Johan S. H. van Leeuwaarden

Bio: Johan S. H. van Leeuwaarden is an academic researcher from Tilburg University. The author has contributed to research in topics: Random graph & Queue. The author has an hindex of 20, co-authored 138 publications receiving 1476 citations. Previous affiliations of Johan S. H. van Leeuwaarden include Eindhoven University of Technology.

Epidemic spreading on complex networks with community structures

Abstract: Many real-world networks display a community structure. We study two random graph models that create a network with similar community structure as a given network. One model preserves the exact community structure of the original network, while the other model only preserves the set of communities and the vertex degrees. These models show that community structure is an important determinant of the behavior of percolation processes on networks, such as information diffusion or virus spreading: the community structure can both enforce as well as inhibit diffusion processes. Our models further show that it is the mesoscopic set of communities that matters. The exact internal structures of communities barely influence the behavior of percolation processes across networks. This insensitivity is likely due to the relative denseness of the communities.

Novel scaling limits for critical inhomogeneous random graphs

Abstract: We flnd scaling limits for the sizes of the largest components at criticality for the rank-1 inho- mogeneous random graphs with power-law degrees with exponent ?. We investigate the case where ? 2 (3;4), so that the degrees have flnite variance but inflnite third moment. The sizes of the largest clusters, rescaled by n i(?i2)=(?i1) , converge to hitting times of a 'thinned' Levy process. This process is intimately connected to the general multiplicative coalescents studied in (1) and (2). In particular, we use the results in (2) to show that, when interpreting the location ‚ inside the critical window as time, the limiting process is a multiplicative process with difiusion constant 0 and the entrance boundary describing the size of relative components in the ‚ ! i1 regime proportional to i i i1=(?i1) ¢ i‚1 . A crucial ingredient is the identiflcation of the scaling of the largest connected components in the barely subcritical regime. Our results should be contrasted to the case where the degree exponent ? satisfles ? > 4, so that the third moment is flnite. There, instead, we see that the sizes of the components rescaled by n i2=3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in (1) for the Erd}os- Renyi random graph and extended to the present setting in (3, 4). The limit again is a multiplicative coalescent, the only difierence with the limit for ? 2 (3;4) being the initial state, corresponding to the barely subcritical regime.

Scaling limits for critical inhomogeneous random graphs with finite third moments

Abstract: We identify the scaling limit for the sizes of the largest components at criticality for inhomogeneous random graphs with weights that have finite third moments. We show that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, which extends results of Aldous (1997) for the critical behavior of Erdos-Renyi random graphs. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme initiated in van der Hofstad (2009) to study the near-critical behavior in inhomogeneous random graphs of so-called rank-1.

Scaling limits for critical inhomogeneous random graphs with finite third moments

Abstract: We identify the scaling limits for the sizes of the largest components at criticality for inhomogeneous random graphs when the degree exponent $\tau$ satisfies $\tau>4$. We see that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, extending results of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme to study the critical behavior in inhomogeneous random graphs of so-called rank-1 initiated in \cite{Hofs09a}.

Scalable load balancing in networked systems: A survey of recent advances.

Abstract: The basic load balancing scenario involves a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. We discuss recent advances on scalable load balancing schemes which provide favorable delay performance when $N$ grows large, and yet only require minimal implementation overhead. Join-the-Shortest-Queue (JSQ) yields vanishing delays as $N$ grows large, as in a centralized queueing arrangement, but involves a prohibitive communication burden. In contrast, power-of-$d$ or JSQ($d$) schemes that assign an incoming task to a server with the shortest queue among $d$ servers selected uniformly at random require little communication, but lead to constant delays. In order to examine this fundamental trade-off between delay performance and implementation overhead, we consider JSQ($d(N)$) schemes where the diversity parameter $d(N)$ depends on $N$ and investigate what growth rate of $d(N)$ is required to asymptotically match the optimal JSQ performance on fluid and diffusion scale. Stochastic coupling techniques and stochastic-process limits play an instrumental role in establishing the asymptotic optimality. We demonstrate how this methodology carries over to infinite-server settings, finite buffers, multiple dispatchers, servers arranged on graph topologies, and token-based load balancing including the popular Join-the-Idle-Queue (JIQ) scheme. In this way we provide a broad overview of the many recent advances in the field. This survey extends the short review presented at ICM 2018 (arXiv:1712.08555).

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

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

Epidemic processes in complex networks

Abstract: Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.