# Birth control for giants

## Abstract

The standard Erdős-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory.

We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v 1, v 2}, {v 3, v 4} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t c so that S(t) → ∞ as t approaches t c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < t c . Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > t c we show the existence of a giant component, so that t = t c may be fairly considered a phase transition.

Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component.

This is a preview of subscription content, access via your institution.

## References

1. [1]

N. Alon and J. Spencer: The Probabilistic Method, 2nd ed., John Wiley, 2000.

2. [2]

T. Bohman and A. Frieze: Avoiding a giant component, Random Structures & Algorithms 19 (2001), 75–85.

3. [3]

T. Bohman, A. Frieze and N. C. Wormald: Avoidance of a giant component in half the edge set of a random graph, Random Structures & Algorithms 25(4) (2004), 432–449.

4. [4]

T. Bohman and D. Kravitz: Creating a giant component, Combinatorics, Probability & Computing 15(4) (2006), 489–511.

5. [5]

A. Flaxman, D. Garmanik and G. B. Sorkin: Embracing the giant component, pp. 69–79 in Latin 2004: Theoretical Informatics, M. Farach-Coltin (ed), Lecture Notes in Computer Science 2976, Springer, 2004.

6. [6]

W. Hurewicz: Lectures on Ordinary Differential Equations, M.I.T. Press, Cambridge, Massachusetts, 1958.

7. [7]

N. C. Wormald: The differential equation method for random graph processes and greedy algorithms, in Lectures on Approximation and Randomized Algorithms, M. Karoński and H. J. Prömel (eds), pp. 73–155, PWN, Warsaw, 1999.

8. [8]

N. C. Wormald: Analysis of greedy algorithms on graphs with bounded degrees, Discrete Mathematics 273 (2003), 235–260.

9. [9]

N. C. Wormald: Random graphs and asymptotics; Section 8.2 in Handbook of Graph Theory, J. L. Gross and J. Yellen (eds), pp. 817–836, CRC, Boca Raton, 2004.

## Author information

Authors

### Corresponding author

Correspondence to Joel Spencer.

Research supported by the Australian Research Council, the Canada Research Chairs Program and NSERC. Research partly carried out while the author was at the Department of Mathematics and Statistics, University of Melbourne.

## Rights and permissions

Reprints and Permissions

Spencer, J., Wormald, N. Birth control for giants. Combinatorica 27, 587–628 (2007). https://doi.org/10.1007/s00493-007-2163-2