Many network solutions and overlay networks utilize probabilistic techniques to reduce information processing and networking costs. This survey article presents a number of frequently used and useful probabilistic techniques. Bloom filters and their variants are of prime importance, and they are heavily used in various distributed systems. This has been reflected in recent research and many new algorithms have been proposed for distributed systems that are either directly or indirectly based on Bloom filters. In this survey, we give an overview of the basic and advanced techniques, reviewing over 20 variants and discussing their application in distributed systems, in particular for caching, peer-to-peer systems, routing and forwarding, and measurement data summarization.

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Theory and Practice of Bloom Filters for Distributed Systems

Routing packets on the growing and changing underlying structure of the Internet is challenging and currently based only on local connectivity. Here, a global Internet map is devised: with a greedy forwarding algorithm, it is robust with respect to network growth, and allows speeds close to the theoretical best.

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Sustaining the Internet with Hyperbolic Mapping

We consider the point-to-point (approximate) shortest-path query problem, which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph). A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible.Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs, query processing on spatial networks, and reachability queries; and theoretical computer scientists analyze distance oracles and sparse spanners. Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well.In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.

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Shortest-path queries in static networks

We demonstrate the utility of a new methodological tool, neural-network word embedding models, for large-scale text analysis, revealing how these models produce richer insights into cultural associations and categories than possible with prior methods. Word embeddings represent semantic relations between words as geometric relationships between vectors in a high-dimensional space, operationalizing a relational model of meaning consistent with contemporary theories of identity and culture. We show that dimensions induced by word differences (e.g. man - woman, rich - poor, black - white, liberal - conservative) in these vector spaces closely correspond to dimensions of cultural meaning, and the projection of words onto these dimensions reflects widely shared cultural connotations when compared to surveyed responses and labeled historical data. We pilot a method for testing the stability of these associations, then demonstrate applications of word embeddings for macro-cultural investigation with a longitudinal analysis of the coevolution of gender and class associations in the United States over the 20th century and a comparative analysis of historic distinctions between markers of gender and class in the U.S. and Britain. We argue that the success of these high-dimensional models motivates a move towards "high-dimensional theorizing" of meanings, identities and cultural processes.

The Geometry of Culture: Analyzing Meaning through Word Embeddings.

Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces: Technical Report

We propose an embedding and routing scheme for arbitrary network connectivity graphs, based on greedy routing and utilizing virtual node coordinates. In dynamic multihop packet-switching communication networks, routing elements can join or leave during network operation or exhibit intermittent failures. We present an algorithm for online greedy graph embedding in the hyperbolic plane that enables incremental embedding of network nodes as they join the network, without disturbing the global embedding. Even a single link or node removal may invalidate the greedy routing success guarantees in network embeddings based on an embedded spanning tree subgraph. As an alternative to frequent reembedding of temporally dynamic network graphs in order to retain the greedy embedding property, we propose a simple but robust generalization of greedy distance routing called Gravity-Pressure (GP) routing. Our routing method always succeeds in finding a route to the destination provided that a path exists, even if a significant fraction of links or nodes is removed subsequent to the embedding. GP routing does not require precomputation or maintenance of special spanning subgraphs and, as demonstrated by our numerical evaluation, is particularly suitable for operation in tandem with our proposed algorithm for online graph embedding.

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Hyperbolic Embedding and Routing for Dynamic Graphs

This paper describes a randomized algorithm for approximate counting that preserves the same modest memory requirements of log(log n) bits per counter as the approximate counting algorithm introduced in the seminal paper of R. Morris (1978), and in addition, is characterized by (i) lower expected number of memory accesses and (ii) lower standard error on more than 99 percent of its counting range. An exact analysis of the relevant statistical properties of the algorithm is carried out. Performance evaluation via simulations is also provided to validate the presented theory. Given its properties, the presented algorithm is suitable as a basic building block of data streaming applications having a large number of simultaneous counters and/or operating at very high speeds. As such, it is applicable to a wide range of measurement and monitoring operations, including performance monitoring of communication hardware, measurements for optimization in large database systems, and gathering statistics for data compression.

An algorithm for approximate counting using limited memory resources

: Multidimensional scaling (MDS) is a class of projective algorithms traditionally used to produce two- or three-dimensional visualizations of datasets consisting of multidimensional objects or interobject distances. Recently, metric MDS has been applied to the problems of graph embedding for the purpose of approximate encoding of edge or path costs using node coordinates in metric space. Several authors have also pointed out that for data with an inherent hierarchical structure, hyperbolic target space may be a more suitable choice for accurate embedding than Euclidean space. In this paper we present the theory and the implementation details of MDS-PD, a metric MDS algorithm designed specifically for the Poincar disk model of the hyperbolic plane. Our construction is based on an approximate hyperbolic line search and exemplifies some of the particulars that need to be addressed when applying iterative optimization methods in a hyperbolic space model. MDS-PD can be used both as a visualization tool and as an embedding algorithm. We provide several examples to illustrate the utility of MDSPD.

Multidimensional Scaling in the Poincare Disk

Edge computing, which is a fundamental component of emerging 5G architectures, involves onloading or offloading multiple virtual network functions from mobile devices to an edge network substrate. In this paper, we present a model for the complete edge function onloading problem, which consists of three main phases: (1) Cyber foraging, which involves discovery of resources monitoring the state of edge resources, (2) edge function mapping, which involves matching requests to available resources, and (3) allocation, which involves assigning resources to mappings. Using optimization theory, we show how these three phases are tightly connected, and how the wide spectrum of existing solutions that either solve a particular phase, or jointly solve two of the phases (along with their interactions), are incomplete and may lead to inefficiencies. Moreover, with extensive simulation experiments we demonstrate that joint optimization of all three phases enables the edge network to host a larger set of constrained edge function requests.

Complete edge function onloading for effective backend-driven cyber foraging

Greedy embedding is a graph embedding that makes the simple greedy packet forwarding scheme successful for every source-destination pair. It is desirable that graph embeddings also yield low hop stretch of the greedy over the shortest paths. In this paper we study how topological and geometric properties of embedded graphs influence the hop stretch. Based on the obtained insights, we construct embedding heuristics that yield minimal hop stretch greedy embeddings.

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Andrej Cvetkovski

Papers

Hyperbolic Embedding and Routing for Dynamic Graphs

An algorithm for approximate counting using limited memory resources

Multidimensional Scaling in the Poincare Disk

Complete edge function onloading for effective backend-driven cyber foraging

Low-stretch greedy embedding heuristics