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 $\left(\right)$ 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 $\left(\right)$ 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
 Zurek, W.H. Algorithmic Information Content, ChurchTuring Thesis, Physical Entropy, and Maxwell’s Demon; Technical Report; Los Alamos National Lab.: Los Alamos, NM, USA, 1990. [Google Scholar]
 Brillouin, L. Maxwell’s demon cannot operate: Information and entropy. I. J. Appl. Phys. 1951, 22, 334–337. [Google Scholar] [CrossRef]
 Herrmann, D. Heron von Alexandria. In Die antike Mathematik; Springer: Berlin/Heidelberg, Germany, 2014; pp. 257–288. [Google Scholar]
 Fouss, F.; Pirotte, A.; Renders, J.M.; Saerens, M. Randomwalk computation of similarities between nodes of a graph with application to collaborative recommendation. IEEE Trans. Knowl. Data Eng. 2007, 19, 355–369. [Google Scholar]
 Sinop, A.K.; Grady, L. A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm. In Proceedings of the 2007 IEEE 11th International Conference on Computer Vision, Rio de Janeiro, Brazil, 2007; pp. 1–8. [Google Scholar]
 Couprie, C.; Grady, L.; Najman, L.; Talbot, H. Power watersheds: A new image segmentation framework extending graph cuts, random walker and optimal spanning forest. In Proceedings of the 2009 IEEE 12th International Conference on Computer Vision, Kyoto, Japan, 27 September–4 October 2009; pp. 731–738. [Google Scholar]
 Blondel, V.D.; Guillaume, J.L.; Lambiotte, R.; Lefebvre, E. Fast unfolding of community hierarchies in large networks. J. Stat. Mech. Theory Exp. 2008. [Google Scholar] [CrossRef]
 Girvan, M.; Newman, M. Community Structure in Social and Biological Networks. Proc. Natl. Acad. Sci. USA 2002, 99, 7821–7826. [Google Scholar] [PubMed]
 Pons, P.; Latapy, M. Computing Communities in Large Networks Using Random Walks. Available online: https://arxiv.org/abs/physics/0512106 (accessed on 28 March 2017).
 Drakopoulos, G.; Kanavos, A.; Makris, C.; Megalooikonomou, V. On converting community detection algorithms for fuzzy graphs in Neo4j. In Proceedings of the 5th International Workshop on Combinations of Intelligent Methods and Applications (CIMA), Vietri sul Mare, Italy, 9–11 November 2015. [Google Scholar]
 Rosvall, M.; Bergstrom, C. Maps of Information Flow Reveal Community Structure in Complex Networks. Technical Report. Available online: https://arxiv.org/abs/0707.0609 (accessed on 28 March 2017).
 Drakopoulos, G.; Kanavos, A. Tensorbased Document Retrieval over Neo4j with an Application to PubMed Mining. In Proceedings of the 7th International Conference of Information, Intelligence, Systems, and Applications (IISA 2016), Chalkidiki, Greece, 13–15 July 2016. [Google Scholar]
 Panzarino, O.P. Learning Cypher; PACKT Publishing: Birmingham, UK, 2014. [Google Scholar]
 Rosenfeld, A. Fuzzy Graphs. Fuzzy Sets Appl. 1975, 513, 77–95. [Google Scholar]
 Drakopoulos, G.; Kanavos, A.; Tsakalidis, A. A Neo4j implementation of fuzzy random walkers. In Proceedings of the 9th Hellenic Conference on Artificial Intelligence (SETN 2016), Thessaloniki, Greece, 18–20 May 2016. [Google Scholar]
 Robinson, I.; Webber, J.; Eifrem, E. Graph Databases; O’Reilly: Sebastopol, CA, USA, 2013. [Google Scholar]
 Fortunato, S. Community Detection in Graphs. Phys. Rep. 2010, 486, 75–174. [Google Scholar] [CrossRef]
 Ng, A.Y.; Jordan, M.I.; Weiss, Y. On Spectral Clustering: Analysis and an algorithm. In Proceedings of the Advances in Neural Information Processing Systems (NIPS 2001), Vancouver, BC, Canada, 3–8 December 2001. [Google Scholar]
 Kernighan, B.; Lin, S. An Efficient Heuristic Procedure for Partitioning Graphs. Bell Syst. Tech. J. 1970, 49, 291–307. [Google Scholar] [CrossRef]
 Shi, J.; Malik, J. Normalized Cuts and Image Segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 2000, 22, 888–905. [Google Scholar]
 Lancichinetti, A.; Fortunato, S. Community Detection Algorithms: A Comparative Analysis. Phys. Rev. E 2009, 80, 056117. [Google Scholar] [CrossRef] [PubMed]
 Leskovec, J.; Lang, K.J.; Mahoney, M.W. Empirical Comparison of Algorithms for Network Community Detection. In Proceedings of the 19th International Conference on World Wide Web (WWW 2010), Raleigh, NC, USA, 26–30 April 2010; pp. 631–640. [Google Scholar]
 Carrington, P.J.; Scott, J.; Wasserman, S. Models and Methods in Social Network Analysis; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
 Scott, J. Social Network Analysis: A Handbook; SAGE Publications: Thousand Oaks, CA, USA, 2000. [Google Scholar]
 Drakopoulos, G.; Kanavos, A.; Tsakalidis, A. Evaluating Twitter Influence Ranking with System Theory. In Proceedings of the 12th International Conference on Web Information Systems and Technologies (WEBIST), Rome, Italy, 23–25 April 2016. [Google Scholar]
 Kontopoulos, S.; Drakopoulos, G. A space efficient scheme for graph representation. In Proceedings of the 26th International Conference on Tools with Artificial Intelligence (ICTAI 2014), Limassol, Cyprus, 10–12 November 2014; pp. 299–303. [Google Scholar]
 Langville, A.; Meyer, C. Google’s PageRank and Beyond: The Science of Search Engine Rankings; Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
 Page, L.; Brin, S.; Motwani, R.; Winograd, T. The PageRank Citation Ranking: Bringing Order to the Web; Stanford InfoLab: Stanford, CA, USA, 1999. [Google Scholar]
 Kleinberg, J.M. Authoritative Sources in a Hyperlinked Environment. In Proceedings of the Symposium of Discrete Algorithms (SODA), San Francisco, CA, USA, 25–27 January 1998; pp. 668–677. [Google Scholar]
 Newman, M. Networks: An Introduction; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
 Newman, M.E. Fast Algorithm for Detecting Community Structure in Networks. Phys. Rev. E 2004, 69, 066133. [Google Scholar] [CrossRef] [PubMed]
 Kafeza, E.; Kanavos, A.; Makris, C.; Chiu, D. Identifying Personalitybased Communities in Social Networks. In Proceedings of the Legal and Social Aspects in Web Modeling (Keynote Speech) in Conjunction with the International Conference on Conceptual Modeling (ER) (LSAWM), Hong Kong, China, 11–13 November 2013. [Google Scholar]
 Kafeza, E.; Kanavos, A.; Makris, C.; Vikatos, P. Predicting Information Diffusion Patterns in Twitter. In Proceedings of the Artificial Intelligence Applications and Innovations (AIAI), Rhodes, Greece, 19–21 September 2014; pp. 79–89. [Google Scholar]
 Kanavos, A.; Perikos, I. Towards Detecting Emotional Communities in Twitter. In Proceedings of the 9th IEEE International Conference on Research Challenges in Information Science (RCIS), Athens, Greece, 13–15 May 2015; pp. 524–525. [Google Scholar]
 Kafeza, E.; Kanavos, A.; Makris, C.; Vikatos, P. TPICE: Twitter Personality based Influential Communities Extraction System. In Proceedings of the IEEE International Congress on Big Data, Anchorage, AK, USA, 27 June–2 July 2014; pp. 212–219. [Google Scholar]
 Zamparas, V.; Kanavos, A.; Makris, C. Real Time Analytics for Measuring User Influence on Twitter. In Proceedings of the 27th IEEE International Conference on Tools with Artificial Intelligence (ICTAI), Vietri sul Mare, Italy, 9–11 November 2015. [Google Scholar]
 Kanavos, A.; Perikos, I.; Vikatos, P.; Hatzilygeroudis, I.; Makris, C.; Tsakalidis, A. Conversation Emotional Modeling in Social Networks. In Proceedings of the 26th IEEE International Conference on Tools with Artificial Intelligence (ICTAI), Limassol, Cyprus, 10–12 November 2014; pp. 478–484. [Google Scholar]
 Kanavos, A.; Perikos, I.; Vikatos, P.; Hatzilygeroudis, I.; Makris, C.; Tsakalidis, A. Modeling Retweet Diffusion using Emotional Content. In Proceedings of the Artificial Intelligence Applications and Innovations (AIAI), Rhodes, Greece, 19–21 September 2014; pp. 101–110. [Google Scholar]
 Kephart, J.O.; White, S.R. Directedgraph epidemiological models of computer viruses. In Proceedings of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, CA, USA, 20–22 May 1991; pp. 343–359. [Google Scholar]
 Ren, J.; Yang, X.; Yang, L.X.; Xu, Y.; Yang, F. A delayed computer virus propagation model and its dynamics. Chaos Solitons Fractals 2012, 45, 74–79. [Google Scholar] [CrossRef]
 Tugnait, J.K.; Luo, W. On channel estimation using superimposed training and firstorder statistics. In Proceedings of the 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’03), Hong Kong, China, 6–10 April 2003; pp. 4–624. [Google Scholar]
 Tong, L.; Xu, G.; Kailath, T. Blind identification and equalization based on secondorder statistics: A time domain approach. IEEE Trans. Inf. Theory 1994, 40, 340–349. [Google Scholar] [CrossRef]
 Belouchrani, A.; AbedMeraim, K.; Cardoso, J.F.; Moulines, E. A blind source separation technique using secondorder statistics. IEEE Trans. Signal Process. 1997, 45, 434–444. [Google Scholar] [CrossRef]
 Mendel, J.M. Tutorial on higherorder statistics (spectra) in signal processing and system theory: Theoretical results and some applications. Proc. IEEE 1991, 79, 278–305. [Google Scholar] [CrossRef]
 Chua, K.C.; Chandran, V.; Acharya, U.R.; Lim, C.M. Application of higher order statistics (spectra) in biomedical signals—A review. Med. Eng. Phys. 2010, 32, 679–689. [Google Scholar] [PubMed]
 Drakopoulos, G.; Megalooikonomou, V. An adaptive higher order scheduling policy with an application to biosignal processing. In Proceedings of the 2016 Symposium Series on Computational Intelligence (SSCI 2016), Athens, Greece, 6–9 December 2016; pp. 921–928. [Google Scholar]
 Comon, P. Independent component analysis, a new concept? Signal Process. 1994, 36, 287–314. [Google Scholar] [CrossRef]
 Delorme, A.; Sejnowski, T.; Makeig, S. Enhanced detection of artifacts in EEG data using higherorder statistics and independent component analysis. Neuroimage 2007, 34, 1443–1449. [Google Scholar] [CrossRef] [PubMed]
 Priest, D.M. Algorithms for arbitrary precision floating point arithmetic. In Proceedings of the 10th IEEE Symposium on Computer Arithmetic, Grenoble, France, 26–28 June 1991; pp. 132–143. [Google Scholar]
 Drakopoulos, G. Tensor fusion of affective Twitter metrics in Neo4j. In Proceedings of the 7th International Conference of Information, Intelligence, Systems, and Applications (IISA 2016), Chalkidiki, Greece, 13–15 July 2016. [Google Scholar]
 Drakopoulos, G.; Megalooikonomou, V. Regularizing Large Biosignals with Finite Differences. In Proceedings of the 7th International Conference of Information, Intelligence, Systems, and Applications (IISA 2016), Chalkidiki, Greece, 13–15 July 2016. [Google Scholar]
 Drakopoulos, G.; Kontopoulos, S.; Makris, C. Eventually consistent cardinality estimation with applications in biodata mining. In Proceedings of the 31st Annual ACM Symposium on Applied Computing, Pisa, Italy, 4–8 April 2016; pp. 941–944. [Google Scholar]
 Hamming, R.W. On the distribution of numbers. Bell Syst. Tech. J. 1970, 49, 1609–1625. [Google Scholar]
 Aggarwal, C.C.; Zhai, C. Mining Text Data; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
 De Jong, K. Learning with genetic algorithms: An overview. Mach. Learn. 1988, 3, 121–138. [Google Scholar] [CrossRef]
 De Jong, K.A.; Spears, W.M. Using Genetic Algorithms to Solve NPComplete Problems. In Proceedings of the ICGA, Fairfax, VA, USA, 4–7 June 1989; pp. 124–132. [Google Scholar]
 Saad, Y.; Schultz, M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 1986, 7, 856–869. [Google Scholar] [CrossRef]
 Morgan, R.B. Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 2000, 21, 1112–1135. [Google Scholar] [CrossRef]
 Leskovec, J.; Chakrabarti, D.; Kleinberg, J.; Faloutsos, C.; Ghahramani, Z. Kronecker graphs: An approach to modeling networks. J. Mach. Learn. Res. 2010, 11, 985–1042. [Google Scholar]
 Leskovec, J.; Kleinberg, J.; Faloutsos, C. Graphs over time: Densification laws, shrinking diameters and possible explanations. In Proceedings of the Eleventh ACM SIGKDD International Conference on Knowledge Discovery in Data Mining (KDD05), Chicago, IL, USA, 21–24 August 2005; pp. 177–187. [Google Scholar]
 Tsourakakis, C.E. Fast counting of triangles in large real networks without counting: Algorithms and laws. In Proceedings of the ICDM, Pisa, Italy, 15–19 December 2008; pp. 608–617. [Google Scholar]
 Drakopoulos, G.; Kanavos, A.; Makris, C.; Megalooikonomou, V. Finding fuzzy communities in Neo4j. In Smart Innovation, Systems, and Technologies; Howlett, R.J., Jain, L.C., Eds.; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
 Drakopoulos, G.; Baroutiadi, A.; Megalooikonomou, V. Higher order graph centrality measures for Neo4j. In Proceedings of the 6th International Conference of Information, Intelligence, Systems, and Applications (IISA 2015), Corfu, Greece, 6–8 July 2015. [Google Scholar]
Symbol  Meaning 

$\stackrel{\u25b5}{=}$  Definition or equality by definition 
⊗  Kronecker tensor product 
$\left(\right)$  Set with elements ${s}_{1},{s}_{2},\dots ,{s}_{n}$ 
$\left\xb7\right$  Set cardinality or path length (depending on the context) 
$\left(\right)$  Path comprised of edges ${e}_{1},\dots \phantom{\rule{0.166667em}{0ex}},{e}_{m}$ 
${K}_{n}$  Complete graph with n vertices and $\left(\right)$ 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 
$\left(\right)$  Sequence of elements ${s}_{k}$ 
$\mathcal{H}\left(\right)open="("\; close=")">{s}_{1},\dots ,{s}_{n}$  Harmonic mean of elements ${s}_{1},\dots ,{s}_{n}$ 
$\mathcal{H}\left(\right)open="("\; close=")">{s}_{1},\dots ,{s}_{n};{\tau}_{0}$  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 $f\left(x\right)$ 
$\left(\right)$  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