Inferring Hidden Structures in Random Graphs

Wasim Huleihel

doi:10.1109/TSIPN.2022.3211208

Inferring Hidden Structures in Random Graphs

Wasim Huleihel

School of Electrical Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

We study the two inference problems of detecting and recovering an isolated community of <italic>general</italic> structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph <inline-formula><tex-math notation="LaTeX">$\mathcal{G(n,q)}$</tex-math></inline-formula> with edge density <inline-formula><tex-math notation="LaTeX">$q\in (0,1)$</tex-math></inline-formula>; under the alternative, there is an unknown structure <inline-formula><tex-math notation="LaTeX">$\Gamma _{k}$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> nodes, planted in <inline-formula><tex-math notation="LaTeX">$\mathcal{G(n,q)}$</tex-math></inline-formula>, such that it appears as an <italic>induced subgraph</italic>. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure <inline-formula><tex-math notation="LaTeX">$\Gamma _{k}$</tex-math></inline-formula> in terms of the parameters <inline-formula><tex-math notation="LaTeX">$(n,k,q)$</tex-math></inline-formula>, as well as certain properties of <inline-formula><tex-math notation="LaTeX">$\Gamma _{k}$</tex-math></inline-formula>, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an “easy-hard-impossible” phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.

Original language	English
Pages (from-to)	1-13
Number of pages	13
Journal	IEEE Transactions on Signal and Information Processing over Networks
DOIs	https://doi.org/10.1109/TSIPN.2022.3211208
State	Accepted/In press - 2022

Keywords

Computational modeling
Hidden structures
Image edge detection
Inference algorithms
Information processing
random graphs and networks
Stars
statistical and computational limits
statistical inference
Task analysis
Testing

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10.1109/TSIPN.2022.3211208

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title = "Inferring Hidden Structures in Random Graphs",

abstract = "We study the two inference problems of detecting and recovering an isolated community of general structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erd{\H o}s-R{\'e}nyi random graph $\mathcal{G(n,q)}$ with edge density $q\in (0,1)$; under the alternative, there is an unknown structure $\Gamma _{k}$ on $k$ nodes, planted in $\mathcal{G(n,q)}$, such that it appears as an induced subgraph. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure $\Gamma _{k}$ in terms of the parameters $(n,k,q)$, as well as certain properties of $\Gamma _{k}$, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an “easy-hard-impossible” phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.",

keywords = "Computational modeling, Hidden structures, Image edge detection, Inference algorithms, Information processing, random graphs and networks, Stars, statistical and computational limits, statistical inference, Task analysis, Testing",

author = "Wasim Huleihel",

note = "Publisher Copyright: IEEE",

year = "2022",

doi = "10.1109/TSIPN.2022.3211208",

language = "אנגלית",

pages = "1--13",

journal = "IEEE Transactions on Signal and Information Processing over Networks",

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T1 - Inferring Hidden Structures in Random Graphs

AU - Huleihel, Wasim

N1 - Publisher Copyright: IEEE

PY - 2022

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N2 - We study the two inference problems of detecting and recovering an isolated community of general structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph $\mathcal{G(n,q)}$ with edge density $q\in (0,1)$; under the alternative, there is an unknown structure $\Gamma _{k}$ on $k$ nodes, planted in $\mathcal{G(n,q)}$, such that it appears as an induced subgraph. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure $\Gamma _{k}$ in terms of the parameters $(n,k,q)$, as well as certain properties of $\Gamma _{k}$, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an “easy-hard-impossible” phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.

AB - We study the two inference problems of detecting and recovering an isolated community of general structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph $\mathcal{G(n,q)}$ with edge density $q\in (0,1)$; under the alternative, there is an unknown structure $\Gamma _{k}$ on $k$ nodes, planted in $\mathcal{G(n,q)}$, such that it appears as an induced subgraph. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure $\Gamma _{k}$ in terms of the parameters $(n,k,q)$, as well as certain properties of $\Gamma _{k}$, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an “easy-hard-impossible” phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.

KW - Computational modeling

KW - Hidden structures

KW - Image edge detection

KW - Inference algorithms

KW - Information processing

KW - random graphs and networks

KW - Stars

KW - statistical and computational limits

KW - statistical inference

KW - Task analysis

KW - Testing

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DO - 10.1109/TSIPN.2022.3211208

M3 - מאמר

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EP - 13

JO - IEEE Transactions on Signal and Information Processing over Networks

JF - IEEE Transactions on Signal and Information Processing over Networks

SN - 2373-776X

ER -

Inferring Hidden Structures in Random Graphs

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