Abstract
We propose the study of graphs that are defined by low-complexity distributed and deterministic agents. We suggest that this viewpoint may help introduce the element of individual choice in models of large scale social networks. This viewpoint may also provide interesting new classes of graphs for which to design algorithms.
We focus largely on the case where the “low complexity” computation is AC 0. We show that this is already a rich class of graphs that includes examples of lossless expanders and power-law graphs. We give evidence that even such low complexity graphs present a formidable challenge to algorithms designers. On the positive side, we show that many algorithms from property testing and data sketching can be adapted to give meaningful results for low-complexity graphs.
Supported by NSF grants 0830673, 0832797, 528414.
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References
Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1(2) (2003)
Hopcroft, J.: Personal communication (2008)
Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Control 56(3), 183–198 (1983)
Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)
Razborov, A.A., Rudich, S.: Natural proofs. J. Comput. Syst. Sci. 55(1), 24–35 (1997)
Arora, S., Barak, B.: Computational Complexity: A modern approach. Cambridge University Press, Cambridge (2009)
Nickel, C.L.M.: Random dot product graphs: A model for social networks. PhD thesis, Johns Hopkins University (2008)
Young, S.J., Scheinerman, E.R.: Random dot product graph models for social networks. In: Bonato, A., Chung, F.R.K. (eds.) WAW 2007. LNCS, vol. 4863, pp. 138–149. Springer, Heidelberg (2007)
Ajtai, M.: \(\sigma_1^1\) formulae on finite structures. Annals of Pure Appl. Logic 24 (1983)
Furst, M.L., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17(1), 13–27 (1984)
Viola, E.: On approximate majority and probabilistic time. In: IEEE Conference on Computational Complexity, pp. 155–168 (2007)
Hastad, J.: Almost optimal lower bounds for small depth circuits. In: Randomness and Computation, pp. 6–20. JAI Press (1989)
Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. AMS (43), 439–561 (2006)
Capalbo, M.R., Reingold, O., Vadhan, S.P., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: STOC, pp. 659–668 (2002)
Alon, N., Schwartz, O., Shapira, A.: An elementary construction of constant-degree expanders. In: SODA, pp. 454–458 (2007)
Reingold, O., Vadhan, S.P., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In: FOCS, pp. 3–13 (2000)
Mihail, M., Papadimitriou, C.H.: On the eigenvalue power law. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 254–262. Springer, Heidelberg (2002)
Babai, L., Fortnow, L., Lund, L.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1, 3–40 (1991); Prelim version FOCS 1990
Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998); Prelim version FOCS 1992
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998); Prelim version FOCS 1992
Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996); Prelim version FOCS 1991
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)
Alon, N., de la Vega, W.F., Kannan, R., Karpinski, M.: Random sampling and approximation of max-csps. J. Comput. Syst. Sci. 67(2), 212–243 (2003)
Alon, N., Fischer, E., Newman, I., Shapira, A.: A combinatorial characterization of the testable graph properties: it’s all about regularity. In: STOC, pp. 251–260 (2006)
Benjamini, I., Schramm, O., Shapira, A.: Every minor-closed property of sparse graphs is testable. In: STOC, pp. 393–402 (2008)
Frieze, A.M., Kannan, R., Vempala, S.: Fast monte-carlo algorithms for finding low-rank approximations. In: FOCS, pp. 370–378 (1998)
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Arora, S., Steurer, D., Wigderson, A. (2009). Towards a Study of Low-Complexity Graphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_12
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