Recovering the structure of random linear graphs

Rocha, Israel; Janssen, Jeannette; Kalyaniwalla, Nauzer

doi:10.1016/j.laa.2018.07.029

Cited by 6 publications

(20 citation statements)

References 27 publications

(29 reference statements)

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“…Very similar random graphs have been extensively studied (see e.g. [2,5,14,36]). Special cases of this noisy seriation problem (with points sometimes embedded on a circle instead of an interval) appear in many areas.…”

Section: Introductionmentioning

confidence: 99%

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Reconstruction of line-embeddings of graphons

Janssen

Smith

2022

Electron. J. Statist.

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Consider a random graph process with n vertices corresponding to points v i i.i.d.∼ Unif[0, 1] embedded randomly in the interval, and where edges are inserted between v i , v j independently with probability given by the graphon w(v i , v j ) ∈ [0, 1]. Following [11], we call a graphon w diagonally increasing if, for each x, w(x, y) decreases as y moves away from x. We call a permutation σ ∈ Sn an ordering of these vertices if v σ(i) < v σ(j) for all i < j, and ask: how can we accurately estimate σ from an observed graph? We present a randomized algorithm with output σ that, for a large class of graphons, achieves error max 1≤i≤n |σ(i) − σ(i)| = Õ( √ n) with high probability; we also show that this is the best-possible convergence rate for a large class of algorithms and proof strategies. Under an additional assumption that is satisfied by some popular graphon models, we break this "barrier" at √ n and obtain the vastly better rate Õ(n ) for any > 0. These improved seriation bounds can be combined with previous work to give more efficient and accurate algorithms for related tasks, including estimating diagonally increasing graphons [20,21] and testing whether a graphon is diagonally increasing [11].

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Section: Introductionmentioning

confidence: 99%

“…A natural idea is to apply a spectral method as used in [1] for the noiseless matrix seriation problem. A related approach was studied in the recent work [36] in the special case that w is of the form (1.4) and d = 0.5, q = 0. Theorem 3 of this paper says that, with high probability, there exists a set I ⊂ {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of line-embeddings of graphons

Janssen

Smith

2022

Electron. J. Statist.

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“…Eigenvectors of random matrices are the main tool we use to construct the layout in our problem. We introduce this idea in [9], where one eigenvector would suffice to recover the structure of a random linear graph. Here, as we will see, one eigenvector alone is not enough to encode the whole layout.…”

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confidence: 99%

“…Fortunately, we can combine two special eigenvectors to find the linear arrangement. Even tough, the use of eigenvectors in the same fashion is a common feature of both methods, here we require some additional technical details that were not present in [9]. Due to the use of angles between subspaces and SVD decomposition, the technique we use here differs significantly from [9].…”

mentioning

confidence: 99%

“…Even tough, the use of eigenvectors in the same fashion is a common feature of both methods, here we require some additional technical details that were not present in [9]. Due to the use of angles between subspaces and SVD decomposition, the technique we use here differs significantly from [9]. There, we pointed out the generality of such method and here it turns out we need a more careful analysis.…”

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confidence: 99%

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Layout of random circulant graphs

Richter

Rocha²

2018

Linear Algebra and its Applications

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A circulant graph H is defined on the set of vertices V = {1, . . . , n}A random circulant graph results from deleting edges of H with probability 1 − p. We provide a polynomial time algorithm that approximates the solution to the minimum linear arrangement problem for random circulant graphs. We then bound the error of the approximation with high probability.2000 Mathematics Subject Classification. 05C50; 05C85; 15A52; 15A18.

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Big Data Processing on Volunteer Computing

Chen

Singh

2021

ACM Trans. Internet Technol.

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In order to calculate the node big data contained in complex networks and realize the efficient calculation of complex networks, based on voluntary computing, taking ICE middleware as the communication medium, the loose coupling distributed framework DCBV based on voluntary computing is proposed. Then, the Master, Worker, and MiddleWare layers in the framework, and the development structure of a DCBV framework are designed. The task allocation and recovery strategy, message passing and communication mode, and fault tolerance processing are discussed. Finally, to calculate and verify parameters such as the average shortest path of the framework and shorten calculation time, an improved accurate shortest path algorithm, the N-SPFA algorithm, is proposed. Under different datasets, the node calculation and performance of the N-SPFA algorithm are explored. The algorithm is compared with four approximate shortest-path algorithms: Combined Link and Attribute (CLA), Lexicographic Breadth First Search (LBFS), Approximate algorithm of shortest path length based on center distance of area division (CDZ), and Hub Vertex of area and Core Expressway (HEA-CE). The results show that when the number of CPU threads is 4, the computation time of the DCBV framework is the shortest (514.63 ms). As the number of CPU cores increases, the overall computation time of the framework decreases gradually. For every 2 additional CPU cores, the number of tasks increases by 1. When the number of Worker nodes is 8 and the number of nodes is 1, the computation time of the framework is the shortest (210,979 ms), and the IO statistics data increase with the increase of Worker nodes. When the datasets are Undirected01 and Undirected02, the computation time of the N-SPFA algorithm is the shortest, which is 4520 ms and 7324 ms, respectively. However, the calculation time in the ca-condmat_undirected dataset is 175,292 ms, and the performance is slightly worse. Overall, however, the performance of the N-SPFA and SPFA algorithms is good. Therefore, the two algorithms are combined. For networks with less complexity, the computational scale coefficient of the SPFA algorithm can be set to 0.06, and for general networks, 0.2. When compared with other algorithms in different datasets, the pretreatment time, average query time, and overall query time of N-SPFA algorithm are the shortest, being 49.67 ms, 5.12 ms, and 94,720 ms, respectively. The accuracy (1.0087) and error rate (0.024) are also the best. In conclusion, voluntary computing can be applied to the processing of big data, which has a good reference significance for the distributed analysis of large-scale complex networks.

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Recovering the structure of random linear graphs

Cited by 6 publications

References 27 publications

Reconstruction of line-embeddings of graphons

Reconstruction of line-embeddings of graphons

Layout of random circulant graphs

Big Data Processing on Volunteer Computing

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