103 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Distribution of Sandpile Groups of Random Graphs with their Pairings
We determine the distribution of the sandpile group (also known as the
Jacobian) of the Erd\H{o}s-R\'{e}nyi random graph along with its
canonical duality pairing as tends to infinity, fully resolving a
conjecture from 2015 due to Clancy, Leake, and Payne and generalizing the
result by Wood on the groups. In particular, we show that a finite abelian
-group equipped with a perfect symmetric pairing appears as the
Sylow -part of the sandpile group and its pairing with frequency inversely
proportional to , where
is the set of automorphisms of preserving the pairing . While this
distribution is related to the Cohen-Lenstra distribution, the two
distributions are not the same on account of the additional algebraic data of
the pairing. The proof utilizes the moment method: we first compute a complete
set of moments for our random variable (the average number of epimorphisms from
our random object to a fixed object in the category of interest) and then show
the moments determine the distribution. To obtain the moments, we prove a
universality result for the moments of cokernels of random symmetric integral
matrices whose dual groups are equipped with symmetric pairings that is strong
enough to handle both the dependence in the diagonal entries and the additional
data of the pairing. We then apply results due to Sawin and Wood to show that
these moments determine a unique distribution.Comment: 43 page
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On learning the structure of clusters in graphs
Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level structure. This is often overlooked in the design and analysis of graph clustering algorithms which make strong simplifying assumptions about the structure of the graph. This thesis addresses the natural question of whether the structure of clusters can be learned efficiently and describes four new algorithmic results for learning such structure in graphs and hypergraphs.
The first part of the thesis studies the classical spectral clustering algorithm, and presents a tighter analysis on its performance. This result explains why it works under a much weaker and more natural condition than the ones studied in the literature, and helps to close the gap between the theoretical guarantees of the spectral clustering algorithm and its excellent empirical performance.
The second part of the thesis builds on the theoretical guarantees of the previous part and shows that, when the clusters of the underlying graph have certain structures, spectral clustering with fewer than k eigenvectors is able to produce better output than classical spectral clustering in which k eigenvectors are employed, where k is the number of clusters. This presents the first work that discusses and analyses the performance of spectral clustering with fewer than k eigenvectors, and shows that general structures of clusters can be learned with spectral methods.
The third part of the thesis considers efficient learning of the structure of clusters with local algorithms, whose runtime depends only on the size of the target clusters and is independent of the underlying input graph. While the objective of classical local clustering algorithms is to find a cluster which is sparsely connected to the rest of the graph, this part of the thesis presents a local algorithm that finds a pair of clusters which are densely connected to each other. This result demonstrates that certain structures of clusters can be learned efficiently in the local setting, even in the massive graphs which are ubiquitous in real-world applications.
The final part of the thesis studies the problem of learning densely connected clusters in hypergraphs. The developed algorithm is based on a new heat diffusion process, whose analysis extends a sequence of recent work on the spectral theory of hypergraphs. It allows the structure of clusters to be learned in datasets modelling higher-order relations of objects and can be applied to efficiently analyse many complex datasets occurring in practice.
All of the presented theoretical results are further extensively evaluated on both synthetic and real-word datasets of different domains, including image classification and segmentation, migration networks, co-authorship networks, and natural language processing. These experimental results demonstrate that the newly developed algorithms are practical, effective, and immediately applicable for learning the structure of clusters in real-world data
Contagion Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis and Network Algorithms
This monograph provides an overview of the mathematical theories and
computational algorithm design for contagion source detection in large
networks. By leveraging network centrality as a tool for statistical inference,
we can accurately identify the source of contagions, trace their spread, and
predict future trajectories. This approach provides fundamental insights into
surveillance capability and asymptotic behavior of contagion spreading in
networks. Mathematical theory and computational algorithms are vital to
understanding contagion dynamics, improving surveillance capabilities, and
developing effective strategies to prevent the spread of infectious diseases
and misinformation.Comment: Suggested Citation: Chee Wei Tan and Pei-Duo Yu (2023), "Contagion
Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis
and Network Algorithms", Foundations and Trends in Networking: Vol. 13: No.
2-3, pp 107-251. http://dx.doi.org/10.1561/130000006
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Fundamentals
Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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