Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mathematical model of information bubbles on networks

Pál Burai, Paweł Pasteczka

Abstract

The main goal of this paper to introduce a new model of evolvement of narratives (common opinions, information bubble) on networks. Our main tools come from invariant mean theory and graph theory. The case, when the root set of the network (influencers, news agencies, etc.) is ergodic is fully discussed. The other possibility, when the root contains more than one component is partially discussed and it could be a motivation for further research.

Mathematical model of information bubbles on networks

Abstract

The main goal of this paper to introduce a new model of evolvement of narratives (common opinions, information bubble) on networks. Our main tools come from invariant mean theory and graph theory. The case, when the root set of the network (influencers, news agencies, etc.) is ergodic is fully discussed. The other possibility, when the root contains more than one component is partially discussed and it could be a motivation for further research.
Paper Structure (11 sections, 7 theorems, 53 equations, 5 figures)

This paper contains 11 sections, 7 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Outline of the paper
  4. Preliminaries
  5. Mathematical tools from graph theory and the concept of the root
  6. Means, mean-type mappings and invariant means
  7. Main result
  8. Further examples and discussion
  9. Markov chains and weighted arithmetic means, when the root is not ergodic
  10. Conclusion and further research
  11. Acknowledgement

Key Result

Lemma 2.1

Let $G=(V,E)$ be an ergodic digraph. Then there exists $q_0$ such that for all $q \ge q_0$, and $v,w \in V$ there exists a walk from $v$ to $w$ of length exactly $q$.

Figures (5)

  • Figure 1: A directed graph $G$ and the corresponding $G^{SCC}$
  • Figure 2: Graph $G_\alpha$ related to Example \ref{['ex2']}.
  • Figure 3: Graph $G_\alpha$ related to Example \ref{['exa:Pas23b-5']}.
  • Figure 4: Graph $G_\alpha$ related to Example \ref{['exa:E1']}.
  • Figure 5: Graph $G_\alpha$ related to Example \ref{['exa:E1']}.

Theorems & Definitions (17)

  • Lemma 2.1: Pas23b, Lemma 1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4: Characterization theorem of $R(G)$
  • proof
  • Example 2.5
  • Theorem 2.6: Pas23b, Theorem 2 (a)-(d)
  • Proposition 2.7: Invariance principle; MatPas21, Theorem 1
  • Theorem 3.1
  • proof
  • ...and 7 more