How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs)

Avin, Chen; Koucký, Michal; Lotker, Zvi

doi:10.1007/978-3-540-70575-8_11

Cited by 156 publications

(202 citation statements)

References 15 publications

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“…Even fundamental properties of classical graphs do not carry over to their temporal counterparts. See, for example, [KKK00] for a violation of Menger's theorem, [MMCS13] for a valid reformulation of Menger's theorem and the definition of several cost optimization metrics for temporal networks, and [AKL08] for the unsuitability of the standard network diameter metric.…”

Section: Introductionmentioning

confidence: 99%

Naming and Counting in Anonymous Unknown Dynamic Networks

Michail

Chatzigiannakis

Spirakis

2013

Lecture Notes in Computer Science

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Abstract. In this work, we study the fundamental naming and counting problems (and some variations) in networks that are anonymous, unknown, and possibly dynamic. In counting, nodes must determine the size of the network n and in naming they must end up with unique identities. By anonymous we mean that all nodes begin from identical states apart possibly from a unique leader node and by unknown that nodes have no a priori knowledge of the network (apart from some minimal knowledge when necessary) including ignorance of n. Network dynamicity is modeled by the 1-interval connectivity model [KLO10], in which communication is synchronous and a (worst-case) adversary chooses the edges of every round subject to the condition that each instance is connected. We first focus on static networks with broadcast where we prove that, without a leader, counting is impossible to solve and that naming is impossible to solve even with a leader and even if nodes know n. These impossibilities carry over to dynamic networks as well. We also show that a unique leader suffices in order to solve counting in linear time. Then we focus on dynamic networks with broadcast. We conjecture that dynamicity renders nontrivial computation impossible. In view of this, we let the nodes know an upper bound on the maximum degree that will ever appear and show that in this case the nodes can obtain an upper bound on n. Finally, we replace broadcast with one-to-each, in which a node may send a different message to each of its neighbors. Interestingly, this natural variation is proved to be computationally equivalent to a full-knowledge model, in which unique names exist and the size of the network is known.

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Section: Introductionmentioning

confidence: 99%

Naming and Counting in Anonymous Unknown Dynamic Networks

Michail

Chatzigiannakis

Spirakis

2013

Lecture Notes in Computer Science

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“…In recent years, there is a growing interest in distributed computing systems that are inherently dynamic [5,6,10,12,13,22,26,27,30,32].…”

Section: Dynamic Distributed Networkmentioning

confidence: 99%

The Complexity of Optimal Design of Temporally Connected Graphs

et al. 2017

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We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u = v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomialtime procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP . On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least n log n labels.

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“…Most notably, in dynamic graphs there can be pairs of nodes whose hitting time is exponential [2], even though in static (connected) graphs it is well-known that the maximum hitting time is at most O(n 3 ) [12]. This is true even under obvious technical restrictions necessary to prevent infinite hitting times, such as requiring the graph to be connected at all times and to have self-loops at all nodes.…”

Section: Random Walksmentioning

confidence: 99%

“…A particularly interesting example is the dynamic star, which was used by Avin et al [2] to prove an exponential lower bound. The dynamic star consists of n vertices {0, 1, .…”

Section: Upper Boundsmentioning

confidence: 99%

Smoothed analysis of dynamic networks

et al. 2017

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We generalize the technique of smoothed analysis to distributed algorithms in dynamic network models. Whereas standard smoothed analysis studies the impact of small random perturbations of input values on algorithm performance metrics, dynamic graph smoothed analysis studies the impact of random perturbations of the underlying changing network graph topologies. Similar to the original application of smoothed analysis, our goal is to study whether known strong lower bounds in dynamic network models are robust or fragile: do they withstand small (random) perturbations, or do such deviations push the graphs far enough from a precise pathological instance to enable much better performance? Fragile lower bounds are likely not relevant for real-world deployment, while robust lower bounds represent a true difficulty caused by dynamic behavior. We apply this technique to three standard dynamic network problems with known strong worst-case lower bounds: random walks, flooding, and aggregation. We prove that these bounds provide a spectrum of robustness when subjected to smoothing-some are extremely fragile (random walks), some are moderately fragile / robust (flooding), and some are extremely robust (aggregation).

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How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs)

Cited by 156 publications

References 15 publications

Naming and Counting in Anonymous Unknown Dynamic Networks

Naming and Counting in Anonymous Unknown Dynamic Networks

The Complexity of Optimal Design of Temporally Connected Graphs

Smoothed analysis of dynamic networks

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