On certain connectivity properties of the internet topology

Mihail, Milena; Papadimitriou, C.; Saberi, Amin

doi:10.1016/j.jcss.2005.06.009

Cited by 86 publications

(79 citation statements)

References 38 publications

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“…It is well known that a random k-regular graph is with high probability a k − 2 − δ expander for all δ > 0 [Kah92]. Also, it is known that for small constant λ, random graphs with a fixed degree sequence with minimum degree 3, and random graphs in preferential attachment model with minimum degree 2 have expansion λ with high probability [GMS03], [MPS06]. The thesis follows from Lemma 1.…”

Section: Rate Of Convergence For Specific Graph Families Proof Of Thmentioning

confidence: 81%

Convergence to Equilibrium in Local Interaction Games

Montanari

Saberi

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

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Abstract-We study a simple game theoretic model for the spread of an innovation in a network. The diffusion of the innovation is modeled as the dynamics of a coordination game in which the adoption of a common strategy between players has a higher payoff.Classical results in game theory provide a simple condition for an innovation to become widespread in the network. The present paper characterizes the rate of convergence as a function of graph structure. In particular, we derive a dichotomy between well-connected (e.g. random) graphs that show slow convergence and poorly connected, low dimensional graphs that show fast convergence.

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Section: Rate Of Convergence For Specific Graph Families Proof Of Thmentioning

confidence: 81%

Convergence to Equilibrium in Local Interaction Games

Montanari

Saberi

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

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“…From (105), (106), Theorem 3, and Corollary 2, we see that the optimal averaging algorithm on any expander graph has -averaging time . The Preferential Connectivity (PC) model [35] is one of the popular models for the Internet. In [35], it is shown that the Internet is an expander under the PC model.…”

Section: B Expander Graphsmentioning

confidence: 99%

“…The Preferential Connectivity (PC) model [35] is one of the popular models for the Internet. In [35], it is shown that the Internet is an expander under the PC model. Using the conclusion above, we obtain the following result for averaging on the Internet.…”

Section: B Expander Graphsmentioning

confidence: 99%

Randomized gossip algorithms

Boyd

Ghosh

Prabhakar

et al. 2006

IEEE Trans. Inform. Theory

2,083

2,229

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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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“…However, experimental evidence shows that optimization is considerably easier in real-world power-law graphs. This suggests that real-world graphs are not "worst-case" instances of power-law graphs, but rather typical instances which may be well modeled by power law random graph models (e.g., [1,6,3,4,7,9]). Combinatorial optimization is generally easier in random graphs and hence from an optimization perspective this somewhat justifies using power law random graphs to model real-world power law graphs.…”

Section: Overview and Resultsmentioning

confidence: 99%

On the Hardness of Optimization in Power Law Graphs

Ferrante

Pandurangan

Park

Lecture Notes in Computer Science

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Abstract. Our motivation for this work is the remarkable discovery that many large-scale real-world graphs ranging from Internet and World Wide Web to social and biological networks exhibit a power-law distribution: the number of nodes yi of a given degree i is proportional to i −β where β > 0 is a constant that depends on the application domain. There is practical evidence that combinatorial optimization in power-law graphs is easier than in general graphs, prompting the basic theoretical question: Is combinatorial optimization in power-law graphs easy? Does the answer depend on the power-law exponent β? Our main result is the proof that many classical NP-hard graph-theoretic optimization problems remain NP-hard on power law graphs for certain values of β. In particular, we show that some classical problems, such as CLIQUE and COLORING, remains NP-hard for all β ≥ 1. Moreover, we show that all the problems that satisfy the so-called "optimal substructure property" remains NP-hard for all β > 0. This includes classical problems such as MIN VERTEX-COVER, MAX INDEPENDENT-SET, and MIN DOMINATING-SET. Our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems.

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On certain connectivity properties of the internet topology

Cited by 86 publications

References 38 publications

Convergence to Equilibrium in Local Interaction Games

Convergence to Equilibrium in Local Interaction Games

Randomized gossip algorithms

On the Hardness of Optimization in Power Law Graphs

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