Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Network-based kinetic models: Emergence of a statistical description of the graph topology

Marco Nurisso, Matteo Raviola, Andrea Tosin

TL;DR

It is shown that for large graphs and specific classes of interactions a statistical description of the graph topology, given in terms of the degree distribution embedded in a Boltzmann-type kinetic equation, is sufficient to capture the collective trends of networked interacting systems.

Abstract

In this paper, we propose a novel approach that employs kinetic equations to describe the collective dynamics emerging from graph-mediated pairwise interactions in multi-agent systems. We formally show that for large graphs and specific classes of interactions a statistical description of the graph topology, given in terms of the degree distribution embedded in a Boltzmann-type kinetic equation, is sufficient to capture the collective trends of networked interacting systems. This proves the validity of a commonly accepted heuristic assumption in statistically structured graph models, namely that the so-called connectivity of the agents is the only relevant parameter to be retained in a statistical description of the graph topology. Then we validate our results by testing them numerically against real social network data.

Network-based kinetic models: Emergence of a statistical description of the graph topology

TL;DR

It is shown that for large graphs and specific classes of interactions a statistical description of the graph topology, given in terms of the degree distribution embedded in a Boltzmann-type kinetic equation, is sufficient to capture the collective trends of networked interacting systems.

Abstract

In this paper, we propose a novel approach that employs kinetic equations to describe the collective dynamics emerging from graph-mediated pairwise interactions in multi-agent systems. We formally show that for large graphs and specific classes of interactions a statistical description of the graph topology, given in terms of the degree distribution embedded in a Boltzmann-type kinetic equation, is sufficient to capture the collective trends of networked interacting systems. This proves the validity of a commonly accepted heuristic assumption in statistically structured graph models, namely that the so-called connectivity of the agents is the only relevant parameter to be retained in a statistical description of the graph topology. Then we validate our results by testing them numerically against real social network data.
Paper Structure (18 sections, 3 theorems, 86 equations, 3 figures, 1 algorithm)

This paper contains 18 sections, 3 theorems, 86 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Kinetic description of graph-mediated interactions
  3. Preliminaries
  4. Interaction algorithms
  5. "Action-reaction" interactions
  6. "Action-action" interactions
  7. Derivation of kinetic equations
  8. Statistical description of the connections
  9. Equivalent Boltzmann-type descriptions
  10. The case of "action-reaction" interactions
  11. The case of "action-action" interactions
  12. More general interactions
  13. On the closure of in the limit
  14. Complete graphs
  15. Rank-one approximation of
  16. ...and 3 more sections

Key Result

Lemma 4.1

It holds that $m^-=m^+$.

Figures (3)

  • Figure 1: a. Graphical representation of the interaction framework considered in this work. Each agent is identified with a vertex in a directed graph and is characterised by a probability distribution of their state which evolves in time. b. In an action-reaction interaction between agents $i,\,j\in\mathcal{I}$ connected by the edge $(i,\,j)\in\mathcal{E}$ the states $v$, $v_\ast$ of both agents are updated. c. In an action-action interaction, the state $v_\ast$ of agent $j$ is updated only if $(j,\,i)\in\mathcal{E}$.
  • Figure 2: Visual representation of the equivalence between the network dynamics and the equivalent Boltzmann one. A situation in which the interactions between the agents are regulated by a network structure (left panel) is replaced, without approximation, by one in which every agent can interact with any other agent (right panel).
  • Figure 3: Numerical validation of the equivalence between the graph-mediated kinetic equations \ref{['eq:AR-f']}, \ref{['eq:AA-f']} and the equivalent Boltzmann-type equations \ref{['eq:std_Boltz-AR']}, \ref{['eq:std_Boltz-AA']} using the "Social circles: Twitter" network dataset snapnetsTwitter_Dataset for different interaction rules. Panels a, b: "action-reaction" and "action-action" Ochrombel opinion formation model, cf. Example \ref{['ex:Ochrombel']}. The solid lines and filled regions represent respectively mean values and 95% confidence intervals computed over 10 repetitions. Both dynamics share the same initial condition. Panel c: separable interaction rule \ref{['eq:separable']}. Panels d, e: inelastic Kac-inspired separable interaction rule \ref{['eq:Kac']}.

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • Theorem 6.1
  • ...and 2 more