Poisson approximation for cycles in the generalised random graph
Abstract: The generalised random graph contains $n$ vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments and study the point process $\mathcal{C}_n$ on ${3,4,\dots}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When $\mathcal{C}_n$ is evaluated on a bounded set $A$, we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets $A$ at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and the longest cycle when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung-Lu model and the Norros-Reittu model.
- O. Angel, R. van der Hofstad and C. Holmgren. Limit laws for self-loops and multiple edges in the configuration model. Ann. Inst. Henri Poincaré Probab. Stat., 55(3):1509–1530, (2019).
- R. Arratia, L. Goldstein and L. Gordon. Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab., 17(1):9–25, (1989).
- S. Bhamidi, R. van der Hofstad and J. S. H. van Leeuwaarden. Scaling limits for critical inhomogeneous random graphs with finite third moments. Electron. J. Probab., 15(54):1682–1703, (2010).
- S. Bhamidi, R. van der Hofstad and J. S. H. van Leeuwaarden. Novel scaling limits for critical inhomogeneous random graphs. Ann. Probab., 40(6):2299–2361, (2012).
- Loops of any size and hamilton cycles in random scale-free networks. JSTAT, (2005).
- S. G. Bobkov, M. A. Danshina and V. V. Ulyanov. Rate of convergence to the Poisson law of the numbers of cycles in the generalized random graphs. Operator Th. Harmonic Anal., OTHA 2020. Springer Proc. Math. Stat., 358:109–133. Springer, Cham, (2021).
- B. Bollobás. Random graphs, Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge, second edition, (2001).
- B. Bollobás, S. Janson and O. Riordan. The phase transition in inhomogeneous random graphs. Random Struct. Alg., 31(1):3–122, (2007).
- T. Britton, M. Deijfen and A. Martin-Löf. Generating simple random graphs with prescribed degree distribution. J. Stat. Phys., 124(6):1377–1397, (2006).
- The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA, 99(25):15879–15882, (2002).
- H. van den Esker, R. van der Hofstad and G. Hooghiemstra. Universality for the distance in finite variance random graphs. J. Stat. Phys., 133(1):169–202, (2008).
- Counting triangles in power-law uniform random graphs. Electron. J. Combin., 27(3):P3.19, (2020).
- R. van der Hofstad. Random graphs and complex networks. Vol. 1, volume [43] of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, (2017).
- R. van der Hofstad, J. S. H. van Leeuwaarden and C. Stegehuis. Optimal subgraph structures in scale-free configuration models. Ann. Appl. Probab., 31(2):501–537, (2021).
- S. Janson. The largest component in a subcritical random graph with a power law degree distribution. Ann. Appl. Probab., 18(4):1651–1668, (2008).
- S. Janson. Asymptotic equivalence and contiguity of some random graphs. Random Struct. Alg., 36(1):26–45, (2010).
- A. J. E. M. Janssen, J. S. H. van Leeuwaarden and S. Shneer. Counting cliques and cycles in scale-free inhomogeneous random graphs. J. Stat. Phys., 175(1):161–184, (2019).
- O. Kallenberg. Random measures, theory and applications, Probability Theory and Stochastic Modelling, vol. 77. Springer, Cham, (2017).
- Limit laws for the number of triangles in the generalized random graphs with random node weights. Stat. & Probab. Letters, 161:108733, (2020).
- Q. Liu, Z. Dong and E. Wang. Moment-based spectral analysis of large-scale generalized random graphs. IEEE Access, 5:9453–9463, (2017).
- On a conditionally Poissonian graph process. Adv. in Appl. Probab., 38(1):59–75, (2006).
- N. Ross. Fundamentals of Stein’s method Probab. Surveys, 8 210–293, (2011).
- R. B. Schinazi. Probability with statistical applications. Birkhäuser/Springer, Cham, third edition, (2022).
- Scale-free networks well done. Phys. Rev. Res., 1:033034, (2019).
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