Abstract
The notion of complex-valued information entropy measure is presented. It applies in particular to directed networks (digraphs). The corresponding statistical physics notions are outlined. The studied network, serving as a case study, in view of illustrating the discussion, concerns citations by agents belonging to two distinct communities which have markedly different opinions: the Neocreationist and Intelligent Design Proponents, on one hand, and the Darwinian Evolution Defenders, on the other hand. The whole, intra- and inter-community adjacency matrices, resulting from quotations of published work by the community agents, are elaborated and eigenvalues calculated. Since eigenvalues can be complex numbers, the information entropy may become also complex-valued. It is calculated for the illustrating case. The role of the imaginary part finiteness is discussed in particular and given some physical sense interpretation through local interaction range consideration. It is concluded that such generalizations are not only interesting and necessary for discussing directed networks, but also may give new insight into conceptual ideas about directed or other networks. Notes on extending the above to Tsallis entropy measure are found in an Appendix.
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References
I. Procaccia, Nature 333, 498 (1988)
N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996)
R. Grobe, F. Haake, H.-J. Sommers, Phys. Rev. Lett. 61, 1899 (1988)
Y.V. Fyodorov, B.A. Khoruzhenko, Phys. Rev. Lett. 83, 65 (1999)
M.A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996)
P. Di Francesco, M. Gaudin, C. Itzykson, F. Lesage, Int. J. Mod. Phys. A 9, 4257 (1994)
B. Jancovici, Mol. Phys. 42, 1251 (1984)
P.J. Forrester, B. Jancovici, Int. J. Mod. Phys. A 11, 941 (1996)
M. Ausloos, J.B. Sousa, M.M. Amado, R.P. Pinto, Appl. Phys. Lett. 43, 927 (1983)
M. Ausloos, in Non Linear Phenomena at Phase Transitions and Instabilities, edited by T. Riste (Plenum Press, New York, 1982), pp. 337–341
H.E. Stanley, Phase Transitions and Critical Phenomena (Clarendon Press, London, 1971)
M.E. Fisher, Rev. Mod. Phys. 70, 653 (1998)
H.E. Stanley, Rev. Mod. Phys. 71, 358 (1999)
C.J. Thompson, Mathematical Statistical Mechanics (Macmillan, London, 1971)
E. Brezin, J.C. LeGuillou, J. Zinn-Justin, Phys. Rev. Lett. 32, 473 (1974)
J.C. Anifrani, C. Le Floc’h, D. Sornette, B. Souillard, J. Phys. I 5, 631 (1995)
D. Sornette, Phys. Rep. 297, 239 (1998)
K. Huang, Statistical Mechanics (Wiley, New York, 1963)
H. Theil, Econometrica 33, 67 (1965)
J. Miśkiewicz, Physica A 387, 6595 (2008)
F.O. Redelico, A.N. Proto, Int. J. Bifurc. Chaos 20, 413 (2010)
Q. Wang, Y. Shen, J.Q. Zhang, Physica D 200, 287 (2005)
I.F. Wilde, Lecture Notes on Complex Analysis (Imperial College Press, London, 2006)
K.W. Boyack, S. Milojević, K. Börner, S. Morris, in Models of Science Dynamics, Understanding Complex Systems, edited by A. Scharnhorst, K. Börner, P. van den Besselaar (Springer-Verlag, Berlin, Heidelberg, 2012), pp. 3–22
Y. Li, H.D. Cao, Y. Tan, Complexity 17, 13 (2011)
R. Sinatra, D. Condorelli, V. Latora, Phys. Rev. Lett. 105, 178702 (2010)
V. Venkatasubramanian, S. Katare, P.R. Patkar, F.-P. Mu, Comput. Chem. Eng. 28, 1789 (2004)
T. Araújo, R. Vilela Mendes, Complex Syst. 12, 357 (2000)
L. Kullmann, J. Kertész, K. Kaski, Phys. Rev. E 64, 057105 (2001)
L. Guo, Xu Cai, Int. J. Mod. Phys. C 19, 1909 (2008)
S. Abe, N. Suzuki, Europhys. Lett. 65, 581 (2004)
B. Tadić, Physica A 286, 509 (2001)
G. Mukherjee, S.S. Manna, Phys. Rev. E 71, 066108 (2005)
L.A. Meyers, M.E.J. Newman, B. Pourbohloul, J. Theor. Biol. 240, 400 (2006)
E.A. Leicht, M.E.J. Newman, Phys. Rev. Lett. 100, 118703 (2008)
A. Ramzanpour, V. Karimipour, Phys. Rev. E 66, 036128 (2002)
A.D. Sánchez, J.M. López, M.A. Rodríguez, Phys. Rev. Lett. 88, 048701 (2002)
F. Radicchi, S. Fortunato, A. Vespignani, in Models of Science Dynamics, Understanding Complex Systems, edited by A. Scharnhorst, K. Börner, P. van den Besselaar (Springer-Verlag, Berlin, Heidelberg, 2012), pp. 233–257
M. Szell, R. Lambiotte, S. Thurner, Proc. Natl. Acad. Sci. 107, 13636 (2010)
A. Garcia Cantù Ross, M. Ausloos, Scientometrics 80, 457 (2009)
G. Rotundo, M. Ausloos, Physica A 389, 5479 (2010)
A. Bermann, N. Shaked-Monderer, Nonnegative Matrices and Digraphs, Encyclopedia of Complexity and System Science (Springer, 2008)
O. Perron, Math. Ann. 64, 248 (1907)
G. Frobenius, S.-B. Prüss, Akad. Wiss. Berlin 456 (1912)
R.A. Brualdi, http://www.math.niu.edu/˜sokolov/la-talks/braudi.pdf
R.B. Bapat, T.E.S. Raghavan, Nonnegative Matrices and Applications, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1997), Vol. 64
A. Berman, M. Neumann, R.J. Stern, Nonnegative Matrices in Dynamic Systems (Wiley-Interscience, New York, 1989)
A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics (SIAM, Philadelphia, 1994)
H. Minc, Nonnegative Matrices (Wiley, New York, 1988)
U. Rothblum, Nonnegative Matrices and Stochastic Matrices, in Handbook of Linear Algebra, edited by L. Hogben (CRC Press, 2006)
E. Senata, Nonnegative Matrices and Markov Chains, Springer Series in Statistics (Springer-Verlag, Berlin, 1981)
C. Meyer, Matrix Analysis and Applied Linear Algebra (2000), Chap. 8.3, p. 670, http://www.matrixanalysis.com/Chapter8.pdf
F.R. Gantmacher, The Theory of Matrices, K.A. Hirsch, transl. 2000 (AMA Chelsea Publ., 2000), Chap. XIII.3, p. 66
R.E. Tarjan, SIAM J. Comput. 1, 146 (1972)
B. Karrer, M.E.J. Newman, Phys. Rev. E 80, 046110 (2009)
W. Zweger, J. Phys. A 18, 2079 (1985)
J.S. Langer, Ann. Phys. 54, 258 (1969)
W. Cook, C. Mounfield, P. Ormerod, L. Smith, Eur. Phys. J. B 27, 189 (2002)
G. Livan, L. Rebecchi, Eur. Phys. J. B 85, 213 (2012)
S. Droźdź, J. Kwapień, A.Z. Gorski, P. Oswieçimka, Acta Phys. Pol. B 37, 3039 (2006)
C. Tsallis, J. Stat. Phys. 52, 479 (1988)
C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998)
C. Tsallis, Braz. J. Phys. 29, 1 (1999)
G. Wilk, Z. Wlodarczyk, Acta Phys. Pol. B 35, 871 (2004)
M. Ausloos, J. Miśkiewicz, Braz. J. Phys. 39, 388 (2009)
J.R. Iglesias, R.M.C. de Almeida, Eur. Phys. J. B 85, 85 (2012)
J. Moody, Social Netw. 20, 291 (1998)
R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, U. Alon, Science 298, 824 (2002)
R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen-Orr, I. Ayzenshtat, M. Sheffer, U. Alon, Science 303, 1538 (2004)
V. Batagelj, A. Mrvar, Pajek, Program for Analysis and Visualization of Large Networks Reference Manual, Fig. 15, p. 52
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Rotundo, G., Ausloos, M. Complex-valued information entropy measure for networks with directed links (digraphs). Application to citations by community agents with opposite opinions. Eur. Phys. J. B 86, 169 (2013). https://doi.org/10.1140/epjb/e2013-30985-6
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DOI: https://doi.org/10.1140/epjb/e2013-30985-6