The Large Deviation Principle for Inhomogeneous Erdős–Rényi Random Graphs

Maarten Markering ()
Additional contact information
Maarten Markering: Leiden University

Journal of Theoretical Probability, 2023, vol. 36, issue 2, 711-727

Abstract: Abstract Consider the inhomogeneous Erdős-Rényi random graph (ERRG) on n vertices for which each pair $$i,j\in \{1,\ldots ,n\}$$ i , j ∈ { 1 , … , n } , $$i\ne j,$$ i ≠ j , is connected independently by an edge with probability $$r_n(\frac{i-1}{n},\frac{j-1}{n})$$ r n ( i - 1 n , j - 1 n ) , where $$(r_n)_{n\in \mathbb {N}}$$ ( r n ) n ∈ N is a sequence of graphons converging to a reference graphon r. As a generalisation of the celebrated large deviation principle (LDP) for ERRGs by Chatterjee and Varadhan (Eur J Comb 32:1000–1017, 2011), Dhara and Sen (Large deviation for uniform graphs with given degrees, 2020. arXiv:1904.07666 ) proved an LDP for a sequence of such graphs under the assumption that r is bounded away from 0 and 1, and with a rate function in the form of a lower semi-continuous envelope. We further extend the results by Dhara and Sen. We relax the conditions on the reference graphon to $$(\log r, \log (1- r))\in L^1([0,1]^2)$$ ( log r , log ( 1 - r ) ) ∈ L 1 ( [ 0 , 1 ] 2 ) . We also show that, under this condition, their rate function equals a different, more tractable rate function. We then apply these results to the large deviation principle for the largest eigenvalue of inhomogeneous ERRGs and weaken the conditions for part of the analysis of the rate function by Chakrabarty et al. (Large deviation principle for the maximal eigenvalue of inhomogeneous Erdoös-Rényi random graphs, 2020. arXiv:2008.08367 ).

Keywords: Inhomogeneous Erdős-Rényi random graph; Large deviations; Rate function; Graphon; Largest eigenvalue; 05C80; 60F10; 60B20 (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-022-01181-1 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:36:y:2023:i:2:d:10.1007_s10959-022-01181-1

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-022-01181-1

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().