On the evolution of topology in dynamic clique complexes

Abstract We consider a time varying analogue of the Erdős–Rényi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous-time Markov chains. Our main result is that when the edge inclusion probability is of the form p=n α, where n is the number of vertices and α∈(-1/k, -1/(k + 1)), then the process of the normalised kth Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n→∞.

[1]  Andrea E. F. Clementi,et al.  Distributed community detection in dynamic graphs , 2013, Theor. Comput. Sci..

[2]  R. Adler,et al.  Random geometric complexes in the thermodynamic regime , 2014, Probability Theory and Related Fields.

[3]  Elizabeth S. Meckes,et al.  Erratum: Limit theorems for Betti numbers of random simplicial complexes , 2010, 1009.4130.

[4]  Matthew Kahle.  Topology of random simplicial complexes: a survey , 2013, 1301.7165.

[5]  J. Norris.  Appendix: probability and measure , 1997 .

[6]  Matthew Kahle,et al.  Sharp vanishing thresholds for cohomology of random flag complexes , 2012, 1207.0149.

[7]  Matthew Kahle,et al.  Random Geometric Complexes , 2009, Discret. Comput. Geom..

[8]  Ram Ramanathan,et al.  Modeling and Analysis of Time-Varying Graphs , 2010, ArXiv.

[9]  Elizabeth S. Meckes,et al.  Limit theorems for Betti numbers of random simplicial complexes , 2010 .

[10]  Andrea E. F. Clementi,et al.  Flooding Time of Edge-Markovian Evolving Graphs , 2010, SIAM J. Discret. Math..

[11]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[12]  David L. Hicks,et al.  Mathematical Methods in Counterterrorism , 2009 .

[13]  R. Meshulam,et al.  Homological connectivity of random k‐dimensional complexes , 2006, Random Struct. Algorithms.

[14]  Matthew Kahle,et al.  Topology of random clique complexes , 2006, Discret. Math..

[15]  Matthew Kahle,et al.  The fundamental group of random 2-complexes , 2007, 0711.2704.

[16]  Matthew Kahle.  The neighborhood complex of a random graph , 2005, J. Comb. Theory, Ser. A.

[17]  Nathan Linial,et al.  Homological Connectivity Of Random 2-Complexes , 2006, Comb..

[18]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[19]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[20]  Cauligi S. Raghavendra,et al.  Spray and wait: an efficient routing scheme for intermittently connected mobile networks , 2005, WDTN '05.

[21]  Cecilia Mascolo,et al.  Adaptive routing for intermittently connected mobile ad hoc networks , 2005, Sixth IEEE International Symposium on a World of Wireless Mobile and Multimedia Networks.

[22]  Allan Gut,et al.  An intermediate course in probability , 1995 .

[23]  Andrzej Rucinski,et al.  A central limit theorem for decomposable random variables with applications to random graphs , 1989, J. Comb. Theory B.

[24]  M. Rosenblatt,et al.  A MARKOVIAN FUNCTION OF A MARKOV CHAIN , 1958 .

[25]  R. Ho.  Algebraic Topology , 2022 .