Distributed coverage in mobile sensor networks without location information

Abstract

With the recent advances in robotic technologies, field coverage using mobile sensors is now possible, so that a small set of sensors can be mounted on mobile robots and move to desired areas. Compared to static settings, area coverage is more complicated in a mobile sensor network due to the dynamics arising from the continuous movement of the sensors. This complication is even higher in the more realistic case where little or no prior metric information is available about the sensor field. In this paper, we consider the problem of self-deployment of a set of mobile sensors which have no knowledge of the area, the number of nodes, their location, and even the distances to each other. In this restricted setting, we formulate the problem as a multi-player game in which each sensor tries to maximize its coverage while considering the overlapping sensing areas by its neighbors. We propose a distributed learning algorithm for coordinating the movement of the sensors in the field, and prove its convergence to the equilibria of the formulated game. Simulation results demonstrate that for moderate density deployments, the proposed algorithm competes with the existing location-dependent mobility strategies, while outperforming location-free algorithms.

Appendices

Appendix A

In this section, we briefly review the concepts of simplicial complexes. In-depth information regarding algebraic topology and simplicial complexes can be found in Hatcher (2002).

Let {v₀, …, v_k} be a geometrically independent set in ${\mathbb{R}}^{N}$, the k-simplex [v₀v₁… v_k] is the set of all points x of ${\mathbb{R}}^{N}$ such that $x = \sum\nolimits_{i = 0}^{k} {\lambda_{i} v_{i} }$, where $\sum\nolimits_{i = 0}^{k} {\lambda_{i} = 1}$, and $\lambda_{i} \ge 0$ for all i. The points v₀, …, v_k are called vertices. Simplex A is a face of simplex B, if the vertices of A form a subset of the vertices of B. A simplicial complex is a finite collection of simplices, K, which are properly joined and have the property that each face of a member of K is also a member of K. An oriented simplicial complex is a simplicial complex with ordering on every k-simplex.

Given a set of points, χ, in a metric space and a fix $\varepsilon > 0$, the Rips complex of χ, R_ϵ(χ), is the abstract simplicial complex whose k-simplices correspond to unordered (k + 1)-tuples of points in χ which are pairwise within distance ε of each other.

Two continuous functions mapping one topological space to another are called homotopic if one can be continuously deformed into the other. Such deformation is called a homotopy between the two functions. Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps f: X → Y and g: Y → X such that gof is homotopic to the identity map id_X and fog is homotopic to id_Y.

A set X is contractible if the identity map on X is homotopic to a constant map. In other words, a contractible space is one that can be continuously shrunk to a point.

Theorem A.1

(Cech theorem) (Bott and Tu 1995) If a collection of sets and all their nonempty finite intersections are contractible, then the union of those sets has the homotopy type as the nerve complex.

Theorem A.2

(Silva and Ghrist 2006) The Rips complex of a sensor network with parameter ε, R_ε, is a subcomplex of the nerve complex corresponding to disks of radius${\varepsilon \mathord{\left/ {\vphantom {\varepsilon {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}$centered at its vertices.

Therefore, $r_{b} \le r_{c} \sqrt 3$ leads to $R_{{r_{b} }} \subseteq C_{{r_{c} }}$.

Appendix B

Definition B.1

Let P⁰ be the transition matrix of a time-homogeneous Markov chain $\{ P_{t}^{0} \}$ on a finite space X, and P^ε be the transition matrix of the perturbed Markov chain $\{ P_{t}^{\varepsilon } \}$. $\{ P_{t}^{\varepsilon } \}$ follows P⁰ with probability $1 - \varepsilon$, and does not follow P⁰ with probability ε. $\{ P_{t}^{\varepsilon } \}$ is a regular perturbation of $\{ P_{t}^{0} \}$ if $\forall x,y \in X$:

• 1. For some $n > 0$, $\forall \varepsilon \in (0,n]$, the Markov chain $\{ P_{t}^{\varepsilon } \}$ is irreducible and aperiodic

• 2. $\mathop {\lim }\nolimits_{{\varepsilon \to 0^{ + } }} P_{xy}^{\varepsilon } = P_{xy}^{0}$

• 3. If $P_{xy}^{\varepsilon } > 0$ for some ε, then there exist a real number $r(x \to y) \ge 0$ such that $0 < \mathop {\lim }\nolimits_{{\varepsilon \to 0^{ + } }} {{P_{xy}^{\varepsilon } } \mathord{\left/ {\vphantom {{P_{xy}^{\varepsilon } } {\varepsilon^{r(x \to y)} }}} \right. \kern-0pt} {\varepsilon^{r(x \to y)} }} < \infty$.

• 4. $r(x \to y)$ is called the resistance of the transition from x to y.

Theorem B.2

(Young 1993) Let$\{ P_{t}^{\varepsilon } \}$be a regular perturbation of$\{ P_{t}^{0} \}$, and$\mu (\varepsilon )$be the unique stationary distribution of$\{ P_{t}^{\varepsilon } \}$for each$\varepsilon > 0$. Then$\mathop {\lim }\nolimits_{{\varepsilon \to 0^{ + } }} \mu_{\varepsilon } = \mu_{0}$exists, and$\mu (0)$is a stationary distribution of$\{ P_{t}^{0} \}$. The stochastically stable states are precisely those states contained in the irreducible classes with minimum stochastic potential.

Definition B.3

The Markov chain (Yuan et al.) is strongly ergodic if there exist a probability distribution µ* on X such that for any initial distribution µ₀ on X and any $m \in {\mathbb{Z}}^{ + }$, $\mathop {\lim }\nolimits_{k \to + \infty } \mu_{0}^{T} P(m,k) = (\mu^{*} )^{T}$, where $P(m,k): = \prod\nolimits_{t = m}^{k - 1} {P(t)}$, $0 \le m < k$. If (Yuan et al.) is strongly ergodic, then (Yuan et al.) in distribution is convergent (Isaacson and Madsen 1976).

Definition B.4

The Markov chain (Yuan et al.) is weakly ergodic if $\forall x,y,z \in X$ and $\forall m \in {\mathbb{Z}}^{ + }$, $\mathop {\lim }\nolimits_{k \to + \infty } \left( {P_{xz} (m,k) - P_{yz} (m,k)} \right) = 0$

Theorem B.5

(Isaacson and Madsen 1976) The Markov chain (Yuan et al.) is weakly ergodic if and only if there is a strictly increasing sequence of positive numbers k_i, such that

$$ \sum\limits_{i = 0}^{ + \infty } {\left( {1 - \lambda P(k_{i} ,k_{i + 1} )} \right) = + \infty } , $$

where $\lambda (P): = 1 - \mathop {\hbox{min} }\nolimits_{1 \le i,j \le n} \sum\nolimits_{k = 1}^{n} {\hbox{min} \left( {P_{ik} ,P_{jk} } \right)}$.

Theorem B.6

(Isaacson and Madsen 1976) A Markov chain (Yuan et al.) is strongly ergodic if

• 1. The Markov chain (Yuan et al.) is weakly ergodic.

• 2. $\forall t$, there exists a stochastic vector µ^(t) on X such that µ^(t) is the left eigenvector of the transition matrix P(t) with eigenvalue 1.

• 3. The eigenvector µ^(t) satisfy $\sum\nolimits_{t = 0}^{ + \infty } \sum\nolimits_{z \in X} \bigg| \mu_{z}^{(t)} - \mu_{z}^{(t + 1)} \bigg| < + \infty$.

Appendix C

In this section, we prove the ergodicity properties of the Markov chain underlying the LFMSD learning algorithm. To this end, we first show that the game’s Markov chain satisfies the “regular perturbation” property (Lemma C.1). This way, we can be sure that all non-zero entries of P^ε are of the order $O\left( {\varepsilon^{r} } \right)$ for some $r \ge 0$. In our problem, $r$ corresponds to the number of sensors which choose their move via exploration rather than via better reply; i.e., $r = \left|\Omega \right|$. Next (in Lemma C.3), we basically follow the standard methodology based on Theorem B.5 for proving the weak ergodicity of the non-stationary Markov chain $P_{t}^{\varepsilon \left( t \right)}$. Basically, the proof entails decomposing the infinite product $\mathop \prod \nolimits_{t} P_{t}^{\varepsilon \left( t \right)}$ into blocks of matrices and showing that the sum of the “ergodic coefficients” of all blocks is infinite. More specifically, the ergodic coefficient of any stochastic matrix ${\mathbb{P}}$ denoted by $erg\left( {\mathbb{P}} \right)$ is given by $erg\left( {\mathbb{P}} \right)\mathop = \limits^{\text{def}} \mathop {\hbox{min} }\nolimits_{ij} \mathop \sum \nolimits_{k} \hbox{min} \left( {{\mathbb{P}}_{ik,} {\mathbb{P}}_{jk} } \right)$ and in order to prove that the ergodic coefficients of all product blocks sums up to infinity, we will use our earlier result (Lemma C.1) to show that for any product block ${\mathbb{P}}$, $\hbox{min} \left( {{\mathbb{P}}_{ik,} {\mathbb{P}}_{jk} } \right)$ is bounded from below by $\varepsilon^{r}$.

Lemma C.1

:$\{ P_{t}^{\varepsilon } \}$is a regular perturbation of$\{ P_{t}^{0} \}$.

Proof

To prove $\{ P_{t}^{\varepsilon } \}$ is a regular perturbation of $\{ P_{t}^{0} \}$, we show that conditions 1, 2, and 3 stated in “Appendix B” hold for all $z^{1} ,z^{2} \in B.$□

Examining condition (1): Since each sensor s_i in location a_i can stay in its position, and its motion to the position with distance 1 from its current position is allowed, so the reachable set from any $z^{0} \in B$ is B. So, the Markov chain $\{ P_{t}^{\varepsilon } \}$ is irreducible on the space B.

Each sensor can stay in its current location, so any state in diag A has period 1. For any state $z: = (a^{0} ,a^{1} )$ the following two paths are feasible:

$$ \begin{aligned} & (a^{0} ,a^{1} ) \to (a^{1} ,a^{0} ) \to (a^{0} ,a^{1} ) \\ & (a^{0} ,a^{1} ) \to (a^{1} ,a^{1} ) \to (a^{1} ,a^{0} ) \to (a^{0} ,a^{1} ) \\ \end{aligned} $$

Therefore, the period of state z is 1. Therefore, $\{ P_{t}^{\varepsilon } \}$ is aperiodic.

Examining condition (2): It is not difficult to see that $\mathop {\lim }\nolimits_{{\varepsilon \to 0^{ + } }} P_{{z^{1} z^{2} }}^{\varepsilon } = P_{{z^{1} z^{2} }}^{0}$.

Examining condition (3): According to (7), it is obvious that $0 < \mathop {\lim }\nolimits_{\varepsilon \to 0} \frac{{P_{{z^{1} z^{2} }}^{\varepsilon } }}{{\varepsilon^{{\left| {\varOmega (z_{1} \to z_{2} )} \right|}} }} < + \infty$. Therefore, the resistance of transition $z^{1} \to z^{2}$ that we show it by $r(z^{1} \to z^{2} )$ is $\varOmega (z^{1} \to z^{2} )$.

Therefore, all the conditions required for being regular perturbation of $\{ P_{t}^{\varepsilon } \}$ are established.

According to (7), it is obvious that $0 < \mathop {\lim }\nolimits_{\varepsilon (t) \to 0} \frac{{P_{t}^{\varepsilon (t)} (z^{1} ,z^{2} )}}{{\varepsilon (t)^{{\left| {\varOmega (z_{1} \to z_{2} )} \right|}} }} < + \infty$. Therefore, for sufficiently large t,

$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } (z^{1} ,z^{2} )\varepsilon_{t}^{{\varOmega (z^{1} ,z^{2} )}} < P_{t}^{\varepsilon (t)} (z^{1} ,z^{2} ) < \bar{\alpha }(z^{1} ,z^{2} )\varepsilon_{t}^{{\varOmega (z^{1} ,z^{2} )}} $$

(10)

Let $\alpha = \mathop {\hbox{min} }\nolimits_{x,y \in B} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } (x,y)} \right)$, then

$$ P_{t}^{\varepsilon (t)} (z^{1} ,z^{2} ) > \alpha \varepsilon_{t}^{{r(z^{1} ,z^{2} )}} $$

(11)

Lemma C.2

Stochastically stable states are exactly the states in${\text{diag}}\left( {\Re (\varGamma_{mc} )} \right)$.

Proof

This Lemma can be proved in the same way as Zhu and Martínez (2013). $\hfill\square$

Lemma C.3

$\{ P^{\varepsilon (t)} \}$ is weakly ergodic.

Proof

Let $z*: = (a*,a*) \in {\text{diag}}\left( {\Re (\varGamma_{mc} )} \right)$ be a stochastically stable state which is in the recurrent class E*. Since $\{ P_{t}^{\varepsilon (t)} \}$ is irreducible, there is a path from any state z to z*. Let $z \Rightarrow z*$ be a path from z to z* in markov chain $\{ P_{t}^{\varepsilon (t)} \}$, and $NT(z \Rightarrow z*)$ be the minimum number of transitions in path $z \Rightarrow z*$. Let $h = \mathop {\hbox{max} }\nolimits_{z} \,\mathop {\hbox{min} }\nolimits_{z \Rightarrow z*} \left( {NT(z \Rightarrow z*)} \right)$. Since there is a transition from z* to itself, there is a path from each state z to z* with length h. Consider path $z \to z^{1} \to \cdots \to z^{h} = z^{*}$. According to (11) $P_{t}^{\varepsilon } (z^{i} ,z^{i + 1} ) > \alpha \varepsilon^{{\varOmega (z^{i} ,z^{i + 1} )}}$, where zⁱ and zⁱ⁺¹ are two consecutive states in the mentioned path. Hence, $\begin{aligned} &P_{(t,t + h)}^{\varepsilon (t)} (z,z^{*}) \ge P_{t}^{\varepsilon (t)} (z,z_{1} )P_{t + 1}^{\varepsilon (t + 1)} (z_{1} ,z_{2} ) \cdots\\ &P_{t + h - 1}^{\varepsilon (t + h - 1)} (z_{h - 1} ,z^{*}) > \alpha^{h} \varepsilon_{t}^{{\varOmega (z,z_{1} )}} \varepsilon_{t + 1}^{{\varOmega (z_{1} ,z_{2} )}} \cdots \varepsilon_{t + h - 1}^{{\varOmega (z_{h - 1} ,z*)}} \\ &\quad > \alpha^{h} \varepsilon_{t + h - 1}^{{\sum\nolimits_{i = 0}^{h - 1} {\varOmega (z_{i} ,z_{i + 1} )} }} \ge \alpha^{h} \varepsilon_{t + h - 1}^{\rho } \\ \end{aligned}$ for all $z \in B$, where $P_{(m,l)}^{{}} : = \mathop \prod \nolimits_{t = m}^{l - 1} P_{t}^{{}} ,0 \le m < l$. Consequently, by choosing a subsequence such that $k_{n + 1} - k_{n} = h$, for sufficiently large n, it holds that

$$ P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (z,z*) > \alpha^{h} \varepsilon_{{k_{n + 1} }}^{\rho } \quad \forall z \in B $$

Therefore,

$$ \begin{aligned} & \hbox{min} \{ P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (x,z*),P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (y,z*)\} > \alpha^{h} \varepsilon_{{k_{n + 1} }}^{\rho } \quad \forall x,y \in B \\ & \sum\limits_{z \in B} {\mathop {\hbox{min} }\limits_{x,y} \{ P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (x,z),P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (y,z)\} } > \alpha^{h} \varepsilon_{{k_{n + 1} }}^{\rho } \\ & \Rightarrow \hbox{min} \sum\limits_{z \in B} {\mathop {\hbox{min} }\limits_{x,y} \{ P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (x,z),P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (y,z)\} } > \alpha^{h} \varepsilon_{{k_{n + 1} }}^{\rho } \\ & \Rightarrow \sum\limits_{n = 0}^{ + \infty } {\hbox{min} \sum\limits_{z \in B} {\mathop {\hbox{min} }\limits_{x,y} \{ P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (x,z),P_{{(k_{n} ,k_{n + 1} )}}^{\varepsilon (t)} (y,z)\} } } \\ & \quad > \alpha^{h} \sum\limits_{n = 0}^{ + \infty } {\varepsilon_{{k_{n + 1} }}^{\rho } = \alpha^{h} \sum\limits_{n = 0}^{ + \infty } {\left( {{{0.1} \mathord{\left/ {\vphantom {{0.1} {\log (k_{n + 1} )}}} \right. \kern-0pt} {\log (k_{n + 1} )}}} \right)}^{\rho } = + \infty } \\ \end{aligned} $$

Therefore, according to Theorem B.5$\{ P_{t}^{\varepsilon (t)} \}$ is weakly ergodic if $\sum\nolimits_{n = 0}^{ + \infty } {\varepsilon_{{k_{n + 1} }}^{\rho } } = + \infty$. For example $\varepsilon (t) = \left( {{1 \mathord{\left/ {\vphantom {1 t}} \right. \kern-0pt} t}} \right)^{{\frac{1}{\rho }}}$ is one choice. $\hfill\square$

Lemma C.4

:$\{ P^{\varepsilon (t)} \}$is strongly ergodic.

Proof

To prove the strong ergodicity of $\{ P^{\varepsilon (t)} \}$ we use Theorem 2 of Anily and Federgruen (1987) and show that the weakly ergodic markov chain $\{ P_{t}^{\varepsilon (t)} \}$ is strongly ergodic. According to this theorem, first, we construct an extension $\bar{\varepsilon }(x)$ of the sequence ε(t); second, we construct a regular extension $\bar{P}^{{\bar{\varepsilon }(x)}} (x)$ of $\{ P_{t}^{\varepsilon } \}$; third, we show that all entries of the regular extension $\bar{P}^{{\bar{\varepsilon }(x)}} (x)$ are members of a closed class of asymptotically monotone functions.

Definition C.5

Let {f(t)} be a sequence with $f(t) \in R^{m}$. The (vector) function $\bar{f}(x):\left( {0,1} \right] \to R^{m}$ is an extension of the sequence if $\bar{f}(x_{t} ) = f(t)$ for some sequence {x_t} with $\mathop {\lim }\nolimits_{t \to \infty } \,x(t) = 0$.

To construct an extension $\bar{\varepsilon }(x)$ of the sequence ε(t), for all $i \in N$, we define $\bar{\varepsilon }_{i} (x)$ as $\bar{\varepsilon }_{i} (x) = 0.1x$, and $x_{t} = {1 \mathord{\left/ {\vphantom {1 {\log (t + 1)}}} \right. \kern-0pt} {\log (t + 1)}}$. Obviously, $\bar{\varepsilon }_{i} (x) = \frac{0.1}{\log (t + 1)} = \varepsilon_{i} (t)$.

Definition C.6

Let $\bar{P}(.)$ be an extension of a nonstationary Markov chain $\{ P_{t}^{\varepsilon (t)} \}$. $\bar{P}(.)$ is a regular extension of $\{ P_{t}^{\varepsilon (t)} \}$ if a positive real number x* exists such that the collection of subchains of $\bar{P}(x)$ is identical for all $x < x*$.

Let $\bar{P}^{{\bar{\varepsilon }_{i} (x)}} (x)$ be an extension to the Markov chain $\{ P_{t}^{\varepsilon (t)} \}$. According to (7), $\overline{P}_{{}}^{{\overline{\varepsilon } (x)}} (z,z^{\prime})$ is positive, so considering x*= 1, it is obvious that the set of transitions with strictly positive probabilities is identical for all x < x*. Now, we should show that every entry function in $\bar{P}^{{\bar{\varepsilon }_{i} (x)}} (x)$ belongs to a closed class of asymptotically monotone functions F.

Definition C.7

A class $F \subset C^{1}$ of functions defined on $\left( {0,1} \right]$ is CAM (closed class of asymptotically monotone functions) if

• 1. $f \in F \Rightarrow f^{\prime} \in F\,{\text{and}}\, - f \in F$

• 2. $f,g \in F \Rightarrow (f + g) \in F\,{\text{and}}\,(f.g) \in F$

• 3. All $f \in F$ change signs finitely often in on (0, 1].

Let F be the class of functions that each $f \in F$ is of the form $\sum\nolimits_{k = 1}^{K} {c_{k} \left( {\log \left( {x + 1} \right)} \right)^{{b_{k} }} V_{k} (x)^{{g_{k} }} }$ where c_k is a real number, and b_k and g_k are integer numbers. It is not difficult to check that the class F is a closed class of asymptotically monotone functions. According to (7), all elements of the transition matrix $P_{t}^{\varepsilon }$ are functions in the class F. Therefore, $\{ P_{t}^{\varepsilon } \}$ is strongly ergodic. $\hfill\square$

Proof of Theorem 2

According to lemma C.4, $\{ P_{t}^{\varepsilon } \}$ is strongly ergodic. Thus, according to theorem B.2 the limiting distribution is $\mu^{*} = \mathop {\lim }\nolimits_{t \to + \infty } \mu^{(t)}$. In addition, we can prove that the stochastically stable states of $\{ P_{t}^{\varepsilon } \}$ are contained in the set ${\text{diag}}\left( {\Re (\varGamma_{mc} )} \right)$ in the same way as Zhu and Martínez (2013). Therefore, the support of $\mu^{*}$ is contained in ${\text{diag}}\left( {\Re (\varGamma_{mc} )} \right)$, that corresponds to $\mathop {\lim }\nolimits_{t \to + \infty } p\left( {z(t) \in diag\Re (\varGamma_{mc} )} \right) = 1$. $\hfill\square$