Sanchayan Sen

Bio: Sanchayan Sen is an academic researcher from Indian Institute of Science. The author has contributed to research in topics: Random graph & Scaling limit. The author has an hindex of 12, co-authored 33 publications receiving 447 citations. Previous affiliations of Sanchayan Sen include Courant Institute of Mathematical Sciences & Eindhoven University of Technology.

Critical window for the configuration model: finite third moment degrees

Abstract: We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Renyi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.

Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph

Abstract: Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time tc which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erd\H{o}s-R\'enyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n2/3) and (b) the structure of components (rescaled by n-1/3) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical Erd\H{o}s-R\'enyi random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions are the same as the Erd\H{o}s-R\'enyi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erd\H{o}s-R\'enyi random graph. As a by product we also get results for component sizes at criticality for a general class of inhomogeneous random graphs.

Continuum limit of critical inhomogeneous random graphs

Abstract: The last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real-world systems. In this study we analyze the maximal components at criticality of one famous class of such models, the rank-one inhomogeneous random graph model (Norros and Reittu, Adv Appl Probab 38(1):59–75, 2006; Bollobas et al., Random Struct Algorithms 31(1):3–122, 2007, Section 16.4). Viewing these components as measured random metric spaces, under finite moment assumptions for the weight distribution, we show that the components in the critical scaling window with distances scaled by $$n^{-1/3}$$ converge in the Gromov–Haussdorf–Prokhorov metric to rescaled versions of the limit objects identified for the Erdős–Renyi random graph components at criticality in Addario-Berry et al. (Probab. Theory Related Fields, 152(3–4):367–406, 2012). A key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called $$\mathbf {p}$$ -trees (Camarri and Pitman, Electron. J. Probab 5(2):1–18, 2000; Aldous et al., Probab Theory Related Fields 129(2):182–218, 2004). This is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics (Braunstein et al., Phys Rev Lett 91(16):168701, 2003) for such graph models. By asymptotic equivalence (Janson, Random Struct Algorithms 36(1):26–45, 2010), the same results are true for the Chung–Lu model (Chung and Lu, Proc Natl Acad Sci 99(25):15879–15882, 2002; Chung and Lu, Ann Combin 6(2):125–145, 2002; Chung and Lu, Complex graphs and networks, 2006) and the Britton–Deijfen–Martin–Lof model (Britton et al., J Stat Phys 124(6):1377–1397, 2006). A crucial ingredient of the proof of independent interest are tail bounds for the height of $$\mathbf {p}$$ -trees. The techniques developed in this paper form the main technical bedrock for the general program developed in Bhamidi et al. (Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős–Renyi random graph. arXiv preprint, 2014) for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels (Bollobas et al., Random Struct Algorithms 31(1):3–122, 2007).

Heavy-tailed configuration models at criticality

Abstract: Nous etudions le comportement critique des tailles des composantes du modele de configuration lorsque la queue de la loi du degre d’un sommet choisi uniformement au hasard est une fonction a variation reguliere d’exposant $\tau -1$, ou $\tau \in (3,4)$ Nous montrons que les tailles des composantes sont d’ordre $n^{(\tau -2)/(\tau -1)}L(n)^{-1}$ ou $L(\cdot )$ est une fonction a variation lente Nous montrons egalement que les tailles des composantes, une fois ordonnees et remises a l’echelle, convergent en loi vers les longueurs ordonnees des excursions d’un processus de Levy rarefie Ceci montre que les limites d’echelle des tailles des composantes pour ces modeles de configuration a queue lourde sont dans une classe d’universalite differente des graphes aleatoires d’Erdos–Renyi De plus nous montrons que le vecteur de ces tailles de composantes remises a l’echelle et de leurs exces respectifs convergent en loi pour une topologie forte Notre approche resout une conjecture de Joseph (Ann Appl Probab 24 (2014) 2560–2594) sur les limites d’echelle de graphes simples uniformes a degres iid dans la fenetre critique, et met en lumiere les relations entre les limites d’echelle obtenues par Joseph et celles considerees dans cet article, qui se revelent tres differentes Par ailleurs, nous utilisons un modele de percolation pour etudier l’evolution des tailles des composantes et des aretes en exces a l’interieur de la fenetre critique, dont nous montrons qu’elle converge au sens des marginales de dimension finie vers le coalescent multiplicatif augmente introduit par Bhamidi et al (Probab Theory Related Fields 160 (2014) 733–796) Les resultats principaux de cet article sont montres sous des hypotheses assez generales sur les degres des sommets, et nous discutons des situations deja considerees ou ces hypotheses sont verifiees

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Random Graphs and Complex Networks

Abstract: This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

Cern european organization for nuclear research

Abstract: Today a new standard, the Gigabyte System Network (GSN), is emerging for computer networking using fast, full-duplex connections with an effective bandwidth of 800 MByte/s in each direction. This paper describes GSN, including the switch structure and its very low latency protocol called Scheduled Transfer (ST). An overview of available components will be given, together with some examples of how this standard can be applied in high end computing and in future high-energy physics data acquisition systems.

Progress in High-Dimensional Percolation and Random Graphs

Abstract: Preface -- 1. Introduction and motivation -- 2. Fixing ideas: Percolation on a tree and branching random walk -- 3. Uniqueness of the phase transition -- 4. Critical exponents and the triangle condition -- 5. Proof of triangle condition -- 6. The derivation of the lace expansion via inclusion-exclusion -- 7. Diagrammatic estimates for the lace expansion -- 8. Bootstrap analysis of the lace expansion -- 9. Proof that δ = 2 and β = 1 under the triangle condition -- 10. The non-backtracking lace expansion -- 11. Further critical exponents -- 12. Kesten's incipient infinite cluster -- 13. Finite-size scaling and random graphs -- 14. Random walks on percolation clusters -- 15. Related results -- 16. Further open problems -- Bibliography.