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Properties and Expressivity of Linear Geometric Centralities

Paolo Boldi, Flavio Furia, Chiara Prezioso

TL;DR

The paper introduces linear geometric centralities, a broad class of centrality measures defined as L_G^a(i)=(C_G·a)_i, where C_G is the distance-count matrix. It develops an axiomatic framework (density, size, score and rank monotonicity) and characterizes when these centralities satisfy them, deriving precise conditions on the coefficient vector a (e.g., a_1>a_2 and a_i≥a_{i+1}≥0 for score monotonicity, and a_k>k(a_2−a_1)+(2a_1−a_2) for density). It then studies expressivity via permutation representability, using Farkas’ lemma and Rouché-Capelli to show that some graphs (e.g., G_n, G'_n) can realize a large fraction or even all permutations, implying high robustness and adversarial relevance. The results illuminate how closely linear centralities approximate the full spectrum of node rankings and provide constructive tools for designing graphs with desired ranking properties. Overall, the work advances understanding of the expressive power and limitations of linear geometric centralities and opens several open problems for further investigation, including necessary conditions for rank monotonicity and the design of strongly connected graphs with maximal representativeness.

Abstract

Centrality indices are used to rank the nodes of a graph by importance: this is a common need in many concrete situations (social networks, citation networks, web graphs, for instance) and it was discussed many times in sociology, psychology, mathematics and computer science, giving rise to a whole zoo of definitions of centrality. Although they differ widely in nature, many centrality measures are based on shortest-path distances: such centralities are often referred to as geometric. Geometric centralities can use the shortest-path-length information in many different ways, but most of the existing geometric centralities can be defined as a linear transformation of the distance-count vector (that is, the vector containing, for every index t, the number of nodes at distance t). In this paper we study this class of centralities, that we call linear (geometric) centralities, in their full generality. In particular, we look at them in the light of the axiomatic approach, and we study their expressivity: we show to what extent linear centralities can be used to distinguish between nodes in a graph, and how many different rankings of nodes can be induced by linear centralities on a given graph. The latter problem (which has a number of possible applications, especially in an adversarial setting) is solved by means of a linear programming formulation, which is based on Farkas' lemma, and is interesting in its own right.

Properties and Expressivity of Linear Geometric Centralities

TL;DR

The paper introduces linear geometric centralities, a broad class of centrality measures defined as L_G^a(i)=(C_G·a)_i, where C_G is the distance-count matrix. It develops an axiomatic framework (density, size, score and rank monotonicity) and characterizes when these centralities satisfy them, deriving precise conditions on the coefficient vector a (e.g., a_1>a_2 and a_i≥a_{i+1}≥0 for score monotonicity, and a_k>k(a_2−a_1)+(2a_1−a_2) for density). It then studies expressivity via permutation representability, using Farkas’ lemma and Rouché-Capelli to show that some graphs (e.g., G_n, G'_n) can realize a large fraction or even all permutations, implying high robustness and adversarial relevance. The results illuminate how closely linear centralities approximate the full spectrum of node rankings and provide constructive tools for designing graphs with desired ranking properties. Overall, the work advances understanding of the expressive power and limitations of linear geometric centralities and opens several open problems for further investigation, including necessary conditions for rank monotonicity and the design of strongly connected graphs with maximal representativeness.

Abstract

Centrality indices are used to rank the nodes of a graph by importance: this is a common need in many concrete situations (social networks, citation networks, web graphs, for instance) and it was discussed many times in sociology, psychology, mathematics and computer science, giving rise to a whole zoo of definitions of centrality. Although they differ widely in nature, many centrality measures are based on shortest-path distances: such centralities are often referred to as geometric. Geometric centralities can use the shortest-path-length information in many different ways, but most of the existing geometric centralities can be defined as a linear transformation of the distance-count vector (that is, the vector containing, for every index t, the number of nodes at distance t). In this paper we study this class of centralities, that we call linear (geometric) centralities, in their full generality. In particular, we look at them in the light of the axiomatic approach, and we study their expressivity: we show to what extent linear centralities can be used to distinguish between nodes in a graph, and how many different rankings of nodes can be induced by linear centralities on a given graph. The latter problem (which has a number of possible applications, especially in an adversarial setting) is solved by means of a linear programming formulation, which is based on Farkas' lemma, and is interesting in its own right.

Paper Structure

This paper contains 29 sections, 25 theorems, 115 equations, 9 figures.

Table of Contents

  1. Introduction and Motivation
  2. Notations and Basic Definitions
  3. Graphs.
  4. Graph homomorphisms.
  5. Matrices and vectors.
  6. Infinite vectors and matrix product.
  7. Permutations.
  8. Signum.
  9. Centralities, Permutations and Agreements
  10. Geometric and Linear Centralities
  11. Lin is not linear
  12. Centralities, Ties and Rigidity
  13. Axioms for Linear Centralities
  14. Size and density axioms
  15. Score monotonicity
  16. ...and 14 more sections

Key Result

theorem 1

All centrality measures have ties on graphs that are not rigid. On the other hand, there exists a centrality measure $\mathfrak{f}$ such that, for all rigid graphs $G$, $\mathfrak{f}$ has no ties on $G$.

Figures (9)

  • Figure 1: A graph with $n$ nodes, and its distance-count matrix $C$.
  • Figure 2: A disconnected undirected graph showing that Lin centrality is not linear.
  • Figure 3: A rigid graph that is not geometrically rigid, its adjacency matrix $A_G$ and its distance-count matrix $C_G$.
  • Figure 4: A $k$-clique (left) and a directed $p$-cycle (right), possibly connected through a bridge $x\,\text{\textemdash}\, y$. We let $S$ be the disconnected graph without the bridge and $S_{xy}$ the strongly connected graph with the bridge.
  • Figure 5: The (family of) graphs $G_{h,k}$ used to distinguish non-proportional sequences when inequality \ref{['eqn:nonprop_cond']} holds. The horizontal dotted line between $x$ and $u$ (resp., $y$ and $v$) represents a path of length $h-1$. Analogously, the vertical dotted line between $f$ and $g$ represents a path of length $k-h-2$. Note that when $h=1$, $u$ gets to coincide with $x$, hence the $s$ nodes in such a case are connected to $x$ (everything holds symmetrically for $y$).
  • ...and 4 more figures

Theorems & Definitions (59)

  • definition 1: Respecting a permutation
  • definition 2: Agreement between centralities
  • definition 3: Distance-count function
  • definition 4: Geometric centralities
  • definition 5: Distance-count matrix
  • definition 6: Linear centrality
  • theorem 1
  • proof
  • definition 7: Geometrically rigid
  • theorem 2
  • ...and 49 more