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Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly "apparent disorder," rather than true "frustration."combinatorial optimization | social network theory O nline social networks are examples of large-scale communities of interacting individuals in which local ties between users (friend, fan, colleague, but also friend/foe, trust/distrust, etc.) give rise to a complex, multidimensional web of aggregated social behavior (1-4). For such complex networks, the emergence of global properties from local interactions is an intriguing subject, so far investigated mostly at structural and topological level (2,(5)(6)(7)(8). In social network theory (9-11), however, the content of the relationships is often even more important than their topology, and this calls for the development of appropriate analytical and computational tools, able to extrapolate content-related features out of the set of interactions of a social community. Obtaining efficient tools is particularly challenging when, as in social networks retrieved from online media, the size of the community is very big, of the order of 10 5 individuals or higher.A global property that has recently attracted some attention (1,(12)(13)(14) is determining the structural balance of a signed social network. Structural (or social) balance theory was first formulated by Heider (15) in order to understand the structure and origin of tensions and conflicts in a network of individuals whose mutual relationships are characterizable in terms of friendship and hostility. It was modeled in terms of signed graphs by Cartwright and Harary (16); see refs. 10 and 11 for an overview of the theory. The nodes of the graph represent users and the positive/ negative edges their friendly/hostile relationships. It has been known for some time how to interpret structural balance on such networks (16): The potential source of tensions are the cycles of the graph (i.e., the closed paths beginning and ending on the same node), notably those of negative sign (i.e., having an odd number of negative edges). It follows that the concept of balance is not related to the actual number of negative edges on the cycles but only to their parity; see Fig. 1 for an illustration on basic graphs. In particular, a signed graph is exactly balanced (i.e., tensions are completely absent) if and only if all its cycles are positive (16). As such, structural balance ...

Abstract-The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedbackdesign problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.

The controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras. Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases.

Various notions from geometric control theory are used to characterize the behavior of the Markovian master equation for N-level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither small-time controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail.