ABSTRACT We consider the network construction process by selfish players. Each player is associated with a vertex of a communication graph and can simultaneously remove one incident edge and add a new incident edge. Each player is interested in a subset of players and the goal of each player is to minimize the average or maximum distance to these players. Starting from a given initial communication graph, a sequence of selfish edge swaps generates an evolution of the communication graph. Due to non‐uniform communication interest, this game may converge to a disconnected Nash equilibrium, which may attain infinite social costs. In this paper, we focus on the dynamics of this game. We first give theoretical analysis such as the existence of a best response cycle and a sufficient condition for keeping connectivity in dynamics. We then present simulation results to show the ratio of Nash equilibria with infinite cost, diameters of Nash equilibria, social cost, price of anarchy, price of stability, and convergence time.

Summary A metamorphic robotic system (MRS) is composed of anonymous, memoryless, and autonomous modules that execute an identical distributed algorithm to move while keeping the connectivity of the modules. For an MRS, the number of modules required to solve a given task is an important complexity measure. Here, we consider evacuation from a finite two‐dimensional square grid field by an MRS. This study aims to establish the minimum number of modules required to solve the evacuation problem under several conditions. We consider a rectangular field surrounded by walls with at least one exit. Our results show that two modules are necessary and sufficient for evacuation from any rectangular field if equipped with a global compass, which provides the modules with a common sense of direction. After that, we focus on the case of modules without a global compass and show that four (resp. seven) modules are necessary and sufficient for restricted (resp. any) initial shapes of an MRS. We also show that two modules are sufficient when an MRS is touching a wall in an initial configuration. Then, we clarify the condition to stop an MRS after evacuation of a rectangular field. Finally, we extend these results to mazes and convex fields.

A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant $\vartheta$ from the robot's current location, where $0<\vartheta\leqπ$ (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if $\vartheta=π$ (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if $\vartheta\leq π/2$, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

A set function $f$ on a finite set $V$ is submodular if $f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y)$ for any pair $X, Y \subseteq V$. The symmetric difference transformation ( SD-transformation) of $f$ by a canonical set $S \subseteq V$ is a set function $g$ given by $g(X) = f(X \vartriangle S)$ for $X \subseteq V$, where $X \vartriangle S = (X \setminus S) \cup (S \setminus X)$ denotes the symmetric difference between $X$ and $S$. Submodularity and SD-transformations are regarded as the counterparts of convexity and affine transformations in a discrete space, respectively. However, submodularity is not preserved under SD-transformations, in contrast to the fact that convexity is invariant under affine transformations. This paper presents a characterization of SD-transformations preserving submodularity. Then, we are concerned with the problem of discovering a canonical set $S$, given the SD-transformation $g$ of a submodular function $f$ by $S$, provided that $g(X)$ is given by a function value oracle. A submodular function $f$ on $V$ is said to be strict if $f(X) + f(Y) > f(X \cup Y) + f(X \cap Y)$ holds whenever both $X \setminus Y$ and $Y \setminus X$ are nonempty. We show that the problem is solved by using $\mathrm{O}(|V|)$ oracle calls when $f$ is strictly submodular, although it requires exponentially many oracle calls in general.

The rotor-router model is a deterministic process analogous to a simple random walk on a graph, and the discrepancy of token configurations between the rotor-router model and its corresponding random walk has been investigated in some contexts. Motivated by general Markov chains beyond simple random walks, this paper investigates a generalized model which imitates a Markov chain (of multiple tokens) possibly containing irrational transition probabilities. We are concerned with the vertexwise discrepancy of the numbers of tokens between the generalized model and its corresponding Markov chain, and present an upper bound of the discrepancy in terms of the mixing time of the Markov chain.

Shape formation (or pattern formation) is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter, where entities are assumed to be small and with severely limited capabilities. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessellation of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization (i.e., particles can flip coins to elect a leader). In this paper we provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of n particles. The characterization is constructive: we provide a universal shape formation algorithm that, for each feasible pair of shapes (S0, SF), allows the particles to form the final shape SF (given in input) starting from the initial shape S0, unknown to the particles. The final configuration will be an appropriate scaled-up copy of SF depending on n. If randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that there are enough particles. Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential. We also consider the complexity of shape formation both in terms of the number of rounds and the total number of moves performed by the particles executing a universal shape formation algorithm. We prove that our solution has a complexity of O(n2) rounds and moves: this number of moves is also asymptotically worst-case optimal.

We consider a distributed system consisting of autonomous mobile computing entities called robots moving in the three-dimensional space (3D-space). The robots are anonymous, oblivious, fully-synchronous and have neither any access to the global coordinate system nor any explicit communication medium. Each robot cooperates with other robots by observing the positions of other robots in its local coordinate system. One of the most fundamental agreement problems in 3D-space is the plane formation problem that requires the robots to land on a common plane, that is not predefined. This problem is not always solvable because of the impossibility of symmetry breaking. While existing results assume that the robots agree on the handedness of their local coordinate systems, we remove the assumption and consider the robots without chirality. The robots without chirality can never break the symmetry consisting of rotation symmetry and reflection symmetry. Such symmetry in 3D-space is fully described by 17 symmetry types each of which forms a group. We extend the notion of symmetricity [Suzuki and Yamashita, SIAM J. Compt. 1999] [Yamauchi et al., PODC 2016] to cover these 17 symmetry groups. Then we give a characterization of initial configurations from which the fully-synchronous robots without chirality can form a plane in terms of symmetricity.

Shape formation is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessellation of the plane, have constant memory, can only communicate with neighboring particles, and can only move from a grid node to an empty neighboring node; their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization. In this paper we provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of n particles. As a byproduct, if randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that n is large enough. Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential. We also consider the complexity of shape formation both in terms of the number of rounds and the total number of moves performed by the particles executing a universal shape formation algorithm. We prove that our solution has a complexity of O(n^2) rounds and moves: this number of moves is also asymptotically optimal.

Creating a swarm of mobile computing entities, frequently called robots, agents, or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three-dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two-dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three-dimensional space . This article presents a necessary and sufficient condition for fully synchronous robots to solve the plane formation problem that does not depend on obliviousness, i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedron.

We investigate the pattern formation problem that requires a swarm of autonomous mobile robots to form a given target pattern in the three-dimensional Euclidean space. We show a necessary and sufficient condition for synchronous robots to form a given target pattern from an initial configuration. We give a pattern formation algorithm for solvable instances that does not need any local memory at each robot.

A connected dominating set (CDS) is useful in forming a virtual backbone in wireless ad hoc or sensor networks because these networks lack a fixed infrastructure and centralized management. Self-stabilization guarantees that the system tolerates any finite number of transient faults and does not need any initialization. The safe convergence property guarantees that the system quickly converges to a feasible safe configuration, and subsequently converges to a legitimate configuration without violating safety. A previous publication on a safely converging algorithm for the minimum CDS assumed a phase clock synchronizer, which is a very strong assumption. In this paper, we propose the first asynchronous self-stabilizing (6+ϵ)-approximation algorithm with safe convergence for the minimum CDS in networks modeled by unit disk graphs (UDGs). We assume that the feasible safe configuration satisfies the condition that a dominating set is constructed. The convergence time to a feasible safe configuration is one round, and the convergence time to a legitimate configuration in which an approximated minimum CDS is constructed is O(max⁡{d2,n}) rounds, and O(n6) steps.

We investigate pattern formation, i.e., self-organization, by a swarm of mobile robots, which is closely related with the agreement problem in distributed computing. Consider a system of anonymous mobile robots in a 2-dimensional Euclidean space in which each robot repeatedly executes a “Look-Compute-Move” cycle, to observe the positions of all the robots, to compute a route to the next position using an algorithm, and then to trace the route, where the algorithm is common to all robots. The robots are said to be fully synchronous if their Look-Compute-Move cycles are completely synchronized, and the $i$th Look, Compute, and Move of all robots start and end simultaneously. They are said to be asynchronous if no assumptions are made on their synchrony. The robots are said to be oblivious if they have no memory to memorize the execution history and hence behave based only on the robots' positions observed during the immediately preceding Look. We show that the set of geometric patterns formable by oblivious asynchronous robots is exactly the set of those formable by nonoblivious fully synchronous robots, except for a point of multiplicity 2, i.e., gathering for two robots. In short, contrary to our intuition, synchrony and memory do not help in pattern formation, except for gathering. Specifically, we propose an algorithm for oblivious asynchronous robots that, given a geometric pattern as input, forms it, as long as it is formable by nonoblivious fully synchronous robots except for a point of multiplicity 2.

Aiming to achieve sensing coverage for a given Area of Interest (AoI) in a People-Centric Sensing (PCS) manner, we propose a concept of (α, T)-coverage of the target field where each point in the field is sensed by at least one node with probability of at least α during the time period T. Our goal is to achieve (α, T)-coverage by a minimal set of mobile sensor nodes for a given AoI, coverage ratio α, and time period T. We model pedestrians as mobile sensor nodes moving according to a discrete Markov chain. Based on this model, we propose two algorithms: the inter-location and inter-meeting-time algorithms, to meet a coverage ratio α in time period T. These algorithms estimate the expected coverage of the specified AoI for a set of selected nodes. The inter-location algorithm selects a minimal number of mobile sensor nodes from nodes inside the AoI taking into account the distance between them. The inter-meeting-time selects nodes taking into account the expected meeting time between the nodes. We conducted a simulation study to evaluate the performance of the proposed algorithms for various parameter setting including a realistic scenario on a specific city map. The simulation results show that our algorithms achieve (α, T)-coverage with good accuracy for various values of α, T, and AoI size.

In this brief announcement, we discuss the trade-off between the locality of information and the optimality of convergence for self-stabilization. We define the optimality of convergence, called monotonic stabilization, and propose a new metrics for the locality of information to achieve monotonic stabilization. Then, we examine the locality of many well-known distributed problems.

Topology embedding enables us to execute a protocol designed for a specific (virtual) topology on another(real) topology by embedding the virtual topology on the real topology. In this paper, we propose a self-stabilizing emulation technique that provides reliable communication on a virtual topology in the presence of transient faults. The proposed protocol improves the execution slowdown of previous protocols and provides adaptive message delivery delay on the emulated channels, which is a new type of adaptability against transient faults.

Reliable and fault-tolerant distributed systems have been attracting more and more attention (see Autonomic Computing Project by IBM, http://www-03.ibm.com/autonomic/). A self-stabilizing protocol is a fault-tolerant protocol that guarantees autonomous recovery from any number of and any type of faults that can affect the data stored locally at some process(es). If the impact of the faults can be contained to the affected process(es) and some of its immediate neighbors, then the protocol is also fault-containing. We present a new method, called causal simulation, which preserves the fault-containing property of ring protocols executed on trees.