## Randomized Gossip Algorithms (2006)

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Venue: | IEEE TRANSACTIONS ON INFORMATION THEORY |

Citations: | 209 - 5 self |

### BibTeX

@ARTICLE{Boyd06randomizedgossip,

author = {Stephen Boyd and Arpita Ghosh and Balaji Prabhakar and Devavrat Shah},

title = {Randomized Gossip Algorithms},

journal = {IEEE TRANSACTIONS ON INFORMATION THEORY},

year = {2006},

volume = {52},

number = {6},

pages = {2508--2530}

}

### Years of Citing Articles

### OpenURL

### Abstract

Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of “gossip ” algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for an arbitrary network graph, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest gossip algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion; however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance and scaling of gossip algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.

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Citation Context ...13s3.3.1 Centralized synchronous algorithm Let P be any n × n doubly-stochastic symmetric matrix corresponding to the probability matrix of the algorithm, as before. By Birkhoff-Von Neumann’s theorem =-=[HJ85]-=-, a non-negative doubly-stochastic matrix P can be decomposed into permutation matrices (equivalently matchings) as P = n 2 � m=1 αmΠm, αm ≥ 0, n 2 � m=1 αm = 1. Define a (matrix) random variable Π wi... |

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Citation Context ...t to the cone of symmetric positive semidefinite matrices. For general background on SDPs, eigenvalue optimization, and associated interior-point methods for solving these problems, see, for example, =-=[7]-=-, [33], [38], [47], and references therein. Interior point methods can be used to solve problems with a thousand edges or so; subgradient methods can be used to solve the problem for larger graphs tha... |

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Citation Context ...d averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], [20], [23], [25], [26], [29], [34], [42], =-=[44]-=-, [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (By a gossip algorithm, we mean specifically... |

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Citation Context ...ributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], [20], [23], [25], [26], [29], [34], =-=[42]-=-, [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (By a gossip algorithm, we mean specif... |

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Citation Context ...g replacements 6.1 Wireless networks n Tave ¦§¦ ¨¨ ¦§¦ log log n log n n Figure 2: Graphical interpretation of Theorem 7. ¢ ¢ £ ¡s¡ ¤ ¤ ¥ ¥ £ The Geometric Random Graph, introduced by Gupta and Kumar =-=[GK00]-=-, has been used successfully to model ad-hoc wireless networks. A d-dimensional Geometric Random Graph on n nodes, denoted G d (n, r), models a wireless ad-hoc network of n nodes with wireless transmi... |

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Citation Context ...t distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], [20], [23], [25], [26], [29], =-=[34]-=-, [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (By a gossip algorithm, we mean ... |

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Citation Context ...on to this problem. The use of the subgradient method to solve eigenvalue problems is well known; see, for example, [1], [31], [32], [39] for material on nonsmooth analysis of spectral functions, and =-=[8]-=-, [5], [19] for more general background on nonsmooth optimization. Recall that a subgradient of at is a symmetric matrix that satisfies the inequality for any feasible, i.e., symmetric stochastic matr... |

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Citation Context ...istributed averaging on a network, see [45]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see =-=[11]-=-, [16], [17], [20], [23], [25], [26], [29], [34], [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected ne... |

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Citation Context ... Lemma 1 For any k ≥ 1, E[Zk] = k/n. Further, for any δ > 0, �� ��� Pr Zk − k � � � � δk n� ≥ n ≤ � 2 exp − δ2 � k . 2 (4) Proof. By definition, E[Zk] = �k j=1 E[Zj − Zj−1] = �k Cramer’s Theorem (see =-=[DZ99]-=-, pp. 30 & 35). As a consequence of Lemma 1, for k ≥ n, Zk = k n � 1 ± j=1 � � 2 log n 1/n = k/n. Equation (4) follows directly from with high probability (i.e., probability at least 1 − 1/n 2 ). In t... |

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Citation Context ... chain, then (106) 1 − λ2(P ∗ ) ≤ dmax(1 − λ2(Pmd)). (107) Thus the averaging times for both random walks are of the same order, and differ by a factor of atmost dmax. For example, the social network =-=[Kle00]-=- is a regular graph with dmax = 5, which is the degree of each node in the graph. For the social network, therefore, the natural random walk (which is the same as the maximum degree chain) leads to an... |

608 | Coordination of groups of mobile autonomous agents using nearest neighbor rules
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Citation Context ...ry sequence drawn i.i.d. from some distribution, and try to characterize the properties the distribution must possess for convergence. Some other results on distributed averaging can be found in [6], =-=[21]-=-, [30], [36], [37]. An interesting result regarding products of random matrices is found in [12]. The authors prove the following result on a sequence of iterations , where the belong to a finite set ... |

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Citation Context ...i.i.d. from some distribution, and try to characterize the properties the distribution must possess for convergence. Some other results on distributed averaging can be found in [6], [21], [30], [36], =-=[37]-=-. An interesting result regarding products of random matrices is found in [12]. The authors prove the following result on a sequence of iterations , where the belong to a finite set of paracontracting... |

430 |
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Citation Context ... problem. The use of the subgradient method to solve eigenvalue problems is well known; see, for example, [1], [31], [32], [39] for material on nonsmooth analysis of spectral functions, and [8], [5], =-=[19]-=- for more general background on nonsmooth optimization. Recall that a subgradient of at is a symmetric matrix that satisfies the inequality for any feasible, i.e., symmetric stochastic matrix (here de... |

299 |
Random Geometric Graphs
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Citation Context ...o nodes that are within distance of each other. An example of a two-dimensional graph, is shown in Fig. 3. The following is a well-known result about the connectivity of (for a proof, see [13], [14], =-=[40]-=-). Lemma 9: For , the is connected with probability at least . We have the following results for averaging algorithms on a wireless sensor network, which are stated at the end of this section as Theor... |

298 | Gossip-based computation of aggregate information
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Citation Context ...zed optimization 2 . For one of the earliest references on distributed averaging on a network, see [Tsi84]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al =-=[KDG03]-=-, for example. For an extensive body of related work, see [KK02, KKD01, HHL88, GvRB01, KEW02, MFHH02, vR00, EGHK99, IEGH02, KSSV00, SMK + 01, RFH + 01]. This paper undertakes an in-depth study of the ... |

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Citation Context ...ce the stationary distribution is almost uniform (it is uniform asymptotically). The rest of this section is the proof of Theorem 8(b). We use a modification of a method developed by Diaconis-Stroock =-=[DS91]-=- to obtain bounds on the second largest eigenvalue using the geometry of the G d (n, r). Note that for d = 1, the proof is rather straightforward. The difficulty arises in the case of d ≥ 2. For ease ... |

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Citation Context ...veraging on a network, see [45]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], =-=[17]-=-, [20], [23], [25], [26], [29], [34], [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nod... |

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Citation Context ...ng on a network, see [45]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], =-=[20]-=-, [23], [25], [26], [29], [34], [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (B... |

195 | Randomized rumor spreading
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Citation Context ...e corresponding quantities when measured in absolute time (for an example, see Corollary 2). 4 ns1.2 Previous results A general lower bound for any graph G and any averaging algorithm was obtained in =-=[KSSV00]-=- in the synchronous setting. Their result is: Theorem 1 For any gossip algorithm on any graph G and for 0 < ɛ < 0.5, the ɛ-averaging time (in synchronous steps) is lower bounded by Ω(log n). The recen... |

193 | Fast Linear Iterations for Distributed Averaging
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Citation Context ...d in [KDG03], with ɛ-averaging time Θ(log ɛ −1 ) for the complete graph (from Corollary 3). The problem of fast distributed averaging without the gossip constraint on an arbitrary graph is studied in =-=[XB03]-=-; here, the matrices W (t) are constant, i.e., W (t) = W for all t. It is shown that the problem of finding the (constant) W that converges fastest to 11 T /n (where 11 T is the matrix of all ones) ca... |

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Citation Context ...in distance r of each other. An example of a two dimensional graph, G 2 (n, r) is shown in Figure3. The following is a well-known result about the connectivity of G d (n, r) (for a proof, see [GK00], =-=[GMPS04]-=-, [Pen03]): Lemma 9 For nr d ≥ 2 log n, the G(n, r) is connected with probability at least 1 − 1/n 2 . We have the following results for averaging algorithms on a wireless sensor network, which are st... |

158 | Gossip algorithms: Design, analysis and applications
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Citation Context ...gorithms by studying the mixing time of the corresponding random walk on the graph. The recent work of Boyd et al [BDX04] shows 1 Preliminary versions of this paper appeared in [BGPS04],[BGPS05b] and =-=[BGPS05a]-=-. 2 The theoretical framework developed in this paper is not restricted merely to averaging algorithms. It easily extends to the computation of other functions which can be computed via pair-wise oper... |

141 | Spatial gossip and resource location protocols
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Citation Context ...ee [45]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], [20], [23], [25], =-=[26]-=-, [29], [34], [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (By a gossip algorit... |

136 | Convex Analysis and Nonlinear Optimization, Theory and Examples
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Citation Context ... this problem. The use of the subgradient method to solve eigenvalue problems is well known; see, for example, [1], [31], [32], [39] for material on nonsmooth analysis of spectral functions, and [8], =-=[5]-=-, [19] for more general background on nonsmooth optimization. Recall that a subgradient of at is a symmetric matrix that satisfies the inequality for any feasible, i.e., symmetric stochastic matrix (h... |

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Citation Context ...ation of autonomous agents, estimation and distributed data fusion on ad-hoc networks, and decentralized optimization 2 . For one of the earliest references on distributed averaging on a network, see =-=[Tsi84]-=-. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al [KDG03], for example. For an extensive body of related work, see [KK02, KKD01, HHL88, GvRB01, KEW02, MFHH0... |

97 | Local control strategies for groups of mobile autonomous agents - Lin, Brouke, et al. - 2004 |

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Citation Context ...the matrix that characterizes the algorithm. This means we can also study averaging algorithms by studying the mixing time of the corresponding random walk on the graph. The recent work of Boyd et al =-=[BDX04]-=- shows 1 Preliminary versions of this paper appeared in [BGPS04],[BGPS05b] and [BGPS05a]. 2 The theoretical framework developed in this paper is not restricted merely to averaging algorithms. It easil... |

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Citation Context ...the cone of symmetric positive semidefinite matrices. For general background on SDPs, eigenvalue optimization, and associated interior-point methods for solving these problems, see, for example, [7], =-=[33]-=-, [38], [47], and references therein. Interior point methods can be used to solve problems with a thousand edges or so; subgradient methods can be used to solve the problem for larger graphs that have... |

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Citation Context ...ne of symmetric positive semidefinite matrices. For general background on SDPs, eigenvalue optimization, and associated interior-point methods for solving these problems, see, for example, [7], [33], =-=[38]-=-, [47], and references therein. Interior point methods can be used to solve problems with a thousand edges or so; subgradient methods can be used to solve the problem for larger graphs that have up to... |

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Citation Context ...uted averaging on a network, see [45]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], =-=[16]-=-, [17], [20], [23], [25], [26], [29], [34], [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network ... |

63 | On certain connectivity properties of the internet topology
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Citation Context ...From (104), (105), Theorem 3 and Corollary 2, we see that the optimal averaging algorithm on any expander graph has ɛ-averaging time Tave(ɛ) = Θ � log ɛ −1� . The Preferential Connectivity (PC) model =-=[MPS03]-=- is one of the popular models for the Internet. In [MPS03], it is shown that the Internet is an expander under the preferential connectivity model. Using the conclusion above, we obtain the following ... |

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Citation Context .... We will use the subgradient method to obtain a distributed solution to this problem. The use of the subgradient method to solve eigenvalue problems is well known; see, for example, [1], [31], [32], =-=[39]-=- for material on nonsmooth analysis of spectral functions, and [8], [5], [19] for more general background on nonsmooth optimization. Recall that a subgradient of at is a symmetric matrix that satisfie... |

56 | Protocols and impossibility results for gossip-based communication mechanisms
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Citation Context ...ork, see [45]. Fast distributed averaging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], [20], [23], =-=[25]-=-, [26], [29], [34], [42], [44], [46]. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (By a gossip a... |

53 | A decentralized algorithm for spectral analysis
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Citation Context ...value in spite of the error incurred during the distributed computation of u at each iteration. The problem of distributed computation of the top-k eigenvectors of a matrix on a graph is discussed in =-=[KM04]-=-. By distributed computation of an eigenvector u of a matrix W , we mean that each node i is aware of the i th row of W , and can only communicate with its immediate neighbors. Given these constraints... |

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Citation Context ... be written as a semidefinite program (under a symmetry constraint), and can therefore be solved numerically. Distributed averaging has also been studied in the context of distributed load balancing (=-=[RSW98]-=-), where nodes (processors) exchange tokens in order to uniformly distribute tokens over all the processors in the network (the number of tokens is constrained to be integral, so exact averaging is no... |

42 | Convex analysis on the hermitian matrices
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Citation Context ...ined in (54). We will use the subgradient method to obtain a distributed solution to this problem. The use of the subgradient method to solve eigenvalue problems is well known; see, for example, [1], =-=[31]-=-, [32], [39] for material on nonsmooth analysis of spectral functions, and [8], [5], [19] for more general background on nonsmooth optimization. Recall that a subgradient of at is a symmetric matrix t... |

35 | Nonsmooth analysis of eigenvalues - Lewis |

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Citation Context ...y averaging algorithms by studying the mixing time of the corresponding random walk on the graph. The recent work of Boyd et al [BDX04] shows 1 Preliminary versions of this paper appeared in [BGPS04],=-=[BGPS05b]-=- and [BGPS05a]. 2 The theoretical framework developed in this paper is not restricted merely to averaging algorithms. It easily extends to the computation of other functions which can be computed via ... |

29 |
Random geometric graphs. Oxford studies in probability
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Citation Context ...e r of each other. An example of a two dimensional graph, G 2 (n, r) is shown in Figure3. The following is a well-known result about the connectivity of G d (n, r) (for a proof, see [GK00], [GMPS04], =-=[Pen03]-=-): Lemma 9 For nr d ≥ 2 log n, the G(n, r) is connected with probability at least 1 − 1/n 2 . We have the following results for averaging algorithms on a wireless sensor network, which are stated at t... |

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Citation Context ...how that the subgradient method converges despite approximation errors in computation of the eigenvector, which spill over into computation of the subgradient. To show this, we will use a result from =-=[Kiw04]-=- on the convergence of approximate subgradient methods. Given an optimization problem with objective function f and feasible set S, the approximate subgradient method generates a sequence {xk } ∞ k=1 ... |

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Citation Context ...bitrary sequence drawn i.i.d. from some distribution, and try to characterize the properties the distribution must possess for convergence. Some other results on distributed averaging can be found in =-=[6]-=-, [21], [30], [36], [37]. An interesting result regarding products of random matrices is found in [12]. The authors prove the following result on a sequence of iterations , where the belong to a finit... |

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Citation Context ...n must possess for convergence. Some other results on distributed averaging can be found in [BS03, Mur03, LBF04, OSM04, JLM03]. An interesting result regarding products of random matrices is found in =-=[EKN90]-=-. The authors prove the following result on a sequence of iterations x(t + 1) = W (t)x(t), where the W (t) belong to a finite set of paracontracting matrices (i.e., W (t)x �= x ⇔ �W (t)x� < �x�). If I... |

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Citation Context ...aging algorithms are also important in other contexts; see Kempe et al. [22], for example. For an extensive body of related work, see [11], [16], [17], [20], [23], [25], [26], [29], [34], [42], [44], =-=[46]-=-. This paper undertakes an in-depth study of the design and analysis of gossip algorithms for averaging in an arbitrarily connected network of nodes. (By a gossip algorithm, we mean specifically an al... |

20 | Analysis and optimization of randomized gossip algorithms - Boyd, Ghosh, et al. - 2004 |

15 | Rapidly Mixing Markov Chains: A Comparison of Techniques
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Citation Context ...fined as �n t 1 j=1 |Pij − n |. Tmix(ɛ) = sup inf{t : ∆i(t i ′ ) ≤ ɛ, ∀ t ′ ≥ t}. (67) Recall also the following well known bounds on the ɛ-mixing time for a Markov chain (see for example, the survey =-=[Gur00]-=-). Lemma 8 The ɛ-mixing time of a Markov chain with doubly stochastic transition matrix P is bounded as: λmax(P ) log(2ɛ) −1 2(1 − λmax(P )) ≤ Tmix(ɛ) ≤ For ɛ = o(1/n), (68) becomes � � λmax(P ) log n... |

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Citation Context ...drawn i.i.d. from some distribution, and try to characterize the properties the distribution must possess for convergence. Some other results on distributed averaging can be found in [6], [21], [30], =-=[36]-=-, [37]. An interesting result regarding products of random matrices is found in [12]. The authors prove the following result on a sequence of iterations , where the belong to a finite set of paracontr... |

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Citation Context ...lowing lemma: Lemma 11 For k, n such that k ≤ n/4, the mixing rate of the fastest-mixing symmetric random walk on Gk cannot be smaller than cos(2πk/n). Proof. It can be shown using symmetry arguments =-=[PXBD03]-=- that the fastest mixing Markov chain on Gk with uniform stationary distribution will have a symmetric and circulant transition matrix. (For this simple graph, this can be easily seen using convexity ... |

2 | algorithms: Design, analysis, and applications - “Gossip - 2005 |

2 |
analysis of eigenvalues
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Citation Context ...n (54). We will use the subgradient method to obtain a distributed solution to this problem. The use of the subgradient method to solve eigenvalue problems is well known; see, for example, [1], [31], =-=[32]-=-, [39] for material on nonsmooth analysis of spectral functions, and [8], [5], [19] for more general background on nonsmooth optimization. Recall that a subgradient of at is a symmetric matrix that sa... |