Inferences for Random Graphs Evolved by Clustering Attachment

TL;DR

The evolution of random undirected graphs by the clustering attachment (CA) both without node and edge deletion and with uniform node or edge deletion is investigated and it is shown that the CA leads to light-tailed distributed node degrees and triangle counts.

Abstract

The evolution of random undirected graphs by the clustering attachment (CA) both without node and edge deletion and with uniform node or edge deletion is investigated. Theoretical results are obtained for the CA without node and edge deletion when a newly appended node is connected to two existing nodes of the graph at each evolution step. Theoretical results concern to (1) the sequence of increments of the consecutive mean clustering coefficients tends to zero; (2) the sequences of node degrees and triangle counts of any fixed node which are proved to be submartingales. These results were obtained for any initial graph. The simulation study is provided for the CA with uniform node or edge deletion and without any deletion. It is shown that (1) the CA leads to light-tailed distributed node degrees and triangle counts; (2) the average clustering coefficient tends to a constant over time; (3) the mean node degree and the mean triangle count increase over time with the rate depending on the parameters of the CA. The exposition is accompanied by a real data study.

Figures (9)

• Figure 1: The graph obtained by the CA with attachment probability (\ref{['CA-norma']}) and parameters $(\alpha, \epsilon) =(1,0)$ and $m_0=2$ from the initial graph in Fig. \ref{['fig:1a']} after $5\cdot10^4$ evolution steps without node and edge deletion (Fig. \ref{['fig:1b']}), with a uniform node deletion (Fig. \ref{['fig:1c']}) and with a uniform edge deletion (Fig. \ref{['fig:1d']}). The node size is proportional to the node degree, and the node color represents the "life time" of the node, i.e. the "older" the node the darker the color.
• Figure 2: Averages of total triangle counts $\Delta_t$ over $10$ graphs against $m_0$ for the CA graphs after $t=5\cdot10^4$ evolution steps with sets of parameters $(\alpha, \epsilon) \in \{(1, 0), (1, 1)\}$ without node and edge deletion (Fig. \ref{['fig:11a']}), with a uniform node deletion (Fig. \ref{['fig:11b']}) and with a uniform edge deletion (Fig. \ref{['fig:11c']}).
• Figure 3: EVI plots of the moment (grey line), mixed moment (black line) and UH (dotted line) estimates $\hat{\gamma}_n(k)$ of node degrees against the parameter $s$ included in the number of the largest order statistics $k=[n^{s}]$, $n = \|V_t\|$, for graphs evolved by the CA with probability (\ref{['CA-norma']}) and parameters $(\alpha, \epsilon)=(1,0)$ (the left column), and $(\alpha, \epsilon)=(1,1)$ (the right column) and with $m_0=2$ after $t=5\cdot10^4$ evolution steps without node and edge deletion (Fig. \ref{['fig:7a']}), with a uniform node deletion (Fig. \ref{['fig:7b']}) and with a uniform edge deletion (Fig. \ref{['fig:7c']}).
• Figure 4: EVI plots of the moment (grey line), mixed moment (black line) and UH (dotted line) estimates $\hat{\gamma}_n(k)$ of node triangle counts against the parameter $s$ included in the number of the largest order statistics $k=[n^{s}]$, $n = \|V_t\|$, for graphs evolved by the CA with probability (\ref{['CA-norma']}) and $m_0=2$ after $t=5\cdot10^4$ evolution steps without node and edge deletion (Fig. \ref{['fig:tr-a']}), with a uniform node deletion (Fig. \ref{['fig:tr-b']}) and with a uniform edge deletion (Fig. \ref{['fig:tr-c']}). Parameters $(\alpha, \epsilon)$ are equal to $(1,0)$ (left) and to $(1,1)$ (right) in Fig. \ref{['fig:tr-a']} , and to $(1,0)$ both in Fig. \ref{['fig:tr-b']}, \ref{['fig:tr-c']}.
• Figure 5: Plots of ${\bar{\delta}}_t\left(G_0 \right)$ with bounds (\ref{['ineqa-01']}) shown by dotted lines for initial graphs $G'_0$ (Fig. \ref{['fig:4.8.2a']}, \ref{['fig:4.8.2b']}) and $G"_0$ (Fig. \ref{['fig:4.8.2c']}, \ref{['fig:4.8.2d']}): $CA^{(\alpha, \epsilon)}$ is provided for $\alpha \in \{0.5, 1, 2\}$ (dark, dashed dark and grey lines, respectively) and $\epsilon \in \{0,1\}$ (left and right columns, respectively).

Theorems & Definitions (9)

• proposition thmcounterproposition
• proposition thmcounterproposition
• corollary thmcountercorollary
• proof
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• proof