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Algebraic gossip: A network coding approach to optimal multiple rumor mongering
 IEEE Transactions on Information Theory
, 2004
"... We study the problem of simultaneously disseminating multiple messages in a large network in a decentralized and distributed manner. We consider a network with n nodes and k (k = O(n)) messages spread throughout the network to start with, but not all nodes have all the messages. Our communication mo ..."
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Cited by 78 (11 self)
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We study the problem of simultaneously disseminating multiple messages in a large network in a decentralized and distributed manner. We consider a network with n nodes and k (k = O(n)) messages spread throughout the network to start with, but not all nodes have all the messages. Our communication model is such that the nodes communicate in discretetime steps, and in every timestep, each node communicates with a random communication partner chosen uniformly from all the nodes (known as the random phone call model). The system is bandwidth limited and in each timestep, only one message can be transmitted. The goal is to disseminate rapidly all the messages among all the nodes. We study the time required for this dissemination to occur with high probability, and also in expectation. We present a protocol based on random linear coding (RLC) that disseminates all the messages among all the nodes in O(n) time, which is order optimal, if we ignore the small overhead associated with each transmission. The overhead does not depend on the size of the messages and is less than 1 % for k = 100 and messages of size 100 KB. We also consider a store and forward mechanism without coding, which is a natural extension of gossipbased dissemination with one message in the network. We show that, such an uncoded scheme can do no better than a sequential approach (instead of doing it simultaneously) of disseminating the messages which takes Θ(n ln(n)) time, since disseminating a single message in a gossip network takes Θ(ln(n)) time. 1
A Decentralized Algorithm for Spectral Analysis
, 2004
"... In many large network settings, such as computer networks, social networks, or hyperlinked text documents, much information can be obtained from the network’s spectral properties. However, traditional centralized approaches for computing eigenvectors struggle with at least two obstacles: the data ma ..."
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Cited by 54 (1 self)
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In many large network settings, such as computer networks, social networks, or hyperlinked text documents, much information can be obtained from the network’s spectral properties. However, traditional centralized approaches for computing eigenvectors struggle with at least two obstacles: the data may be difficult to obtain (both due to technical reasons and because of privacy concerns), and the sheer size of the networks makes the computation expensive. A decentralized, distributed algorithm addresses both of these obstacles: it utilizes the computational power of all nodes in the network and their ability to communicate, thus speeding up the computation with the network size. And as each node knows its incident edges, the data collection problem is avoided as well. Our main result is a simple decentralized algorithm for computing the top k eigenvectors of a symmetric weighted adjacency matrix, and a proof that it converges essentially in O(τmix log 2 n) rounds of communication and computation, where τmix is the mixing time of a random walk on the network. An additional contribution of our work is a decentralized way of actually detecting convergence, and diagnosing the current error. Our protocol scales well, in that the amount of computation performed at any node in any one round, and the sizes of messages sent, depend polynomially on k, but not at all on the (typically much larger) number n of nodes.
On the cover time and mixing time of random geometric graphs
 Theor. Comput. Sci
, 2007
"... The cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties ..."
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Cited by 25 (2 self)
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The cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius ropt such that for any r ≥ ropt G(n, r) has optimal cover time of Θ(n log n) with high probability, and, importantly, ropt = Θ(rcon) where rcon denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(rcon). On the other hand, the radius required for rapid mixing rrapid = ω(rcon), and, in particular, rrapid = Θ(1/poly(log n)). We are able to draw our results by giving a tight bound on the electrical resistance and conductance of G(n, r) via certain constructed flows.
On the Cover Time of Random Geometric Graphs
 In: ICALP. (2005
, 2005
"... Abstract. The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have be ..."
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Cited by 17 (4 self)
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Abstract. The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of adhoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius ropt such that for any r ≥ ropt G(n, r) has optimal cover time of Θ(n log n) with high probability, and, importantly, ropt = Θ(rcon) where rcon denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(rcon). We are able to draw our results by giving a tight bound on the electrical resistance of G(n, r) via the power of certain constructed flows. 1
Bounds on the Mixing Time and Partial Cover of AdHoc and Sensor Networks
, 2004
"... In [1], the authors proposed the partial cover of a random walk on a broadcast network to be used to gather information and supported their proposal with experimental results. In this paper, we demonstrate analytically that for sufficiently large broadcast radius, the partial cover of a random walk ..."
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Cited by 13 (3 self)
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In [1], the authors proposed the partial cover of a random walk on a broadcast network to be used to gather information and supported their proposal with experimental results. In this paper, we demonstrate analytically that for sufficiently large broadcast radius, the partial cover of a random walk on a broadcast network is in fact efficient and generates a good distribution of the visited nodes. Our result is based on bounding the conductance, which intuitively measures the amount of bottlenecks in a graph. We show that the conductance of a random broadcast network in a unit square is #(R), and this bound allows us to analyze properties of the random walk such as mixing time and load balancing. We find that for the random walk to be both efficient and have a high quality cover and partial cover (i.e. rapid mixing), radius R = O(1/poly(logN)) is sufficient. Experimental results on the random unit disk graphs that resemble the conductance of the 3D grid indicate that the analytical bounds on efficiency, namely cover time and partial cover time, are not tight. In particular, R = O(1/N ) is sufficient radius to obtain optimal cover time and partial cover time, if one is not concerned about the quality of the distribution of the visited nodes (for example in a query based on majority vote).
Distributed averaging on asynchronous communication networks
 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference (CDCECC ’05
, 2005
"... Abstract — Distributed algorithms for averaging have attracted interest in the control and sensing literature. However, previous works have not addressed some practical concerns that will arise in actual implementations on packetswitched communication networks such as the Internet. In this paper, w ..."
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Cited by 8 (3 self)
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Abstract — Distributed algorithms for averaging have attracted interest in the control and sensing literature. However, previous works have not addressed some practical concerns that will arise in actual implementations on packetswitched communication networks such as the Internet. In this paper, we present several implementable algorithms that are robust to asynchronism and dynamic topology changes. The algorithms do not require global coordination and can be proven to converge under very general asynchronous timing assumptions. Our results are verified by both simulation and experiments on a realworld TCP/IP network. I.
Information survival threshold in sensor and P2P networks
 Proceedings of 26th Annual IEEE ICC
, 2007
"... Abstract—Consider a network of, say, sensors, or P2P nodes, or bluetoothenabled cellphones, where nodes transmit information to each other and where links and nodes can go up or down. Consider also a ‘datum’, that is, a piece of information, like a report of an emergency condition in a sensor netw ..."
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Cited by 8 (3 self)
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Abstract—Consider a network of, say, sensors, or P2P nodes, or bluetoothenabled cellphones, where nodes transmit information to each other and where links and nodes can go up or down. Consider also a ‘datum’, that is, a piece of information, like a report of an emergency condition in a sensor network, a national traditional song, or a mobile phone virus. How often should nodes transmit the datum to each other, so that the datum can survive (or, in the virus case, under what conditions will the virus die out)? Clearly, the link and node fault probabilities are important — what else is needed to ascertain the survivability of the datum? We propose and solve the problem using nonlinear dynamical systems and fixed point stability theorems. We provide a closedform formula that, surprisingly, depends on only one additional parameter, the largest eigenvalue of the connectivity matrix. We illustrate the accuracy of our analysis on realistic and real settings, like mote sensor networks from Intel and MIT, as well as Gnutella and P2P networks. I.
Bounding fastest mixing
 Electron. Comm. Probab
, 2005
"... In a series of recent works, Boyd, Diaconis, and their coauthors have introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixingtime analysis tec ..."
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Cited by 5 (1 self)
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In a series of recent works, Boyd, Diaconis, and their coauthors have introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixingtime analysis techniques—variational characterizations, conductance, canonical paths—can be used to give simple, nontrivial lower and upper bounds on the fastest mixing time. To test the applicability of this idea, we consider several detailed examples including the Glauber dynamics of the Ising model—and get sharp bounds.
Minimizing Weighted Sum Finish Time for OnetoMany File Transfer in PeertoPeer Networks
"... Abstract — This paper considers the problem of transferring a file from one source node to multiple receivers in a peertopeer (P2P) network. The objective is to minimize the weighted sum finish time (WSFT) for the onetomany file transfer where peers have both uplink and downlink bandwidth constr ..."
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Abstract — This paper considers the problem of transferring a file from one source node to multiple receivers in a peertopeer (P2P) network. The objective is to minimize the weighted sum finish time (WSFT) for the onetomany file transfer where peers have both uplink and downlink bandwidth constraints specified. The static scenario is a filetransfer scheme in which the constructed network topology and the network resource (link throughput) allocation remains static until all receivers finish downloading. This paper first shows that the static scenario can be optimized in polynomial time by convex optimization, and the associated optimal static WSFT can be achieved by linear network coding. This paper also proposes a static ratelesscodingbased scheme which has almostoptimal empirical performance. The dynamic scenario is a filetransfer scheme which can reconstruct the network topology and reallocate the network resource during the file transfer. This paper proposes a dynamic ratelesscodingbased scheme, which provides significantly smaller WSFT than the optimal static scheme does. Index Terms — P2P network, network coding, rateless code, static scenario, dynamic scenario.