Modal logics for reasoning about social networks is currently an active field of research. There is still a gap, however, between the state of the art in logical formalisations of concepts related to social networks and the much more mature field of social network analysis. In this paper we take a step to bridge that gap. One of the key foundations of social network analysis is balance theory , which is used to analyse signed social networks where agents can have positive (“friends”) or negative (“enemies”) relationships. Certain combinations of positive and negative relationships are considered to be unbalanced, or unstable—in particular the occurrence of cycles with an odd number of negative relationships. Especially relatively short cycles with an odd number of negative relationships are thought to put pressure on the agents to change one or more of the involved relationships from negative to positive or the other way around. Most existing logics for reasoning about social networks are defined for unsigned networks. In this paper we develop a modal logic for reasoning about structural properties of signed social networks, and give a sound and complete Hilbert-style axiomatic system. Furthermore, we completely axiomatise classes of signed social networks that are balanced to a certain degree n , in the sense that there are no cycles of length up to n with an odd number of negative relationships. Finally, we completely axiomatise the class of all fully balanced complete signed social networks, i.e., networks where everyone is connected with everyone else. Axiomatic completeness is non-trivial because neither the balance properties, nor the dichotomy between positive and negative relations, are modally definable. The paper thus provides a logical basis for reasoning about signed social networks in general and balanced networks in particular.

中文翻译：

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社交网络中的平衡逻辑

用于推理社交网络的模态逻辑目前是一个活跃的研究领域。然而，与社交网络相关概念的逻辑形式化的最新技术与更成熟的社交网络分析领域之间仍然存在差距。在本文中，我们采取了措施来弥合这一差距。社交网络分析的关键基础之一是平衡理论，它用于分析代理可以具有积极（“朋友”）或消极（“敌人”）关系的签名社交网络。某些正面和负面关系的组合被认为是不平衡的或不稳定的——特别是出现具有奇数个负面关系的循环。特别是具有奇数个负面关系的相对较短的周期被认为会给代理施加压力，以将一个或多个相关关系从负面变为正面或相反。大多数用于推理社交网络的现有逻辑都是为未签名的网络定义的。在本文中，我们开发了一种模态逻辑来推理有符号社交网络的结构特性，并给出了一个健全而完整的希尔伯特式公理系统。此外，我们完全公理化了平衡到一定程度 n 的签名社交网络的类别，从这个意义上说，不存在长度达到 n 的循环具有奇数个负关系。最后，我们完全公理化所有完全平衡的完全签名社交网络的类，即，每个人都与其他人相连的网络。公理完备性是不平凡的，因为无论是平衡属性还是正负关系之间的二分法，都不是模态可定义的。因此，本文为一般的签名社交网络和特别是平衡网络的推理提供了逻辑基础。