectivity based schemes seem of greater interest since they provide an exact mathematical description of coverage without any geographical (location or distance) information. In [8], the authors introduced the Vietoris-Rips complex, based on the proximity graph of a wireless network, as a tool to compute its topology. Coverage computation via simplicial homology comes down to linear algebra computations. It is for instance used in [4] as a tool for a network operator to evaluate the quality of its network.

In this paper, we present a homology based algorithm for disaster recovery of wireless networks. We represent wireless networks with C̆ech simplicial complexes characterizing their coverage. Given a set of vertices and their coverage radius, our algorithm first adds supernumerary vertices in order to patch every existing coverage hole and connect every components, then runs an improved version of the reduction algorithm presented in [13] in order to reach an optimal result with a minimum number of added vertices. At the end, we obtain the locations in which to put new nodes. For the addition of new vertices, we first compared two usual methods presenting low complexity: grid positioning and uniform positioning. Then, we propose the use of a determinantal point process: the Ginibre point process. This process has the ability to create repulsion between vertices, and therefore has the inherent ability to locate areas with low density of vertices: namely coverage holes. Therefore using this process, we will optimally patch the damaged wireless network. The use and simulation of determinantal point processes in wireless networks is new, and it provides tremendous results compared to classic methods.We finally compared our whole distaster recovery algorithm performance to the classic recovery algorithm performance: the greedy algorithm for the set cover problem.

This is the first algorithm that we know of that adds too many vertices then remove them to reach an optimal result instead of adding the exact needed number of vertices. This, first, allows flexibility in the choice of the new vertices positions, which can be useful when running the algorithm in a real life scenario. Indeed, in case of a disaster, every location is not always available for installing new nodes and preferring some areas or locations can be done with our algorithm. The originality of our work lies also in the choice of the vertices addition method we suggest. On top of flexibility, our algorithm provides a more reliable repaired wireless network than other algorithms. Indeed, adding the exact needed number of vertices can be optimal mathematically speaking but it is very sensitive to the adherence of the nodes positions chosen by the algorithm. To compare our work to literature, we can see that the disaster recovery problem can be viewed as a set cover problem. It suffices to define the universe as the area to be covered and the subsets as the balls of radii the coverage radii. Then the question is to find the optimal set of subsets that cover the universe, considering there are already balls centered on the existing vertices. A greedy algorithm can solve this problem as explained in [3]. We can see in [10] that -nets also provide an algorithm for the set cover problem via a sampling of the universe. We can also cite landmark-based routing, seen in [2], which, using furthest point sampling, provides a set of nodes for optimal routing that we can interpret as a minimal set of vertices to cover an area.

The remainder of this paper is structured as follows: after a section on related work we present the main idea of our disaster recovery algorithm in Section III using some definitions from simplicial homology. Then in Section IV, we compare usual vertices addition methods. In Section V, we expose the determinantal method for new vertices addition. Section VI is devoted to the reduction algorithm description. Finally in Section VII we compare the performance of the whole disaster recovery algorithm with the greedy algorithm for the set cover problem. We conclude in Section VIII.

Ii Recovery in cellular networks

The first step of recovery in cellular networks is the detection of failures. The detection of the failure of a cell occurs when its performance is considerably and abnormally reduced. In [12], the authors distinguish three stages of cell outage: degraded, crippled and catatonic. This last stage matches with the event of a disaster when there is complete outage of the damaged cells. After detection, compensation from other nodes can occur through relay assisted handover for ongoing calls, adjustments of neighboring cell sizes via power compensation or antenna tilt. In [1], the authors not only propose a cell outage management description but also describe compensation schemes. These steps of monitoring and detection, then compensation of nodes failures are comprised under the self-healing functions of future cellular networks.

In this work, we are interested in what happens when self-healing is not sufficient. In case of serious disasters, the compensation from remaining nodes and traffic rerouting might not be sufficient to provide service everywhere. In this case, the cellular network needs a manual intervention: the adding of new nodes to compensate the failures of former nodes. However a traditional restoration with brick-and-mortar base stations could take a long time, when efficient communication is particularly needed. In these cases, a recovery trailer fleet of base stations can be deployed by operators [11], it has been for example used by AT&T after 9/11 events. But a question remains: where to place the trailers carrying the recovery base stations. An ideal location would be adjacent to the failed node. However, these locations are not always available because of the disaster, and the recovery base stations may not have the same coverage radii than the former ones. Therefore a new deployment for the recovery base stations has to be decided, in which one of the main goal is complete coverage of the damaged area. This becomes a mathematical set cover problem. It can been solved by a greedy algorithm [3], -nets [10], or furthest point sampling [2]. But these mathematical solutions provide an optimal mathematical result that do not consider any flexibility at all in the choosing of the new nodes positions, and that can be really sensitive to imprecisions in the nodes positions.

Iii Main idea

When representing a wireless network, one’s first idea will be a geometric graph, where sensors are represented by vertices, and an edge is drawn whenever two sensors can communicate with each other. However, the graph representation has some limitations; first of all there is no notion of coverage. Graphs can be generalized to more generic combinatorial objects known as simplicial complexes. While graphs model binary relations, simplicial complexes represent higher order relations. A simplicial complex is a combinatorial object made up of vertices, edges, triangles, tetrahedra, and their -dimensional counterparts. Given a set of vertices and an integer , a -simplex is an unordered subset of vertices where and for all . Thus, a -simplex is a vertex, a -simplex an edge, a -simplex a triangle, a -simplex a tetrahedron, etc.

Any subset of vertices included in the set of the vertices of a -simplex is a face of this -simplex. Thus, a -simplex has exactly -faces, which are -simplices. For example, a tetrahedron has four -faces which are triangles. An abstract simplicial complex is a collection of simplices which is closed with respect to the inclusion of faces, i.e. all faces of a simplex are in the set of simplices. For details about algebraic topology, we refer to [9].

We consider as inputs the set of existing vertices: the nodes of a damaged wireless network, and their coverage radii. We also need a list of boundary nodes, these nodes can be fictional, but we need to know the whole area that is to be covered. We restrict ourselves to wireless networks with a fixed communication radius , but it is possible to build the C̆ech complex of a wireless network with different coverage radii using the intersection of different size coverage balls. The construction of the C̆ech abstract simplicial complex for a fixed radius is given:

Definition 1 (C̆ech complex)

Given a metric space, a finite set of points in , and a real positive number. The C̆ech complex of , denoted , is the abstract simplicial complex whose -simplices correspond to -tuples of vertices in for which the intersection of the balls of radius centered at the vertices is non-empty.

The C̆ech complex characterizes the coverage of the wireless network. The -th Betti numbers of an abstract simplicial complex are defined as the number of -th dimensional holes in and are computed via linear algebra computations. For example, counts the number of -dimensional holes, that is the number of connected components. And counts the number of holes in the plane. Therefore the Betti number of the C̆ech complex counts the number of coverage holes of the wireless network it represents.

The algorithm begins by adding new vertices in addition to the set of existing vertices presenting coverage holes. We suggest here the use of two common methods, and the new determinantal addition method. As we can see in Section IV, it is possible to consider deterministic or random based vertices addition methods: flexibility is one of the greatest advantage of our algorithm. In particular, it is possible to consider a method with pre-defined positions for some of the vertices in real-life scenarios. For any non-deterministic method, we choose that the number of added vertices, that we denote by , is determined as follows. First, it is set to be the minimum number of vertices needed to cover the whole area minus the number of existing vertices: , being the area to cover. This way, we take into account the number of existing vertices . Then the Betti numbers and are computed via linear algebra thanks to the simplicial complex representation. If there is still more than one connected component, and coverage holes, then the number of added vertices is incremented with a random variable following an exponential growth: , and .

The next step of our approach is to run the coverage reduction algorithm from [13] which maintains the topology of the wireless network: the algorithm removes vertices from the simplicial complex without modifying its Betti numbers. At this step, we remove some of the supernumerary vertices we just added in order to achieve an optimal result with a minimum number of added vertices. We give in Algorithm 1 the outline of the algorithm.

0:  Set of vertices , radius , boundary vertices
  Computation of the C̆ech complex
  Addition of vertices to following chosen method 
  Computation of and
  while  or  do
     Addition of vertices to following chosen method 
     Computation of and
  end while
  Coverage reduction algorithm on
  return  List of kept added vertices.
Algorithm 1 Disaster recovery algorithm

Iv Vertices addition methods

In this section, we propose two vertices addition methods. The aim of this part of the algorithm is to add enough vertices to patch the coverage of the simplicial complex, but the less vertices the better since the results will be closer to the optimal solution. We consider grid and uniform positioning which require minimum simulation capacities and are well known in wireless networks management.

The first method we suggest is deterministic: the number of added vertices and their positions are set and are independent from the initial configuration. It thus ensures perfect coverage, the new vertices are positioned along a square lattice grid of parameter . In the second method we propose, the number of added vertices is computed accordingly to the method presented in Section III, taking into account the number of existing vertices . Then the vertices are sampled following a uniform law on the entire domain.

We can compare the vertices addition methods presented here along two variables: their complexity and their efficiency. First, we compare the complexities of the two methods. They both are of complexity : computations of positions. For the uniform method we have to add the complexity of computing the coverage via the Betti numbers, which is of the order of the number of triangles times the number of edges that is for a square of side according to [5].

To compare the methods efficiency we count the number of vertices each method adds on average to reach complete coverage. The grid method being determinist, the number of added vertices is constant: for a C̆ech complex or for a Vietoris-Rips complex which is an approximation of the C̆ech complex easier to simulate. We can see in TableI the mean number of added vertices on simulations for each method in different scenarios on a square of side with coverage radius , and a Vietoris-Rips complex. Scenarios are defined by the mean percentage of area covered before running the recovery algorithm: if there are many or few existing vertices, and thus few or many vertices to add. We need to note that number of added vertices is computed following our incrementation method presented in Section III and these results only concern the vertices addition methods before the reduction algorithm runs. The grid method is mathematically optimal for the number of added vertices to cover the whole area, however it is not optimal in a real life scenario where positions can not be defined with such precision, and any imprecision leads to a coverage hole. This method fares even or better both in complexity and in number of added vertices.

V Determinantal addition method

The most common point process in wireless network representation is the Poisson point process. However in this process, conditionally to the number of vertices, their positions are independent from each other (as in the uniform positioning method presented in Section IV). This independence creates some aggregations of vertices, that is not convenient for our application. That is why we introduce the use of determinantal point processes, in which the vertices positions are not independent anymore. General point processes can be characterized by their so-called Papangelou intensity. Informally speaking, for a location, and a realization of a given point process, that is a set of vertices, is the probability to have a vertex in an infinitesimal region around knowing the set of vertices . For Poisson process, for any and any . A point process is said to be repulsive (resp. attractive) whenever (resp. ) as soon as . For repulsive point process, that means that the greater the set of vertices, the smaller the probability to have an other vertex.

Among repulsive point processes, we are in particular interested in determinantal processes:

Definition 2 (Determinantal point process)

Given a Polish space equipped with the Radon measure , and a measurable complex function on , we say that is a determinantal point process on with kernel if it is a point process on with correlation functions for every and .

We can see that when two vertices and tends to be close to each other for , the determinant tends to zero, and so does the correlation function. That means that the vertices of repel each other. There exist as many determinantal point processes as functions . We are interested in the following:

Definition 3 (Ginibre point process)

The Ginibre point process is the determinantal point process with kernel , where , are independent Bernoulli variables and for and .

The Ginibre point process is invariant with respect to translations and rotations, making it relatively easy to simulate on a compact set. Moreover, the repulsion induced by a Ginibre point process is of electrostatic type. The principle behind the repulsion lies in the probability density used to draw vertices positions. The probability to draw a vertex at the exact same position of an already drawn vertex is zero. Then, the probability increases with increasing distance from every existing vertices. Therefore the probability to draw a vertex is greater in areas the furthest away from every existing vertices, that is to say in coverage holes. Therefore, added vertices are almost automatically located in coverage holes thus reducing the number of superfluous vertices.

Using determinantal point processes allows us to not only take into account the number of existing vertices, via the computation of , but also take into account the existing vertices positions, then every new vertex position as it is added. It suffices to consider the existing vertices as the first vertices sampled in the process, then each vertex is taken into account as it is drawn. The Ginibre process is usually defined on the whole plane thus we needed to construct a process with the same repulsive characteristics but which could be restricted to a compact set. We also needed to be able to set the number of vertices to draw. Due to space limitations, we will not delve into these technicalities but they are developed in [6].

We can compare the determinantal vertices addition method to the methods presented in Section IV. As for the complexity, since the determinantal method takes into account the position of both existing vertices and randomly added vertices, it is the more complex. First taking into account the existing vertices positions is of complexity , then the position drawing with the rejection sampling is of complexity at most. Thus we have a final complexity of . To which we add the Betti numbers computation complexity: . We also give the comparison between the mean number of added vertices for the three methods, simulation parameters being the same as in Section IV. The determinantal method is the best method among all for the almost covered scenario.

% of area initially covered 20% 40% 60% 80%
Grid method
Uniform method
Determinantal method
TABLE I: Mean number of added vertices

Vi Reduction algorithm

In this section, we recall the steps of the reduction algorithm for simplicial complexes presented in [13]. The algorithm takes as input an abstract simplicial complex: here it is the C̆ech complex of the wireless network plus the added vertices, and a list of boundary vertices. At this step we have ensured that we have one connected component , and no coverage hole . The first step is to characterize the superfluousness of -simplices for the coverage with a degree defined to be the size of the largest simplex a -simplex is the face of. Then to transmit the superfluousness of its -simplices to a vertex, an index of a vertex is defined to be the minimum of the degrees of the -simplices it is a face of. The indices give an order for an optimal removal of vertices: the greater the index of a vertex, the more likely it is superfluous for the coverage of its C̆ech simplicial complex. In our disaster recovery case, we do not want to remove the remaining vertices of our damaged network. So these remaining vertices are given a negative index to flag them as unremovable, and only the newly added vertices are considered for removing. So, the vertices with the greatest index are candidates for removal: one is chosen randomly. If its removal does not change the homology, then it is effectively removed, otherwise it is flagged as unremovable with a negative index. The algorithm goes on until every remaining vertex is unremovable, thus achieving optimal result. For more information on the reduction algorithm we refer to [13]. We can see in the Fig. 1 an execution of the homology based disaster recovery algorithm on a damaged network. Existing vertices are black circles, kept added vertices are red plusses, and removed vertices are blue diamonds.

Fig. 1: Execution of the homology algorithm

Vii Performance comparison

We now compare the performance results of the whole disaster recovery algorithm to the most known coverage recovery algorithm: the greedy algorithm for the set cover problem. First, we compare their complexities. The greedy algorithm method lays a lattice grid of parameter of potential new vertices. Then the first added vertex is the furthest from all existing vertices. The algorithm goes on adding the furthest potential vertex of the grid from all vertices (existing+added). It stops when the furthest vertex is in the coverage ball of an existing or added vertex. Then for the -th vertex addition, the greedy algorithm computes the distances from the existing vertices to the potential vertices. Therefore its complexity is in . For our algorithm, we consider first the complexity of building the abstract simplicial complex of the network which is in , where is the clique number. This complexity seems really high since can only be upper bounded by in the general case but it is the only way to compute the coverage when vertices position are not defined along a given pattern. Then the complexity of the coverage reduction algorithm is in (see [13]). So the greedy algorithm is less complex than ours in the general case, however when is small before , the trend is reversed.

Then, we compare the mean number of added vertices in the final state. The final number of added vertices is the number of added vertices for the greedy algorithm, and the number of kept added vertices after the reduction for the homology algorithm. It is important to note that our algorithm with the grid method gives the exact same result as the greedy algorithm, number of added vertices and their positions being exactly the same. Simulations are done in the same conditions as in Section IV.

% of area initially covered 20% 40% 60% 80%
Greedy algorithm
Homology algorithm
TABLE II: Mean final number of added vertices

The mean numbers of vertices added in the final state both with our recovery algorithm and the greedy algorithm are presented in Table II. They are roughly the same, they both tend to the minimum number of vertices required to cover the uncovered area depending on the initial configuration. Nonetheless, we can see that our algorithm performs a little bit worse than the greedy algorithm in the less covered area scenarios because the vertices are not optimally positioned and it can be seen when just a small percentage of area is covered, and whole parts of the grid from the greedy algorithm are used, instead of isolated vertices. In compensation, our homology algorithm performs better in more covered scenarios. The greedy algorithm is not flexible at all: its success depends highly on the precision of the chosen positions: a coverage hole appears as soon as a vertex is slightly moved. Therefore our algorithm seems more fitted to the disaster recovery case when a recovery network is deployed in emergency both indoor, via Femtocells, and outdoor, via a trailer fleet, where GPS locations are not always available, and sticking to positioning not always exact.

Viii Conclusion

In this paper, we adopt the simplicial homology representation for wireless networks which characterizes both the connectivity and the coverage of a given network. Based on that representation, we write an algorithm which patches coverage holes of damaged wireless networks by giving the positions in which to put new nodes. Our recovery algorithm first adds enough new nodes to cover the whole domain, then runs a reduction algorithm on the newly added nodes to reach an optimal result. The originality of the algorithm lies in the fact that we do not only add the needed nodes, thus providing a mathematically optimal but not reliable result, but adds too many nodes before removing the superfluous ones thus providing a stronger coverage that is less sensitive to the imprecisions of following approximatively GPS locations. Moreover, the vertices addition methods can be adapted to any particular situation. We compare classic positioning methods to the new determinantal method that is more efficient at positioning new vertices where they are needed.


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