The spatial preferential attachment (SPA) is a model for complex networks. In the SPA model, nodes are embedded in a metric space, and each node has a sphere of influence whose size increases if the node gains an in-link, and otherwise decreases with time. In this paper, we study the behaviour of the SPA model when the distribution of the nodes is non-uniform. Specifically, the space is divided into dense and sparse regions, where it is assumed that the dense regions correspond to coherent communities. We prove precise theoretical results regarding the degree of a node, the number of common neighbours, and the average out-degree in a region. Moreover, we show how these theoretically derived results about the graph properties of the model can be used to formulate a reliable estimator for the distance between certain pairs of nodes, and to estimate the density of the region containing a given node.
An earlier version of this work, containing no proofs, was presented at the workshop WAW 2013. An extended abstract can be found in @cite_11 .
@cite_11
2088751297 1987890787 2899382568 1931447968
This paper is a reprint of the original work by A. P. Calderon published by the Brazilian Mathematical Society (SBM) in ATAS of SBM (Rio de Janeiro), pp. 65-73, 1980. The original paper had no abstract, so this reprint to be truthful to the original work is published with no abstract. The wide applicability of zero-knowledge interactive proofs comes from the possibility of using these proofs as subroutines in cryptographic protocols. A basic question concerning this use is whether the (sequential and or parallel) composition of zero-knowledge protocols is zero-knowledge too. We demonstrate the limitations of the composition of zero-knowledge protocols by proving that the original definition of zero-knowledge is not closed under sequential composition; and that even the strong formulations of zero-knowledge (e.g., black-box simulation) are not closed under parallel execution. We present lower bounds on the round complexity of zero-knowledge proofs, with significant implications for the parallelization of zero-knowledge protocols. We prove that three-round interactive proofs and constant-round Arthur--Merlin proofs that are black-box simulation zero-knowledge exist only for languages in BPP. In particular, it follows that the "parallel versions" of the first interactive proofs systems presented for quadratic residuosity, graph isomorphism, and any language in NP, are not black-box simulation zero-knowledge, unless the corresponding languages are in BPP. Whether these parallel versions constitute zero-knowledge proofs was an intriguing open questions arising from the early works on zero-knowledge. Other consequences are a proof of optimality for the round complexity of various known zero-knowledge protocols and the necessity of using secret coins in the design of "parallelizable" constant-round zero-knowledge proofs. We prove that a class of fundamental shared memory tasks are not amenable to certain standard proof techniques in the field. We formally define a class of extension-based proofs, which contains impossibility arguments like the valency proof by Fisher, Lynch and Paterson of the impossibility of wait-free consensus in an asynchronous system. We introduce a framework which models such proofs as an interaction between a prover and an adversarial protocol. Our main contribution is showing that extension-based proofs are inherently limited in power: for instance, they cannot establish the impossibility of solving (n-1)-set agreement among n > 2 processes in a wait-free manner. This impossibility result does have proofs based on combinatorial topology. However, it was unknown whether proofs based on simpler techniques were possible. For more than three and one-half decades, beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems studied range from the trivial to the deep, even including some that corresponded to open questions. Often the paradigm asks for a theorem whose proof is in hand but that cannot be obtained in a fully automated manner by the program in use. The theorem whose hypothesis consists solely of the Meredith single axiom for two-valued sentential (or propositional) calculus and whose conclusion is the Lukasiewicz three-axiom system for that area of formal logic was just such a theorem. Featured in this article is the methodology that enabled the program OTTER to find the first fully automated proof of the cited theorem, a proof with the intriguing property that none of its steps contains a term of the form i>n(i>n(i>t)) for any term i>t. As evidence of the power of the new methodology, the article also discusses OTTER's success in obtaining the first known proof of a theorem concerning a single axiom of Lukasiewicz.