Large deviation principles for empirical measures of colored random graphs

Doku-Amponsah, Kwabena; Mörters, Peter

doi:10.1214/09-aap647

Cited by 18 publications

(21 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From this large deviation results we obtain LDPs for graph quantities such as number of edges per vertex, the degree distribution and the proportion of isolated vertices of geometric random graphs in the intermediate case. Our results are similar to those in O’Connell ( 1998 ), Biggins and Penman ( 2009 ), Doku-Amponsah and Mörters ( 2010 ), Doku-Amponsah ( 2006 , ( 2014 ), Bordenave and Caputo ( 2015 ) and Mukherjee ( 2014 ) for the Erdö–Renyi graph except that the rate functions of the LDPs in our current setting is bigger as a result of the effect of the geometric in the model.…”

Section: Introductionsupporting

confidence: 88%

“…In this article we study the coloured geometric random graph CGRG, where n points or vertices or nodes are picked uniformly at random in colours or spins are assigned independently from a finite alphabet and any two points with colours distance at most apart are connected. This random graph models, which has the geometric random graph (see Penrose 2003 ) as special case, has been suggested by Cannings and Penman ( 2003 ) as a possible extension to the coloured random graph studied in Biggins and Penman ( 2009 ), Doku-Amponsah and Mörters ( 2010 ), Doku-Amponsah ( 2006 ), Bordenave and Caputo ( 2015 ), Mukherjee ( 2014 ) and Doku-Amponsah ( 2014 ).…”

Section: Introductionmentioning

confidence: 89%

“…Several large deviation results about the coloured random graphs and hence Erdős–Rényi graph have been established recently. See O’Connell ( 1998 ), Biggins and Penman ( 2009 ), Doku-Amponsah and Mörters ( 2010 ), Doku-Amponsah ( 2006 , 2014 ), Bordenave and Caputo ( 2015 ), Mukherjee ( 2014 ).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Joint large deviation result for empirical measures of the coloured random geometric graphs

Doku-Amponsah

2016

SpringerPlus

Self Cite

View full text Add to dashboard Cite

We prove joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate coloured random geometric graph models on n points picked uniformly in a d-dimensional torus of a unit circumference. From this result we obtain large deviation principles for the number of edges per vertex, the degree distribution and the proportion of isolated vertices for the near intermediate random geometric graph models.

show abstract

Section: Introductionsupporting

confidence: 88%

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

Joint large deviation result for empirical measures of the coloured random geometric graphs

Doku-Amponsah

2016

SpringerPlus

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the next subsection, we review the symbolled random graph model as in (Doku-Amponsah et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

Exponential Approximation, Method of Types for Empirical Neighbourhood Distributions of Random Graphs by Random Allocations

Doku-Amponsah¹

2014

IJSP

View full text Add to dashboard Cite

In this article we find exponential good approximation of the empirical neigbourhood distribution of symbolled random graphs conditioned to a given empirical symbol distribution and empirical pair distribution. Using this approximation we shorten or simplify the proof of (Doku-Amponsah & Morters, 2010, Theorem 2.5); the large deviation principle (LDP) for empirical neigbourhood distribution of symbolled random graphs. We also show that the LDP for the empirical degree measure of the classical Erdős-Rényi graph is a special case of (Doku-Amponsah & Moerters, 2010, Theorem 2.5). From the LDP for the empirical degree measure, we derive an LDP for the the proportion of isolated vertices in the classical Erdős-Rényi graph.

show abstract

“…i.e. [15], [2], [9], [3], [7], [5]. The main technique use to prove our main result is rooted in spectral potential theory.…”

Section: Introductionmentioning

confidence: 93%

Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

Doku-Amponsah¹

2017

Journal of Mathematics and Statistics

Self Cite

View full text Add to dashboard Cite

For a finite typed graph on n nodes and with type law µ, we define the socalled spectral potential ρ λ ( ·, µ), of the graph.From the ρ λ ( ·, µ) we obtain Kullback action or the deviation function, H λ (π ν), with respect to an empirical pair measure, π, as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure π and empirical type measure µ, we prove a Local large deviation principle (LLDP), with rate function H λ (π ν) and speed n. We deduce from this LLDP, a full conditional large deviation principle and a weak variant of the classical McMillian Theorem for the typed random graphs. Given the typical empirical link measure, λµ ⊗ µ, the number of typed random graphs is approximately equal e n λµ⊗µ H λµ⊗µ/ λµ⊗µ . Note that we do not require any topological restrictions on the space of finite graphs for these LLDPs. Now, some Large deviation principles(LDPs) and Coding Theorems exists for networked data structures modelled as the typed random graph (TRG) models.See,[15], [2], [9], [3],[7], [5] and the reference therein. O'Connor [15] proved large deviation principle (LDP) for the relative size of the largest connected component in the random graph with small edge probability. Biggin and Penman [2] have found LDPs for the number of edges of TRG models, where the link probabilities are independent of the number of nodes, using the Garten-Ellis Theorem, see [12].[9] proved LDPs for the empirical measures of the TRG where the link probabilities are dependent on the number of nodes of the graph. In [3], LDP for the empirical neighborhood distribution in sparse random graphs was proved using a technique that relies on the typical behavior within the framework of the local weak convergence of finite graph sequences. Asymptotic Equipartition Properties including the Lossy version have been found in [7] and [5], by the techniques of exponential change of measure and random allocation, respectively. Mathematics Subject Classification : 94A15, 94A24, 60F10, 05C80

show abstract

Large deviation principles for empirical measures of colored random graphs

Cited by 18 publications

References 19 publications

Joint large deviation result for empirical measures of the coloured random geometric graphs

Joint large deviation result for empirical measures of the coloured random geometric graphs

Exponential Approximation, Method of Types for Empirical Neighbourhood Distributions of Random Graphs by Random Allocations

Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

Contact Info

Product

Resources

About