A logic for networks

arXiv:1003.3629v1 [cs.LO] 18 Mar 2010 Submitted to: GandALF 2010 A logic for networks Massimo Franceschet Department of Mathematics and Computer Science, University of Udine Via delle Scienze 206, 33100 Udine, Italy Phone: +39 0432 558754 / Fax: +39 0432 558499 massimo.franceschet@dimi.uniud.it Networks are pervasive in the real world. Nature, society, economy, and technology are supported by ostensibly different networks that in fact share an amazing number of interesting structural properties. Network thinking exploded in the last decade, boosted by the availability of large databases on the topology of various real networks, mainly the Web and biological networks, and converged to the new discipline of network analysis – the holistic analysis of complex systems through the study of the network that wires their components. Physicists mainly drove the investigation, studying the structure and function of networks using methods and tools of statistical mechanics. Here, we give an alternative perspective on network analysis, proposing a logic for specifying general properties of networks and a modular algorithm for checking these properties. The logic borrows from two intertwined computing fields: XML databases and model checking. 1 Introduction “Networks are present everywhere. All we need is an eye for them”, says Albert-L´aszl´o Barab´asi, in the introduction of his captivating, playful, and elegantly written book about network science [7]. He is not far from truth. Networks are fundamental tools for modelling and understanding social, linguistic, biological, technological, and economic complex systems. A complex system is made up of a large number of components, or agents, interacting in such a way that their collective behaviour in not a simple combination of their individual behaviour [47]. Craig Reynolds, an artificial life and computer graphics expert, expressed it as “A flock is not a big bird, but the sum of the birds plus the interactions between the birds” [52]. For decades, we assumed that the components of such complex systems are randomly wired together. In the last ten years, thanks to the wide availability of large databases on the topology of various real networks, many researchers independently showed that such an assumption is wrong: real networks have similar architectures, regardless of their age, function, and scope, that elude the random world [8]. This provoked the fast growth of the new research field of network analysis, the holistic study of structural properties of real networks. This scientific revolution was driven mainly by physicists, because the methods and tools of statistical mechanics are particularly well suited to analyse the patterns of interactions in networks [53]. Pioneers in this endeavor were Albert-L´aszl´o Barab´asi and Mark Newman [1, 47]. Network analysis addresses questions at three levels of granularity [15]: element-level analysis, where methods to identify the most important nodes of the network are investigated, group-level analysis, that involves methods for defining and finding cohesive groups of nodes in the network, and network- level analysis, that focuses on topological properties of networks as a whole as well as on theoretical models explaining the generation of empirical networks with certain properties. In this work, we throw logic in the arena of network analysis. Kripke structures – networks in which nodes are labelled with a set of properties that hold at the node – have, in fact, a long tradition in logic: 2 A logic for networks they are the models for the interpretation of modal and temporal formulas [40]. Since the behaviour of a nondeterministic finite state machine can be modelled as a Kripke structure, modal and temporal logics are extensively used for the formal specification of properties of hardware and software systems [24], and many algorithms and heuristics have being developed to automatically check these properties against the modelled behaviour [20]. Here, we devise a combined logic for the specification of meaningful properties of real networks. The proposed logic combines XML Path Language (XPath) properties that look inside the local nodes of the network with Computation Tree Logic (CTL) statements that browse the topology of the network. XPath, a simple and elegant node-retrieval language for Extensible Markup Language (XML) documents, is one of the most successful technologies for XML data management [62]. CTL is a blockbuster among logics for formal verification of computer systems [19], effectively applied in the formal verification of safety- and security-critical systems [44]. We furthermore discuss an approach to build a modular model checker that verifies the properties specified in the proposed logic. The model checker exploits an XPath query processor and a CTL model checker. Efficient and scalable implementations exist for both such modules. The outline of the paper is as follows. In Section 2 we give some remarkable examples

…(Full text truncated)…

This content is AI-processed based on ArXiv data.