On the expected maximum degree of Gabriel and Yao graphs

Advances in Applied Probability

Volumn 41, Issue 4, 2009, Pages 1123-1140

  Devroye, Luc ^a   Gudmundsson, Joachim ^b   Morin, Pat ^c  

^a MCGILL UNIVERSITY   (Canada)

^b UNIVERSITY OF SYDNEY   (Australia)

^c CARLETON UNIVERSITY   (Canada)

Author keywords

Gabriel graph; Maximum degree; Random geometric graph; Yao graph

Indexed keywords

GABRIEL GRAPH; MAXIMUM DEGREE; RANDOM GEOMETRIC GRAPHS; UNIT SQUARES; YAO GRAPHS;

TELECOMMUNICATION NETWORKS; WIRELESS AD HOC NETWORKS;

AD HOC NETWORKS;

EID: 77949308784     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1261669589     Document Type: Article

References (20)

1. Asmussen, S. and Hering, H. (1983). Branching Processes (Progress Prob. Statist. 3), Birkhäuser, Boston, MA.
2. Athreya, K. B. and Karlin, S. (1971). Branching processes with random environments. II. Limit theorems. Ann. Math. Statist. 42, 1843-1858.
3. Athreya, K. B. and Karlin, S. (1971). On branching processes with random environments. I. Extinction probabilities. Ann. Math. Statist. 42, 1499-1520.
4. Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover Publications, Mineola, NY.
5. Athreya, K. B. and Kang, H.-J. (1998). Some limit theorems for positive recurrent branching Markov chains. I. Adv. Appl. Prob. 30, 693-710.
6. Athreya, K. and Kang, H.-J. (1998). Some limit theorems for positive recurrent branching Markov chains. II. Adv. Appl. Prob. 30, 711-722.
7. Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Criticality for branching processes in a random environment. Ann. Prob. 33, 645-673.
8. Bansaye, V. (2008). Proliferating parasites in dividing cells: Kimmel's branching model revisited. Ann. Appl. Prob. 18, 967-996.
9. Benjamini, I. and Peres, Y. (1994). Markov chains indexed by trees. Ann. Prob. 22, 219-243.
10. Feller, W. (1966). An Introduction to Probability Theory and its Applications, Vol. II. JohnWiley, NewYork.
11. Geiger, J., Kersting, G. and Vatutin, V. A. (2003). Limit theorems for subcritical branching processes in a random environment. Ann. Inst. H. Poincaré Prob. Statist. 39, 593-620.
12. Guyon, J. (2007). Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Prob. 17, 1538-1569.
13. Key, E. S. (1987). Limiting distributions and regeneration times for multitype branching processes with immigration in a random environment. Ann. Prob. 15, 344-353.
14. Kendall, D. G. (1959). Unitary dilations of Markov transition operators, and the corresponding integral representations for transition-probability matrices. In Probability and Statistics: The Harald Cramér Volume, JohnWiley, NewYork, pp. 139-161.
15. Kimmel, M. (1997). Quasistationarity in a branching model of division-within-division. In Classical and Modern Branching Processes (Minneapolis, MN, 1994; IMA Vol. Math. Appl. 84), Springer, NewYork, pp. 157-164.
16. Kozlov, M. V. (1976). The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment. Theory Prob. Appl. 21, 813-825.
17. Lyons, R. and Peres, Y. (2005). Probability onTrees and Networks.Available at http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.
18. Roitershtein, A. (2007).A note on multitype branching processes with immigration in a random environment. Ann. Prob. 35, 1573-1592.
19. Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814-827.
20. Stewart, E. J., Madden, R., Paul, G. and Taddei, F. (2005). Aging and death in an organism that reproduces by morphologically symmetric division. PLoS Biol. 3, e45.