Some Models of Reproducing Graphs: II Age Capped Vertices

ABSTRACT

In the prequel to this paper we introduced eight reproducing graph models. The simple idea behind these models is that graphs grow because the vertices within reproduce. In this paper we make our models more realistic by adding the idea that vertices have a finite life span. The resulting models capture aspects of systems like social networks and biological networks where reproducing entities die after some amount of time. In the 1940’s Leslie introduced a population model where the reproduction and survival rates of individuals depends upon their ages. Our models may be viewed as extensions of Leslie’s model-adding the idea of network joining the reproducing individuals. By exploiting connections with Leslie’s model we are to describe how many aspects of graphs evolve under our systems. Many features such as degree distributions, number of edges and distance structure are described by the golden ratio or its higher order generalisations.

In the prequel to this paper we introduced eight reproducing graph models. The simple idea behind these models is that graphs grow because the vertices within reproduce. In this paper we make our models more realistic by adding the idea that vertices have a finite life span. The resulting models capture aspects of systems like social networks and biological networks where reproducing entities die after some amount of time. In the 1940’s Leslie introduced a population model where the reproduction and survival rates of individuals depends upon their ages. Our models may be viewed as extensions of Leslie’s model-adding the idea of network joining the reproducing individuals. By exploiting connections with Leslie’s model we are to describe how many aspects of graphs evolve under our systems. Many features such as degree distributions, number of edges and distance structure are described by the golden ratio or its higher order generalisations.

Cite this paper

nullR. Southwell and C. Cannings, "Some Models of Reproducing Graphs: II Age Capped Vertices,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 251-259. doi: 10.4236/am.2010.14031.

nullR. Southwell and C. Cannings, "Some Models of Reproducing Graphs: II Age Capped Vertices,"

References

[1] P. Erd?s and A. Rényi, “On Random Graphs. I,” Publicationes Mathematicae, Vol. 6, 1959, pp. 290-297.

[2] G. U. Yule, “A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F. R. S.,” Philosophical Transactions of the Royal Society of London, B, Vol. 213, 1925, pp. 21-87.

[3] D. J. Watts and S. H. Strogatz, “Collective Dynamics of ‘Small-World’ Networks,” Nature, Vol. 393, No. 6684, 1998, pp. 440-442.

[4] R. Southwell and C. Cannings, “Games on Graphs that Grow Deterministically,” Proceedings of International Conference on Game Theory for Networks GameNets ‘09, Istanbul, Turkey, 2009, pp. 347-356.

[5] R. Southwell and C. Cannings, “Some Models of Reproducing Graphs: I Pure Reproduction,” Journal of Applied Mathematics, Vol. 1, No. 3, 2010, pp. 137-145.

[6] A. Bonato, N. Hadi, P. Horn, P. Praalat and C. Wand, “Models of On-Line Social Networks,” To appear in Internet Mathematics, 2010.

[7] P. H. Leslie, “The Use of Matrices in Certain Population Mathematics,” Biometrika, Vol. 30, 1945, pp. 183-212.

[8] P. H. Leslie, “Some Further Notes on the Use of Matrices in Population Mathematics,” Biometrika, Vol. 35, No. 3-4, 1948, pp. 213-245.

[9] T. D. Noe, “Primes in Fibonacci n-step and Lucas n-step Sequences,” Journal of Integer Sequences, Vol. 8, 2005, pp. 1-12

[1] P. Erd?s and A. Rényi, “On Random Graphs. I,” Publicationes Mathematicae, Vol. 6, 1959, pp. 290-297.

[2] G. U. Yule, “A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F. R. S.,” Philosophical Transactions of the Royal Society of London, B, Vol. 213, 1925, pp. 21-87.

[3] D. J. Watts and S. H. Strogatz, “Collective Dynamics of ‘Small-World’ Networks,” Nature, Vol. 393, No. 6684, 1998, pp. 440-442.

[4] R. Southwell and C. Cannings, “Games on Graphs that Grow Deterministically,” Proceedings of International Conference on Game Theory for Networks GameNets ‘09, Istanbul, Turkey, 2009, pp. 347-356.

[5] R. Southwell and C. Cannings, “Some Models of Reproducing Graphs: I Pure Reproduction,” Journal of Applied Mathematics, Vol. 1, No. 3, 2010, pp. 137-145.

[6] A. Bonato, N. Hadi, P. Horn, P. Praalat and C. Wand, “Models of On-Line Social Networks,” To appear in Internet Mathematics, 2010.

[7] P. H. Leslie, “The Use of Matrices in Certain Population Mathematics,” Biometrika, Vol. 30, 1945, pp. 183-212.

[8] P. H. Leslie, “Some Further Notes on the Use of Matrices in Population Mathematics,” Biometrika, Vol. 35, No. 3-4, 1948, pp. 213-245.

[9] T. D. Noe, “Primes in Fibonacci n-step and Lucas n-step Sequences,” Journal of Integer Sequences, Vol. 8, 2005, pp. 1-12