Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Reporting Durable Patterns in Temporal Proximity Graphs

Pankaj K. Agarwal, Xiao Hu, Stavros Sintos, Jun Yang

TL;DR

An implicit representation of the proximity graph is worked with, where nodes are additionally annotated by time intervals, and near-linear-time algorithms for finding (approximately) durable patterns above a given durability threshold are designed.

Abstract

Finding patterns in graphs is a fundamental problem in databases and data mining. In many applications, graphs are temporal and evolve over time, so we are interested in finding durable patterns, such as triangles and paths, which persist over a long time. While there has been work on finding durable simple patterns, existing algorithms do not have provable guarantees and run in strictly super-linear time. The paper leverages the observation that many graphs arising in practice are naturally proximity graphs or can be approximated as such, where nodes are embedded as points in some high-dimensional space, and two nodes are connected by an edge if they are close to each other. We work with an implicit representation of the proximity graph, where nodes are additionally annotated by time intervals, and design near-linear-time algorithms for finding (approximately) durable patterns above a given durability threshold. We also consider an interactive setting where a client experiments with different durability thresholds in a sequence of queries; we show how to compute incremental changes to result patterns efficiently in time near-linear to the size of the changes.

On Reporting Durable Patterns in Temporal Proximity Graphs

TL;DR

An implicit representation of the proximity graph is worked with, where nodes are additionally annotated by time intervals, and near-linear-time algorithms for finding (approximately) durable patterns above a given durability threshold are designed.

Abstract

Finding patterns in graphs is a fundamental problem in databases and data mining. In many applications, graphs are temporal and evolve over time, so we are interested in finding durable patterns, such as triangles and paths, which persist over a long time. While there has been work on finding durable simple patterns, existing algorithms do not have provable guarantees and run in strictly super-linear time. The paper leverages the observation that many graphs arising in practice are naturally proximity graphs or can be approximated as such, where nodes are embedded as points in some high-dimensional space, and two nodes are connected by an edge if they are close to each other. We work with an implicit representation of the proximity graph, where nodes are additionally annotated by time intervals, and design near-linear-time algorithms for finding (approximately) durable patterns above a given durability threshold. We also consider an interactive setting where a client experiments with different durability thresholds in a sequence of queries; we show how to compute incremental changes to result patterns efficiently in time near-linear to the size of the changes.
Paper Structure (50 sections, 13 theorems, 7 equations, 3 figures, 2 tables, 8 algorithms)

This paper contains 50 sections, 13 theorems, 7 equations, 3 figures, 2 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definitions
  3. Our Results and Approach
  4. Extensions.
  5. Preliminaries
  6. Basic concepts and data structures
  7. Doubling dimension.
  8. Interval tree.
  9. Cover tree.
  10. Durable ball query
  11. Data structure.
  12. Reporting Durable Triangles
  13. Algorithm.
  14. Incremental Reporting When Varying $\pmb\tau$
  15. Reporting for each activated point
  16. ...and 35 more sections

Key Result

lemma 1

Given a set $P$ of $n$ points, a data structure can be built in $O(n\log^2 n)$ time with $O(n\log n)$ space, that supports an $\varepsilon$-approximate $\tau$-durable ball query, computing a family of $O(\varepsilon^{-d})$ canonical subsets in $O(\varepsilon^{-d}\log n)$ time.

Figures (3)

  • Figure 1: Illustration of Algorithm \ref{['alg:offline-2']}: $p$ is visited. The small (possibly overlapping) balls represent the canonical nodes returned from $\mathcal{D}$. Each point within distance $1$ from $p$ lies in exactly one such ball. We report the triangles formed by $p$ and the points in red and blue balls that satisfy the durability constraint. We do not report triangles formed by $p$ and the points in blue and green balls because they are well separated.
  • Figure 2: An illustration of $\mathcal{C}_{p,j} = \Lambda_{p,j} \cup {\overline{\Lambda}}_{p,j}$. Here $q_1 \in \Lambda_{p,j}$ and $q_2 \in {\overline{\Lambda}}_{p,j}$.
  • Figure 3: An illustration of Algorithm \ref{['alg:report-unit']}: $p$ is visited.

Theorems & Definitions (19)

  • definition 1: DurableTriangle
  • definition 2: IncrDurableTriangle
  • definition 3: AggDurablePair
  • definition 4: $\varepsilon$-approximate $\tau$-durable ball query
  • lemma 1
  • theorem 1
  • definition 5: Activation threshold
  • theorem 2
  • theorem 3
  • theorem 4
  • ...and 9 more