arXiv subjects

Joseph G. Peters

Publications and source records attributed to Joseph G. Peters.

4 recordsLinked to original sources

arXiv2021

Temporal Cliques Admit Sparse Spanners

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $λ:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of integer labels (times). Such a graph is {\em temporally connected} if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar \cite{KKK02} asked whether, given such a temporal graph, a {\em sparse} subset of edges always exists whose labels suffice to preserve temporal connectivity -- a {\em temporal spanner}. Axiotis and Fotakis \cite{AF16} answered negatively by exhibiting a family of $Θ(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. In this paper, we give the first positive answer as to the existence of $o(n^2)$-sparse spanners in a dense class of temporal graphs, by showing (constructively) that if $G$ is a complete graph, then one can always find a temporal spanner of density $O(n \log n)$.

cs.DM↗

arXiv2017

Efficiently Testing T-Interval Connectivity in Dynamic Graphs

Many types of dynamic networks are made up of durable entities whose links evolve over time. When considered from a {\em global} and {\em discrete} standpoint, these networks are often modelled as evolving graphs, i.e. a sequence of graphs ${\cal G}=(G_1,G_2,...,G_δ)$ such that $G_i=(V,E_i)$ represents the network topology at time step $i$. Such a sequence is said to be $T$-interval connected if for any $t\in [1, δ-T+1]$ all graphs in $\{G_t,G_{t+1},...,G_{t+T-1}\}$ share a common connected spanning subgraph. In this paper, we consider the problem of deciding whether a given sequence ${\cal G}$ is $T$-interval connected for a given $T$. We also consider the related problem of finding the largest $T$ for which a given ${\cal G}$ is $T$-interval connected. We assume that the changes between two consecutive graphs are arbitrary, and that two operations, {\em binary intersection} and {\em connectivity testing}, are available to solve the problems. We show that $Ω(δ)$ such operations are required to solve both problems, and we present optimal $O(δ)$ online algorithms for both problems. We extend our online algorithms to a dynamic setting in which connectivity is based on the recent evolution of the network.

cs.DS↗

arXiv2015

An Algebra to Merge Heterogeneous Classifiers

In distributed classification, each learner observes its environment and deduces a classifier. As a learner has only a local view of its environment, classifiers can be exchanged among the learners and integrated, or merged, to improve accuracy. However, the operation of merging is not defined for most classifiers. Furthermore, the classifiers that have to be merged may be of different types in settings such as ad-hoc networks in which several generations of sensors may be creating classifiers. We introduce decision spaces as a framework for merging possibly different classifiers. We formally study the merging operation as an algebra, and prove that it satisfies a desirable set of properties. The impact of time is discussed for the two main data mining settings. Firstly, decision spaces can naturally be used with non-stationary distributions, such as the data collected by sensor networks, as the impact of a model decays over time. Secondly, we introduce an approach for stationary distributions, such as homogeneous databases partitioned over different learners, which ensures that all models have the same impact. We also present a method that uses storage flexibly to achieve different types of decay for non-stationary distributions. Finally, we show that the algebraic approach developed for merging can also be used to analyze the behaviour of other operators.

cs.DM↗

arXiv2013

Influence Diffusion in Social Networks under Time Window Constraints

We study a combinatorial model of the spread of influence in networks that generalizes existing schemata recently proposed in the literature. In our model, agents change behaviors/opinions on the basis of information collected from their neighbors in a time interval of bounded size whereas agents are assumed to have unbounded memory in previously studied scenarios. In our mathematical framework, one is given a network $G=(V,E)$, an integer value $t(v)$ for each node $v\in V$, and a time window size $λ$. The goal is to determine a small set of nodes (target set) that influences the whole graph. The spread of influence proceeds in rounds as follows: initially all nodes in the target set are influenced; subsequently, in each round, any uninfluenced node $v$ becomes influenced if the number of its neighbors that have been influenced in the previous $λ$ rounds is greater than or equal to $t(v)$. We prove that the problem of finding a minimum cardinality target set that influences the whole network $G$ is hard to approximate within a polylogarithmic factor. On the positive side, we design exact polynomial time algorithms for paths, rings, trees, and complete graphs.

cs.DS↗