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On the complexity of reasoning about opinion diffusion under majority dynamics. (English) Zbl 07228995
This paper studies complexity analysis of opinion diffusion problems under majority dynamics in social networks. A key problems studied in this paper is consensus problem. Given a graph \(G=(N,E)\) and a rational number \(\alpha\) such that \(\alpha\in(0,1)\), this problem requires to compute a set \(S\subseteq N\) of seeds such that \(|S|\le\alpha|N|\) and there is a dynamic \(c\) tends to \(1\) with \(N_{1/c}=S\) or check that there is no set of seeds satisfying the above conditions. Here \(N_{1/c}\) is the set of nodes with opinion \(1\) under a given configuration \(c:N\rightarrow\{0,1\}\). It is shown that consensus problem is a NP-hard problem. Two other problems called double problem and plural problem are also proved to be NP-hard. Double problem refers to computing a configuration \(c\) such that \(|N_{1/c}|\le\varepsilon|N|\) and there exists a dynamic \(c\) tending to \(c^*\) with \(|N_{1/c^*}|>2\varepsilon |N|\) or check that there is no such configuration satisfying the above conditions. Plural problem refers to computing a configuration \(c\) such that \(c\) is stable and \(N_{0/c}\not=\emptyset\) and \(N_{0/c}\not=N\) or check that there is no configuration satisfying the above conditions.
MSC:
91D30 Social networks; opinion dynamics
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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[1] Auletta, V.; Ferraioli, D.; Greco, G., Reasoning about consensus when opinions diffuse through majority dynamics, (Proc. of IJCAI’18 (2018)), 49-55
[2] Granovetter, M., Threshold models of collective behavior, Am. J. Sociol., 83, 6, 1420-1443 (1978)
[3] Degroot, M. H., Reaching a consensus, J. Am. Stat. Assoc., 69, 345, 118-121 (1974) · Zbl 0282.92011
[4] Memon, N.; Alhajj, R., From Sociology to Computing in Social Networks: Theory, Foundations and Applications (2010), Springer Publishing Company, Incorporated
[5] Easley, D.; Kleinberg, J., Networks, Crowds, and Markets: Reasoning About a Highly Connected World (2010), Cambridge University Press · Zbl 1205.91007
[6] Grandi, U., Social choice and social networks, Trends in Computational Social Choice. Trends in Computational Social Choice, AI Access, 169-184 (2017)
[7] Salehi-Abari, A.; Boutilier, C.; Larson, K., Empathetic decision making in social networks, Artif. Intell., 275, 174-203 (2019) · Zbl 07099227
[8] Christoff, Z.; Grossi, D., Stability in binary opinion diffusion, (Proc. of LORI ‘17 (2017)), 166-180 · Zbl 06810780
[9] Tambe, M.; Rice, E., Artificial Intelligence and Social Work (2018), Cambridge University Press
[10] Domingos, P.; Richardson, M., Mining the network value of customers, (Proc. of KDD’01 (2001)), 57-66
[11] Richardson, M.; Domingos, P., Mining knowledge-sharing sites for viral marketing, (Proc. of KDD ‘02 (2002)), 61-70
[12] Long, C.; Wong, R. C.-W., Viral marketing for dedicated customers, Inf. Sci., 46, 1-23 (2014)
[13] Misner, I. R.; Devine, V., The World’s Best-Known Marketing Secret: Building Your Business with Word-of-Mouth Marketing (1999), Bard Press
[14] Li, Y.; Fan, J.; Wang, Y.; Tan, K., Influence maximization on social graphs: a survey, IEEE Trans. Knowl. Data Eng., 30, 10, 1852-1872 (2018)
[15] Kempe, D.; Kleinberg, J.; Tardos, E., Influential nodes in a diffusion model for social networks, (Proc. of ICALP’05 (2005)), 1127-1138 · Zbl 1084.91053
[16] Schelling, T. C., Micromotives and Macrobehavior (1978), W. W. Norton & Company
[17] Granovetter, M., The strength of weak ties, Am. J. Sociol., 78, 6, 1360-1380 (1973)
[18] Shakarian, P., A multidisciplinary survey of social network diffusion models, IEEE Intel. Inform. Bull., 16, 1, 3-7 (2015)
[19] Kempe, D.; Kleinberg, J.; Tardos, E., Maximizing the spread of influence through a social network, Theory Comput., 11, 4, 105-147 (2015) · Zbl 1337.91069
[20] Gursoy, F.; Gunnec, D., Influence maximization in social networks under deterministic linear threshold model, Knowl.-Based Syst., 161, 111-123 (2018)
[21] Goyal, A.; Bonchi, F.; Lakshmanan, L. V., Learning influence probabilities in social networks, (Proc. of WSDM ‘10 (2010)), 241-250
[22] Shakarian, P.; Eyre, S.; Paulo, D., A scalable heuristic for viral marketing under the tipping model, Soc. Netw. Anal. Min., 3, 4, 1225-1248 (2013)
[23] Borodin, A.; Filmus, Y.; Oren, J., Threshold models for competitive influence in social networks, (Saberi, A., Proc. of WINE’10 (2010)), 539-550
[24] Bharathi, S.; Kempe, D.; Salek, M., Competitive influence maximization in social networks, (Proc. of WINE’07 (2007)), 306-311
[25] Budak, C.; Agrawal, D.; El Abbadi, A., Limiting the spread of misinformation in social networks, (Proc. of WWW ‘11 (2011)), 665-674
[26] Chen, W.; Collins, A.; Cummings, R.; Ke, T.; Liu, Z.; Rincon, D.; Sun, X.; Wei, W.; Wang, Y.; Yuan, Y., Influence maximization in social networks when negative opinions may emerge and propagate, (Proc. of SDM’11 (2011)), 379-390
[27] Fazli, M.; Ghodsi, M.; Habibi, J.; Jalaly, P.; Mirrokni, V.; Sadeghian, S., On non-progressive spread of influence through social networks, Theor. Comput. Sci., 550, 36-50 (2014) · Zbl 1360.91120
[28] Goles, E.; Olivos, J., Periodic behaviour of generalized threshold functions, Discrete Math., 30, 2, 187-189 (1980) · Zbl 0444.94038
[29] Frischknecht, S.; Keller, B.; Wattenhofer, R., Convergence in (social) influence networks, (Afek, Y., Proc. of DISC’13 (2013)), 433-446 · Zbl 1400.91461
[30] Lou, V. Y.; Bhagat, S.; Lakshmanan, L. V.; Vaswani, S., Modeling non-progressive phenomena for influence propagation, (Proc. of COSN ‘14 (2014)), 131-138
[31] V. Tzoumas, C. Amanatidis, E. Markakis, A game-theoretic analysis of a competitive diffusion process over social networks, in: Proc. of WINE’12, pp. 1-14.
[32] Bredereck, R.; Elkind, E., Manipulating opinion diffusion in social networks, (Proc. of IJCAI’17 (2017)), 894-900
[33] Dong, Y.; Zha, Q.; Zhang, H.; Kou, G.; Fujita, H.; Chiclana, F.; Herrera-Viedma, E., Consensus reaching in social network group decision making: research paradigms and challenges, Knowl.-Based Syst., 162, 3-13 (2018)
[34] Osborne, M. J.; Rubinstein, A., A Course in Game Theory (1994), The MIT Press: The MIT Press Cambridge, USA · Zbl 1194.91003
[35] Brandt, F.; Conitzer, V.; Endriss, U.; Lang, J.; Procaccia, A. D., Handbook of Computational Social Choice (2016), Cambridge University Press: Cambridge University Press New York, NY, USA · Zbl 1436.91001
[36] Endriss, U., Trends in Computational Social Choice. Trends in Computational Social Choice, AI Access (2017)
[37] R. Angell, G. Schoenebeck, Proc. of WINE’17, pp. 16-29.
[38] Dreyer, P.; Roberts, F. S., Irreversible k-threshold processes: graph-theoretical threshold models of the spread of disease and of opinion, Discrete Appl. Math., 157, 1615-1627 (2009) · Zbl 1163.92035
[39] Auletta, V.; Caragiannis, I.; Ferraioli, D.; Galdi, C.; Persiano, G., Minority becomes majority in social networks, (Proc. of WINE’15 (2015)), 74-88 · Zbl 1406.91317
[40] Ferraioli, D.; Ventre, C., Social pressure in opinion games, (Proc. of IJCAI’17 (2017)), 3661-3667
[41] Robertson, N.; Seymour, P., Graph minors III: planar tree-width, J. Comb. Theory, Ser. B, 36, 1, 49-64 (1984) · Zbl 0548.05025
[42] Arnborg, S.; Lagergren, J.; Seese, D., Easy problems for tree-decomposable graphs, J. Algorithms, 12, 2, 308-340 (1991) · Zbl 0734.68073
[43] Gottlob, G.; Greco, G.; Leone, N.; Scarcello, F., Hypertree decompositions: questions and answers, (Proc. of PODS’16 (2016)), 57-74
[44] Dechter, R., Constraint Processing (2003), Morgan Kaufmann Publishers Inc.: Morgan Kaufmann Publishers Inc. San Francisco, CA, USA
[45] Gottlob, G.; Greco, G.; Scarcello, F., Treewidth and hypertree width, (Tractability: Practical Approaches to Hard Problems (2013), Cambridge University Press)
[46] Chen, N., On the approximability of influence in social networks, SIAM J. Discrete Math., 23, 3, 1400-1415 (2009) · Zbl 1203.68314
[47] Khoshkhah, K.; Soltani, H.; Zaker, M., On dynamic monopolies of graphs: the average and strict majority thresholds, Discrete Optim., 9, 2, 77-83 (2012) · Zbl 1246.91115
[48] Peleg, D., Local majorities, coalitions and monopolies in graphs: a review, Theor. Comput. Sci., 282, 2, 231-257 (2002) · Zbl 0997.68088
[49] Peleg, D., Size bounds for dynamic monopolies, Discrete Appl. Math., 86, 2, 263-273 (1998) · Zbl 0910.90286
[50] Chen, N., On the approximability of influence in social networks, (Proc. of SODA’08 (2008)), 1029-1037 · Zbl 1192.91169
[51] Garey, M. R.; Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness (1979), W. H. Freeman & Co. · Zbl 0411.68039
[52] Greenlaw, R.; Petreschi, R., Cubic graphs, ACM Comput. Surv., 27, 4, 471-495 (1995)
[53] Bodlaender, H. L., A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM J. Comput., 25, 6, 1305-1317 (1996) · Zbl 0864.68074
[54] Gaifman, H., On local and non-local properties, (Stern, J., Proceedings of the Herbrand Symposium. Proceedings of the Herbrand Symposium, Studies in Logic and the Foundations of Mathematics, vol. 107 (1982), Elsevier), 105-135
[55] Rossi, F.; v. Beek, P.; Walsh, T., Handbook of Constraint Programming (Foundations of Artificial Intelligence) (2006), Elsevier Science Inc.: Elsevier Science Inc. New York, NY, USA
[56] Maniu, S.; Senellart, P.; Jog, S., An experimental study of the treewidth of real-world graph data, (Proc. of ICDT’19 (2019)), 12:1-12:18
[57] Greco, G.; Lupia, F.; Scarcello, F., The tractability of the shapley value over bounded treewidth matching games, (Proc. of IJCAI’17 (2017)), 1046-1052
[58] Demetrescu, C.; Lupia, F.; Mendicelli, A.; Ribichini, A.; Scarcello, F.; Schaerf, M., On the shapley value and its application to the italian vqr research assessment exercise, J. Informetr., 13, 1, 87-104 (2019)
[59] Grohe, M., Local tree-width, excluded minors, and approximation algorithms, Combinatorica, 23, 4, 613-632 (2003) · Zbl 1089.05503
[60] Demaine, E. D.; Hajiaghayi, M. T.; Kawarabayashi, K., Algorithmic graph minor theory: decomposition, approximation, and coloring, (Proc. of FOCS’05 (2005)), 637-646
[61] Baker, B. S., Approximation algorithms for np-complete problems on planar graphs, J. ACM, 41, 1, 153-180 (1994) · Zbl 0807.68067
[62] Bazgan, C.; Chopin, M.; Nichterlein, A.; Sikora, F., Parameterized approximability of maximizing the spread of influence in networks, J. Discret. Algorithms, 27, 54-65 (2014) · Zbl 1361.68105
[63] Bazgan, C.; Chopin, M.; Nichterlein, A.; Sikora, F., Parameterized inapproximability of target set selection and generalizations, Computability, 3, 2, 135-145 (2014) · Zbl 1320.68088
[64] Nichterlein, A.; Niedermeier, R.; Uhlmann, J.; Weller, M., On tractable cases of target set selection, Soc. Netw. Anal. Min., 3, 2, 233-256 (2013)
[65] Cordasco, G.; Gargano, L.; Mecchia, M.; Rescigno, A. A.; Vaccaro, U., Discovering small target sets in social networks: a fast and effective algorithm, Algorithmica, 80, 6, 1804-1833 (2018) · Zbl 1390.05224
[66] Ben-Zwi, O.; Hermelin, D.; Lokshtanov, D.; Newman, I., Treewidth governs the complexity of target set selection, Discrete Optim., 8, 1, 87-96 (2011) · Zbl 1248.90068
[67] Robertson, N.; Seymour, P., Graph minors. II. Algorithmic aspects of tree-width, J. Algorithms, 7, 3, 309-322 (1986) · Zbl 0611.05017
[68] Ackerman, E.; Ben-Zwi, O.; Wolfovitz, G., Combinatorial model and bounds for target set selection, Theor. Comput. Sci., 411, 44, 4017-4022 (2010) · Zbl 1235.90168
[69] Reichman, D., New bounds for contagious sets, Discrete Math., 312, 10, 1812-1814 (2012) · Zbl 1242.05265
[70] Chang, C.; Lyuu, Y., On irreversible dynamic monopolies in general graphs (2009), CoRR
[71] Chang, C.-L.; Lyuu, Y.-D., Bounding the sizes of dynamic monopolies and convergent sets for threshold-based cascades, Theor. Comput. Sci., 468, 37-49 (2013) · Zbl 1258.05041
[72] Flocchini, P.; Královič, R.; Ružička, P.; Roncato, A.; Santoro, N., On time versus size for monotone dynamic monopolies in regular topologies, J. Discret. Algorithms, 1, 2, 129-150 (2003) · Zbl 1074.68045
[73] Khoshkhah, K.; Nemati, M.; Soltani, H.; Zaker, M., A study of monopolies in graphs, Graphs Comb., 29, 5, 1417-1427 (2013) · Zbl 1272.05149
[74] Sigarreta, J. M.; Rodríguez, J. A., On the global offensive alliance number of a graph, Discrete Appl. Math., 157, 2, 219-226 (2009) · Zbl 1191.05072
[75] Cicalese, F.; Cordasco, G.; Gargano, L.; Milanič, M.; Peters, J.; Vaccaro, U., Spread of influence in weighted networks under time and budget constraints, Theor. Comput. Sci., 586, C, 40-58 (2015) · Zbl 1327.68175
[76] Rautenbach, D.; dos Santos, V. F.; Schäfer, P. M., Irreversible conversion processes with deadlines, J. Discret. Algorithms, 26, 69-76 (2014) · Zbl 1298.05282
[77] Mossel, E.; Neeman, J.; Tamuz, O., Majority dynamics and aggregation of information in social networks, Auton. Agents Multi-Agent Syst., 28, 3, 408-429 (2014)
[78] Feldman, M.; Immorlica, N.; Lucier, B.; Weinberg, S. M., Reaching consensus via non-Bayesian asynchronous learning in social networks, (Proc. of APPROX/RANDOM’14 (2014)), 192-208 · Zbl 1360.91121
[79] Auletta, V.; Caragiannis, I.; Ferraioli, D.; Galdi, C.; Persiano, G., Information retention in heterogeneous majority dynamics, (Proc. of WINE’17 (2017)), 30-43 · Zbl 1405.91515
[80] Auletta, V.; Ferraioli, D.; Fionda, V.; Greco, G., Maximizing the spread of an opinion when tertium datur est, (Proc. of AAMAS’19 (2019)), 1207-1215
[81] Fortunato, S., On the consensus threshold for the opinion dynamics of krause-hegselmann, Int. J. Mod. Phys. C, 16, 02, 259-270 (2005) · Zbl 1112.91353
[82] Lorenz, J.; Urbig, D., About the power to enforce and prevent consensus by manipulating communication rules, Adv. Complex Syst., 10, 02, 251-269 (2007) · Zbl 1135.91029
[83] Carvalho, A.; Larson, K., A consensual linear opinion pool, (Proc. of IJCAI’13 (2013)), 2518-2524
[84] Auletta, V.; Fanelli, A.; Ferraioli, D., Consensus in opinion formation processes in fully evolving environments, (Proc. of AAAI’19 (2019)), 6022-6029
[85] Yoshinaka, R., Higher-order matching in the linear lambda calculus in the absence of constants is np-complete, (Proc. or RTA ‘05 (2005)), 235-249 · Zbl 1078.68045
[86] Chierichetti, F.; Kleinberg, J.; Oren, S., On discrete preferences and coordination, J. Comput. Syst. Sci., 93, 11-29 (2018) · Zbl 1408.91041
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