A Bubble-breaking Phenomenon in the Variation of a Swarm Communication Network

Communication among individuals in a group of agents (fish/birds/robots/people) is the foundation for forming system behaviours. The “communication” here can be any exchange/flow of information in any form. In flocking/schooling of birds/fish it may refer to the sensing (by the group members) of changes in positions/velocities of nearby companions [1]. In a multirobot system it can mean the transmission of signals among the robots. In epidemic spreading it is then the transfer of viruses/bacteria.

Representing with vertices the agents and with edges the established communications yields a communication network of the group. These networks serve as basic mathematical structures upon which system dynamics are built [5, 3, 2, 4]. In nature as well as in practise, the ability of an agent to set up communication with other individuals is usually limited. For the case of interest in this note, it is the maximal distance to set up message channels between two robots. As a consequence, the motion of the group may in turn cause changes to the topology of the network.

Put an iron ring (rigid) into a balloon (not inflated) and seal the valve. No matter how hard you try, it is impossible to stretch and attach the entire balloon onto the ring without breaking it. In other words, if you force as large a part of the balloon as possible to be attached onto the ring, then the rest part of the balloon will be terribly stretched and the continuum of the material may eventually break down. We call this issue a $bubble-breaking$ phenomenon, and its mathematical essense can have implications on various problems from different contexts. For example, in stability theory it implies the fact that, on any sphere enclosing an asymptotically stable limit cycle $\mathcal{P}$ of a (smooth) flow $\varphi$ in $\mathbb{R}^{3}$, there always exists some point $p$ such that the trajectory $t\mapsto\varphi^{t}(p)$ does not converge to $\mathcal{P}$. We encoutered this fact in [6] when studying path-following control. In this note, we demonstrate a case in which the topology of the communication network of a robotic swarm $has$ $to$ change during its movement. As we will see, the mathematics behind also reflects the essense of the bubble-breaking phenomenon.

The description of the case is detailed in Subsection 2.1 where the central problem of this study is stated as Question 1, and then Proposition 2 is proved in Subsection 2.2 as answer to the question. The analysis in Section 2 assumes a special structure from the initial positions of the robots. In Section 3 we show a natural condition on the intitial positions under which the analysis can be applied, and the main result in this part is demonstrated as Proposition 7. In the end, by combining Propositions 2 and 7 we draw the final conclusion as Theorem 11.

Imagine that there is a facility $\mathrm{\mathfrak{P}}$ (floating in the space) which occupies an area of a solid torus

$$\mathrm{\mathfrak{P}}=\big{\{}(r\cos\theta,r\sin\theta,z)\big{|}\theta\in[0,2\pi],\,z^{2}+(r-\frac{1}{10})^{2}\leq\epsilon^{2}\big{\}}$$

with $\epsilon<\frac{1}{100}$. Note that the circle

$$\mathrm{\mathcal{P}}=\{x^{2}+y^{2}=\frac{1}{10},z=0\}$$

is the central axis of the solid torus $\mathrm{\mathfrak{P}}$. Suppose that the robots can only move in the area out of $\mathrm{\mathfrak{P}}$, and each robot can only communicate with those within the distance $\delta<\epsilon$.

A movement of a group of $n$ robots can be represented as a continuous map

$$\mathcal{R}:\mathbb{R}\ni t\mapsto\big{(}r_{1}(t),...,r_{n}(t)\big{)}\in(\mathbb{R}^{3})^{n}$$

with $r_{i}(t)$ being the position of the $i$th robot at the moment $t$. The communication network among the individuals of the group can be represented by an $n\times n$ “$0$-$1$” matrix $[p_{ij}]$. To be precise, if there is a message channel built up between the $i$ th and the $j$ th robots with $i<j$, then we set $p_{ij}=1$ and otherwise $p_{ij}=0$. Since (the structure of) the network may change over time, we use the symbol $[p_{ij}]_{t}:=[p_{ij}^{t}]$ to denote the network at the moment $t$.

The answer is not surprising: it has to change. An explanation is given in the next subsection.

Associated to the matrix $[p_{ij}]_{0}$ there is a $2$ dimensional simplicial complex in $\mathbb{R}^{3}$. For $i,j,k$ such that $p_{ik}^{0}=p_{ij}^{0}=p_{jk}^{0}=1$, let $\Delta_{ijk}$ be the triangle in $\mathbb{R}^{3}$ with vertices $o_{i}$, $o_{j}$ and $o_{k}$. $\Delta_{ijk}$ can be seen as a linear approximation to the curved triangle $\sigma_{ijk}$ on $S^{2}$, and

$$K=\bigcup\Delta_{ijk}$$

is a polytope with a triagulization given by $\{\Delta_{ijk}\}$. Moreover, $\Delta_{ijk}$ is homeomorphic to $\sigma_{ijk}$ via the radial projection. That is, for each $p$ on $\Delta_{ijk}$, there exists a unique $b_{p}\geq 1$ such that $b_{p}\cdot p\in\sigma_{ijk}$, and the map

$$\Delta_{ijk}\ni p\xrightarrow{\tau_{ijk}}b_{p}\cdot p\in\sigma_{ijk}$$

is a homeomorphism. Piecing together the maps $\tau_{ijk}$ we get a homeomorphism $\tau$ from $K$ to $S^{2}$ with $\tau\big{|}_{\Delta_{ijk}}=\tau_{ijk}$. With small $\delta$, $K$ also encloses $\mathcal{P}$. Then

$$h_{t}(w)=(1-t)\cdot w+t\cdot\tau^{-1}(w)$$

(1)

is a continuous map from $S^{2}\times[0,1]$ to $\mathbb{R}^{3}-\mathcal{P}$ and it is a homotopy between the inclusion $\iota_{S^{2}}=h_{0}$ and $\tau^{-1}=h_{1}$.

The movement $\mathcal{R}$ induces a homotopy

$$\mathcal{K}:K\times[0,1]\rightarrow\mathbb{R}^{3}$$

(2)

by sending each point $p=s_{i}o_{i}+s_{j}o_{i}+s_{k}o_{i}$ in $\Delta_{ijk}$ to the point $s_{i}r_{i}(t)+s_{j}r_{j}(t)+s_{k}r_{k}(t)$. Here $s_{i},s_{j},s_{k}\geq 0$ are the weights with $s_{i}+s_{j}+s_{k}=1$, and $\mathcal{K}$ is well-defined since $\{\Delta_{ijk}\}$ is a triangulation of $K$. Note that $\Delta_{ijk}^{t}=\mathcal{K}_{t}(\Delta_{ijk})$ is also a “triangle”: it is the convex hull of the points $r_{i}(t)$, $r_{j}(t)$ and $r_{k}(t)$ in $\mathbb{R}^{3}$, and its diameter equals to the largest distance between these points.

The $condition$ of the set $\{o_{1},...,o_{n}\}$ being a $\frac{\delta}{6}$-net on $S^{2}$ means that every point on $S^{2}$ is at a distance less than $\frac{\delta}{6}$ from some point (robot) $o_{i}$. Here we choose the distance to be the (restriction of the) Euclidean metric from $\mathbb{R}^{3}$ (on $S^{2}$). That is, the distance between any two points $o$ and $o^{\prime}$ on $S^{2}$ is measured by the length of the vector $o-o^{\prime}$ in $\mathbb{R}^{3}$. With this metric, the $\frac{\delta}{6}$-neighbourhood $D(o,\frac{\delta}{6})$ of $o$ on $S^{2}$ is simply the intersection of $S^{2}$ with the $3$-dimension ball $B(o,\frac{\delta}{6})$ in $\mathbb{R}^{3}$ (with center $o$ and radius $\frac{\delta}{6}$). Here both $D(o,\frac{\delta}{6})$ and $B(o,\frac{\delta}{6})$ are taken as open sets in $S^{2}$ and $\mathbb{R}^{3}$, respectively. The $condition$ is equivalent to saying the neighbourhoods $D(o_{i},\frac{\delta}{6})$ constitute an open cover of $S^{2}$, and we consider it to be natural since it merely gives a description on the density of the robots on $S^{2}$. Note that this condition may be coarse in the sense that we could have taken a (much) larger radius than $\frac{\delta}{6}$, or, say, a (much) smaller density of the robots. However, giving a finer estimation on how sparse the robots can be (for inducing a triangulation) is beyond the scope of this note.

For any $1\leq i,j,k\leq n$, define a function $f_{ijk}$ on $(S^{2})^{n}$ with

$$f_{ijk}(a_{1},...,a_{n}):=\det[a_{i};a_{j};a_{k}].$$

(5)

As a consequence of Lemma 8, we get a general position $(o^{\prime}_{1},...,o^{\prime}_{n})$ from any given $\frac{\delta}{6}$-net $\{o_{1},...,o_{n}\}$ on $S^{2}$ via an arbitrarily small perturbation. Furthermore, when $(o^{\prime}_{1},...,o^{\prime}_{n})$ is sufficiently closed to $(o_{1},...,o_{n})$, i.e. the displacement

$$\delta o:=\max_{1\leq i\leq n}\{|o_{i}-o^{\prime}_{i}|\}$$

(6)

is small enough, $(o^{\prime}_{1},...,o^{\prime}_{n})$ is an admissible perturbation from $(o_{1},...,o_{n})$, and $\{o^{\prime}_{1},...,o^{\prime}_{n}\}$ is also a $\frac{\delta}{6}$-net on $S^{2}$. We give this an explanation in the following.

Since there are only finitely many pairs of robots with initial conditions $|o_{i}-o_{j}|<\delta$, the number

$$l_{o}:=\max\{|o_{i}-o_{j}|\big{|}|o_{i}-o_{j}|<\delta\}$$

is strictly smaller than $\delta$. If $\delta o<l_{o}$, then the “straightline” movement of the swarm (out of $\mathfrak{P}$) defined by

$$\tilde{r}_{i}(t):=(1-t)\cdot o^{\prime}_{i}+t\cdot o_{i}$$

gives an admissible perturbation.

Since the neighbourhoods $D(o_{i},\frac{\delta}{6})$ form a finite cover of $S^{2}$, for a number $\delta^{\prime}$ slightly smaller than $\delta$, the neighbourhoods $D(o_{i},\frac{\delta^{\prime}}{6})$ also cover $S^{2}$. To see this, we first take a look at the compact set

$$A_{1}=S^{2}-\bigcup_{j\neq 1}D(o_{j},\frac{\delta}{6}).$$

Since $A_{1}$ has no intersection with $D(o_{j},\frac{\delta}{6})$ for all $j>1$, it is contained in $D(o_{1},\frac{\delta}{6})$. Due to the compactness of $A_{1}$, with $\delta^{\prime}_{1}$ slightly but strictly smaller than $\delta$, $D(o_{1},\frac{\delta^{\prime}_{1}}{6})$ also contains $A_{1}$. Therefore, the sets $D(o_{1},\frac{\delta^{\prime}_{1}}{6})$ and $D(o_{j},\frac{\delta}{6})$ for $j>1$ constitute an open cover of $S^{2}$. Now apply this process inductively to the rest $D(o_{i},\frac{\delta}{6})$. Suppose that we already get an open cover consisting of the sets $U_{i}=D(o_{i},\frac{\delta^{\prime}_{i}}{6})$ for $i\leq k$ and $U_{i}=D(o_{i},\frac{\delta}{6})$ for $i\geq k+1$. The compact set

$$A_{k+1}=S^{2}-\bigcup_{j\neq k+1}U_{j}$$

is then contained in $U_{k+1}=D(o_{k+1},\frac{\delta}{6})$. Replace it with $D(o_{k+1},\frac{\delta^{\prime}_{k+1}}{6})$ and continue until we eventually get an open cover $D(o_{i},\frac{\delta^{\prime}_{i}}{6})$ with $\delta^{\prime}_{i}<\delta$ for $i=1,...,n$. Take

$$\delta^{\prime}=\max\{\delta^{\prime}_{i}\}$$

and then the neighbourhoods $D(o_{i},\frac{\delta^{\prime}}{6})$ again cover $S^{2}$ with $\delta^{\prime}<\delta$, i.e. $\{o_{1},...,o_{n}\}$ is also a $\frac{\delta^{\prime}}{6}$-net.

Now we take $\delta o<\min\{\frac{\delta-\delta^{\prime}}{6},l_{o}\}$. Since $\{o_{1},...,o_{n}\}$ is a $\frac{\delta^{\prime}}{6}$-net, each $p\in S^{2}$ is contained in some $D(o_{i},\frac{\delta^{\prime}}{6})$, i.e. $|p-o_{i}|<\frac{\delta^{\prime}}{6}$. Therefore, it holds that

$$|p-o^{\prime}_{i}|<\frac{\delta^{\prime}}{6}+\delta o<\frac{\delta}{6},$$

As a $\frac{\delta}{6}$-net, $\{o_{1},..,o_{n}\}$ is also a a $\frac{\delta^{\prime}}{6}$-net on $S^{2}$ for some $\delta^{\prime}<\delta$. According to Conclusion 9, we can perturb $(o_{1},...,o_{n})$ to a general position $(o^{\prime}_{1},...,o^{\prime}_{n})$ such that $\{o^{\prime}_{1},...,o^{\prime}_{n}\}$ is a $\frac{\delta^{\prime}}{6}$-net, and, the displacement $\delta o$ defined by (6) is smaller than $\frac{\delta-\delta^{\prime}}{2}$. By the triangle inequality, if a subgraph with vertices from $\{o^{\prime}_{1},...,o^{\prime}_{n}\}$ has all the edges shorter than $\delta^{\prime}$, these edges will then remain shorter than $\delta$ during the swarm moves back to the actual position $(o_{1},...,o_{n})$ along the straightlines $\overline{o^{\prime}_{i}o_{i}}$.

Instead of directly triangulate $S^{2}$ with $\{o^{\prime}_{1},...,o^{\prime}_{n}\}$ as the vertices, we will build a triangulation with the vertices from another set $\mathcal{V}$, and show that the points in $\mathcal{V}$ are sufficiently close to $\{o^{\prime}_{1},...,o^{\prime}_{n}\}$. More specifically, each point $q$ in $\mathcal{V}$ is obaitned from a point $o^{\prime}$ in $\{o^{\prime}_{1},...,o^{\prime}_{n}\}$ with $|o^{\prime}-q|<\frac{\delta^{\prime}}{6}$, and the mapping $q\mapsto o^{\prime}$ is injective.