Abstract
Graph theory has been developed in regard to its applications in travel, biology, energetics, and many other fields. The general optimization mindset dominating these researches has addressed to graph theory the questions which were often related to finding a shortest path between nodes, as being of the minimum time delay for information transmission and of the minimum cost for connection maintenance. Not surprisingly, the very definition of distance between two vertices in a graph is given as the geodesic distance, i.e., the shortest path connecting them. With respect to the graph metric, a complex network of weighted edges is rather considered as a minimum weight spanning tree of the underlying graph, i.e., a subset of paths that has no cycles but still connects to every vertices at the lowest total cost. However, in many problems of practical interest found in biology, sociology, and economics, the existence of many paths of different lengths as well as a nexus of cycles traversing the nodes in complex interaction graphs do matter.
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References
A.G. Ableton, Live 8 [computer software] (Berlin, 2009)
M. Abramovitz, I.A. Stegun, The Editors of the Handbook of Mathematical Functions (Dover, New York, 1986)
L.Ts. Adzhemyan, N.V. Antonov, A.N. Vasiliev, Field Theoretic Renormalization Group in Fully Developed Turbulence (Gordon and Breach, London, 1998)
R.P. Agaev, P.Yu. Chebotarev, On determining the eigenprojection and components of a matrix. Automat. Rem. Contr. 63(10), 1537 (2002)
R.K. Ahuja, T.L. Magnanti, J.B. Orlin, T.L. Magnanti, Network Flows: Theory, Algorithms, and Applications (Prentice Hall, United States, 1993)
R. Albert, H. Jeong, A.-L. Barabási, The diameter of the World Wide Web. Nature 401, 130–131 (1999)
R. Albert, A.-L. Barabási, Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47 (2002)
D.J. Aldous, J. Pitman, Brownian bridge asymptotic for random mappings. Random. Struct. Algor. 5, 487 (1994)
D.J. Aldous, J.A. Fill, Reversible Markov Chains and Random Walks on Graphs (University of California, Berkeley, 2002)
A. Anderson, Y. Sinoto, New radiocarbon ages for colonization sites in East Polynesia. Asian Perspect. 41, 242 (2002)
N.V. Antonov, J. Honkonen, Field theoretic renormalization group for a nonlinear diffusion equation. Phys. Rev. E 66, 046105 (2002)
A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469, 93 (2008)
N. Aubry, On the hidden beauty of the proper orthogonal decomposition. Theor. Comp. Fluid Dyn. 2, 339 (1991)
N. Aubry, R. Guyonnet, R. Lima, J. Stat. Phys. 64, 683 (1991)
N. Aubry, L. Lima, Spatio-temporal Symmetries (Centre de Physique Theorique, Luminy, Marseille, France, 1993), Preprint CPT-93/P.2923
F.R. Bach, M.I. Jordan, Learning spectral clustering. Technical report, UC Berkeley, http://www.cs.berkeley.edu/fbach. Accessed 2003; Tutorial given at ICML 2004 international conference on machine learning, Banff, Alberta, Canada, 2004
J. Balakrishnan, Spatial curvature effects on molecular transport by diffusion. Phys.Rev. E 61, 4648 (2000)
R.B. Bapat, T.E.S. Raghavan, in Nonnegative Matrices and Applications, Encyclopedia of Mathematics and its Applications, ed. by R. Doran, P. Flajolet, M. Ismail, T.-Y. Lam, E. Lutwak, (Cambridge University Press, New York, 1997)
R.B. Bapat, I. Gutman, W. Xiao, A simple method for computing resistance distance. Z. Naturforsch. 58a, 494 (2003)
N.T.J. Bailey, The Mathematical Theory of Infectious Diseases, 2nd edn. (Griffin, London, 1975)
Ph. Baldi, in The Foundations of Latin, Trends in Linguistics: Studies and Monographs, vol. 117, ed. by Gast, Volker, (Mouton de Gruyter, Berlin, New York, 2002)
A.-L. Barabási, R. Albert, Emergence of scaling in random networks. Science 286, 509 (1999)
G.P. Basharin, A.N. Langville, V.A. Naumov, The life and work of A.A. Markov. Linear Algebra Appl. 386, 3–26 (2004)
V. Batagelj, T. Pisanski, D. Keržic, Automatic clustering of languages. Comput. Linguist. 18(3), 339 (1992)
M. Batty, A New Theory of Space Syntax, CASA Working Paper, vol. 75 (UCL Centre For Advanced Spatial Analysis Publications, London, UK, 2004)
P. Bellwood, P. Koon, Lapita colonists leave boats unburned! Antiquity 63(240), 613 (1989)
C.M. Bender, S. Boettcher, P.N. Meisinger, Universality in random walk models with birth and death. Phys. Rev. Lett. 75, 3210 (1995)
M. Ben Hamed, F. Wang, Stuck in the forest: Trees, networks and Chinese dialects. Diachronica 23(1), 29–60 (2006)
A. Ben-Israel, A. Charnes, Contributions to the theory of generalized inverses. J. Soc. Ind. Appl. Math. 11(3), 667 (1963)
A. Ben-Israel, TH.N.E. Greville, Generalized inverses: Theory and applications, 2nd edn. (Springer, New York, 2003)
S.N. Bernstein, Sur lextension du théoremé limite du calcul des probabilies. Math. Ann. Bd. 97, 1–59 (1926)
G. Besson, Sur la multiplicité des valeurs propres du laplacien. Séminaire de théorie spectrale et géométrie (Grenoble) 5, 107–132 (1986–1987)
W. Bialek, Stability and noise in biochemical switches, in Advances in Neural Information Processing Systems, vol. 13, ed. by T.K. Leen, V. Tresp. Papers from Neural Information Processing Systems (NIPS) 2000 (MIT, Denver, CO, USA, 2001), pp. 75–81
G. Bianconi, N. Gulbahce, A.E. Motter, Local structure of directed networks. Phys. Rev. Lett. 100, 118701 (2008)
N. Biggs, Algebraic Graph Theory, 2nd edn. (Cambridge Mathematical Library, Cambridge, 1993)
N.L. Biggs, E.K. Lloyd, R.J. Wilson, Handbook of Combinatorics, vol. 2 (MIT, Cambridge, MA, USA, 1996)
S. Bilke, Shuffling Yeast Gene Expression Data (2000), Lund University - preprint TP 00-18; arXiv:physics/0006050 (2000)
S. Bilke, C. Peterson, Topological properties of citation and metabolic networks. Phys. Rev. E 64, 036106 (2001)
S. Bilke, T. Breslin, M. Sigvardsson, Probabilistic estimation of microarray data reliability and underlying gene expression. BMC Bioinformatics 4, 40 (2003)
T. Biyikoğlu, W. Hordijk, J. Leydold, T. Pisanski, P.F. Stadler, Graph Laplacians, nodal domains, and hyperplane arrangements. Linear Algebra Appl. 390, 155 (2004)
T. Biyikoglu, J. Leydold, P.F. Stadler, Laplacian eigenvectors of graphs – Perron-Frobenius and Faber-Krahn type theorems, in Springer Lecture Notes in Mathematics, vol. 1915, ed. by J.-M. Morel, B. Teissier (Springer-verlag, Berlin, Heidelberg, 2007)
A. Björner, L. Lovász, P. Shor, Chip-firing games on graphs. Eur. J. Combin. 12, 283 (1991)
A. Björner, L. Lovász, Chip-firing games on directed graphs. J. Algebr. Comb. 1, 305 (1992)
Ph. Blanchard, D. Volchenkov, Intelligibility and first passage times in complex urban networks. Proc. R. Soc. A 464, 2153 (2008). doi:10.1098/rspa.2007.0329
Ph. Blanchard, D. Volchenkov, in Mathematical Analysis of Urban Spatial Networks, Springer series: Understanding Complex Systems, vol. XIV, ed. by Kelso, J.A. Scott, (Springer-verlag, Berlin, Heidelberg, 2009)
Ph. Blanchard, D. Volchenkov, Probabilistic embedding of discrete sets as continuous metric spaces. Stochastics: Int. J. Prob. Stoch. Proc. (formerly: Stochastics and Stochastic Reports) 81(3), 259 (2009)
Ph. Blanchard, F. Petroni, M. Serva, D. Volchenkov, Geometric representations of language taxonomies. Comput. Speech Lang. (2010). doi: 10.1016/j.csl.2010.05.003
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006)
S. Boettcher, M. Moshe, Statistical models on spherical geometries. Phys. Rev. Lett. 74, 2410 (1995)
M. Boguñá, F. Papadopoulos, D. Krioukov, Sustaining the internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010)
B. Bollobas, Graph Theory (Springer-Verlag, New York, 1979)
B. Bollobas, Modern Graph Theory (Springer-Verlag, New York, 1998)
B. Bollobas, A. Thomason, Combinatorics, Geometry and Probability (Cambridge University Press, Cambridge 2004)
R.P. Bolton, Building For Profit (Reginald Pelham Bolton, New York, 1922)
M. Bona, Combinatorics of Permutations (Chapman Hall-CRC, Boca Raton, FL, 2004). ISBN 1-58488-434-7
S. Bornholdt, Less is more in modeling large genetic networks. Science 310(5747), 449 (2005)
G. Brightwell, I. Leader, A. Scott, A. Thomason, Combinatorics and Probability (Cambridge University Press, Cambridge, 2007)
J. Bricmont, A. Kupiainen, Renormalization group and the Ginzbug-Landau equation. Commun. Math. Phys. 150, 193 (1992)
J. Bricmont, A. Kupiainen, G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun. Pur. Appl. Math. 47, 893 (1994)
J. Bricmont, A. Kupiainen, High temperature expansions and dynamical systems. Commun. Math. Phys. 178, 703 (1996)
J. Bricmont, A. Kupiainen, J. Xin, Global large time self-similarityof a thermal-diffusive combustion system with critical nonlinearity. J. Differ. Equ. 130(1), 9 (1996)
R. Brooks, E. Makover, Random Construction of Riemann Surfaces, Preprint (Department of Mathematics, Technion, 2001)
C. Brown, A Portrait of Mendelssohn (Yale University Press, New Haven and London, 2003)
E. Bryant, The Quest for the Origins of Vedic Culture: The Indo-Aryan Migration Debate (Oxford University Press, New York, 2001)
T.S. Bunch, L. Parker, Feynman propagator in curved spacetime: A momentum-space representation. Phys. Rev. D 20(10), 2499 (1979)
E.M. Burns, Intervals, scales, and tuning, in The Psychology of Music, 2nd edn. ed. by D. Deutsch (Academic, San Diego, CA, 1999)
P. Buser, On the bipartition of graphs. Discrete Appl. Math. 9, 105 (1984)
T. Brylawski, J.G. Oxley, The tutte polynomial and its applications, in Matroid Applications, ed. by N. White (Cambridge University Press, Cambridge, 1992), pp. 123–225
S. Butler, Interlacing for weighted graphs using the normalized Laplacian. Electron. J. Linear. Al. 16, 90 (2007)
S.L. Campbell, C.D. Meyer, N.J. Rose, Applications of the drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31(3), 411 (1976)
S.L. Campbell, C.D. Meyer Jr., Generalized Inverses of Linear transformations (Dover Publications, New York, 1979)
J.W. Cannon, W.J. Floyd, R. Kenyon, W.R. Parry, Hyperbolic geometry, in Flavors of Geometry, vol. 31, ed. by S. Levy (Cambridge Univ. Press, Cambridge, 1997)
D. Cassi, S. Regina, Random walks on bundled structures. Phys. Rev. Lett. 76, 2914 (1996)
L.L. Cavalli-Sforza, Genes, Peoples, and Languages (North Point Press, New York, 2000)
P. Chan, M. Schlag, J. Zien, IEEE Trans. CAD-Integrated Circuits and Syst. 13, 1088 (1994)
A. Chan, C. Godsil, Symmetry and eigenvectors, in Graph Symmetry, Algebraic Methods and Applications, ed. by G. Hahn, G. Sabidussi (Kluwer, Dordrecht, The Netherlands, 1997), p. 75
A.K. Chandra, P. Raghavan, W.L. Ruzzo, R. Smolensky, P. Tiwari, The electrical resistance of a graph captures its commute and cover times. Comput. Complex. 6(4), 312 (1996)
G. Chartrand, Introductory Graph Theory (Dover, New York, 1985). ISBN 0-486-24775-9
H. Chaté, P. Manneville, Spatiotemporal intermittency in coupled map lattices. Physica D 32, 409–422 (1988)
H. Chaté, P. Manneville, Transition to turbulence via spatio-temporal intermittency. Europhys. Lett. 6, 591–595 (1988)
H. Chaté, P. Manneville, Collective behaviors in coupled map lattices with local and nonlocal connections. Chaos 2, 307–313 (1992)
H. Chaté, P. Manneville, Collective behaviors in spatially extended systems: Interactions and synchronous updating. Prog. Theor. Phys. 87, 1 (1992)
J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, Papers dedicated to Salomon Bochner, ed. by Gunning, Robert C., (Princeton University Press, Princeton, 1969), p. 195
H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum. Discrete Appl. Math. 155, 654 (2007)
S.Y. Cheng, Eigenfunctions and nodal sets. Commun. Math. Helv. 51, 43 (1976)
F.R.K. Chung, Lecture Notes on Spectral Graph Theory (AMS Publications Providence, 1997)
F. Chung, L. Lu, V. Vu, Spectra of random graphs with given expected degrees. Proc. Natl. Acad. Sci. U.S.A. 100(11), 6313 (2003)
F. Chung, Laplacians and the cheeger inequality for directed graphs. Ann. Comb. 9, 1 (2005)
P. Collet, J.-P. Eckmann, The number of large graphs with a positive density of triangles. J. Stat. Phys. 109(5–6), 923 (2002)
G.C. Conant, A. Wagner, Convergent evolution in gene circuits. Nat. Genet. 34(3), 264 (2003)
J.H. Conway, R.K. Guy, Arrangement numbers, in The Book of Numbers, ed. by J.H. Conway, (Springer-Verlag, New York, 1996)
D. Coppersmith, P. Tetali, P. Winkler. Collisions among random walks on a graph. SIAM J. Discrete Math. 6(3), 363 (1993)
R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function. Adv. Comput. Math. 5, 329 (1996)
T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 2nd edn. Chapter 21: Data Structures for Disjoint Sets (The MIT Press, Cambridge, Massachusetts, 2001), pp. 498–524. ISBN 0-262-03293-7
M.G. Cosenza, R. Kapral, Coupled maps on fractal lattices. Phys. Rev. A 46(4), 1850 (1992)
M.G. Cosenza, R. Kapral, Spatiotemporal intermittency on fractal lattices. Chaos 4, 99 (1994)
M.G. Cosenza, A. Parravano, Turbulence in globally coupled maps. Phys. Rev. E 53, 6032 (1996)
M.G. Cosenza, K. Tucci, Transition to turbulence in coupled maps on hierarchical lattices. Chaos Soliton. Fract. 11, 2039–2044 (2000)
M.G. Cosenza, K. Tucci, Turbulence in small-world networks. Phys. Rev. E 65, 036223 (2002)
L.D.F. Costa, F.A. Rodrigues, G. Travieso, P.R.V. Boas, Characterization of complex networks: A survey of measurements. Adv. Phys. 56, 167 (2007)
L.D.F. Costa, G. Travieso, Exploring complex networks through random walks. Phys. Rev. E 75, 016102 (2007)
R. Coutinho, B. Fernandez, R. Lima, A. Meyroneinc, Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52(4), 0303–6812 (Print), 1432–1416 (Online) (2006)
T.M. Cover, J.A. Thomas, Elements of Information Theory, (Wiley, New York, 1991)
P. Crucitti, V. Latora, S. Porta, Centrality in networks of urban streets. Chaos 16, 015113 (2006)
D.M. Cvetkovic, P. Rowlinson, S. Simic, Eigenspaces of Graphs, in series Encyclopedia of Mathematics and Its Applications, vol. 66, ed. by R. Doran, P. Flajolet, M. Ismail, T.-Y. Lam, E. Lutwak, (Cambridge University Press, Cambridge, UK, 1997)
D.M. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs, 3rd Rev. edn. (Academic, New York, 1980)
D. d’Urville, Sur les îles du Grand Océan. Bulletin de la Société de Géographie 17, 1 (1832)
O.C. Dahl, Avhandlinger utgitt av Egede-Instituttet, vol. 3 (Arne Gimnes Forlag, Oslo, 1951)
C. Dahlhaus, Harmony, in Grove Music Online, ed. by L. Macy (2007), http://www.grovemusic.com . Accessed 24 Feb 2007
G.B. Dantzig, R. Fulkerson, S.M. Johnson, Solution of a large-scale traveling salesman problem. Oper. Res. 2, 393 (1954)
Database. The database of 200 words most resistant to changes is available at http://univaq.it/~serva/languages/languages.html
E.B. Davis, G.M.L. Gladwell, J. Leydold, P.F. Stadler, Discrete nodal domain theorems. Linear Algebra Appl. 336, 51 (2001)
C. de Dominicis, Techniques de renormalisation de la théorie des champs et dynamique des phénomène critiques. J. Phys. (Paris) 37, Colloq. C1, C1-247 (1976)
H. de Jong, R. Lima, Modeling the dynamics of genetic regulatory networks: Continuous and discrete approaches, in Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, vol. 671, ed. by J.-R. Chazottes, B. Fernandez. Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2005), p. 307
Y.C. de Verdiére, Construction de laplaciens dont une partie finie du spectre est donnée. Ann. Sci. Ecole. Norm. S. 20, 599 (1987)
Y.C. de Verdiére, Spectres de Graphes, Cours Spécialisés 4, (Société Mathématique de France (1998) (in French)
B.S. De Witt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965)
B.S. De Witt, Quantum-field theory in curved space–time. Phys. Rep. 19, 295 (1975)
M. Dyer, A. Frieze, R.Kannan, A random polynomial-time algorithm for approximating the volume of convex bodies, Journal of the ACM 38 (1), 1 (1991)
I. Dhillon, Y. Guan, B. Kulis, A Unified View of Kernel k-means, Spectral clustering and Graph Cuts, Technical Report TR-04-25, University of Texas at Austin, 2004
I. Dhillon, Y. Guan, B. Kulis, Kernel k-means: spectral clustering and normalized cuts, in Proceedings of the 10th ACM SIGKDD international conference on Knowledge discovery and data mining, Seattle, WA, USA, 2004
P. Diaconis, Group Representations in Probability and Statistics (Institute of Mathematical Statistics, Hayward, CA, 1988)
J.M. Diamond, Express train to polynesia. Nature 336, 307 (1988)
R. Diestel, Graph Theory (Springer, Berlin, 2005)
R. Dickson, J. Ableson, W. Barnes, W. Reznikoff, Genetic regulation: the Lac control region. Science 187, 27 (1975)
C. Ding, X. He, K-means clustering via principal component analysis, in Proceedings of International Conference Machine Learning (ICML’2004), ACM, New York, July 2004, pp. 225–232
S. Dorogovtsev, J.F.F. Mendes, Evolution of networks. Adv. Phys. 51, 1079 (2002)
S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275 (2008)
S.N. Dorogovtsev, Lectures on Complex Networks (Oxford University Press, Oxford, 2010)
P.G. Doyle, J.L. Snell, Random Walks and Electrical Networks (Mathematical Association of America, Washington, DC, 1984); freely redistributable under the terms of the GNU General Public License (2000)
M.P. Drazin, Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506 (1958)
I. Dyen, J.B. Kruskal, P. Black An Indo-european classification: A lexicostatistical experiment. Trans. Am. Philos. Soc. 82(5), 1 (1992)
I. Dyen, J. Kruskal, P. Black, Comparative Indo-European Database, collected by Isidore Dyen. (Available at http://www.wordgumbo.com/ie/cmp/iedata.txt) Copyright (C) 1997 by Isidore Dyen, Joseph Kruskal, and Paul Black. The file was last modified on Feb 5 (1997). Redistributable for academic, non-commercial purposes
H. Dym, Linear Algebra in Action, in Series Graduate Studies in Mathematics, vol. 78 (AMS, 2007)
J.-P. Eckman, E. Moses, Curvature of co-links uncovers hidden thematic layers in the World Wide Web. Proc. Natl. Acad. Sci. U.S.A. 99, 5825 (2002)
T.M. Ellison, S. Kirby, Measuring language divergence by intra-lexical comparison, in Proceedings of the 21st International Conference on Computational Linguistics & 44th Annual Meeting of the Association for Computational Linguistics, Sydney, Australia, 2006
S.M. Embelton, Statistics in Historical Linguistics (Brockmeyer, Bochum, 1986)
D. Eppstein, Spanning trees and spanners, in Handbook of Computational Geometry, ed. by J.R. Sack, J. Urrutia (Elsevier, Amsterdam, 1999), pp. 425–461
I. Erdélyi, On the matrix equation Ax = λBx. J. Math. Anal. Appl. 17, 119 (1967)
P. Erdös, A. Rényi, On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17 (1960)
P. Erdös and A. Rényi, Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 14 295 (1963)
E. Estrada, J.A. Rodríguez-Velázquez, Subgraph centrality in complex networks. Phys. Rev. E 71, 056103 (2005)
E. Estrada, J.A. Rodríguez-Velázquez, Spectral measures of bipartivity. Phys. Rev. E 72, 046105 (2005)
European Environment Agency report Urban sprawl in Europe. The ignored challenge (2006). ISBN 92-9167-887-2
K. Fan, On a theorem of weyl concerning eigenvalues of linear transformations. Proc. Natl. Acad. Sci. U.S.A. 35, 652 (1949)
I.J. Farkas, I. Derenyi, A.-L. Barabasi, T. Vicsek, Spectra of “real-world” graphs: Beyond the semicircle law. Phys. Rev. E 64, 026704:1 (2001)
I. Farkas, I. Derényi, H. Jeong, Z. Neda, Z.N. Oltvai, E. Ravasz, A. Schubert, A.-L. Barabási, T. Vicsek, Networks in life: Scaling properties and eigenvalue spectra. Physica A 314, 25 (2002)
I.J. Farkas, H. Jeong, T. Vicsek, A.-L. Barabasi, Z.N. Oltvai, The topology of the transcription regulatory network in the yeast, S. cerevisiae. Physica A 318, 601 (2003)
M. Fiedler, Algebraic connectivity of graphs. Czech. Math. J. 23(98), 298–305 (1973); 25(146), (1975)
P. Forster, A. Toth, Toward a phylogenetic chronology of ancient Gaulish, Celtic, and Indo-European. Proc. Natl. Acad. Sci. U.S.A. 100(15), 9079 (2003)
P. Fouracre, The New Cambridge Medieval History (Cambridge University Press, Cambridge, UK, 1995–2007)
D.M. Franz, Markov Chains as Tools for Jazz Improvisation Aanalysis, Master’s Thesis, Industrial and Systems Engineering Department, Virginia Tech, 1998
L.C. Freeman, Centrality in social networks: Conceptual clarification. Soc. Networks 1, 215 (1979)
E. Friedgut, Sharp thresholds of graph properties, and the k-sat problem. J. Am. Math. Soc. 12, 1017 (1999)
J.S. Friedländer et al., Genetic structure of Pacific Islanders. PLoS Genet. 4(1), e19 (2008)
M. Friendly, Corrgrams: Exploratory displays for correlation matrices. Am. Stat. 56(4), 316 (2002)
P.M. Gade, H.A. Cerdeira, R. Ramaswamy, Coupled maps on trees. Phys. Rev. E 52, 2478 (1995)
P. Gade, Synchronization of oscillators with random nonlocal connectivity. Phys. Rev. E 54, 64 (1996)
Th.V. Gamkrelidze, V.V. Ivanov, The early history of Indo-European languages. Sci. Am. 262(3), 110 (1990)
Th.V. Gamkrelidze, V.V. Ivanov, Indo-European and the Indo-Europeans: A Reconstruction and Historical Analysis of a Proto-Language and a Proto-Culture, in series Trends in Linguistics: Studies and Monographs, vol. 80, ed. by Gast, Volker, (Mouton de Gruyter, Berlin, New York, 1995)
F.R. Gantmacher, The Theory of Matrices. Trans. from the Russian by K. A. Hirsch, vols. I and II, (Chelsea, New York, 1959)
T.S. Gardner, D. di Bernardo, D. Lorenz, J.J. Collins, Inferring genetic networks and identifying compound mode of action via expression profiling. Science 301, 102 (2003)
G. Gielis, R.S. MacKay, Coupled map lattices with phase transition. Nonlinearity 13, 867 (2000)
P. Gilkey, The spectral geometry of a Riemannian manifold. J. Differ. Geom. 10, 601 (1975)
M. Gimbutas, Old Europe in the fifth millenium B.C.: The European situation on the arrival of Indo-Europeans, in The Indo-Europeans in the Fourth and Third Millennia, ed. by E.C. Polomé (Karoma Publishers, Ann Arbor, 1982), pp. 1–60
M. Girvan, M.E.J. Newman, Community structure in social and biological networks. Proc. Natl. Acad. Sci. U.S.A. 99 (12), 7821 (2002)
E.L. Glaeser, J. Gyourko, Why is Manhattan So Expensive?, vol. 39, Manhattan Institute for Policy Research, Civic Report (2003)
L. Glass, S.A. Kauffman, The logical analysis of continuous non–linear biochemical control networks. J. Theor. Biol. 39, 103 (1973)
P.M. Gleiss, P.F. Stadler, A. Wagner, D.A. Fell, Relevant cycles in chemical reaction network. Adv. Complex Syst. 4, 207 (2001)
Ch. Godsil, G. Royle, Algebraic Graph Theory, in Springer Series: Graduate Texts in Mathematics, vol. 207 (Springer-Verlag, New York, 2001)
N. Goldenfeld, O. Martin, Y. Oono, F. Liu, Anomalous dimensions and the renormalization group in a nonlinear diffusion process. Phys. Rev. Lett. 64(12), 1361 (1990)
G.H. Golub, Ch.F. Van Loan, Matrix Computations, 3rd edn. Johns Hopkins Studies in Mathematical Sciences (The Johns Hopkins University Press, Baltimore, MD, 1996)
J. Gomez-Gardenes, V. Latora, Entropy Rate of diffusion processes on complex networks. Phys. Rev. E 78, 065102(R) (2008)
R.G. Gordon Jr. (ed.), Ethnologue: Languages of the World, 15th edn. (SIL International, Dallas, TX, 2005). Online version: http://www.ethnologue.com/
R. Gould, Graph Theory (Benjamin/Cummings, Menlo Park, CA, 1988)
A. Graham, Nonnegative Matrices and Applicable Topics in Linear Algebra (John Wiley & Sons, New York, 1987)
R.L. Graham, M. Grötschel and L. Lovász (eds.), Handbook of Combinatorics (Elsevier Science B.V., MIT, Amsterdam, Cambridge, MA 1995)
R.D. Gray, F.M. Jordan, Language trees support the express-train sequence of Austronesian expansion. Nature 405, 1052 (2000)
R.D. Gray, Q.D. Atkinson, Language-tree divergence times support the Anatolian theory of Indo-European origin. Nature 426, 435 (2003)
R.D. Gray, S.J. Greenhill, R.M. Ross, The pleasures and perils of darwinizing culture (with phylogenies). Biol. Theor. 2(4), 360 (2007)
R.D. Gray, A.J. Drummond, S.J. Greenhill, Language phylogenies reveal expansion pulses and pauses in pacific settlement. Science 323, 479 (2009)
P. Green, The Greco-Persian Wars (University of California Press, Berkeley, Los Angeles, London, 1996)
S.J. Greenhill, R. Blust, R.D. Gray, The austronesian basic vocabulary database: From bioinformatics to lexomics. Evol. Bioinform. 4, 271 (2008). The Austronesian Basic Vocabulary Database is available at http://language.psy.auckland.ac.nz/austronesian
W.H. Greub, Linear Algebra, 4th edn. Graduate Texts in Mathematics (Springer, New York, 1981)
J. Gross, Graph Theory, Textbooks and Resources at http://www.graphtheory.com/
A. Groenlund, Networking genetic regulations and neural computation: Directed network topology and its effect on the dynamics. Phys. Rev. E 70, 061908 (2004)
M.J. Hall Jr., Combinatorial Theory, 2nd edn. (Wiley, New York, 1986; 1998)
W.G. Hansen, How accessibility shapes land use. J. Am. Inst. Planners 25, 73 (1959)
F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969)
J.M. Harris, J.L. Hirst, M.J. Mossinghoff, Combinatorics and Graph Theory (Springer, New York, 2005)
R.E. Hartwig, More on the souriau-frame algorithm and the drazin inverse. SIAM J. Appl. Math. 31(1), 42 (1976)
M.B. Hastings, An ε-expansion for small-world networks. Euro Phys. J. B. 42, 297 (2004)
J. Hasty, J. Pradines, M. Dolnik, J.J. Collins, Noise-based switches and amplifiers for gene expression. Proc. Natl. Acad. Sci. U.S.A. 97, 2075–2080 (2000)
P. Heggarty, Interdisciplinary indiscipline? Can phylogenetic methods meaningfully be applied to language data and to dating language? in Phylogenetic Methods and the Prehistory of Languages, ed. by P. Forster, C. Renfrew (McDonald Institute for Archaeological Research, Cambridge, 2006), p. 183
P. Heggarty, Splits or waves? Trees or webs? Network analysis of language divergence. AHRC Conference on Cultural and Linguistic Diversity, Great Missenden, 9–13 December 2008
H.W. Hethcote, The mathematics of infectious diseases. Soc. Indus. Appl. Math. 42, 599 (2000)
L.A. Hiller, L.M. Isaacson, Experimental Music-Composition with an Electronic Computer (McGraw–Hill, New York, 1959)
B. Hillier, J. Hanson, The Social Logic of Space (Cambridge University Press, Cambridge, 1984). ISBN 0-521-36784-0
B. Hillier, Space is the Machine: A Configurational Theory of Architecture (Cambridge University Press, Cambridge, 1999). ISBN 0-521-64528-X
B. Hillier, The Common Language of Space: A Way of Looking at the Social, Economic and Environmental Functioning of Cities on a Common Basis (Bartlett School of Graduate Studies, London, 2004)
R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1990)
J.M. Houlrik, I. Webman, M.H. Jensen, Mean-field theory and critical behavior of coupled map lattices. Phys. Rev. A 41, 4210 (1990)
B.A. Huberman, L.A. Adamic, Growth dynamics of the world wide web. Nature 401, 131 (1999)
B.D. Hughes, Random Walks and Random Environments (Oxford University Press, New York, 1996)
M.E. Hurles, E. Matisoo-Smith, R.D. Gray, D. Penny, Untangling pacific settlement: The edge of the knowable. Trends Ecol. Evol. 18, 531 (2003)
M.E. Hurles, B.C. Sykes, M.A. Jobling, P. Forster, The dual origins of the Malagasy in Island Southeast Asia and East Africa: Evidence from maternal and paternal lineages. Am. J. Hum. Gen. 76, 894 (2005)
A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis (Wiley, New York, 2001)
F. Jacob, J. Monod, Genetic regulatory mechanisms in th synthesis of proteins. J. Mol. Biol. 3, 318 (1961)
S. Janson, T. Łuszak, A. Rucinski, Random Graphs (Wiley, New York 2000), 333 p
M.R. Jerrum, A. Sinclair, Approximating the permanent. SIAM J. Comput. 18(6), 1149 (1989)
M. Jiang, Ya.B. Pesin, Equilibrium measures for coupled map lattices. Commun. Math. Phys. 193, 677 (1998)
B. Jiang, C. Claramunt, Topological analysis of urban street networks. Environ. Plann. B 31, 151 (2004)
I.T. Jolliffe, Principal Component Analysis, vol. XXIX, 2nd edn. Springer Series in Statistics (Springer, New York, 2002)
K. Jones, Compositional applications of stochastic processes. Comput. Music J. 5(2), (1981)
P.E.T. Jorgensen, E.P.J. Pearse, Operator theory of electrical resistance networks, arXiv:0806.3881; (2008)
P.E.T. Jorgensen, E.P.J. Pearse, A Hilbert space approach to effective resistance metric. Complex Anal. Oper. Th. 4(4), 975–1030 (2009). doi: 10.1007/s11785-009-0041-1
M. Kac, On the notion of recurrence in discrete stochastic processes. Bull. Am. Math. Soc. 53, 1002 (1947) [Reprinted in M. Kac Probability, Number Theory, and Statistical Physics: Selected Papers, K. Baclawski, M.D. Donsker (eds.), Cambridge, Mass: MIT Press, Series: Mathematicians of our time Vol. 14, 231 (1979)]
K. Kaneko, Period-doubling of kink-antikink patterns, quasi-periodicity in antiferro-like structures and spatial intermittency in coupled map lattices — toward a prelude to a “field theory of chaos”. Prog. Theor. Phys. 72, 480–486 (1984)
K. Kaneko, Spatiotemporal intermittency in coupled map lattices. Prog. Theor. Phys. 74, 1033–1044 (1985)
K. Kaneko, Lyapunov analysis and information flow in coupled map lattices. Physica D 23, 436–447 (1986)
K. Kaneko, Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements. Physica D 41, 137–172 (1990)
K. Kaneko (ed.), Theory and applications of coupled map lattices, Nonlinear science: theory and applications (Wiley, New York, Chichester, 1993)
K. Karhunen, Zur spektraltheorie stochatischer prozesse. Ann. Acad. Sci. Fenn. A:1, 34 (1944)
S. Kauffman, C. Peterson, B. Samuelsson, C. Troein, Random boolean network models and the yeast transcriptional network. Proc. Natl. Acad. Sci. U.S.A. 100, 14796 (2003)
M. Kayser et al., Melanesian and asian origins of polynesians: mtDNA and Y chromosome gradients across the pacific. Mol. Biol. Evol. 23, 2234 (2006)
S.M. Keane, Stock Market Efficiency (Philip Allan Ltd, Oxford, 1983)
A. Keller, Model genetic circuits encoding autoregulatory transcription factors. J. Theor. Biol. 172, 169 (1995)
F. Kelly, Reversibility and stochastic networks (Wiley, New York, 1979)
B.Kessler, Phonetic comparison algorithms. T. Philol. Soc. 103(2), 243 (2005)
L. Kim, A. Kyrikou, M. Desbrun, G. Sukhatme, An implicit based haptic rendering technique. Intelligent Robots and Systems, 3, 2943 (2002)
B.M. Kim, J. Rossignac, Localized bi-laplacian solver on a triangle mesh and its applications. GVU Technical Report Number: GIT-GVU-04-12, College of Computing, Georgia Tech. (2004).
P.V. Kirch, The Lapita Peoples: Ancestors of the Oceanic World (Blackwell, Cambridge, Mass, 1997)
P.V. Kirch, On the road of the winds: An archaeological history of the Pacific Islands before European contact (University of California Press, Berkley, CA, 2000)
D.J. Klein, M. Randić, Resistance distance. J. Math. Chem. 12(4), 81 (1993)
K. Klemm, S. Bornholdt, Robust gene regulation: Deterministic dynamics from asynchronous networks with delay. Phys. Rev. E 72, 055101 (2005)
K. Klemm, S. Bornholdt, Topology of biological networks and reliability of information processing. Proc. Natl. Acad. Sci. U.S.A. 102, 18414 (2005)
V.F. Kolchin, Random Mappings (Optimization Software, New York, 1986)
B. Kolman, D.R. Hill, Elementary Linear Algebra with Applications, 9th edn. (Prentice Hall, New Jersey, USA, 2007)
B.O. Koopman, Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. U.S.A. 17, 315 (1931)
B. Kozma, M.B. Hastings, G. Korniss, Roughness scaling for Edwards-Wilkinson relaxation in small-world networks. Phys. Rev. Lett. 98(10), 108701 (2004)
K.S. Krell, Gimbutas kurgan-PIE homeland hypothesis: A linguistic critique, in Archaeology and Language, vol. II, ed. by R. Blench, M. Spriggs (Routledge, London 1998)
D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, M. Boguñá, Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)
D. Krioukov, F. Papadopoulos, A. Vahdat, M. Boguñá, Curvature and temperature of complex networks. Phys. Rev. E 80, 035101(R) (2009)
J. Lamping, R. Rao, P. Pirolli, A focus + context technique based on hyperbolic geometry for visualizing large hierarchies, CHI ’95 Proceedings of the SIGCHI conference on Human factors in computing systems, ed. by I.R. Katz, R. Mack, L. Marks, M.B. Rosson, J. Nielsen, (ACM Press/Addison-Wesley Publishing Co. New York, NY, USA, 1995)
J.-L. Lagrange, Œuvres, vol. 1, (Gauthier-Villars, Paris, 1867), pp. 72–79 (in French)
G. Larson et al., Phylogeny and ancient DNA of Sus provides insights into neolithic expansion in Island Southeast Asia and Oceania. Proc. Natl. Acad. Sci. U.S.A. 104(12), 4834 (2007)
V. Latora, M. Marchiori, Efficient behavior of small-world networks. Phys. Rev. Lett. 87, 198701 (2001)
V. Latora, M. Marchiori, Vulnerability and protection of infrastructure networks. Phys. Rev. E 71, 015103(R) (2005)
E.A. Leicht, P. Holme, M.E.J. Newman, Vertex similarity in networks. Phys. Rev. E 73, 026120 (2006)
A. Lemaître, H. Chaté, Nonperturbative renormalization group for chaotic coupled map lattices. Phys. Rev. Let. 80, 5528 (1998)
V.I. Levenshtein, Binary codes capable of correcting deletions, insertions and reversals. Sov. Phys. Doklady 10, 707 (1966)
D.A. Levitt, A representation for musical dialects, in Machine Models of Music, ed. by S. Schwanauer, D. Levitt (MIT, Cambridge, Massachusetts, 1993), pp. 456-469
L. Lewin, Polylogarithms and Associated Functions, (North Holland, New York, 1981)
P.J. Li, Types of lexical derivation of men’s speech in Mayrinax. B. Inst. Hist. Philol. 54(3), 1 (1983)
W. Li, On the relationship between complexity and entropy for markov chains and regular languages. Com. Sys. 5, 381 (1991)
P.J. Li, The dispersal of the Formosan Aborigines in Taiwan. Lang. Linguist. 2(1), 271 (2001)
L. Li, D. Alderson, W. Willinger, J. Doyle, A first principles approach to understanding the internet router-level topology, in Proceedings of the ACM SIGCOMM’04, Portland, August 2004
Y. Limoge, J.L. Bocquet, Temperature behavior of tracer diffusion in amorphous materials: A random-walk approach. Phys. Rev. Lett. 65, 60 (1990)
N. Lin, The Study of Human Communication (The Bobbs-Merrill Company, Indianapolis, 1973)
D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge 1995)
M. Loève, Probability Theory (van Nostrand, New York, 1955)
L. Lovász, Random walks on graphs: A survey, in Bolyai Society Mathematical Studies, vol. 2: Combinatorics, Paul Erdös is Eighty, (Keszthely, Hungary, 1993), p. 1
L. Lovász, P. Winkler, in Mixing of Random Walks and Other Diffusions on a Graph, Surveys in Combinatorics, Stirling. London Mathematical Society Lecture Note Series, vol. 218 (Cambridge University Press, 1995), p. 119
J.K. Lum, L.B. Jorde, W. Schiefenhovel, Affinities among melanesians, micronesians, and polynesians: A neutral, biparental genetic perspective. Hum. Biol. 74, 413 (2002)
M.C. Mackey, Time’s Arrow: The Origins of Thermodynamic Behavior (Springer, New York, 1991)
P. Mahadevan, D. Krioukov, K. Fall, A. Vahdat, A basis for systematic analysis of network topologies, in Proceedings of the ACM SIGCOMM Conference, Pisa, Italy, September 2006
J.P. Mallory, In Search of the Indo-Europeans: Language, Archaeology, and Myth (Thames & Hudson, London, 1991)
D. Mangoubi, Riemann Surfaces and 3-regular Graphs, Research MS Thesis, Technion-Israel Institute of Technology, Haifa, 2001
S.C. Manrubia, A.S. Mikhailov, Mutual synchronization and clustering in randomly coupled chaotic dynamical networks. Phys. Rev. E 60, 1579 (1999)
A.A. Markov, Extension of the limit theorems of probability theory to a sum of variables connected in a chain, reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, vol. 1: Markov Chains, (Wiley, 1971)
Y. Marom, Improvising Jazz With Markov Chains, The report for the Honor Program of the Department of Computer Science, The University of Western Australia, 1997
P.C. Martin, E.D. Siggia, H.A. Rose, Statistical dynamics of classical systems. Phys. Rev. A 8, 423 (1973)
S. Maslov, K. Sneppen, Specificity and stability in topology of protein networks. Science 296(5569), 910 (2002)
E. Matisoo-Smith, J.H. Robins, Origins and dispersals of Pacific peoples: Evidence from mtDNA phylogenies of the pacific rat. Proc. Natl. Acad. Sci. U.S.A. 101(24), 9167 (2004)
H.H. McAdams, L. Shapiro, Circuit simulation of genetic networks. Science 269, 650 (1995)
K. McCann, A. Hastings, G.R. Huxel, Weak trophic interactions and the balance of nature. Nature 395, 794 (1998)
J. McLeod, The History of India (Greenwood Publishing Group, Westport, CT, 2002)
A. McMahon, R. McMahon, Language Classification by Numbers (Oxford University Press, Oxford, UK, 2005)
A. McMahon, P. Heggarty, R. McMahon, N. Slaska, Swadesh sublists and the benefits of borrowing: An andean case study. T. Philol. Soc. 103(2), 147 (2005)
C.D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17, 443 (1975)
C.D. Meyer, Analysis of finite Markov chains by group inversion techniques. Recent Applications of Generalized Inverses, in Research Notes in Mathematics, vol. 66, ed. by S.L. Campbell (Pitman, Boston, 1982), pp. 50–81
C.D. Michener, R.R. Sokal, A quantitative approach to a problem in classification. Evolution 11, 130 (1957)
H. Minc, Nonnegative Matrices (Wiley, New York, 1988). ISBN 0-471-83966-3
A. Möbius, Der Barycentrische Calcul (Johann Ambrosius Barth, Leipzig, 1827)
R. Monason, Diffusion, localization and dispersion relations on small-world lattices. Eur. Phys. J. B 12, 555 (1999)
J.A. Moorer, Music and computer composition, in Machine Models of Music, ed. by S. Schwanauer, D. Levitt (MIT, Cambridge, Massachusetts, 1993)
T. Morris, Computer Vision and Image Processing (Palgrave Macmillan, Basingstoke, 2004). ISBN 0-333-99451-5
T. Muir, Treatise on the Theory of Determinants (revised and enlarged by W. H. Metzler), (Dover, New York, 1960)
J.D. Murray, Mathematical Biology (Springer-Verlag, Berlin, 1993)
Mutopia. All music in the Mutopia Project free to download, print out, perform and distribute is available at http://www.mutopiaproject.org, While collecting the data, we have also used the following free resources: http://windy.vis.ne.jp/art/englib/berg.htm (for Alban Berg), http://www.classicalmidi.co.uk/page7.htm, http://www.jacksirulnikoff.com/
N. Nadirashvili, Multiple eigenvalues of laplace operators. Math. USSR Sbornik 61, 325 (1973)
B.S. Nelson, P. Panangaden, Scaling behavior of interacting quantum fields in curved spacetime. Phys. Rev. D 25, 1019 (1982)
J. Nerbonne, W. Heeringa, P. Kleiweg, Edit distance and dialect proximity, in Time Warps, String Edits and Macromolecules: The Theory and Practice of Sequence Comparison, ed. by D. Sankoff, J. Kruskal (CSLI Press, Stanford, 1999), pp. 5–15
J. Nesetril, E. Milková, H. Nesetrilová, Otakar Boruvka on Minimum Spanning Tree Problem (translation of the both 1926 papers, comments, history), CiteSeer, DMATH: Discrete Mathematics (2000)
M.E.J. Newman, S.H. Strogatz, D.J. Watts, Random graphs with arbitrary degree distribution and their applications. Phys. Rev. E 64, 026118 (2001)
M.E.J. Newman, Assortative mixing in networks. Phys. Rev. Lett. 89, 208702 (2002)
M.E.J. Newman, The structure and function of complex networks. SIAM Review 45, 167 (2003)
M.E.J. Newman, Communities, clustering phase transitions, and hysteresis: Pitfalls in constructing network ensembles. Phys. Rev. E 68, 026121 (2003)
M.E.J. Newman, Modularity and community structure in networks. Proc. Natl. Acad. Sci. U.S.A. 103, 8577 (2006)
J. Nichols, T. Warnow, Tutorial on computational linguistic phylogeny. Lang. Linguist. Compass 2(5), 760 (2008)
J. Nieminen, On centrality in a graph. Scand. J. Psychol. 15, 322 (1974)
F. Ninio, A simple proof of the Perron-Frobenius theorem for positive symmetric matrices. J. Phys. A-Math. Gen. 9(8), 1281 (1976)
H. Noguchi, Mozart: Musical game in C K.516f http://www.asahi-net.or.jp/~rb5h-ngc/e/k516f.htm. Accessed 1996
P. Novotná, V. Blažek, Glottochronolgy and its application to the Balto-Slavic languages. Baltistica XLII(2), 185 (2007)
E. Nummelin, General irreducible Markov chains and non-negative operators (Cambridge University Press, Cambridge, 2004). ISBN 0-521-60494-X
R.T. Paine, Food-web analysis through field measurement of per capita interaction strength. Nature 355, 73 (1992)
L. Parker, in Recent Developments in Gravitation: Cargese Lectures 1978, ed. by M. Levy, S.Deser (Plenum Press, New York, 1979)
L.Parker, D.J. Toms, New form for the coincidence limit of the Feynman propagator, or heat kernel, in curved spacetime. Phys. Rev. D 31, 953 (1985)
L.Parker, D.J. Toms, Explicit curvature dependence of coupling constants. Phys. Rev. D 31, 2424 (1985)
R. Pastor-Satorras, A. Vespignani, Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)
R. Pastor-Satorras, A. Vespignani, Epidemic dynamics in finite size scale-free networks. Phys. Rev. E 65, 035108 (2002)
A. Penn, Space syntax and spatial cognition. or, why the axial line? in Proceedings of the Space Syntax 3rd International Symposium, ed. by J. Peponis, J. Wineman, S. Bafna (Georgia Institute of Technology, Atlanta, 2001), pp. 11.1–11.17
R. Penrose, A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 406 (1955)
Perl MIDI. The software that allows to read, compose, modify, and write MIDI files is freely available at the web-page http://search.cpan.org/~sburke/MIDI-Perl-0.8
F. Petroni, M. Serva, Language distance and tree reconstruction. J. Stat. Mech. 2008, P08012 (2008)
E. Prisner, Graph Dynamics (CRC Press, Boca Raton, FL, 1995)
M. Ptashne, A. Gann, Genes and Signals (Cold Spring Harbour Laboratory Press, Cold SpringHarbour, NY, 2002)
J.G. Ratcliffe, Foundations of hyperbolic manifolds, Springer series: Graduate Texts in Mathematics (Springer-Verlag, New York, London, 1994)
M. Ravallion, Urban poverty. Financ. Dev. 44(3), 141–167 (2007)
Native Instruments Software Synthesis GmbH, Reaktor 5.1 [computer software] (Berlin, 2005)
Propellerhead Software, Reason 4 [computer software] (Stockholm, 2007)
C. Renfrew, Archaeology and Language: The Puzzle of Indo-European Origins (Cambridge University Press, New York, 1987)
C. Renfrew, Time depth, convergence theory, and innovation in Proto-Indo-European, in Proceedings of the Conference Languages in Prehistoric Europe, Eichstätt University, 4–6 October 1999, Heidelberg, published in 2003, p. 227
K. Rho, H. Jeong, B. Kahng, Identification of lethal cluster of genes in the yeast transcription network. Physica A 364, 557 (2006)
P. Robert, On the group inverse of a linear transformation. J. Math. Anal. Appl. 22, 658 (1968)
S. Roman, Advanced Linear Algebra, 2nd edn., in Springer series: Graduate Texts in Mathematics, (Springer, New York, 2005)
D. Ruelle, Thermodynamic Formalism, in 5 of Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, Mass, 1978)
L.G. Sabidussi, The centrality index of a graph. Psychometrica 31, 581 (1966)
V.N. Sachkov, Combinatorial Methods in Discrete Mathematics, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1996)
S. Saitoh, Theory of Reproducing Kernels and its Applications (Longman Scientific and Technical, Harlow, UK, 1988)
N. Saitou, N. Masatoshi, The neighborhood joining method: A new method of constructing phylogenetic trees. Mol. Biol. Evol. 4, 406 (1987)
K. Salisburym D. Brock, T. Massie, N. Swarup, C. Zilles, Haptic rendering: Programming touch interaction with virtual objects, in Proceedings of the 1995 Symposium on Interactive 3D Graphics (ACM, New York, 1995), p. 123
L. Saloff-Coste, Lectures on Finite Markov Chains, Ecole d’Été, Saint-Flour, Lecture Notes in Mathematics, vol. 1664 (Springer, Berlin, 1997), pp. 301–413
R. Shaw, The Dripping Faucet as a Model Chaotic System (CA Aerial Press, Santa Cruz, 1984)
R. Schneider, L. Kobbelt, Generating fair meshes with G1 boundary conditions. Comput. Aided Geom. D. 4(18), 159 (2001)
B. Schölkopf, A.J. Smola, K.-R. Müller, Nonlinear component analysis as a kernel eigenvalue problem. Neur. Comput. 10, 1299 (1998)
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency (Springer, Berlin, 2002)
T. Schürmann, P. Grassberger, Entropy estimation of symbol sequences. Chaos 6(3), 414 (1996)
Ch. Seeger, Reflections upon a given topic: Music in universal perspective. Ethnomusicology 15(3), 385 (1971)
D. Sergi, Random graph model with power-law distributed triangle subgraphs. Phys. Rev. E 72, 025103 (2005)
M. Serva, F. Petroni, Indo-European languages tree by Levenshtein distance. Europhys. Lett. 81, 68005 (2008)
M.F. Shlesinger, First encounters. Nature 450(1), 40 (2007)
R. Sedgewick, Permutation generation methods. Comput. Surv. 9, 137 (1977)
D. Volchenkov, S. Sequeira, Ph. Blanchard, M.G. Cosenza, Transitions to intermittency and collective behavior in randomly coupled map networks. Stoch. Dynam. 2(2), 203 (2002)
B. Sévennec, Multiplicité du spectre des surfaces: Une approche topologique (Preprint ENS, Lyon, 1994)
Ya.G. Sinai, Gibbs measures in ergodic theory. Usp. Mat. Nauk 27(4), 21-64 (1972) (in Russian); MR, 53, # 3265; Russ. Math. Surv. 27(4), 21-69 (1972) (in English)
Ya.G. Sinai, A.B. Soshnikov, A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl. 32(2), 114 (1998)
S. Skiena, Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathe- matica. Reading, MA: Addison-Wesley, (1990)
C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379; 623 (1948)
C.E. Shannon, Prediction and entropy of printed english. Bell Syst. Tech. J. 30, 50 (1951)
S. Shaw, Evidence of scale-free topology and dynamics in gene regulatory networks, in Proceedings of the ISCA 12th International Conference on Intelligent and Adaptive Systems and Software Engineering, ed. by A. Satyadas and S. Dascalu, (San Francisco, California, USA) 2003, p. 37
J. Shi, J. Malik, Normalized cuts and image segmentation. IEEE Trans. PAMI 22(8), 888–905 (2000)
M. Shlesinger, G.M. Zaslavsky, U. Frisch (eds.), Lévy Flights and Related Topics in Physics (Springer-Verlag, New York, 1995)
T.S. Shores, Applied Linear Algebra and Matrix Analysis, in Springer series: Undergraduate Texts in Mathematics (Springer, 2006)
G.E. Shilov, B.L. Gurevich, Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman (trans. from Russian) (Dover Publications, New York, 1978)
A. Smola, R.I. Kondor, Kernels and regularization on graphs, in Learning Theory and Kernel Machines, ed. by B. Scholkopf, M.K. Warmuth (Springer, Berlin, New York, 2003), pp. 144–158
P. Smolen, D.A. Baxter, J.H. Byrne, Frequency selectivity, multistability and oscillations emerge from models of genetic regulatory systems. Am. J. Physiol.–Cell Ph. 43, C531 (1998)
E.H. Snoussi, R. Thomas, Logical identification of all steady states: The concept of feedback loop caracteristic states. Bull. Math. Biol. 55(5), 973 (1993)
D. Stassinopoulos, P. Alstrøm, Coupled maps: An approach to spatiotemporal chaos. Phys. Rev. A 45, 675 (1992)
B. Su, L. Jin, P. Underhill, J. Martinson, N. Saha, S.T. McGarveyi, M.D. Shriver, J. Chu, P. Oefner, R. Chakraborty, R. Deka, Polynesian origins: Insights from the Y chromosome. Proc. Natl. Acad. Sci. U.S.A. 97(15), 8225 (2000)
M. Swadesh, Lexico-statistic dating of prehistoric ethnic contacts. Proc. Natl. Acad. Sci. U.S.A. 96, 452 (1952)
K. Symanzik, Schrödinger representation and casimir effect in renormalizable quantum field theory. Nucl. Phys. B 190, 1 (1981)
B. Tadic, Exploring complex graphs by random walks. in Modeling of complex systems: Seventh Granada Lectures, ed. by P. Garrido, J. Marro, Granada, Spain, AIP Conference Proceedings, vol. 661 (American Institute of Physics, Melville, 2002), pp. 24–26
H. Tangmunarunkit, J. Doyle, R. Govindan, S. Jamin, W. Willinger, S. Shenker, Does AS size determine AS degree? in ACM SIGCOMM Computer Communication Review (ACM, New York, 2001)
E.V. Teodorovich, The renormalization group method in the problem of transport in the presence of nonlinear sources and sinks. J. Eksp. Theor. Phys. (Sov. JETP) 115, 1497 (1999) [JETP 88, 826 (1999)]
P. Tetali, Random walks and the effective resistance of networks. J. Theor. Probab. 4(1), 101 (1991)
D. Thieffry, D. Romero, The modularity of biological regulatory networks. BioSystems 50, 49 (1999)
D. Thieffry, E.H. Snoussi, J. Richelle, R. Thomas, Positive loops and differentiation. J. Biol. Syst. 3(2), 457 (1995)
D. Thieffry, Qualitative Analysis of Gene Networks in the Memoire pour l’obtention d’Agrege de l’Eseignement Superieur, Universite Libre de Bruxelles (2000)
R. Thomas, On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Series Syne. 9, 180 (1981)
R. Thomas, The role of feedback circuits] positive feedback circuits are a necessary condition for positive real eigenvalues in the jacobian matrix. Ber. Brunzen. Phys. Chem. 98, 1148 (1994)
R. Thomas, M. Kaufman, Logical analysis of regulatory networks in terms of feedback circuits. Chaos 11(1), 180 (2001)
W. Thomson, Tonality in Music: A General Theory (Everett Books, San Marino, CA, 1999)
D.J. Toms, Renormalization of interacting scalar field theories in curved space-time. Phys. Rev. D 26, 2713 (1982)
W.T. Trotter, Combinatorics and Partially Ordered Sets (The Johns Hopkins University Press, Baltimore, MD, 2001)
W.T. Tutte, A contribution to the theory of chromatic polynomials. Canadian J. Math. 6, 80 (1954)
W.T. Tutte, Graph Theory (Cambridge University Press, Cambridge, 2001)
Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat, World population prospects: The 2006 revision, Dataset on CD-ROM. (United Nations, New York, 2007)
US Bureau of Census Data of Urbanized Areas, available at http://www.sprawlcity.org/
N. Utsurikawa, A Genealogical and Classificatory Study of the Formosan Native Tribes (Toko shoin, Tokyo, 1935)
L. Vaughan, The relationship between physical segregation and social marginalisation in the urban environment. World Archit. 185, 88 (2005)
L. Vaughan, D. Chatford, O. Sahbaz, Space and exclusion: The relationship between physical segregation. Economic marginalization and povetry in the city. Paper presented to Fifth International Space Syntax Symposium, Delft, Holland, 2005
M.U. Vera, D.J. Durian, The angular distribution of diffusely transmitted light. Phys. Rev. E 53, 3215 (1996)
D. Volchenkov, L. Volchenkova, Ph. Blanchard, Epidemic spreading in a variety of scale free networks. Phys. Rev. E 66(4), 046137 (2002); Virtual J. Biol. Phys. Res. 4(9), (2002)
D. Volchenkov, Ph. Blanchard, An algorithm generating scale free graphs. Physica A 315, 677 (2002)
D.Volchenkov, R. Lima, Random shuffling of switching parameters in a model of gene expression regulatory network. Stoch. Dynam. 5(1), 75–95 (2005)
D. Volchenkov, Ph. Blanchard, Nonlinear diffusion through large complex networks with regular subgraphs. J. Stat. Phys. 127(4), 677 (2007)
D. Volchenkov, Ph. Blanchard, Random walks along the streets and channels in compact cities: Spectral analysis, dynamical modularity, information, and statistical mechanics. Phys. Rev. E 75, 026104 (2007)
D. Volchenkov, Ph. Blanchard, Scaling and universality in city space syntax: Between Zipf and Matthew. Physica A 387(10), 2353 (2008)
D. Volchenkov, R. Lima, Asymptotic series in dynamics of fluid flows: Diffusion versus bifurcations. Commun. Nonlin. Sci. Num. Simulat. 13, 1329 (2008)
D. Volchenkov, Renormalization group and instantons in stochastic nonlinear dynamics. Eur. Phys. J. Spec. Top. 170(1), 1–142 (2009)
D. Volchenkov, Random walks and flights over connected graphs and complex networks. Commun. Nonlin. Sci. Num. Simul. (2010), http://dx.doi.org/10.1016/j.cnsns.2010.02.016
P.H. von Hippel, Integrated model of the transcription complex in elongation, termination, and editing. Science 281, 660 (1998)
U. von Luzburg, M. Belkin. O. Bousquet, Consistency of Spectral Clustering. Technical Report Number TR-134, Max-Planck-Institut fuer biologische Kybernetik, 2004
I. Vragovic, E. Louis, A. Diaz-Guilera, Efficiency of informational transfer in regular and complex networks. Phys. Rev. E 71, 036122 (2005)
G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59 (SIAM, Philadelphia, 1990)
A. Wagner, Evolution of gene networks by gene duplications: A mathematical model and its implications. Proc. Natl. Acad. Sci. U.S.A. 91, 4387–4391 (1994)
A. Wagner, Inferring lifestyle from gene expression patterns. Mol. Biol. Evol. 17, 1985–1987 (2000)
A. Wagner, How large protein interaction networks evolve. Proc. R. Soc. Lond. B 270, 457–466 (2003)
A.M. Walczaka, M. Sasaic, P.G. Wolynes, Self-consistent proteomic field theory of stochastic gene switches. Biophys. J. 88, 828–850 (2005)
W.S.-Y. Wang, J.W. Minett, Vertical and horizontal transmission in language evolution. Trans. Philol. Soc. 103(2), 121 (2005)
T. Warnow, S.N. Evans, D.A. Ringe Jr., L. Nakhleh, A stochastic model of language evolution that incorporates homoplasy and borrowing, in Phylogenetic Methods and the Prehistory of Languages, ed. by P. Forster, C. Renfrew (McDonald Institute for Archaeological Research, Cambridge, 2006), p. 75
S. Wasserman, K. Faust, Social Network Analysis (Cambridge University Press, Cambridge, England, 1994)
D.J. Watts, S.H. Strogatz, Collective dynamics of ’small-world’ networks. Nature 393(6684), 440 (1998)
S. Wichmann, A. Müller, V. Velupillai, Homelands of the world’s language families. Diachronica 27(2), 247 (2010)
H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17 (1947)
A.G. Wilson, Entropy in Urban and Regional Modeling (Pion Press, London, 1970)
H. Whitney, A logical expansion in mathematics. Bull. Am. Math. Soc. 38, 572 (1932)
L. Wirth, The Ghetto (edition 1988), Studies in Ethnicity, (Transaction Publishers, New Brunswick, USA, London, UK, 1928)
D.M. Wolf, F.H. Eeckman, On the relationship between genomic regulatory element organization and gene regulatory dynamics. J. Theor. Biol. 195, 167 (1998)
J. Wolfe, Speech and music, acoustics and coding, and what music might be ’for’, in Proceedings of the 7th International Conference on Music Perception and Cognition, Sydney, ed. by C. Stevens, D. Burnham, G. McPherson, E. Schubert, J. Renwick (Causal Productions, Adelaide, 2002)
F.Y. Wu, Theory of resistor networks: The two-point resistance. J. Phys. A: Math. Gen. 37, 6653 (2004)
B.Y. Wu, K.-M. Chao, Spanning Trees and Optimization Problems (CRC Press, Boca Raton, 2004)
I. Xenakis, Formalized Music (Indiana University Press, Bloomington, 1971)
W. Xiao, I. Gutman, Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284 (2003)
W. Xiao, I. Gutman, On resistance matrices. MATCH: Commun. Math. Co. 49, 67 (2003)
W. Xiao, I. Gutman, Relations between resistance and Laplacian matrices and their applications. MATCH Commun. Math. Co. 51, 119 (2004)
S.-J. Yang, Exploring complex networks by walking on them. Phys. Rev. E 71, 016107, (2005)
S.X. Yu, J. Shi, Multiclass spectral clustering, in Proceedings of International Conference on Computer Vision (IEEE Computer Society, Washington, DC, USA, 2003), pp. 313–319
D.H. Zanette, P.A. Alemany, Thermodynamics of anomalous diffusion. Phys. Rev. Lett. 75, 366 (1995)
H. Zha, C. Ding, M. Gu, X. He, H. Simon, Neural Information Processing Systems (NIPS 2001), vol. 14 (Vancouver, Canada, 2001)
H.-H. Zhang, W.-B. Yan, X.-S. Li Trace formulae of characteristic polynomial and Cayley-Hamilton’s theorem, and applications to chiral perturbation theory and general relativity. Commun. Theor. Phys. 49, 801 (2008)
D. Zicarelli, M and jam factory. Comp. Music J. 11(4), 1329 (1987)
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Blanchard, P., Volchenkov, D. (2011). When Feedbacks Matter: Epidemics, Synchronization, and Self-regulation in Complex Networks. In: Random Walks and Diffusions on Graphs and Databases. Springer Series in Synergetics, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19592-1_9
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