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Proportional Selection in Networks

Georgios Papasotiropoulos, Oskar Skibski, Piotr Skowron, Tomasz Wąs

TL;DR

The paper addresses selecting a fixed number $k$ of nodes from a network to both maximize influence and reflect the network's diversity proportionally. It develops two families of methods that extend centrality measures into proportional group selection: an Election-Based approach using the Method of Equal Shares to yield MesRank/MesKatz, and Absorbing rules such as AbsorbRank/AbsorbKatz that base scores on remaining support after absorbing chosen nodes. The authors present axioms of proportionality (e.g., Clique-Entitlement, Subgraph-Entitlement) and show theoretical properties, including NP-hardness for AbsorbRank/AbsorbKatz and polynomial-time computability for the MES-based variants in key graph classes. Empirically, MesRank/MesKatz and especially the BOS variants produce more representative committees across bipartite, functional, and real-world networks, with PageRank-based methods generally offering better proportionality than Katz in many scenarios. The results highlight a robust, versatile framework for proportional representation in networked elections and group selection, with practical implications for selecting diverse yet influential sets of nodes.

Abstract

We address the problem of selecting $k$ representative nodes from a network, aiming to achieve two objectives: identifying the most influential nodes and ensuring the selection proportionally reflects the network's diversity. We propose two approaches to accomplish this, analyze them theoretically, and demonstrate their effectiveness through a series of experiments.

Proportional Selection in Networks

TL;DR

The paper addresses selecting a fixed number of nodes from a network to both maximize influence and reflect the network's diversity proportionally. It develops two families of methods that extend centrality measures into proportional group selection: an Election-Based approach using the Method of Equal Shares to yield MesRank/MesKatz, and Absorbing rules such as AbsorbRank/AbsorbKatz that base scores on remaining support after absorbing chosen nodes. The authors present axioms of proportionality (e.g., Clique-Entitlement, Subgraph-Entitlement) and show theoretical properties, including NP-hardness for AbsorbRank/AbsorbKatz and polynomial-time computability for the MES-based variants in key graph classes. Empirically, MesRank/MesKatz and especially the BOS variants produce more representative committees across bipartite, functional, and real-world networks, with PageRank-based methods generally offering better proportionality than Katz in many scenarios. The results highlight a robust, versatile framework for proportional representation in networked elections and group selection, with practical implications for selecting diverse yet influential sets of nodes.

Abstract

We address the problem of selecting representative nodes from a network, aiming to achieve two objectives: identifying the most influential nodes and ensuring the selection proportionally reflects the network's diversity. We propose two approaches to accomplish this, analyze them theoretically, and demonstrate their effectiveness through a series of experiments.

Paper Structure

This paper contains 27 sections, 5 theorems, 24 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Centrality Measures
  4. Election Rules
  5. Group Selection Rules for Networks
  6. Election-Based Group Selection
  7. Absorbing Rules
  8. Case Studies
  9. Bipartite Graphs
  10. Functional Graphs
  11. Axioms of Proportionality
  12. Experiments
  13. College Football Network
  14. Political Blogs Network
  15. Further Real-World Data
  16. ...and 12 more sections

Key Result

Theorem 1

Given an input $(G,k)$ it is NP-hard to compute ${\textsc{AbsorbRank}}\xspace(G,k)$ and ${\textsc{AbsorbKatz}}\xspace(G,k)$.

Figures (10)

  • Figure 1: Selecting $k$ nodes from a bipartite graph. TopRank (red double lines) and AbsorbRank (green shading) select $k$ nodes from the first group of $V_2$. MesRank (blue pattern) selects $0.4k$ of nodes from the first group and $0.3k$ from each other group of $V_2$.
  • Figure 2: Selecting $k$ nodes from the $n$-path ($n$ divisible by $k$). Top-Rank (red double lines) and MesRank (blue pattern) choose the last $k$ nodes. AbsorbRank (green shading) selects nodes evenly splitting the path.
  • Figure 3: Selecting $k=5$ nodes from a (directed in-)tree with two unbalanced branches of equal size. TopRank (red double lines) selects mostly from the right-hand side. MesRank (blue pattern) selects two nodes from each side. AbsorbRank (green shading) splits the tree in subtrees of size $3$.
  • Figure 4: The College Football Network, where each group of nodes represents one of 11 conferences or a group of independent teams. For $k=8$, TopRank (red double lines) selects nodes only from 4 conferences, while both MesRank (blue pattern) and SeqAbsorbRank (green shading) select at most 1 team per conference.
  • Figure 5: Maximum number of nodes from one conference that are selected by our rules for a given committee size in the College Football Network.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Theorem 5