Causality, influence, and computation in possibly disconnected synchronous dynamic networks,☆☆

https://doi.org/10.1016/j.jpdc.2013.07.007Get rights and content

Highlights

  • We study computation in possibly disconnected dynamic distributed systems.

  • We replace continuous connectivity by minimal temporal connectivity conditions.

  • We propose metrics capturing the speed of information spreading in dynamic networks.

  • We give efficient protocols for the counting and token dissemination problems.

Abstract

In this work, we study the propagation of influence and computation in dynamic distributed computing systems that are possibly disconnected at every instant. We focus on a synchronous message-passing communication model with broadcast and bidirectional links. Our network dynamicity assumption is a worst-case dynamicity controlled by an adversary scheduler, which has received much attention recently. We replace the usual (in worst-case dynamic networks) assumption that the network is connected at every instant by minimal temporal connectivity conditions. Our conditions only require that another causal influence occurs within every time window of some given length. Based on this basic idea, we define several novel metrics for capturing the speed of information spreading in a dynamic network. We present several results that correlate these metrics. Moreover, we investigate termination criteria in networks in which an upper bound on any of these metrics is known. We exploit our termination criteria to provide efficient (and optimal in some cases) protocols that solve the fundamental counting and all-to-all token dissemination (or gossip) problems.

Introduction

Distributed computing systems are becoming increasingly dynamic. The static and relatively stable models of computation can no longer represent the plethora of recently established and rapidly emerging information and communication technologies. In recent years, we have seen a tremendous increase in the number of new mobile computing devices. Most of these devices are equipped with some sort of communication, sensing, and mobility capabilities. Even the Internet has become mobile. The design is now focused on complex collections of heterogeneous devices that should be robust, adaptive, and self-organizing, possibly moving around and serving requests that vary with time. Delay-tolerant networks are highly dynamic, infrastructureless networks whose essential characteristic is a possible absence of end-to-end communication routes at any instant. Mobility may be active, when the devices control and plan their mobility pattern (e.g. mobile robots), or passive, in opportunistic-mobility networks, where mobility stems from the mobility of the carriers of the devices (e.g. humans carrying cell phones) or a combination of both (e.g. the devices have partial control over the mobility pattern, like for example when GPS devices provide route instructions to their carriers). Thus, it can vary from being completely predictable to being completely unpredictable. Gossip-based communication mechanisms, e-mail exchanges, peer-to-peer networks, and many other contemporary communication networks all assume or induce some sort of highly dynamic communication network.

The formal study of dynamic communication networks is hardly a new area of research. There is a huge amount of work in distributed computing that deals with causes of dynamicity such as failures and changes in the topology that are rather slow and usually eventually stabilize (like, for example, in self-stabilizing systems [11]). However, the low rate of topological changes that is usually assumed there is unsuitable for reasoning about truly dynamic networks. Even graph-theoretic techniques need to be revisited: the suitable graph model is now that of a dynamic graph (also known as a temporal graph or time-varying graph) (see e.g. [25], [15], [8]), in which each edge has an associated set of time labels indicating availability times. Even fundamental properties of classical graphs do not easily carry over to their temporal counterparts. For example, Kempe, Kleinberg, and Kumar  [15] found that there is no analog of Menger’s theorem (see e.g.  [7] for a definition) for arbitrary temporal networks with one label on every edge, which additionally renders the computation of the number of node-disjoint st paths NP-complete. Very recently, the authors of [25] achieved a reformulation of Menger’s theorem which is valid for all temporal networks, and additionally they introduced several interesting cost-minimization parameters for optimal temporal network design and gave some first results on them. Even the standard network diameter metric is no longer suitable, and it has to be replaced by a dynamic/temporal version. In a dynamic star graph in which all leaf nodes but one go to the center one after the other in a modular way, any message from the node that last enters the center to the node that never enters the center needs n1 steps to be delivered, where n is the size (number of nodes) of the network; that is, the dynamic diameter is n1 while, one the other hand, the classical diameter is just 2  [3] (see also  [18]).

Section snippets

Related work

Distributed systems with worst-case dynamicity were first studied in  [26]. Their outstanding novelty was to assume a communication network that may change arbitrarily from time to time subject to the condition that each instance of the network is connected. They studied asynchronous communication and considered nodes that can detect local neighborhood changes; these changes cannot happen faster than it takes for a message to transmit. They studied flooding (in which one node wants to

Contribution

In this work, we study worst-case dynamic networks that are free of any connectivity assumption about their instances. Our dynamic network model is formally defined in Section  4.1. We only impose some temporal connectivity conditions on the adversary guaranteeing that another causal influence occurs within every time window of some given length, meaning that, in that time, another node first hears of the state that some node u had at some time t (see Section  4.3 for a formal definition of

The dynamic network model

A dynamic network is modeled by a dynamic graph G=(V,E), where V is a set of n nodes (or processors) and E:NP(E) (wherever we use N we mean N1) is a function mapping a round number rN to a set E(r) of bidirectional links drawn from E={{u,v}:u,vV}.1  Intuitively, a dynamic graph G is an infinite sequence G(1),G(2), of instantaneous graphs, whose edge sets are subsets of E chosen by a worst-case adversary. A

Our metrics

As already stated, in this work we aim to deal with dynamic networks that are allowed to have disconnected instances. To this end, we define some novel generic metrics that are particularly suitable for capturing the speed of information propagation in such networks.

Fast propagation of information under continuous disconnectivity

In Section  5.1, we presented a simple example of an always-disconnected dynamic graph, namely, the alternating matchings dynamic graph, with optimal oit (i.e. unit oit). Note that the alternating matchings dynamic graph may be conceived as simple, as it has small fer (equal to 2). We now pose an interesting question: Is there an always-disconnected dynamic graph with unit oit and fer as big as n1? Note that this is harder to achieve, as it allows no edge to ever reappear in less than n1

Termination and computation

We now turn our attention to termination criteria that we exploit to solve the fundamental counting and all-to-all token dissemination problems. First observe that, if nodes know an upper bound H on the iit, then there is a straightforward optimal termination criterion taking time D+H, where D is the dynamic diameter. In every round, all nodes forward all ids that they have heard of so far. If a node does not hear of a new id for H rounds, then it must have already heard from all nodes. We

Conclusions

To the best of our knowledge, this is the first study of worst-case dynamic networks that are free of any connectivity assumption about their instances. To enable a quantitative study, we proposed some novel generic metrics that capture the speed of information propagation in a dynamic network. We proved that fast dissemination and computation are possible even under continuous disconnectivity. In particular, we presented optimal termination conditions and protocols based on them for the

Acknowledgments

We would like to thank the anonymous reviewers for carefully reading the manuscript and for providing valuable comments that have helped us to improve our work substantially.

Othon Michail, born in 1984, obtained his Ph.D. from the Computer Engineering & Informatics Department of the University of Patras in 2010. He is currently a Postdoc in Research Unit 1 of the Computer Technology Institute and Press “Diophantus” (CTI) and an Adjunct Faculty at the Computer Engineering & Informatics Department of the University of Patras. He is one of the authors of the book “New Models for Population Protocols”. Since 2008, when he obtained his Diploma, he has coauthored over 20

References (28)

  • I. Chatzigiannakis et al.

    Passively mobile communicating machines that use restricted space

    Theoretical Computer Science

    (2011)
  • O. Michail et al.

    Mediated population protocols

    Theoretical Computer Science

    (2011)
  • D. Angluin et al.

    Computation in networks of passively mobile finite-state sensors

    Distributed Computing

    (2006)
  • D. Angluin et al.

    The computational power of population protocols

    Distributed Computing

    (2007)
  • C. Avin et al.

    How to explore a fast-changing world (cover time of a simple random walk on evolving graphs)

  • J. Augustine et al.

    Towards robust and efficient computation in dynamic peer-to-peer networks

  • H. Attiya et al.

    Distributed Computing: Fundamentals, Simulations, and Advanced Topics, Vol. 19

    (2004)
  • H. Baumann et al.

    Parsimonious flooding in dynamic graphs

  • B. Bollobás

    Modern Graph Theory

    (1998)
  • A. Casteigts et al.

    Time-varying graphs and dynamic networks

    International Journal of Parallel, Emergent and Distributed Systems

    (2012)
  • A.E. Clementi et al.

    Flooding time in edge-markovian dynamic graphs

  • S. Dolev

    Self-stabilization

    (2000)
  • C. Dutta et al.

    On the complexity of information spreading in dynamic networks

  • B. Haeupler

    Analyzing network coding gossip made easy

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    Othon Michail, born in 1984, obtained his Ph.D. from the Computer Engineering & Informatics Department of the University of Patras in 2010. He is currently a Postdoc in Research Unit 1 of the Computer Technology Institute and Press “Diophantus” (CTI) and an Adjunct Faculty at the Computer Engineering & Informatics Department of the University of Patras. He is one of the authors of the book “New Models for Population Protocols”. Since 2008, when he obtained his Diploma, he has coauthored over 20 scientific publications. His research interests include distributed and mobile computing, modeling and analyzing novel ICT systems, computational complexity, and algorithms.

    Ioannis Chatzigiannakis obtained his Ph.D. from the Department of Computer Engineering & Informatics of the University of Patras in 2003. He is currently Adjunct Faculty at the Computer Engineering & Informatics Department of the University of Patras (since October 2005). He has been the Director of Research Unit 1 of CTI since July 2007. He has coauthored over 70 scientific publications. His main research interests include distributed and mobile computing, wireless sensor networks, algorithm engineering, and software systems. He has served as a consultant to major Greek computing industries. He has been the Secretary of the European Association for Theoretical Computer Science since July 2008.

    Paul G. Spirakis, born in 1955, obtained his Ph.D. from Harvard University, in 1982. He is currently the Director of the CTI and a Full Professor at Patras University, Greece. He was acknowledged between the top 50 scientists worldwide in Computer Science with respect to The best Nurturers in Computer Science Research, published by B. Kumar and Y.N. Srikant, ACM Data Mining, 2005. His research interests are algorithms and complexity and interaction of complexity and game theory. Paul Spirakis has extensively published in most of the important Computer Science journals and most of the significant refereed conferences, contributing to over 300 scientific publications. He was elected unanimously as one of the two Vice Presidents of the Council of the EATCS. He is a member of the ACM Europe Council and also a member of Academia Europaea. Recently, he was elected as Chair in Computer Science at the Department of Computer Science of the University of Liverpool (effective from September 2013).

    Supported in part by the project “Foundations of Dynamic Distributed Computing Systems” (FOCUS) which is implemented under the “ARISTEIA” Action of the Operational Programme “Education and Lifelong Learning” and is co-funded by the European Union (European Social Fund) and Greek National Resources..

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    A preliminary version of the results in this paper has appeared in  [24].

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