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Information dissemination and confusion in signed networks

Ligang Jin, Eckhard Steffen

TL;DR

Addressing information dissemination in signed networks, the paper defines two related problems (IDP and rIDP) and develops formal measures of confusion. It provides constructions and upper bounds showing that large fractions of vertices can become confused, even in balanced graphs. It proves invariance of the relaxed confusion number under switching and connects these notions to balance, antibalance, and the frustration index. It concludes with conjectures linking confusion numbers to burning numbers and the frustration index, highlighting theoretical challenges and potential cyber-physical network applications.

Abstract

We introduce a model of information dissemination in signed networks. It is a discrete-time process in which uninformed actors incrementally receive information from their informed neighbors or from the outside. Our goal is to minimize the number of confused actors - that is, the number of actors who receive contradictory information. We prove upper bounds for the number of confused actors in signed networks and in equivalence classes of signed networks. In particular, we show that there are signed networks where, for any information placement strategy, almost 60\% of the actors are confused. Furthermore, this is also the case when considering the minimum number of confused actors within an equivalence class of signed graphs.

Information dissemination and confusion in signed networks

TL;DR

Addressing information dissemination in signed networks, the paper defines two related problems (IDP and rIDP) and develops formal measures of confusion. It provides constructions and upper bounds showing that large fractions of vertices can become confused, even in balanced graphs. It proves invariance of the relaxed confusion number under switching and connects these notions to balance, antibalance, and the frustration index. It concludes with conjectures linking confusion numbers to burning numbers and the frustration index, highlighting theoretical challenges and potential cyber-physical network applications.

Abstract

We introduce a model of information dissemination in signed networks. It is a discrete-time process in which uninformed actors incrementally receive information from their informed neighbors or from the outside. Our goal is to minimize the number of confused actors - that is, the number of actors who receive contradictory information. We prove upper bounds for the number of confused actors in signed networks and in equivalence classes of signed networks. In particular, we show that there are signed networks where, for any information placement strategy, almost 60\% of the actors are confused. Furthermore, this is also the case when considering the minimum number of confused actors within an equivalence class of signed graphs.
Paper Structure (5 sections, 15 theorems, 5 equations, 2 figures)

This paper contains 5 sections, 15 theorems, 5 equations, 2 figures.

Table of Contents

  1. Introduction and basics
  2. Basics on graphs and signed graphs
  3. Information dissemination on signed graphs
  4. Relaxed $IDP$ and equivalence classes of signed graphs
  5. Problems and conjectures

Key Result

Theorem 1.1

A signed graph $(G,\sigma)$ is balanced if and only if $V(G)$ can be partitioned into two sets $U_1$ and $U_2$ (possibly empty) such that all edges of $E(G[U_1]) \cup E(G[U_2])$ are positive and all edges of $\partial_G(U_1)$ are negative.

Figures (2)

  • Figure 1: The signed graphs $(K_{5,5},\tau_5)$ (on the left-side) and $(K_{5,5},-\tau_5)$ (on the right-side), in which red lines (resp., black lines) represents negative edges (resp., positive edges).
  • Figure 2: A member of the signed graph family $\mathcal{G}_{5,3}$

Theorems & Definitions (29)

  • Theorem 1.1: Balance
  • Theorem 1.2: SignedGraphs
  • Lemma 1.3
  • Definition 2.1: Algorithm for Information Dissemination ($ID$) on a Signed Graph $(G,\sigma)$
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 19 more