We propose a two dimensional \social network plausibility framework" to model doxastic in uence through communication in a social network. To do so, we combine two approaches: on the one hand, a hybrid logic setting, to model the social network itself (who is related to whom), and on the other hand, dynamic epistemic logic, to model the distribution of beliefs among agents (who believes what) and belief changes induced by communication events (what is said to whom and how do the hearers revise their beliefs). Combining both, we show how to design some particular communication protocols in this new framework to represent some level of social doxastic in uence, assuming that the communicating agents are sincere and trust each other.
Agents involved in a social network typically interact with the agents they are related to. By exchanging information, they in uence those related agents and are in uenced by them. If we consider the example of online social networks, agents communicate mainly with their \friends" or \followers". The same seems to apply in other examples of social relationships such as being colleagues or family members: people are in uenced by people they interact with and the structure of who can interact with whom speci es a social network structure. In other words, agents who communicate are typically related by some social relationship (and hence, are part of s social network structure) and agents in a social network typically communicate with the agents they are related to according to the structure of a social network. Therefore, it seems quite natural to try and combine explicitly a social network structure with communication events in a unique framework to model the e ects of social in uence.
We propose a logical setting in which di erent scenarios of \social in uence
via communication" can be modeled. We are interested in reasoning about the
doxastic states of agents in a social network and focus in particular on the
communication protocols that can express di erent levels of social doxastic in uence.
Formally we use the tools and techniques of dynamic epistemic logic1, combining
1 See for instance [
First, we build on the hybrid logic framework of Liu, Seligman and Girard [
Second, we adopt the perspective of Baltag and Smets in [
Both the above mentioned logical frameworks are concerned with representing the belief revision induced by what the other agents believe and with the reachability of a certain type of stable states. Moreover, in both settings, once all agents of a group agree, nothing changes anymore. However, while the rst setting assumes direct (i.e., without explicit communication) bilateral and synchronic in uence between all related agents, the second one assumes unilateral and diachronic in uence on all agents through sequential public communication events. Moreover, while only the rst setting allows to model agents and the social network explicitly, only the second setting allows to model beliefs and belief revision explicitly in terms of their underlying plausibility structures. We aspire to design a framework which can incorporate both aspects.
Combining both approaches, we propose a uni ed general \social network
plausibility framework" in which the agents, their plausibility orderings and their
social relationships are modeled explicitly and on which di erent communication
protocols can be de ned to represent di erent types of belief change under peer
pressure. In the next sections, we will introduce the logical tools needed to build
this new framework. In section 2 we introduce the two dimensional social network
plausibility static framework. In section 3 we add the dynamics to our framework
and show how the notion of strong in uence from [
A two dimensional social network plausibility
framework
Our formal setting consists of two dimensions: a doxastic dimension and a social
network dimension. To model the beliefs of agents, we follow [
De nition 1 (Syntax). The social network plausibility static language is the following, where p ∈ Φ is an atomic proposition and n ∈ N is an agent nominal:
Our two dimensional models are simply the result of embedding one dimension into the other one: a social network plausibility model is a ( nite, multiagent, pointed) plausibility model in which a social network frame is associated to each possible state.
De nition 2 (Social network plausibility model). A social network plausibility model is a tuple M = (S, A, ≤a∈A, ∥·∥, s0, ≍s∈S ), such that: { S is a ( nite) set of possible states, { A is a ( nite) set of agents, { ≤a⊆ S × S is a locally connected preorder, interpreted as the subjective plausibility relation of agent a, for each agent a ∈ A, { s0 ∈ S is a designated state, interpreted as the actual state, { ≍s⊆ A×A is an irre exive and symmetric relation, interpreted as friendship, for each state s ∈ S, { ∥·∥ : N ∪ Φ → P(S × A) is a valuation, assigning a set ∥p∥ ⊆ S × A to every element p of some given set Φ of \atomic sentences" and assigning a set ∥n∥ = S × {a} for some unique a ∈ A to every element n of some given set N of \nominals".
We inherited from [
M, w, a
p i ⟨w, a⟩ ∈ ∥p∥ M, w, a M, w, a M, w, a M, w, a M, w, a M, w, a n i ⟨w, a⟩ ∈ ∥n∥ i a = n ¬ϕ i M, w, a 2 ϕ ϕ ∧ ψ i M, w, a F ϕ i M, w, b @n ϕ i M, w, n Bϕ i M, v, a
ϕ and M, w, a ψ ϕ for all b such that a ≍ b
ϕ ϕ for all v ∈ S such that v ∈ bestaw(a) a c w b p d
c a, b, d
a
p c
v
b p
d
Let us now consider how models change under social in uence through
communication. We start by imposing some simplifying assumptions. First, only friends
communicate.4 What change is induced by communication between friends? In
general, depending on how much an agent trusts the source of new information,
she can transform her belief state in di erent ways.5 We assume that friends trust
their friends: whatever any of my friends announces, I revise my beliefs with it.6
More precisely, we assume that friends trust their friends strongly enough to
perform a radical revision with any formula ϕ announced. The operation on
plausibility models corresponding to this strong level of trust is radical upgrade
or \lexicographic upgrade" ⇑ ϕ [
De nition 4 (Joint radical upgrade). ⇑ ϕ is a model transformer which takes as input M= (S, A, ≤a∈A, ∥·∥, s0, ≍s∈S ) and outputs M′=(S, A, ≤′a∈A, ∥·∥, s0, ≍s∈S ) such that: s ≤′a t i either (s, t ̸∈ ∥ϕ∥and s ≤a t)or (s, t ∈ ∥ϕ∥and s ≤a t)or (t ∈ s(a)and s ̸∈ ∥ϕ∥and t ∈ ∥ϕ∥).
How do agents with such a level of trust have to communicate to reach a state
where they all have the same beliefs? The assumption in [
We adapt this protocol to accommodate the indexicality or our setting, i.e., we make sure that when an agent a announces ϕ, the hearers revise with @aϕ and not directly with ϕ. This seems to re ect in a natural way the indexicality of real-life communication. Note also that this protocol trivially requires that all agents are friends with each other.
De nition 5 (Beliefs lexicographic merge indexical protocol). ρa := ∏{⇑ @aϕ : ∥@aϕ∥ ⊆ S × A such that M, w, a |= Bϕ}
etc for all c ∈ A where ∏ is a sequential composition operator and M[ a] is the new model after all agents have performed a radical upgrade with each formula announced by a.
In the reminder, we will show social in uence as given in [
Let us consider the simplest example of strong in uence: some agent a is the other agents' only friend, has the highest expertise rank, and believes that ϕ. Translating this into a communication setting, a will announce that ϕ and the others will be (strongly) in uenced into revising their beliefs with ϕ, evaluated at agent a. For instance, if a announces \I am blonde"(p), which is equivalent to her announcing \a is blonde" (@ap), her friends come to believe that a is blonde (@ap). They now agree on whatever was announced (they all believe @ap). This is, therefore, at the same time, the simplest case of beliefs merge, and the simplest case of strong in uence (as long as we only consider the case of a's friends and ignore what happens to agent a for now). This can be represented by a one step version of the beliefs lexicographic merge indexical protocol given above: De nition 6 (One-to-the others unilateral strong in uence protocol).
where ∏is a sequential composition operator
Assume now that all of agent a's friends are friends with each others. Agent a is in a state of strong in uence with ϕ if and only if everybody else agrees on believing ϕ, in which case she will revise her belief with ϕ. But before she revises her beliefs, a rst needs to be aware of what her friends believe. We de ne the following protocol, in which, unlike in the above, agents have to announce that they believe something, whenever they did initially believe it. This will result in a believing that all of her friends believe something if they actually did. We still need a to revise her beliefs with ϕ. So we add the last step of the protocol, fundamentally di erent from the rest of the protocol, representing the reasoning of agent a, the conclusion she reaches, announcing to herself (and thereby to her friends) that ϕ.
De nition 7 (The others-to-one unilateral strong in uence protocol).
etc, for all d ∈ A such that M, w, d |= ⟨F ⟩a
M[ b; c;:::], w, a |= BF Bϕ} where ∏is a sequential composition operator and M[ b; c;:::] is the model resulting from the successive revisions (by all friends) with each of the formulas announced by each of them.8 8 The de nition of strong in uence has the counterintuitive consequence that if ϕ means \I am blonde" and if all of my friends believe that they (themselves) are
So far we have considered only the case in which agents are under maximal
and unilateral in uence. We have ignored weaker cases where not all but only
a signi cant proportion of friends believe something, and we have ignored the
fact that while being in uenced by my friends, I in uence them too. This is a
crucial di erence between the way the communication protocols from [
As mentioned in the introduction, in the setting of [
Acknowledgments I would like to thank Johan van Benthem, Jens Ulrik
Hansen, Emiliano Lorini, and Sonja Smets for their suggestions and comments
during the elaboration of this short paper. An early version of this work has been
presented at the Seventh Workshop in Decisions, Games & Logic and I would
blonde, I end up believing that I am blonde too. One solution is to restrict the
above protocol to formulas ϕ whose truth do not depend on the evaluation agent,
characterized by making the following valid: @aϕ ⇔ @bϕ.
9 For a dynamic hybrid framework allowing such a discrepancy between what an agent
believes and what she seems to believe, see [
The research leading to these results has received funding from the European Research Council under the European Communitys Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 283963.