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Incremental Measurement of Structural Entropy for Dynamic Graphs

Runze Yang, Hao Peng, Chunyang Liu, Angsheng Li

TL;DR

Introducing Incre-2dSE, a novel incremental measurement framework that dynamically adjusts the community partitioning and efficiently computes the updated structural entropy for each updated graph, which effectively capture the dynamic evolution of the communities, reduce time consumption, and provide great interpretability.

Abstract

Structural entropy is a metric that measures the amount of information embedded in graph structure data under a strategy of hierarchical abstracting. To measure the structural entropy of a dynamic graph, we need to decode the optimal encoding tree corresponding to the best community partitioning for each snapshot. However, the current methods do not support dynamic encoding tree updating and incremental structural entropy computation. To address this issue, we propose Incre-2dSE, a novel incremental measurement framework that dynamically adjusts the community partitioning and efficiently computes the updated structural entropy for each updated graph. Specifically, Incre-2dSE includes incremental algorithms based on two dynamic adjustment strategies for two-dimensional encoding trees, i.e., the naive adjustment strategy and the node-shifting adjustment strategy, which support theoretical analysis of updated structural entropy and incrementally optimize community partitioning towards a lower structural entropy. We conduct extensive experiments on 3 artificial datasets generated by Hawkes Process and 3 real-world datasets. Experimental results confirm that our incremental algorithms effectively capture the dynamic evolution of the communities, reduce time consumption, and provide great interpretability.

Incremental Measurement of Structural Entropy for Dynamic Graphs

TL;DR

Introducing Incre-2dSE, a novel incremental measurement framework that dynamically adjusts the community partitioning and efficiently computes the updated structural entropy for each updated graph, which effectively capture the dynamic evolution of the communities, reduce time consumption, and provide great interpretability.

Abstract

Structural entropy is a metric that measures the amount of information embedded in graph structure data under a strategy of hierarchical abstracting. To measure the structural entropy of a dynamic graph, we need to decode the optimal encoding tree corresponding to the best community partitioning for each snapshot. However, the current methods do not support dynamic encoding tree updating and incremental structural entropy computation. To address this issue, we propose Incre-2dSE, a novel incremental measurement framework that dynamically adjusts the community partitioning and efficiently computes the updated structural entropy for each updated graph. Specifically, Incre-2dSE includes incremental algorithms based on two dynamic adjustment strategies for two-dimensional encoding trees, i.e., the naive adjustment strategy and the node-shifting adjustment strategy, which support theoretical analysis of updated structural entropy and incrementally optimize community partitioning towards a lower structural entropy. We conduct extensive experiments on 3 artificial datasets generated by Hawkes Process and 3 real-world datasets. Experimental results confirm that our incremental algorithms effectively capture the dynamic evolution of the communities, reduce time consumption, and provide great interpretability.
Paper Structure (41 sections, 5 theorems, 62 equations, 18 figures, 6 tables, 4 algorithms)

This paper contains 41 sections, 5 theorems, 62 equations, 18 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and Notations
  3. Dynamic Adjustment Strategies for Two-Dimensional Encoding Trees
  4. Naive Adjustment Strategy
  5. Strategy Description
  6. Global Invariant and Local Difference
  7. Boundedness Analysis
  8. Convergence Analysis
  9. Limitations
  10. Node-Shifting Adjustment Strategy
  11. Strategy Description
  12. Theoretical Proof
  13. Limitations
  14. Further Discussion between the Two Dynamic Adjustment Strategies
  15. Incre-2dSE: an Incremental Measurement Framework of the Updated Two-Dimensional Structural Entropy
  16. ...and 26 more sections

Key Result

Theorem 1

Suppose that a new graph node $v$ is connected to an existing node $u$, where $\{ u\} \subseteq T_\alpha$. If $\frac{2m+2}{V_\alpha+2} \ge e$, we have: where $H^{\mathcal{T}'}_{v\rightarrow \alpha}(G')$ denotes the updated structural entropy when the new node $v$ is assigned to $u$'s community $T_\alpha$, i.e., $\{v\} \subset T_\alpha$, and $H^{\mathcal{T}'}_{v\rightarrow \beta \neq \alpha}(G')$

Figures (18)

  • Figure 1: a) A graph containing three communities A, B, and C, where A is divided into two sub-communities A.1 and A.2. b) An encoding tree of the left graph. Each leaf node corresponds to a single graph node. Each branch node corresponds to a community. The root node corresponds to the graph node set.
  • Figure 2: An example graph with its two equivalent two-dimensional encoding trees.
  • Figure 3: An example of the node strategy for adjusting two-dimensional encoding trees.
  • Figure 4: An example of the node-shifting adjustment strategy for adding new edges.
  • Figure 5: An example of the node-shifting adjustment strategy for adding new nodes.
  • ...and 13 more figures

Theorems & Definitions (21)

  • Definition 1: Incremental Sequence
  • Definition 2: Dynamic Graph
  • Definition 3: Encoding Tree
  • Definition 4: Structural Entropy
  • Theorem 1
  • Definition 5: Global Invariant
  • Definition 6: Local Difference
  • Theorem 2
  • Definition 7
  • Theorem 3
  • ...and 11 more