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 (...) or negative 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. (shrink)

This paper is part of an on-going programme on the study of the logical aspects of social network formation. It recalls the so-called social network model, discussing the properties of a notion of closeness between agents ; then introduces an extended social network model in which different agents might assign different values to different traits, discussing the properties of the notion of weighted closeness that arises. These notions are used to define social network creation operations by means of a threshold (...) strategy. The paper studies the properties of the social networks the updates create, providing sound and complete axiom systems for formal languages describing these updates’ effects. (shrink)

Epistemic logic is usually employed to model two aspects of a situation: the factual and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with these situations, where agents have only knowledge about the available information, which can be incomplete or conflicting, but not explicitly about facts. This layer of available (...) information or evidence, which is the object of the agents’ knowledge, can be seen as a database. By adopting this sceptical posture in our semantics, we prepare the ground for logics where the notion of knowledge—or more appropriately, belief—is entirely based on evidence. The technical results include a set of reduction axioms for public announcements, correspondence proofs, and a complete tableau system. In summary, our contributions are twofold: on the one hand we present an intuition and possible application for many-valued modal logics, and on the other hand we develop a logic that models the dynamics of evidence in a simple and intuitively clear fashion. (shrink)

Reasoning about equilibria in normal form games involves the study of players’ incentives to deviate unilaterally from any profile. In the case of large anonymous games, the pattern of reasoning is different. Payoffs are determined by strategy distributions rather than strategy profiles. In such a game each player would strategise based on expectations of what fraction of the population makes some choice, rather than respond to individual choices by other players. A player may not even know how many players there (...) are in the game. Logicising such strategisation involves many challenges as the set of players is potentially unbounded. This suggests a logic of quantification over player variables and modalities for player deviation, but such a logic is easily seen to be undecidable. Instead, we propose a propositional modal logic using player types as names and implicit quantification over players. With modalities for player deviation and transitive closure, the logic can be used to specify game equilibrium and interesting patterns of reasoning in large games. We show that the logic is decidable and present a complete axiomatisation of the valid formulas. (shrink)

Traditionally, the formulae in modal logic express properties of possible worlds. Prior introduced “egocentric” logics that capture properties of agents rather than of possible worlds. In such a setting, the article proposes the modality “know how to tell apart” and gives a complete logical system describing the interplay between this modality and the knowledge modality. An important contribution of this work is a new matrix-based technique for proving completeness theorems in an egocentric setting.

Diffusion in social networks is a result of agents’ natural desires to conform to the behavioral patterns of their peers. In this article we show that the recently proposed “propositional opinion diffusion model” could be used to model an agent’s conformity to different social groups that the same agent might belong to, rather than conformity to the society as whole. The main technical contribution of this article is a sound and complete logical system describing the properties of the influence relation (...) in this model. The logical system is an extension of Armstrong’s axioms from database theory by one new axiom that captures the topological structure of the network. (shrink)

In many social contexts, social influence seems to be inescapable: the behavior of others influences us to modify ours, and vice-versa. However, social psychology is full of examples of phenomena where individuals experience a discrepancy between their public behavior and their private opinion. This raises two central questions. First, how does an individual reason about the behavior of others and their private opinions in situations of social influence? And second, what are the laws of the resulting information dynamics? In this (...) paper, we address these questions by introducing a formal framework for representing reasoning about an individual’s private opinions and public behavior under the dynamics of social influence in social networks. Moreover, we dig deeper into the involved information dynamics by modeling how individuals can learn about each other based on this reasoning. This compels us to introduce a new formal notion of reflective social influence. Finally, we initialize the work on proof theory and automated reasoning for our framework by introducing a sound and complete tableaux system for a fragment of our logic. Furthermore, this constitutes the first tableau system for the “Facebook logic” of J. Seligman, F. Liu, and P. Girard. (shrink)

We take a logical approach to threshold models, used to study the diffusion of opinions, new technologies, infections, or behaviors in social networks. Threshold models consist of a network graph of agents connected by a social relationship and a threshold value which regulates the diffusion process. Agents adopt a new behavior/product/opinion when the proportion of their neighbors who have already adopted it meets the threshold. Under this diffusion policy, threshold models develop dynamically towards a guaranteed fixed point. We construct a (...) minimal dynamic propositional logic to describe the threshold dynamics and show that the logic is sound and complete. We then extend this framework with an epistemic dimension and investigate how information about more distant neighbors’ behavior allows agents to anticipate changes in behavior of their closer neighbors. Overall, our logical formalism captures the interplay between the epistemic and social dimensions in social networks. (shrink)

This thesis is on information dynamics modeled using *dynamic epistemic logic*. It takes the simple perspective of identifying models with maps, which under a suitable topology may be analyzed as *topological dynamical systems*. It is composed of an introduction and six papers. The introduction situates DEL in the field of formal epistemology, exemplifies its use and summarizes the main contributions of the papers.Paper I models the information dynamics of the *bystander effect* from social psychology. It shows how augmenting the standard (...) machinery of DEL with a decision making framework yields mathematically self-contained models of dynamic processes, a prerequisite for rigid model comparison.Paper II extrapolates from Paper I's construction, showing how the augmentation and its natural peers may be construed as maps. It argues that under the restriction of dynamics produced by DEL dynamical systems still falls a collection rich enough to be of interest. Paper III compares the approach of Paper II with *extensional protocols*, the main alternative augmentation to DEL. It concludes that both have benefits, depending on application. In favor of the DEL dynamical systems, it shows that extensional protocols designed to mimic simple, DEL dynamical systems require infinite representations. Paper IV focuses on *topological dynamical systems*. It argues that the *Stone topology* is a natural topology for investigating logical dynamics as, in it, *logical convergence* coinsides with topological convergence. It investigates the recurrent behavior of the maps of Papers II and III, providing novel insigths on their long-term behavior, thus providing a proof of concept for the approach.Paper V lays the background for Paper IV, starting from the construction of metrics generalizing the Hamming distance to infinite strings, inducing the Stone topology. It shows that the Stone topology is unique in making logical and topological convergens coinside, making it the natural topology for logical dynamics. It further includes a metric-based proof that the hitherto analyzed maps are continuous with respect to the Stone topology. Paper VI presents two characterization theorems for the existence of *reduction laws*, a common tool in obtaining complete dynamic logics. In the compact case, continuity in the Stone topology characterizes existence, while a strengthening is required in the non-compact case. The results allow the recasting of many logical dynamics of contemporary interest as topological dynamical systems. (shrink)