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On the Logic of Balance in Social Networks

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Abstract

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.

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Notes

  1. The creation of a relation between two people who both know the same person (Simmel 1908).

  2. Not to be confused with weak balance (Davis 1967), which allows triangles with three negative relationships but not with only one.

  3. Here we represent an undirected graph with two types of edges using two disjoint (directed) symmetric binary relations, a representation that is natural to work with from the perspective of modal logic. It is, of course, equivalent to the standard definition in graph theory.

  4. Notation: for binary relations R we use aRb to denote the fact that \((a,b) \in R\).

  5. See, e.g., Easley and Kleinberg (2010, Ch. 5) for a modern overview and discussion.

  6. We henceforth use \(\mathbb {N}\) for the set of natural numbers (non-negative integers) including 0, and \(\mathbb {N}^+\) for \(\mathbb {N}\setminus \{0\}\).

  7. \(G^{\prime }= (A^{\prime }, {R^+}^{\prime }, {R^-}^{\prime })\) is a subgraph of \(G= (A, {R^+}, {R^-})\) if \(A^{\prime } \subseteq A\), \({R^+}^{\prime } = {R^+}\cap (A^{\prime } \times A^{\prime })\) and \({R^-}^{\prime } = {R^-}\cap (A^{\prime } \times A^{\prime })\).

  8. Sometimes we will abuse notation and use \(M = (A,{R^+},{R^-}, V)\) to denote a model.

  9. The assumption makes some technical definitions and proofs easier. It is possible in the general signed graph model of social networks that an agent has a negative relation to herself, and this could be of interest in the study of certain phenomena (“I don’t trust myself” or “you are your own worst enemy”). For balance, however, it would be an anomaly that is not of practical or theoretical interest. Indeed, in many papers on social networks it is assumed that the relations are irreflexive. Allowing, even requiring, positive reflexivity does not change anything when it comes to the concept of balance.

  10. If \(\varphi \) defines the non-overlapping property, then there is a V such that \((\mathcal {F}_2,V),s \not \models \varphi \) for some s since \(\mathcal {F}_2\) does not have this property. \((\mathcal {F}_1,V),a_1\) satisfies the same formulas as \((\mathcal {F}_2,V),a^{\prime }\), and the same for \(b_1\) (and \(b_2\)) vs. \(b^{\prime }\) (the logic is a normal modal logic and there is a bisimulation between the two models, linking \(a_1\) and \(a^{\prime }\), \(a_2\) and \(a^{\prime }\), \(b_1\) and \(b^{\prime }\) and \(b_2\) and \(b^{\prime }\)), so that means that \((\mathcal {F}_1,V),s^{\prime } \not \models \varphi \) for some \(s^{\prime }\), and thus that \(\mathcal {F}_1\not \models \varphi \), but \(\mathcal {F}_1\) has the non-overlapping property.

  11. There are simpler formulas that are sufficient for the mentioned property but do not serve our purpose in the inference rule.

  12. The step-by-step technique is very general and has been widely used, in particular in the case of modally undefinable properties (like in our case). We apply the standard technique as described in Blackburn et al. (2001), Chapter 4; details such as name-formulas, defects and the inference rule of course differ.

  13. \(\mathcal {L}_{{\textsc {pnl}}}\) is a normal modal logic with two diamonds: . It can easily be seen that for any formula \(\alpha ,\beta \in \mathcal {L}_{{\textsc {pnl}}}\), is consistent iff is consistent iff is consistent iff is consistent.

  14. It is a subset of the set (the four types of defects, respectively).

  15. Every network \(\mathcal {N}_i\) has defects, since it is finite. By “the minimal defect in \(\mathcal {N}_i\)” we mean the minimal defect with respect to the enumeration from the set of D1 defects \((s,\varphi )\), D2 defects , D3 defects and D4 defects s, where \(s \in N^l_i \cup N^r_i\).

  16. Repairing a defect might, of course, introduce new defects. For example, adding nodes while repairing D2 defects will create new D1 defects. Each of these will eventually be the minimal defect in \(\mathcal {N}_k\) for some k.

  17. In \({{\mathbf {\mathsf{{pnl}}}}}_1\), the role of the rule \({{\mathbf {\mathsf{{Nb}}}}}_1\) is to ensure the non-overlapping property.

  18. By the standard definition in modal logic, see, e.g., Blackburn et al. (2001, Ch. 2): W is the set of states in \(M^c_A\) reachable from \(\Xi \) by the reflexive and transitive closure of the relation \(({R^+}_A \cup {R^-}_A)\); \({R^+}, {R^-}\) and V are \({R^+}_A\), \({R^-}_A\) and \(V_A\), respectively, restricted to W.

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Acknowledgements

We have benefited from very helpful comments from Zoé Christoff and Sonja Smets based on an earlier draft of this manuscript. We also sincerely thank the anonymous referees for valuable and insightful comments and suggestions that helped us significantly improve the article. Zuojun Xiong is supported by the Major Program of the National Social Science Foundation of China (Grant No. 18ZDA032).

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Xiong, Z., Ågotnes, T. On the Logic of Balance in Social Networks. J of Log Lang and Inf 29, 53–75 (2020). https://doi.org/10.1007/s10849-019-09297-0

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