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Age-dependent random connection models with arc reciprocity: clustering and connectivity

Lukas Lüchtrath, Christian Mönch

Abstract

We introduce a model for directed spatial networks. Starting from an age-based preferential attachment model in which all arcs point from younger to older vertices, we add \emph{reciprocal} connections whose probabilities depend on the age difference between their end-vertices. This yields a directed graph with reciprocal correlations, a power-law indegree distribution, and a tunable outdegree distribution. We consider two versions of the model: an infinite version embedded in $\mathbb{R}^d$, which can be constructed as a weight-dependent random connection model with a non-symmetric kernel, and a growing sequence of graphs on the unit torus that converges locally to the infinite model. Besides establishing the local limit result linking the two models, we investigate degree distributions, various directed clustering metrics, and directed percolation.

Age-dependent random connection models with arc reciprocity: clustering and connectivity

Abstract

We introduce a model for directed spatial networks. Starting from an age-based preferential attachment model in which all arcs point from younger to older vertices, we add \emph{reciprocal} connections whose probabilities depend on the age difference between their end-vertices. This yields a directed graph with reciprocal correlations, a power-law indegree distribution, and a tunable outdegree distribution. We consider two versions of the model: an infinite version embedded in , which can be constructed as a weight-dependent random connection model with a non-symmetric kernel, and a growing sequence of graphs on the unit torus that converges locally to the infinite model. Besides establishing the local limit result linking the two models, we investigate degree distributions, various directed clustering metrics, and directed percolation.
Paper Structure (35 sections, 16 theorems, 151 equations, 3 figures)

This paper contains 35 sections, 16 theorems, 151 equations, 3 figures.

Table of Contents

  1. Introduction and overview of results
  2. Our contribution
  3. Summary of results.
  4. Relevance to network and data science.
  5. Organisation of the paper.
  6. Model description
  7. Notation.
  8. Degree distributions of the DARCM
  9. Out-percolation in the DARCM
  10. Directed age-based preferential attachment
  11. Empirical degree distribution
  12. Clustering in the DARCM and the directed age-based PA model
  13. Friendship clustering.
  14. Interest clustering.
  15. Conclusion and future work
  16. ...and 20 more sections

Key Result

Theorem 3.1

Let $\beta>0$, $\gamma\in(0,1)$, $\delta>1$, and $\Gamma\geq 0$. Almost surely, the root $\boldsymbol{o}$ in $\mathscr{D}_o$ has finite degree. More precisely, for $k\in\mathbb{N}_0$, we have

Figures (3)

  • Figure 1: Simulation of a finitary toroidal variant of the DARCM (cf. Section \ref{['sec:DPA']}) on the unit torus $[-1/2,1/2)^2$ with $N=60$ vertices, $\beta=0.4$, $\gamma=0.35$, $\delta=2.5$, and $\Gamma=1$. Vertex size is proportional to indegree; colour indicates birth time (dark = old, light = young). Gray arcs are non-reciprocated forward arcs (younger $\to$ older); blue arcs are forward arcs that were reciprocated; red arcs are the corresponding reciprocal arcs. Torus-wrapping arcs are suppressed for clarity.
  • Figure 2: Depiction of the two clustering metrics. The labeled vertices $\boldsymbol{x}$ and $\boldsymbol{y}$ refer to the local versions.
  • Figure 3: A directed path where a vertex's birth time is denoted on the $t$-axis. The vertices of the skeleton with running minimum birth time are in black. We successively remove all local maxima, starting with the youngest, and replace them by directed edges until the directed path, only containing the skeleton vertices, is left.

Theorems & Definitions (34)

  • Theorem 3.1: Neighbourhoods in DARCM
  • Theorem 4.1: Existence vs. non existence of an out-percolation phase
  • Theorem 5.1: DARCM as local limit
  • Remark 5.2
  • Theorem 5.3
  • Theorem 5.4
  • Theorem 6.1: Average friend clustering
  • Theorem 6.2: Global friend clustering
  • Remark 6.3
  • Theorem 6.4: Average interest clustering
  • ...and 24 more