Distinguishing Views in Symmetric Networks: A Tight Lower Bound

The view of a node in a port-labeled network is an infinite tree encoding all walks in the network originating from this node. We prove that for any integers $n\geq D\geq 1$, there exists a port-labeled network with at most $n$ nodes and diameter at most $D$ which contains a pair of nodes whose (infinite) views are different, but whose views truncated to depth $\Omega(D\log (n/D))$ are identical

Self-Stabilizing Clock Synchronization in Dynamic Networks

We consider the fundamental problem of periodic clock synchronization in a synchronous multi-agent system. Each agent holds a clock with an arbitrary initial value, and clocks must eventually be congruent, modulo some positive integer P. Previous algorithms worked in static networks with drastic connectivity properties and assumed that global informations are available at each node. In this paper, we propose a finite-state algorithm for time-varying topologies that does not require any global knowledge on the network. The only assumption is the existence of some integer D such that any two nodes can communicate in each sequence of D consecutive rounds, which extends the notion of strong connectivity in static network to dynamic communication patterns. The smallest such D is called the dynamic diameter of the network. If an upper bound on the diameter is provided, then our algorithm achieves synchronization within 3D rounds, whatever the value of the upper bound. Otherwise, using an adaptive mechanism, synchronization is achieved with little performance overhead. Our algorithm is parameterized by a function g, which can be tuned to favor either time or space complexity. Then, we explore a further relaxation of the connectivity requirement: our algorithm still works if there exists a positive integer R such that the network is rooted over each sequence of R consecutive rounds, and if eventually the set of roots is stable. In particular, it works in any rooted static network

Termination Detection of Local Computations

Contrary to the sequential world, the processes involved in a distributed system do not necessarily know when a computation is globally finished. This paper investigates the problem of the detection of the termination of local computations. We define four types of termination detection: no detection, detection of the local termination, detection by a distributed observer, detection of the global termination. We give a complete characterisation (except in the local termination detection case where a partial one is given) for each of this termination detection and show that they define a strict hierarchy. These results emphasise the difference between computability of a distributed task and termination detection. Furthermore, these characterisations encompass all standard criteria that are usually formulated : topological restriction (tree, rings, or triangu- lated networks ...), topological knowledge (size, diameter ...), and local knowledge to distinguish nodes (identities, sense of direction). These results are now presented as corollaries of generalising theorems. As a very special and important case, the techniques are also applied to the election problem. Though given in the model of local computations, these results can give qualitative insight for similar results in other standard models. The necessary conditions involve graphs covering and quasi-covering; the sufficient conditions (constructive local computations) are based upon an enumeration algorithm of Mazurkiewicz and a stable properties detection algorithm of Szymanski, Shi and Prywes

Know your audience

Distributed function computation is the problem, for a networked system of $n$ autonomous agents, to collectively compute the value $f(v_1, \ldots, v_n)$ of some input values, each initially private to one agent in the network. Here, we study and organize results pertaining to distributed function computation in anonymous networks, both for the static and the dynamic case, under a communication model of directed and synchronous message exchanges, but with varying assumptions in the degree of awareness or control that a single agent has over its outneighbors. Our main argument is three-fold. First, in the "blind broadcast" model, where in each round an agent merely casts out a unique message without any knowledge or control over its addressees, the computable functions are those that only depend on the set of the input values, but not on their multiplicities or relative frequencies in the input. Second, in contrast, when we assume either that a) in each round, the agents know how many outneighbors they have; b) all communications links in the network are bidirectional; or c) the agents may address each of their outneighbors individually, then the set of computable functions grows to contain all functions that depend on the relative frequencies of each value in the input - such as the average - but not on their multiplicities - thus, not the sum. Third, however, if one or several agents are distinguished as leaders, or if the cardinality of the network is known, then under any of the above three assumptions it becomes possible to recover the complete multiset of the input values, and thus compute any function of the distributed input as long as it is invariant under permutation of its arguments. In the case of dynamic networks, we also discuss the impact of multiple connectivity assumptions

Duplicate Address Detection in Wireless Ad Hoc Networks Using Wireless Nature

We consider duplicate address detection in wireless ad hoc networks under the assumption that addresses are unique in two hops neighborhood. Our approaches are based on the concepts of physical neighborhood views, the information of physically connected nodes, and logical neighborhood views, which are built on neighborhood information that is propagated in networks. Since neighborhood information is identified by addresses, inconsistency of these two views might be caused due to duplicate addresses. It is obvious that consistency of physical and logical views on each node's neighborhood is necessary for a network to have unique addresses, while the sufficiency depends on the types of information contained in views of neighborhood. We investigate different definitions of neighborhood views. Our results show that the traditional neighborhood information, neighboring addresses, is not sufficient for duplication detetion, while the wireless nature of ad hoc networks provides powerful neighborhood information in detecting duplication

Fibration symmetries uncover the building blocks of biological networks

A major ambition of systems science is to uncover the building blocks of any biological network to decipher how cellular function emerges from their interactions. Here, we introduce a graph representation of the information flow in these networks as a set of input trees, one for each node, which contains all pathways along which information can be transmitted in the network. In this representation, we find remarkable symmetries in the input trees that deconstruct the network into functional building blocks called fibers. Nodes in a fiber have isomorphic input trees and thus process equivalent dynamics and synchronize their activity. Each fiber can then be collapsed into a single representative base node through an information-preserving transformation called 'symmetry fibration', introduced by Grothendieck in the context of algebraic geometry. We exemplify the symmetry fibrations in gene regulatory networks and then show that they universally apply across species and domains from biology to social and infrastructure networks. The building blocks are classified into topological classes of input trees characterized by integer branching ratios and fractal golden ratios of Fibonacci sequences representing cycles of information. Thus, symmetry fibrations describe how complex networks are built from the bottom up to process information through the synchronization of their constitutive building blocks

Algorithmique distribuée, calculs locaux et homomorphismes de graphes

Dans cette thèse, on étudie ce qui est calculable dans différents modèles d’algorithmique distribuée. Les modèles considérés correspondent à différents niveaux d’abstraction et à différents niveaux de synchronisation entre les processus d’un système distribué. On s’intéresse en particulier au problèmes de l’élection et du nommage dans ces différents modèles. Pour chaque modèle, on caractérise les systèmes distribués dans lesquels on peut résoudre ces problèmes et on étudie la complexité des problèmes de décision correspondants. Nos caractérisations utilisent des homomorphismes de graphes qui préservent certaines propriétés locales. Nos preuves sont constructives : quand on peut résoudre l’élection (ou le nommage) dans un réseau, on présente un algorithme d’élection (ou de nommage) pour ce réseau. Ces problèmes permettent de mettre en évidence les différences entre les puissances de calculs des différents modèles considérés. De plus, l’étude de ces problèmes permet de mettre à jour les bons outils qui permettent d’étudier ce qui est calculable de manière distribuée dans les différents modèles.In this thesis, we consider different models of distributed computations. These models correspond to different levels of abstraction and they encode different levels of synchronization between processes in a distributed system. In these different models, we particularly focus on two classical problems in distributed computing : election and naming. For each model, we present a characterization of distributed systems where these problems can be solved and we study the complexity of the corresponding decision problems. Our characterizations are expressed in terms of graph homomorphisms that preserve some local properties. Our proofs are constructive : when a network admits an election (or a naming) algorithm, we present such an algorithm for this network. These problems enable to highlight the differences between the computation powers of the different models we consider. Moreover, studying these problems enable to introduce some combinatorial and algorithmic tools that can be used to study what can be computed in a distributed way in these different models

Universal dynamic synchronous self-stabilization

We prove the existence of a "universal" synchronous self-stabilizing protocol, that is, a protocol that allows a distributed system to stabilize to a desired nonreactive behaviour (as long as a protocol stabilizing to that behaviour exists). Previous proposals required drastic increases in asymmetry and knowledge to work, whereas our protocol does not use any additional knowledge, and does not require more symmetry-breaking conditions than available; thus, it is also stabilizing with respect to dynamic changes in the topology. We prove an optimal quiescence time n + D for a synchronous network of n processors and diameter D; the protocol can be made finite state with a negligible loss in quiescence time. Moreover, an optimal D + 1 protocol is given for the case of unique identifiers. As a consequence, we provide an effective proof technique that allows to show whether self-stabilization to a certain behaviour is possible under a wide range of models