Dynamic Logics for Threshold Models and their Epistemic Extension ZOÉ CHRISTOFF ⋆ AND RASMUS K. RENDSVIG ⋆⋆ Accepted for ELISIEM 2014 Abstract We take a logical approach to threshold models, used to study the diffusion of e.g. new technologies or behaviours in social net- works. In short, threshold models consist of a network graph of agents connected by a social relationship and a threshold to adopt a possibly cascading behaviour. Agents adopt new behaviour when the proportion of their neighbours who have already adopted it meets the threshold. Under this adoption policy, threshold models develop dynamically with 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 investi- gate how information about more distant neighbours’ behaviours allows agents to anticipate changes in behaviour of their closer neighbours. It is shown that this epistemic prediction dynamics is equivalent to the non-epistemic threshold model dynamics if and only if agents know exactly their neighbours’ behaviour. We fur- ther show results regarding ﬁxed points and convergence speed, and provide a partial set of reduction laws, venues for further re- search, and graphical representations of the dynamics. 1 Introduction An individual’s actions or opinions may be inﬂuenced by the actions of people around her [7]. The way a new product or fashion gets adopted by a population depends on how agents are inﬂuenced by others, which in turn de- pends both on the way the population is structured and on how inﬂuenceable agents are. This paper focuses on one particular account of so- cial inﬂuence, the notion of “threshold inﬂuence” as pre- sented in e.g. [5]. Threshold inﬂuence relies on a sim- ple imitation or conformity pressure effect: agents adopt a behaviour/product/like/fashion whenever some given threshold of the agents they are related to in their social network, their neighbours, have adopted it already. In this sense, investigating diffusion is investigating how agents are locally inﬂuenced and how they tend to become more similar to their neighbours. The so-called threshold mod- els, ﬁrst introduced by [6,13], are used precisely to repre- sent the dynamics of diffusion under threshold inﬂuence. This type of models has received much attention in recent literature [5,8,11,16]. The paper has two main goals. The ﬁrst one is to de- sign a logic to represent the traditional view of threshold inﬂuence and to reason about diffusion phenomena in so- cial networks. This is the topic of section 2. After recalling standard threshold models in Subsection 2.1, a dynamic logic for modeling threshold inﬂuence within social net- works is introduced in Subsection 2.2. While conceptu- ally in line with [4,12,14,15,22] in using logic to model social inﬂuence effects within networks structures, this framework differs by avoiding the use of static modali- ties or hybrid logic tools. In this sense, the logic intro- duced is “minimal”: propositional logic is used to specify both network structure and agent behaviour, and a single dynamic modality is used to represent “threshold inﬂu- ence update”. Moreover, while [4,12,14,15,22] focus on the limit thresholds of 100% (all neighbours) and non- 0% (at least one neighbour), here any (uniform) adop- tion threshold can be used, as is standard with respect to threshold models. Subsections 2.3 and 2.4 exemplify how the logic allows to reason about clusters, cascading effects and the minimal seed problem. The second goal of the paper is to extend threshold models and their dynamic logic with an epistemic dimen- sion. This is done in Section 3. Subsection 3.1 introduces epistemic threshold models and their update procedure. This corresponds to a conceptual jump from a minimal modeling of inﬂuence as “blind adoption” to a more so- ⋆ University of Amsterdam, zoe.christoff@gmail.com ⋆⋆ Lund University, rendsvig@gmail.com