Introduction
Background
RN
RE
RN Versus RE
Inclusive random sampling
Inclusive RN (IRN)
IRN versus RN
Inclusive RE (IRE)
IRE versus RE
IRN versus IRE
The \(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}\) and \(\frac{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{N}}]}{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{E}}]}\) ratios are unbounded
Bounding \(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}\) as a function of \({\varvec{n}}\)
The \(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}\) and \(\frac{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{E}}]}{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{N}}]}\) ratios are unbounded
Bounding \(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}\) as a function of \({\varvec{n}}\)
\(\frac{{\mathbb{E}}\left[IRN\right]}{{\mathbb{E}}\left[RE\right]}\) and \(\frac{{\mathbb{E}}\left[IRE\right]}{{\mathbb{E}}\left[RN\right]}\)
Random sampling in trees
\(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}\) and \(\frac{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{N}}]}{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{E}}]}\)
\(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}\) and \(\frac{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{E}}]}{{\mathbb{E}}[{\varvec{I}}{\varvec{R}}{\varvec{N}}]}\) are unbounded in trees
Bounding \(\frac{{\mathbb{E}}[{\varvec{R}}{\varvec{E}}]}{{\mathbb{E}}[{\varvec{R}}{\varvec{N}}]}\) as a function of \({\varvec{n}}\)
Experimental analysis
Synthetic graphs
Inclusive sampling in synthetic graphs
RV | Erdős Rényi Graphs, \(\mathrm{n}\) = 6000 | Barabási Albert Graphs, \(\mathrm{n}\) = 6000 | ||||||
---|---|---|---|---|---|---|---|---|
RN | RE | IRN | IRE | RN | RE | IRN | IRE | |
6 | 6.9952 | 6.9946 | 7.9227 | 8.361 | 19.54 | 17.68 | 21.34 | 29.63 |
10 | 10.9883 | 10.9882 | 12.3023 | 12.755 | 27.87 | 26.18 | 30.7 | 42.76 |
16 | 16.973 | 16.9714 | 18.7509 | 19.2119 | 38.9 | 37.43 | 43.24 | 59.46 |
30 | 30.922 | 30.9212 | 33.525 | 33.9967 | 63.89 | 62.75 | 71.64 | 96.63 |
60 | 60.6866 | 60.6864 | 64.5381 | 65.0121 | 113.3 | 112.55 | 128.18 | 167.78 |
129 | 129.5657 | 129.565 | 135.4022 | 135.8766 | 216.32 | 216.42 | 246.99 | 310.69 |
Real-world networks
Category | Pct \(\mathrm{RN }>\mathrm{ RE}\) (%) | Pct \(\mathrm{IRN }>\mathrm{ IRE}\) (%) | \(\mathrm{IRN}/\mathrm{RN}\) | \(\mathrm{IRE}/\mathrm{RE}\) |
---|---|---|---|---|
Affiliation | 100 | 17 | 1.05 | 1.68 |
Animal | 75 | 0 | 1.09 | 1.13 |
Authorship | 99 | 67 | 1.01 | 1.94 |
Citation | 50 | 0 | 1.08 | 1.58 |
Cocitation | 0 | 0 | 1.1 | 1.47 |
Communication | 83 | 25 | 1.04 | 1.7 |
Computer | 64 | 0 | 1.07 | 1.60 |
Feature | 83 | 50 | 1.02 | 1.88 |
Human Contact | 86 | 14 | 1.12 | 1.31 |
Human Social | 55 | 0 | 1.12 | 1.21 |
Hyperlink | 71 | 14 | 1.02 | 1.84 |
Infrastructure | 48 | 0 | 1.1 | 1.2 |
Interaction | 81 | 62 | 1.04 | 1.71 |
Lexical | 67 | 33 | 1.08 | 1.66 |
Metabolic | 75 | 0 | 1.07 | 1.59 |
Misc | 67 | 0 | 1.08 | 1.55 |
Neural | 100 | 0 | 1.11 | 1.45 |
Online Contact | 75 | 13 | 1.03 | 1.69 |
Rating | 100 | 57 | 1.02 | 1.87 |
Social | 71 | 31 | 1.03 | 1.76 |
Software | 100 | 67 | 1.003 | 1.98 |
Text | 83 | 0 | 1.04 | 1.58 |
Trophic | 100 | 0 | 1.14 | 1.33 |