Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs
We consider an inhomogeneous Erdős-Rényi random graph G N with vertex set [ N ] = { 1 , ⋯ , N } for which the pair of vertices i , j ∈ [ N ] , i ≠ j , is connected by an edge with probability r ( i N , j N ) , independently of other pairs of vertices. Here, r : [ 0 , 1 ] 2 → ( 0 , 1 ) is a symmetric...
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| Veröffentlicht in: | Journal of theoretical probability 2022-12, Vol.35 (4), p.2413-2441 |
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| container_title | Journal of theoretical probability |
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| creator | Chakrabarty, Arijit Hazra, Rajat Subhra Hollander, Frank den Sfragara, Matteo |
| description | We consider an inhomogeneous Erdős-Rényi random graph
G
N
with vertex set
[
N
]
=
{
1
,
⋯
,
N
}
for which the pair of vertices
i
,
j
∈
[
N
]
,
i
≠
j
, is connected by an edge with probability
r
(
i
N
,
j
N
)
, independently of other pairs of vertices. Here,
r
:
[
0
,
1
]
2
→
(
0
,
1
)
is a symmetric function that plays the role of a reference graphon. Let
λ
N
be the maximal eigenvalue of the adjacency matrix of
G
N
. It is known that
λ
N
/
N
satisfies a large deviation principle as
N
→
∞
. The associated rate function
ψ
r
is given by a variational formula that involves the rate function
I
r
of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of
ψ
r
, specially when the reference graphon is of rank 1. |
| doi_str_mv | 10.1007/s10959-021-01138-w |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2736325986</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2736325986</sourcerecordid><originalsourceid>FETCH-LOGICAL-c363t-298543493acb9fbce9fab9f38854b5b445e761592240407ee899d785a04943dc3</originalsourceid><addsrcrecordid>eNp9kEtOwzAQhi0EEqVwAVaWWBvGjzTxEpVSKhWBKlhbTuK0qdI42ElLj8ExOAfiXhiCxI7VjEb_Q_MhdE7hkgLEV56CjCQBRglQyhOyO0ADGsWMSMbhEA0gkYLIRMAxOvF-DQBSAgyQnmu3NPjGbEvdlrbGj66ss7KpDC6sw-3K4Hv9Wm50hSfl0tRbXXUG2wLP6pXd2HAxtvN44vLPN08WH-_1vsQLXed2g6dONyt_io4KXXlz9juH6Pl28jS-I_OH6Wx8PScZH_GWMJlEggvJdZbKIs2MLHRYeBLOaZQKEZl4RCPJmAABsTGJlHmcRBqEFDzP-BBd9LmNsy-d8a1a287VoVKxOFSwSCajoGK9KnPWe2cK1bjwndsrCuobpepRqoBS_aBUu2DivckHcb007i_6H9cXHYJ4QQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2736325986</pqid></control><display><type>article</type><title>Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs</title><source>SpringerLink</source><creator>Chakrabarty, Arijit ; Hazra, Rajat Subhra ; Hollander, Frank den ; Sfragara, Matteo</creator><creatorcontrib>Chakrabarty, Arijit ; Hazra, Rajat Subhra ; Hollander, Frank den ; Sfragara, Matteo</creatorcontrib><description>We consider an inhomogeneous Erdős-Rényi random graph
G
N
with vertex set
[
N
]
=
{
1
,
⋯
,
N
}
for which the pair of vertices
i
,
j
∈
[
N
]
,
i
≠
j
, is connected by an edge with probability
r
(
i
N
,
j
N
)
, independently of other pairs of vertices. Here,
r
:
[
0
,
1
]
2
→
(
0
,
1
)
is a symmetric function that plays the role of a reference graphon. Let
λ
N
be the maximal eigenvalue of the adjacency matrix of
G
N
. It is known that
λ
N
/
N
satisfies a large deviation principle as
N
→
∞
. The associated rate function
ψ
r
is given by a variational formula that involves the rate function
I
r
of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of
ψ
r
, specially when the reference graphon is of rank 1.</description><identifier>ISSN: 0894-9840</identifier><identifier>EISSN: 1572-9230</identifier><identifier>DOI: 10.1007/s10959-021-01138-w</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Apexes ; Deviation ; Eigenvalues ; Graph theory ; Mathematics ; Mathematics and Statistics ; Principles ; Probability Theory and Stochastic Processes ; Statistics ; Vertex sets</subject><ispartof>Journal of theoretical probability, 2022-12, Vol.35 (4), p.2413-2441</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-298543493acb9fbce9fab9f38854b5b445e761592240407ee899d785a04943dc3</citedby><cites>FETCH-LOGICAL-c363t-298543493acb9fbce9fab9f38854b5b445e761592240407ee899d785a04943dc3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10959-021-01138-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10959-021-01138-w$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Chakrabarty, Arijit</creatorcontrib><creatorcontrib>Hazra, Rajat Subhra</creatorcontrib><creatorcontrib>Hollander, Frank den</creatorcontrib><creatorcontrib>Sfragara, Matteo</creatorcontrib><title>Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs</title><title>Journal of theoretical probability</title><addtitle>J Theor Probab</addtitle><description>We consider an inhomogeneous Erdős-Rényi random graph
G
N
with vertex set
[
N
]
=
{
1
,
⋯
,
N
}
for which the pair of vertices
i
,
j
∈
[
N
]
,
i
≠
j
, is connected by an edge with probability
r
(
i
N
,
j
N
)
, independently of other pairs of vertices. Here,
r
:
[
0
,
1
]
2
→
(
0
,
1
)
is a symmetric function that plays the role of a reference graphon. Let
λ
N
be the maximal eigenvalue of the adjacency matrix of
G
N
. It is known that
λ
N
/
N
satisfies a large deviation principle as
N
→
∞
. The associated rate function
ψ
r
is given by a variational formula that involves the rate function
I
r
of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of
ψ
r
, specially when the reference graphon is of rank 1.</description><subject>Apexes</subject><subject>Deviation</subject><subject>Eigenvalues</subject><subject>Graph theory</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Principles</subject><subject>Probability Theory and Stochastic Processes</subject><subject>Statistics</subject><subject>Vertex sets</subject><issn>0894-9840</issn><issn>1572-9230</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kEtOwzAQhi0EEqVwAVaWWBvGjzTxEpVSKhWBKlhbTuK0qdI42ElLj8ExOAfiXhiCxI7VjEb_Q_MhdE7hkgLEV56CjCQBRglQyhOyO0ADGsWMSMbhEA0gkYLIRMAxOvF-DQBSAgyQnmu3NPjGbEvdlrbGj66ss7KpDC6sw-3K4Hv9Wm50hSfl0tRbXXUG2wLP6pXd2HAxtvN44vLPN08WH-_1vsQLXed2g6dONyt_io4KXXlz9juH6Pl28jS-I_OH6Wx8PScZH_GWMJlEggvJdZbKIs2MLHRYeBLOaZQKEZl4RCPJmAABsTGJlHmcRBqEFDzP-BBd9LmNsy-d8a1a287VoVKxOFSwSCajoGK9KnPWe2cK1bjwndsrCuobpepRqoBS_aBUu2DivckHcb007i_6H9cXHYJ4QQ</recordid><startdate>20221201</startdate><enddate>20221201</enddate><creator>Chakrabarty, Arijit</creator><creator>Hazra, Rajat Subhra</creator><creator>Hollander, Frank den</creator><creator>Sfragara, Matteo</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20221201</creationdate><title>Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs</title><author>Chakrabarty, Arijit ; Hazra, Rajat Subhra ; Hollander, Frank den ; Sfragara, Matteo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-298543493acb9fbce9fab9f38854b5b445e761592240407ee899d785a04943dc3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Apexes</topic><topic>Deviation</topic><topic>Eigenvalues</topic><topic>Graph theory</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Principles</topic><topic>Probability Theory and Stochastic Processes</topic><topic>Statistics</topic><topic>Vertex sets</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chakrabarty, Arijit</creatorcontrib><creatorcontrib>Hazra, Rajat Subhra</creatorcontrib><creatorcontrib>Hollander, Frank den</creatorcontrib><creatorcontrib>Sfragara, Matteo</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of theoretical probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chakrabarty, Arijit</au><au>Hazra, Rajat Subhra</au><au>Hollander, Frank den</au><au>Sfragara, Matteo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs</atitle><jtitle>Journal of theoretical probability</jtitle><stitle>J Theor Probab</stitle><date>2022-12-01</date><risdate>2022</risdate><volume>35</volume><issue>4</issue><spage>2413</spage><epage>2441</epage><pages>2413-2441</pages><issn>0894-9840</issn><eissn>1572-9230</eissn><abstract>We consider an inhomogeneous Erdős-Rényi random graph
G
N
with vertex set
[
N
]
=
{
1
,
⋯
,
N
}
for which the pair of vertices
i
,
j
∈
[
N
]
,
i
≠
j
, is connected by an edge with probability
r
(
i
N
,
j
N
)
, independently of other pairs of vertices. Here,
r
:
[
0
,
1
]
2
→
(
0
,
1
)
is a symmetric function that plays the role of a reference graphon. Let
λ
N
be the maximal eigenvalue of the adjacency matrix of
G
N
. It is known that
λ
N
/
N
satisfies a large deviation principle as
N
→
∞
. The associated rate function
ψ
r
is given by a variational formula that involves the rate function
I
r
of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of
ψ
r
, specially when the reference graphon is of rank 1.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10959-021-01138-w</doi><tpages>29</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0894-9840 |
| ispartof | Journal of theoretical probability, 2022-12, Vol.35 (4), p.2413-2441 |
| issn | 0894-9840 1572-9230 |
| language | eng |
| recordid | cdi_proquest_journals_2736325986 |
| source | SpringerLink |
| subjects | Apexes Deviation Eigenvalues Graph theory Mathematics Mathematics and Statistics Principles Probability Theory and Stochastic Processes Statistics Vertex sets |
| title | Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs |
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