On the Fairness of Probabilistic Schedulers for Population Protocols

We propose a novel, generic definition of emph{probabilistic schedulers} for population protocols. We design two new schedulers, the emph{State Scheduler} and the emph{Transition Function Scheduler}. Both possess the significant capability of being emph{protocol-aware}, i.e. they can assign transition probabilities based on information concerning the underlying protocol. We prove that the proposed schedulers, and also the emph{Random Scheduler} that was defined by Angluin et al. cite{AADFP04}, are all fair with probability $1$. We also define and study emph{equivalence} between schedulers w.r.t. emph{performance} and emph{correctness} and prove that there exist fair probabilistic schedulers that are not equivalent w.r.t. to performance and others that are not equivalent w.r.t. correctness. We implement our schedulers using a new tool for simulating population protocols and evaluate their performance from the viewpoint of experimental analysis and verification. We study three representative protocols to verify stability, and compare the experimental time to convergence with the known complexity bounds. We run our experiments from very small to extremely large populations (of up to $10^{8}$ agents). We get very promising results both of theoretical and practical interest

Passively Mobile Communicating Logarithmic Space Machines

We propose a new theoretical model for passively mobile Wireless Sensor Networks. We call it the PALOMA model, standing for PAssively mobile LOgarithmic space MAchines. The main modification w.r.t. the Population Protocol model is that agents now, instead of being automata, are Turing Machines whose memory is logarithmic in the population size n. Note that the new model is still easily implementable with current technology. We focus on complete communication graphs. We define the complexity class PLM, consisting of all symmetric predicates on input assignments that are stably computable by the PALOMA model. We assume that the agents are initially identical. Surprisingly, it turns out that the PALOMA model can assign unique consecutive ids to the agents and inform them of the population size! This allows us to give a direct simulation of a Deterministic Turing Machine of O(nlogn) space, thus, establishing that any symmetric predicate in SPACE(nlogn) also belongs to PLM. We next prove that the PALOMA model can simulate the Community Protocol model, thus, improving the previous lower bound to all symmetric predicates in NSPACE(nlogn). Going one step further, we generalize the simulation of the deterministic TM to prove that the PALOMA model can simulate a Nondeterministic TM of O(nlogn) space. Although providing the same lower bound, the important remark here is that the bound is now obtained in a direct manner, in the sense that it does not depend on the simulation of a TM by a Pointer Machine. Finally, by showing that a Nondeterministic TM of O(nlogn) space decides any language stably computable by the PALOMA model, we end up with an exact characterization for PLM: it is precisely the class of all symmetric predicates in NSPACE(nlogn).Comment: 22 page

NETCS: A New Simulator of Population Protocols and Network Constructors

Network Constructors are an extension of the standard population protocol model in which finite-state agents interact in pairs under the control of an adversary scheduler. In this work we present NETCS, a simulator designed to evaluate the performance of various network constructors and population protocols under different schedulers and network configurations. Our simulator provides researchers with an intuitive user interface and a quick experimentation environment to evaluate their work. It also harnesses the power of the cloud, as experiments are executed remotely and scheduled through the web interface provided. To prove the validity and quality of our simulator we provide an extensive evaluation of multiple protocols with more than 100000 experiments for different network sizes and configurations that validate the correctness of the theoretical analysis of existing protocols and estimate the real values of the hidden asymptotic coefficients. We also show experimentally (with more than 40000 experiments) that a probabilistic algorithm is capable of counting the actual size of the network in bounded time given a unique leader

How Many Cooks Spoil the Soup?

In this work, we study the following basic question: "How much parallelism does a distributed task permit?" Our definition of parallelism (or symmetry) here is not in terms of speed, but in terms of identical roles that processes have at the same time in the execution. We initiate this study in population protocols, a very simple model that not only allows for a straightforward definition of what a role is, but also encloses the challenge of isolating the properties that are due to the protocol from those that are due to the adversary scheduler, who controls the interactions between the processes. We (i) give a partial characterization of the set of predicates on input assignments that can be stably computed with maximum symmetry, i.e., $\Theta(N_{min})$, where $N_{min}$ is the minimum multiplicity of a state in the initial configuration, and (ii) we turn our attention to the remaining predicates and prove a strong impossibility result for the parity predicate: the inherent symmetry of any protocol that stably computes it is upper bounded by a constant that depends on the size of the protocol.Comment: 19 page

Time-Varying Graphs and Dynamic Networks

The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems -- delay-tolerant networks, opportunistic-mobility networks, social networks -- obtaining closely related insights. Indeed, the concepts discovered in these investigations can be viewed as parts of the same conceptual universe; and the formal models proposed so far to express some specific concepts are components of a larger formal description of this universe. The main contribution of this paper is to integrate the vast collection of concepts, formalisms, and results found in the literature into a unified framework, which we call TVG (for time-varying graphs). Using this framework, it is possible to express directly in the same formalism not only the concepts common to all those different areas, but also those specific to each. Based on this definitional work, employing both existing results and original observations, we present a hierarchical classification of TVGs; each class corresponds to a significant property examined in the distributed computing literature. We then examine how TVGs can be used to study the evolution of network properties, and propose different techniques, depending on whether the indicators for these properties are a-temporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.Comment: A short version appeared in ADHOC-NOW'11. This version is to be published in Internation Journal of Parallel, Emergent and Distributed System

Leader Election in Anonymous Rings: Franklin Goes Probabilistic

We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size

Simple and Efficient Local Codes for Distributed Stable Network Construction

In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of the their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks.Comment: 43 pages, 7 figure

Mediated population protocols

We extend here the Population Protocol (PP) model of Angluin et al. (2004, 2006) [2,4] in order to model more powerful networks of resource-limited agents that are possibly mobile. The main feature of our extended model, called the Mediated Population Protocol (MPP) model, is to allow the edges of the interaction graph to have states that belong to a constant-size set. We then allow the protocol rules for pairwise interactions to modify the corresponding edge state. The descriptions of our protocols preserve both the uniformity and anonymity properties of PPs, that is, they do not depend on the size of the population and do not use unique identifiers. We focus on the computational power of the MPP model on complete interaction graphs and initially identical edges. We provide the following exact characterization of the class MPS of stably computable predicates: a predicate is in MPS iff it is symmetric and is in NSPACE(n2). © 2010 Elsevier B.V. All rights reserved