Analysis of Large Graphs

The Laplacian Spectrum of Large Graphs Sampled from Graphons

IEEE Transactions on Network Science and Engineering ◽

10.1109/tnse.2021.3069675 ◽

2021 ◽

pp. 1-1

Author(s):

Renato Vizuete ◽

Federica Garin ◽

Paolo Frasca

Keyword(s):

Laplacian Spectrum ◽

Large Graphs

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VColor*: a practical approach for coloring large graphs

Frontiers of Computer Science ◽

10.1007/s11704-020-9205-y ◽

2021 ◽

Vol 15 (4) ◽

Author(s):

Yun Peng ◽

Xin Lin ◽

Byron Choi ◽

Bingsheng He

Keyword(s):

Practical Approach ◽

Large Graphs

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Faster Motif Counting via Succinct Color Coding and Adaptive Sampling

ACM Transactions on Knowledge Discovery from Data ◽

10.1145/3447397 ◽

2021 ◽

Vol 15 (6) ◽

pp. 1-27

Author(s):

Marco Bressan ◽

Stefano Leucci ◽

Alessandro Panconesi

Keyword(s):

Adaptive Sampling ◽

Relative Frequency ◽

State Of The Art ◽

Color Coding ◽

Input Graph ◽

Large Graphs ◽

Running Time ◽

Uniform Sampling ◽

Current State ◽

Connected Subgraphs

We address the problem of computing the distribution of induced connected subgraphs, aka graphlets or motifs , in large graphs. The current state-of-the-art algorithms estimate the motif counts via uniform sampling by leveraging the color coding technique by Alon, Yuster, and Zwick. In this work, we extend the applicability of this approach by introducing a set of algorithmic optimizations and techniques that reduce the running time and space usage of color coding and improve the accuracy of the counts. To this end, we first show how to optimize color coding to efficiently build a compact table of a representative subsample of all graphlets in the input graph. For 8-node motifs, we can build such a table in one hour for a graph with 65M nodes and 1.8B edges, which is times larger than the state of the art. We then introduce a novel adaptive sampling scheme that breaks the “additive error barrier” of uniform sampling, guaranteeing multiplicative approximations instead of just additive ones. This allows us to count not only the most frequent motifs, but also extremely rare ones. For instance, on one graph we accurately count nearly 10.000 distinct 8-node motifs whose relative frequency is so small that uniform sampling would literally take centuries to find them. Our results show that color coding is still the most promising approach to scalable motif counting.

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Tiered Sampling

ACM Transactions on Knowledge Discovery from Data ◽

10.1145/3441299 ◽

2021 ◽

Vol 15 (5) ◽

pp. 1-52

Author(s):

Lorenzo De Stefani ◽

Erisa Terolli ◽

Eli Upfal

Keyword(s):

Large Scale ◽

Analysis Of Algorithms ◽

Base Layer ◽

Single Edge ◽

Real World Data ◽

High Quality ◽

Large Graphs ◽

Massive Graphs ◽

Variance Estimate ◽

Low Probability

We introduce Tiered Sampling , a novel technique for estimating the count of sparse motifs in massive graphs whose edges are observed in a stream. Our technique requires only a single pass on the data and uses a memory of fixed size M , which can be magnitudes smaller than the number of edges. Our methods address the challenging task of counting sparse motifs—sub-graph patterns—that have a low probability of appearing in a sample of M edges in the graph, which is the maximum amount of data available to the algorithms in each step. To obtain an unbiased and low variance estimate of the count, we partition the available memory into tiers (layers) of reservoir samples. While the base layer is a standard reservoir sample of edges, other layers are reservoir samples of sub-structures of the desired motif. By storing more frequent sub-structures of the motif, we increase the probability of detecting an occurrence of the sparse motif we are counting, thus decreasing the variance and error of the estimate. While we focus on the designing and analysis of algorithms for counting 4-cliques, we present a method which allows generalizing Tiered Sampling to obtain high-quality estimates for the number of occurrence of any sub-graph of interest, while reducing the analysis effort due to specific properties of the pattern of interest. We present a complete analytical analysis and extensive experimental evaluation of our proposed method using both synthetic and real-world data. Our results demonstrate the advantage of our method in obtaining high-quality approximations for the number of 4 and 5-cliques for large graphs using a very limited amount of memory, significantly outperforming the single edge sample approach for counting sparse motifs in large scale graphs.

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Isoscattering strings of concatenating graphs and networks

Scientific Reports ◽

10.1038/s41598-020-80950-6 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Michał Ławniczak ◽

Adam Sawicki ◽

Małgorzata Białous ◽

Leszek Sirko

Keyword(s):

Trace Function ◽

Quantum Graphs ◽

Mathematical Approach ◽

Large Graphs ◽

Graphs And Networks ◽

Scattering Matrices ◽

Theoretical Predictions ◽

Infinite Strings ◽

Microwave Networks ◽

Insight Into

AbstractWe identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for $$n \rightarrow \infty $$ n → ∞ . The theoretical predictions are confirmed experimentally using $$n=2$$ n = 2 units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the $$2n \times 2n $$ 2 n × 2 n scattering matrices $${\hat{S}}$$ S ^ of the systems to 2n diagonal elements, while the old measures of isoscattering require all $$(2n)^2$$ ( 2 n ) 2 entries. The studied problem generalizes a famous question of Mark Kac “Can one hear the shape of a drum?”, originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.

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