Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis
Abstract
:1. Introduction
2. Related Work
3. EdgeFuzzy Graphs
3.1. Definitions
 the vertex set V is fixed, namely they belong to G with probability one,
 the distribution h is the same for each edge,
 the existence probability of ${e}_{k}$ is drawn independently for each edge.
3.2. Reciprocal Random Variables
4. Family of Walktrap Heuristics
4.1. Deterministic Walktrap
Algorithm 1: Deterministic Walktrap 
Require: graph $G(V,E)$, termination criterion ${\tau}_{0}$ Ensure: vertex pair sequence $\u2329{s}_{k},{s}_{k}^{*}\u232a$ is generated

4.2. Fuzzy Walktrap
Algorithm 2: Fuzzy Walktrap 
Require: fuzzy graph $G(V,\tilde{E},h)$, termination criterion ${\tau}_{0}$ Ensure: vertex pair sequence $\u2329{s}_{k},{s}_{k}^{*}\u232a$ is generated

4.3. Markov Walktrap and Chebyshev Walktrap
4.4. Escape Strategies
5. Analysis
5.1. Data
5.2. Time and Memory Requirements
5.3. Community Coherence
5.4. Relocations
6. Conclusions
Author Contributions
Conflicts of Interest
References
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Symbol  Meaning 

$\stackrel{\u25b5}{=}$  Definition or equality by definition 
⊗  Kronecker tensor product 
$\left\{{s}_{1},\dots ,{s}_{n}\right\}$  Set with elements ${s}_{1},{s}_{2},\dots ,{s}_{n}$ 
$\left\xb7\right$  Set cardinality or path length (depending on the context) 
$\left({e}_{1},\dots ,{e}_{m}\right)$  Path comprised of edges ${e}_{1},\dots \phantom{\rule{0.166667em}{0ex}},{e}_{m}$ 
${K}_{n}$  Complete graph with n vertices and $\left(\genfrac{}{}{0pt}{}{n}{2}\right)$ edges 
$\mathrm{E}\left[X\right]$  Mean value of random variable X 
$Var\left[X\right]$  Variance of random variable X 
${\tau}_{T,V}$  Tanimoto similarity coefficient for sets T and V 
${\nu}_{T,V}$  Asymmetric Tversky index for sets T and V 
${S}_{1}\setminus {S}_{2}$  Asymmetric set difference ${S}_{1}$ minus ${S}_{2}$ 
$\tilde{S}$  Fuzzy set S 
$\u2329{s}_{k}\u232a$  Sequence of elements ${s}_{k}$ 
$\mathcal{H}\left({s}_{1},\dots ,{s}_{n}\right)$  Harmonic mean of elements ${s}_{1},\dots ,{s}_{n}$ 
$\mathcal{H}\left({s}_{1},\dots ,{s}_{n};{\tau}_{0}\right)$  Thresholded or effective harmonic mean of ${s}_{1},\dots ,{s}_{n}$ 
${\underline{\mathbf{1}}}_{n}$  $n\times 1$ vector with ones 
${\underline{\mathbf{e}}}_{n}^{k}$  $n\times 1$ zero vector with a single one at the kth entry 
${f}^{\left(n\right)}\left(x\right)$  nth order derivative of $f\left(x\right)$ 
$\u2329p\phantom{\rule{0.222222em}{0ex}}\left\right\phantom{\rule{0.222222em}{0ex}}q\u232a$  Kullback–Leibler divergence between distributions p and q 
Property  Value  Property  Value  Property  Value  Property  Value 

Vertices  117,649  ${\sigma}_{0}$  7.68 × 10${}^{4}$  Triangles  37127  Squares  1981 
Edges  5,315,625  ${\sigma}_{0}^{\prime}$  $0.6631$  min cost  $5.1891$  max cost  $10.7232$ 
${\rho}_{0}$  $45.1821$  Diameter length  19  avg cost  $72.1149$  avg cost  $106.0012$ 
${\rho}_{0}^{\prime}$  $1.3263$  Diameter cost  $124.4021$  max cost  $143.2716$  max cost  $198.2221$ 
Algorithm  Distribution  RW (s)  CB (s)  Total (s)  Visits 

CW  Poisson  $128.1381$  $2000.9831$  $2129.1212$  471,858 
CW  Binomial  $131.2182$  $2000.0304$  $2131.2486$  470,342 
CW + I  Poisson  $144.4752$  $1911.9118$  $2056.3870$  477,423 
CW + I  Binomial  $139.9916$  $1904.0216$  $2044.0132$  477,216 
CW + R  Poisson  $157.0104$  $1840.0013$  $1997.0117$  476,997 
CW + R  Binomial  $157.6633$  $1850.1003$  $2007.7636$  476,957 
CW + IR  Poisson  $164.3779$  $2002.7745$  $2167.1524$  477,762 
CW + IR  Binomial  $162.0222$  $1911.8664$  $2073.8886$  477,002 
MW  Poisson  $157.5248$  $2104.9918$  $2262.5166$  475,308 
MW  Binomial  $156.9924$  $2099.5256$  $2256.5180$  475,121 
MW + I  Poisson  $165.3374$  $1908.6612$  $2073.9986$  478,313 
MW + I  Binomial  $163.0015$  $1902.4319$  $2065.4334$  478,102 
MW + R  Poisson  $173.9016$  $1850.0971$  $2023.9987$  477,916 
MW + R  Binomial  $171.0017$  $1840.0054$  $2011.0071$  477,831 
MW + IR  Poisson  $181.0013$  $2000.9917$  $2181.9930$  478,514 
MW + IR  Binomial  $178.0017$  $2000.8585$  $2178.8602$  478,333 
FW  Poisson  $191.9989$  $2425.1121$  $2617.1110$  515,444 
FW  Binomial  $184.0451$  $2417.3376$  $2601.3827$  514,312 
FNG  Poisson  –  –  $9322.9514$  – 
FNG  Binomial  –  –  $9344.7778$  – 
Algorithm  Distribution  min  max  mean  std 

CW  Poisson  4128  6742  4892  322 
CW  Binomial  4128  6744  4880  324 
CW + I  Poisson  4128  6739  4886  335 
CW + I  Binomial  4128  6744  4881  338 
CW + R  Poisson  4128  8002  5113  427 
CW + R  Binomial  4128  8000  5121  422 
CW + IR  Poisson  4128  8001  5208  345 
CW + IR  Binomial  4128  8002  5214  339 
MW  Poisson  4128  6744  4800  331 
MW  Binomial  4128  6740  4881  325 
MW + I  Poisson  4128  6739  4879  336 
MW + I  Binomial  4128  6751  4883  335 
MW + R  Poisson  4128  8004  5112  428 
MW + R  Binomial  4128  8002  5108  428 
MW + IR  Poisson  4128  8002  5200  343 
MW + IR  Binomial  4128  8000  5204  341 
FW  Poisson  4128  6750  4912  281 
FW  Binomial  4128  6742  4910  280 
FNG  Poisson  8192  12,402  11,012  278 
FNG  Binomial  8192  12,464  11,121  280 
CW  CW + I  CW + R  CW + IR  MW  MW + I  MW + R  MW + IR  FW  FNG  

Poisson  13,751  137,58  13,332  13,443  13,761  13,789  13,456  13,804  14,127  12,816 
Binomial  18,841  18,912  16,090  17,621  18,877  18,801  17,002  18,811  18,891  15,117 
CW  CW + I  CW + R  CW + IR  MW  MW + I  MW + R  MW + IR  FW  

Poisson  $0.6409$  $0.6311$  $0.2766$  $0.4504$  $0.3826$  $0.3911$  $0.3281$  $0.4519$  $0.8012$ 
Binomial  $0.3885$  $0.3977$  $0.3519$  $0.3700$  $0.5804$  $0.5687$  $0.6140$  $0.5626$  $0.6748$ 
Linear  CW  CW + I  CW + R  CW + IR  MW  MW + I  MW + R  MW + IR  FW 
Poisson  $0.6199$  $0.6126$  $0.6616$  $0.6012$  $0.5881$  $0.5902$  $0.6211$  $0.5577$  $0.2742$ 
Binomial  $0.4323$  $0.4153$  $0.6059$  $0.5853$  $0.5648$  $0.5702$  $0.5814$  $0.5332$  $0.2131$ 
Exponential  CW  CW + I  CW + R  CW + IR  MW  MW + I  MW + R  MW + IR  FW 
Poisson  $0.5505$  $0.5345$  $0.7503$  $0.5462$  $0.5271$  $0.5383$  $0.7354$  $0.4811$  $0.1703$ 
Binomial  $0.5230$  $0.4841$  $0.6992$  $0.4737$  $0.5102$  $0.4890$  $0.6772$  $0.4283$  $0.1523$ 
CW + R  MW + R  

Number of relocations  18  14 
First relocation step  101  44 
min between relocations  17,812  32,991 
max between relocations  27,099  64,818 
mean between relocations  21,002  38,002 
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Drakopoulos, G.; Kanavos, A.; Tsakalidis, K. Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis. Algorithms 2017, 10, 40. https://doi.org/10.3390/a10020040
Drakopoulos G, Kanavos A, Tsakalidis K. Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis. Algorithms. 2017; 10(2):40. https://doi.org/10.3390/a10020040
Chicago/Turabian StyleDrakopoulos, Georgios, Andreas Kanavos, and Konstantinos Tsakalidis. 2017. "Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis" Algorithms 10, no. 2: 40. https://doi.org/10.3390/a10020040