Susceptibility in inhomogeneous random graphs

We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives a way of determining the critical point by solving certain linear equations. Here we relate the susceptibility of suitable random graphs to a quantity associated to the corresponding branching process, and study both quantities in various natural examples.Comment: 51 page

Susceptibility of random graphs with given vertex degrees

We study the susceptibility, i.e., the mean cluster size, in random graphs with given vertex degrees. We show, under weak assumptions, that the susceptibility converges to the expected cluster size in the corresponding branching process. In the supercritical case, a corresponding result holds for the modified susceptibility ignoring the giant component and the expected size of a finite cluster in the branching process; this is proved using a duality theorem. The critical behaviour is studied. Examples are given where the critical exponents differ on the subcritical and supercritical sides.Comment: 25 page

Duality in inhomogeneous random graphs, and the cut metric

The classical random graph model $G(n,\lambda/n)$ satisfies a `duality principle', in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric.Comment: 13 page

Random minimum spanning tree and dense graph limits

A theorem of Frieze from 1985 asserts that the total weight of the minimum spanning tree of the complete graph $K_n$ whose edges get independent weights from the distribution $UNIFORM[0,1]$ converges to Ap\'ery's constant in probability, as $n\to\infty$. We generalize this result to sequences of graphs $G_n$ that converge to a graphon $W$. Further, we allow the weights of the edges to be drawn from different distributions (subject to moderate conditions). The limiting total weight $\kappa(W)$ of the minimum spanning tree is expressed in terms of a certain branching process defined on $W$, which was studied previously by Bollob\'as, Janson and Riordan in connection with the giant component in inhomogeneous random graphs.Comment: 20 pages, 1 figur

The giant in random graphs is almost local

Local convergence techniques have become a key methodology to study random graphs in sparse settings where the average degree remains bounded. However, many random graph properties do not directly converge when the random graph converges locally. A notable, and important, random graph property that does not follow from local convergence is the size and uniqueness of the giant component. We provide a simple criterion that guarantees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant are also described by the local limit. We apply this novel method to the configuration model as a proof of concept, reproving a result that is well-established. As a side result this proof, we show that the proof also implies the small-world nature of the configuration model.Comment: 23 page

Mesoscopic scales in hierarchical configuration models

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. The connections between the communities are formed as in a configuration model. We study the component sizes of the hierarchical configuration model at criticality when the inter-community degrees have a finite third moment. We find the conditions on the community sizes such that the critical component sizes of the HCM behave similarly as in the configuration model. Furthermore, we study critical bond percolation on the HCM. We show that the ordered components of a critical HCM on $N$ vertices are of sizes $O(N^{2/3})$. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift, as for the scaling limit for the configuration model under a finite third moment condition

ULTRA-FAST AND MEMORY-EFFICIENT LOOKUPS FOR CLOUD, NETWORKED SYSTEMS, AND MASSIVE DATA MANAGEMENT

Systems that process big data (e.g., high-traffic networks and large-scale storage) prefer data structures and algorithms with small memory and fast processing speed. Efficient and fast algorithms play an essential role in system design, despite the improvement of hardware. This dissertation is organized around a novel algorithm called Othello Hashing. Othello Hashing supports ultra-fast and memory-efficient key-value lookup, and it fits the requirements of the core algorithms of many large-scale systems and big data applications. Using Othello hashing, combined with domain expertise in cloud, computer networks, big data, and bioinformatics, I developed the following applications that resolve several major challenges in the area. Concise: Forwarding Information Base. A Forwarding Information Base is a data structure used by the data plane of a forwarding device to determine the proper forwarding actions for packets. The polymorphic property of Othello Hashing the separation of its query and control functionalities, which is a perfect match to the programmable networks such as Software Defined Networks. Using Othello Hashing, we built a fast and scalable FIB named \textit{Concise}. Extensive evaluation results on three different platforms show that Concise outperforms other FIB designs. SDLB: Cloud Load Balancer. In a cloud network, the layer-4 load balancer servers is a device that acts as a reverse proxy and distributes network or application traffic across a number of servers. We built a software load balancer with Othello Hashing techniques named SDLB. SDLB is able to accomplish two functionalities of the SDLB using one Othello query: to find the designated server for packets of ongoing sessions and to distribute new or session-free packets. MetaOthello: Taxonomic Classification of Metagenomic Sequences. Metagenomic read classification is a critical step in the identification and quantification of microbial species sampled by high-throughput sequencing. Due to the growing popularity of metagenomic data in both basic science and clinical applications, as well as the increasing volume of data being generated, efficient and accurate algorithms are in high demand. We built a system to support efficient classification of taxonomic sequences using its k-mer signatures. SeqOthello: RNA-seq Sequence Search Engine. Advances in the study of functional genomics produced a vast supply of RNA-seq datasets. However, how to quickly query and extract information from sequencing resources remains a challenging problem and has been the bottleneck for the broader dissemination of sequencing efforts. The challenge resides in both the sheer volume of the data and its nature of unstructured representation. Using the Othello Hashing techniques, we built the SeqOthello sequence search engine. SeqOthello is a reference-free, alignment-free, and parameter-free sequence search system that supports arbitrary sequence query against large collections of RNA-seq experiments, which enables large-scale integrative studies using sequence-level data

Critical Percolation on Random Networks with Prescribed Degrees

Random graphs have played an instrumental role in modelling real-world networks arising from the internet topology, social networks, or even protein-interaction networks within cells. Percolation, on the other hand, has been the fundamental model for understanding robustness and spread of epidemics on these networks. From a mathematical perspective, percolation is one of the simplest models that exhibits phase transition, and fascinating features are observed around the critical point. In this thesis, we prove limit theorems about structural properties of the connected components obtained from percolation on random graphs at criticality. The results are obtained for random graphs with general degree sequence, and we identify different universality classes for the critical behavior based on moment assumptions on the degree distribution.Comment: Ph.D. thesi