Trust and distrust in contradictory information transmission
 Giuseppe Primiero^{1}Email authorView ORCID ID profile,
 Franco Raimondi^{1},
 Michele Bottone^{1} and
 Jacopo Tagliabue^{2}
Received: 24 February 2017
Accepted: 29 April 2017
Published: 5 June 2017
Abstract
We analyse the problem of contradictory information distribution in networks of agents with positive and negative trust. The networks of interest are built by ranked agents with different epistemic attitudes. In this context, positive trust is a property of the communication between agents required when message passing is executed bottomup in the hierarchy, or as a result of a sceptic agent checking information. These two situations are associated with a confirmation procedure that has an epistemic cost. Negative trust results from refusing verification, either of contradictory information or because of a lazy attitude. We offer first a natural deduction system called SecureND^{sim} to model these interactions and consider some metatheoretical properties of its derivations. We then implement it in a NetLogo simulation to test experimentally its formal properties. Our analysis concerns in particular: conditions for consensusreaching transmissions; epistemic costs induced by confirmation and rejection operations; the influence of ranking of the initially labelled nodes on consensus and costs; complexity results.
Keywords
Introduction
Trusted information is an essential feature of computational contexts where agents might have to rely on external sources to execute decisions effectively and securely. With increasingly large networks, trust becomes a method to acquire information that would otherwise be unavailable or that is effectively hard to produce. Similarly, trust applies in contexts where a hierarchical structure is in place, defined for example by privileges in an access control model: here trust can be either a property of topdown communications, where information is not required to be confirmed; or the result of confirmation procedures in bottomup transfers.
Besides network size and structure, another important factor that influences the result of trusted information sharing processes is the attitude of the network’s nodes, when these are understood as epistemic agents. Sceptic agents can be characterised by a requirement to check information; lazy ones by an attitude to reject it. This is of even greater relevance where contradictory information is allowed, e.g. in the context of opinion diffusion, belief propagation and its extreme cases, like fake news.
Hence, understanding conditions of information propagation and the costs related to topological and epistemic factors is crucial for dynamic (social) network analysis and access control models, with applications in mathematics, computer science, economics and biology, but also in concrete, less formal scenarios, e.g. (van de Bunt et al. 2005).
 1.
in contexts with contradictory information like social networks or insecure access control systems, understanding how positive and negative trust help or hinder the data flow;
 2.
the epistemic costs of (negative) trust transitivity.

sceptic agents pay an epistemic cost for performing a checking operation before trusting the received information;

lazy agents accept without checking information consistent with their current knowledge, while they distrust inconsistent messages.
In this context, positive trust is a property of the communication between agents required when message passing is executed bottomup in the hierarchy, or as a result of a sceptic agent checking information. Negative trust is instead the result of rejecting received contradictory information. These two situations are associated with epistemic costs, essential to determine if a network that resolves contradictory transmissions by rejecting information is more or less costly than one which facilitates message passing by straightforward acceptance. We focus in particular on networks that preserve memory of previously obtained trusted communications.
 1.
the computation of the trust value in a given derivation;
 2.
the resolution of derivations with contradictory information;
 3.
the convergence between valid formulae, order on agents and the application of specific rules.
The logic is an extension of the calculus for trust developed in (Primiero and Raimondi 2014) and extended in (Primiero 2016) with the semantics for negative trust from (Primiero and Kosolosky 2016). The calculus has been applied to problems in software management in (Boender et al. 2015a).
 1.
changes in the final distribution of contradictory information in view of network topology;
 2.
changes in the final distribution of contradictory information in view of ranking and epistemic role of seeding agents;
 3.
the quantification of the epistemic costs for trust and distrust operations.
The experimental analysis offered here extends the initial results presented in (Primiero et al. 2017).
The paper is organised as follows. In “Related work” section we overview the related work, both formal and experimental. In “The logic (un)SecureND^{ sim } ” section we introduce the calculus SecureND^{sim} and provide the metatheoretical results. In “Design and implementation” section we introduce the principles underlying the graph construction and analyse the algorithms at the basis of the simulation. In “Experimental results” section we describe our experimental results on consensus, rankings, costs and complexity. Finally, “Conclusions” section presents general observations on our analysis and future work.
Related work
Our work is at the confluence of several different research areas. We focus in particular on the role of trust in computational environments and the several approaches to its propagation. In relation to contradictory information, we overview the distinction between controversial users vs. controversial trust values. In qualifying our own semantic notion of trust, we report on models using binary and continuous trust values. Finally, we consider how local vs. global trust methods differ.
The first research area of interest concerns the treatment of computational trust for models of access control and network analysis. This area includes several logical model and algorithmic treatments, with an eye on applications and it spans several disciplines, including logics, cryptography, network theory, security protocols. Since the BellLaPadula Model in access control theory (Bell and LaPadula 1973), trust is intended as a property of agents, as opposed to security as a property of the system: trusted subjects are allowed to violate security constraints and trustworthiness of resources corresponds to prevention of unauthorised change. In more recent resourcebased access control models, trustworthiness is either defined by temporalspatial constraints (Chandran and Joshi 2005), or by userdefined constraints (Chakraborty and Ray 2006; Oleshchuk 2012). In authentication logics, trust is coupled to beliefs with application to distributed settings (Barker and Genovese 2011).
Trust propagation, interference and distrust blocking in uncertain environments and autonomous systems are receiving increasing attention (Carbone et al. 2003; Guha et al. 2004; Ziegler and Lausen 2005; Marsh and Dibben 2005; Jøsang and Pope 2005), with applications to internetbased services (Grandison and Sloman 2000), componentbased (Yan and Prehofer 2011) and software management systems for security and reputation (Bugiel et al. 2011), or accuracy (Ali et al. 2013). A costsefficient analysis in terms of modal logic is to be found in (Anderson et al. 2013). In the context of propagation, transitivity is a natural property to study. The problem of trust transitivity has received much attention, see (Christianson and Harbison 1997; Jøsang and Pope 2005) for an older and a more recent approach. An analysis for trusted communications in terms of dependent types is given in (Primiero and Taddeo 2012). In (Primiero and Raimondi 2014), trust is defined prooftheoretically as a function on resources rather than a relation between agents: this allows transitivity of writing privileges only under satisfaction of a consistency constraint. The calculus introduced in “The logic (un)SecureND^{ sim } ” section is an extension thereof. Unfortunately, none of these previous logicbased approaches consider either the case of contradictory information, nor the aspects of epistemic costs associated with such a scenario. This task is the aim of the translation of logical principles in an experimental analysis, presented in “Design and implementation” section. Transitivity of trust is also analysed in the context of cryptographic applications, see e.g. (Maurer and Schmid 1996).
A related area of analysis is that of belief diffusion, especially in social networks. Continuous models assume a continuous numerical value on agents’ opinions, with updates depending on weights (Lehrer and Wagner 1981), where the latter can also vary over time (Chatterjee and Seneta 1977) or where influence is admitted only below a certain distance (Hegselmann and Krause 2002). A model that combines opinion diffusion with influencing power is presented in (Grandi et al. 2015). Another aspect of information diffusion in social networks which is gaining much attention is related to the propagation of misinformation or ‘fake news’, the resulting polarisation of communities and the possibility of cascades, see e.g. (del Vicario et al. 2016). Our model combines the comparative ranking value of agents with both their distinct epistemic attitudes and a majority selection in the case of conflicting information, which we take to indicate the presence of both a correct and an incorrect interpretation of facts.
In (Massa and Avesani 2005) controversial users are those generating a disagreement on their trustworthiness, either as the minimum between trust and distrust evaluations by other users, or as the difference in the number of trust and distrust judgements. The work in (Zicari et al. 2016) considers, instead, controversial trust values between two nodes, determined either as the trust weight of their edge, or as a fixed negative value when no path exists, or as a continuous value t∈[0,1] when there is no direct edge. Similarly, in our logic trust is a function on formulas obtained by verification, encoded in the network model by a property of edges when a node is labelled. Differently from the above, our model uses discrete values but it combines the comparative ranking of agents with both their epistemic attitudes and a majority selection in the case of conflicting information. The approach in (Massa and Avesani 2005) also uses a binary classification for users, so do several models for belief diffusion in social networks, with binary opinions for agents, considering neighbours’ influence (Granovetter 1978; Kempe et al. 2005) or majority (Raghavan et al. 2007). Trust defined by global methods is a value attached to a user and appropriate for a reputation evaluation at network level; in local methods, trust is inferred instead as a value between source and sink nodes, i.e., it is a feature of an edge. As it appears clearly from the above, our approach uses a local trust method in the case of nonconflicting information, resorting to a computation of trust using path lengths to determine which elements need to be distrusted in the case of conflicting information. This combination of features recalls the two controversial cases discussed in (Zicari et al. 2016): the ToTrustOrNotToTrust case resembles our binary choice, but moderated by continuous trust values, while we rely on ranking and epistemic attitudes; the Asymmetric Controversy case resorts to path lengths with preference for shortest paths, while we base our result on the number of distrustful edges present in each path.
The modelling and computer simulation of trust and its management for large (social) networks sharing data are also becoming widespread across various scientific communities. The exposure of users in social networks to ideologically diverse contents is a recent object of study in networks theories, see (Bakshy et al. 2015), in particular to qualify the assumption that such networks facilitate the creation of filter bubbles and echo chambers. Another example is given by the (standard and digital) scientific networks, which count trustworthiness as a parameter to select citations and coauthorship (Quattrociocchi et al. 2012). Accordingly, the role of trust in these types of networks is receiving much attention in academia, industry and policymaking. An overview of this research area with a focus on trust evolution is available from (Netrvalová 2006). Another extensive literature review of computational modelling of trust with a focus on evolutionary games and social networks is given in (Mui 2002). The role of trust in virtual societies is also analysed in (Coelho 2002) and (Corritore et al. 2003). A more recent analysis for social relations is provided in (Sutcliffe and Wang 2012). In (Iyer and Thuraisingham 2007) a Java simulation for trust negotiation and confidentiality between agents with common goals and only partial information is introduced. Transaction costs economics in intergroups relations has been extended in view of trust in (Nooteboom et al. 2000): here agents attach a metric on trust relative to potential profit. Joint work in teams is also analysed in terms of trust to quantify performance at the individual and team level in (MartínezMiranda and Pavón 2009).
As far as we are aware, none of the previous works combines a rulebased semantics with an experimental analysis. Moreover, the combination of characteristics of the model we analyse and simulate seem to have been ignored so far. These include: a characterisation of agents as sceptic or lazy towards epistemic contents; their structural ranking, typical of access control systems; the quantification of epistemic costs of trust and distrust; and the conditions for consensus in the presence of contradictory information.
The logic (un)SecureND^{ sim }
In this section we introduce the logic (un)SecureND^{ sim }, a natural deduction calculus whose rules define how agents can execute access operations on (atomic) formulas and their negations. The prooftheoretic setting of the language has several advantages. First, it allows us to clearly express the algorithmic protocol introduced in “Design and implementation” section through pairs of rules that fully describe the semantics of each operation available to agents. Second, it provides the means to explore metatheoretical properties of the model which the simulation cannot offer. Finally, it lays down preparatory work for the possible formal verification of the protocol.

sceptic agents: they pay an epistemic cost by performing a checking operation before trusting received information;

lazy agent: they distrust the information when this is not consistent with their current knowledge.
Obviously, this distinction does not cover the whole spectrum of possible epistemic attitudes towards information received and it could be offered in a graded scale. We limit ourselves here to these two basic cases. While the logic is rigid with respect to how these attitudes are executed, in the simulation model developed in “Design and implementation” section, we allow for slight changes of behaviour, i.e. where we establish that only a certain fraction of sceptic agents check information.

verification: it is required either by a topdown reading operation, i.e., when message passing is executed from below in the hierarchy; or by a reading operation performed by a sceptic agent;

falsification: it is formulated as closure of verification under negation and it follows from reading contents that are inconsistent with the current knowledge of the receiver; or from a reading operation performed by a lazy agent;

trust: is a function that follows from verification, when the content passed is consistent with the knowledge of the receiver;

distrust: it is formulated as closure of trust under negation and it follows from falsification.
It is important to stress that, according to this operational semantics, agents do verify or falsify information on the basis of a contextual evaluation. Agents are here presented in a contextually empty process of information transmission and their evaluation is purely based on the evaluation of trustfulness and distrustfulness on the basis of criteria of majority and origin. In this sense, our logic, the algorithm and the simulation focus on the role of trust and distrust as independent from truthfulness criteria, while consistency requirements are crucial. In other words, we do not establish beforehand which of two contradictory atoms of information is true.
Alice trusts Bob and Bob trusts Carol; therefore Alice trusts Carol.
In (Boender et al. 2015b), the logic is formally verified through translation to a Coq protocol and applied to a problem of trust transitivity in software management. In (Primiero 2016), the calculus is further extended with negation to define the logic (un)SecureND, which includes two negative trust protocols: one for misplacement of trust (mistrust), on for betrayal (distrust). (un)SecureND^{ sim } models the latter in the context of message passing in a network of ordered agents with contradictory information. The extension to a verification protocol is a new property added to this family of logics by the present fragment.
We first introduce the syntax of the system and the main properties of the underlying access control model. We further illustrate in some details the rules and some minimal metatheoretical properties of the related derivability relation. Finally we illustrate some structural properties of (un)SecureND^{ sim } derivations concerning especially validity and the role of trust instances in the length of such derivations. These formal properties are later experimentally tested in “Experimental results” section.
Formal preliminaries
Definition 1
V ^{<} indicates the set of lazy and sceptic agents, each denoted by \(\phantom {\dot {i}\!}v_{i},{v{j}},\dots \). The apex works as a formal reminder that agents are ordered according to an order relation <. The order relation < over V×V models the dominance relation between agents: v _{ i }<v _{ j } means that agent v _{ i } has higher relevance (e.g., in terms of security privileges) than agent v _{ j }. ϕ ^{ V } is a metavariable for boolean atomic formulae closed under negation and functions for reading, writing, verification and trust. It should be stressed that the current formal model and the subsequent algorithmic model refer to atomic information for simplicity, but the complexity of the formula representing the transmitted information is entirely irrelevant for our results. We use \(\phantom {\dot {i}\!}\Gamma ^{v_{i}}\) to express a set of formulae typed by one agent v _{ i }∈V, typically the sender, in which a given formula ϕ ^{ V } is derivable. \(\phantom {\dot {i}\!}\Gamma ^{v_{i}}\) is called the context in which \(\phantom {\dot {i}\!}\phi ^{v_{i}}\) is derived. We denote an empty context by ·⊩.
Definition 2
(Judgement) A judgement \(\Gamma ^{v_{i}} \vdash \phi ^{v_{j}}\) states that a formula ϕ is valid for agent v _{ j } in the context Γ of formulas (including operations) of agent v _{ i }.
Our judgements express thus some operation that the agent on the lefthand side of the derivability sign performs on information typed the agent on the righthand side of the same sign. When message passing includes more than one agent, this is encoded in the system by an extension of the context, denoted as \(\phantom {\dot {i}\!}\Gamma ^{v_{i}};\Gamma ^{v_{j}}\). A judgement stating the validity of a formula for one agent under a (possibly extended) context of formulas of (an)other agent(s) matches the procedure Transmission introduced below in “Design and implementation” section to extend a given graph G with a newly labelled vertex.

\(\Gamma ^{v_{j}}\vdash Read(\phi ^{v_{i}})\): reading is always allowed when messages come from up in the order relation; this is not always the case in access control models, where one might establish reading privileges to hold only upwards, e.g. in the case where strict security is applied; we model a less strict scenario, where agents can always read information that is passed topdown.

If \(\Gamma ^{v_{i}}\vdash Read(\phi ^{v_{j}})\) then \(\Gamma ^{v_{i}}\vdash Verify(\phi ^{v_{j}})\): messages coming upwards from below in the order relation are passed on under a verification function.

If sceptic( v _{ j } ), and \(\Gamma ^{v_{j}}\vdash Read(\phi ^{v_{i}})\), then \(\Gamma ^{v_{j}}\vdash Verify(\phi ^{v_{i}})\): we further wish to enhance the structure of the model by requiring that the message passing is qualified by verification in one additional case: not only because the transmission is executed upwards in the dominance relation, but also because a sceptic agent is on the receiving side and it verifies the information.

If lazy( v _{ j } ), and \(\Gamma ^{v_{j}}\vdash Read(\phi ^{v_{i}})\), then \(\Gamma ^{v_{j}}\vdash \neg Verify(\phi ^{v_{i}})\): when a lazy agent is on the receiving side, information is not verified.

If \(\Gamma ^{v_{i}}\vdash Read(\phi ^{v_{j}})\) and \(\Gamma ^{v_{i}}\vdash \neg \phi \), then \(\Gamma ^{v_{i}}\vdash \neg Verify(\phi ^{v_{j}})\): when a content read from below contradicts current knowledge, refutation is modelled as negation of verification.
Example 1
A simple derivation of message passing (assuming v _{ i }<v _{ j }):
This derivation illustrates a message ϕ written by agent v _{ j }, read by agent v _{ i }, verified and written to be passed further on. This standard case holds also if v _{ i }∈sceptic_node (even assuming v _{ j }<v _{ i }); if v _{ i }∈lazy_node, then the verification passage is skipped to infer write directly.
Proposition 1
Any successful (un)SecureND^{ sim } messagepassing operation is a derivation tree including a WriteRead(VerifyTrust)Write series of sequents.
Standard logical notions can be formulated as follows:
Definition 3
(Satisfiability) An (un)SecureND^{ sim } judgement \(\Gamma ^{v_{i}}\vdash \phi ^{v_{i}}\) is satisfied if there is a derivation D and a branch D ^{′}⊆D with a final step terminating with such a judgement.
Definition 4
(Validity) An (un)SecureND^{ sim } judgement Γ ^{ V }⊩ϕ ^{ V } is valid if there is a derivation D and for all branches D ^{′}⊆D and for all agents v _{ i }∈V, there is a final step terminating with such a judgement.
Structural properties on derivations
By Proposition 1, verification and trust are optional steps in a derivation if the message is received by a lazy agent. This suggests that each derivation (or branch thereof) can be analysed in view of its length to count the number of trust rule instances occurring in it.^{1} This allows us to identify the number of times an atomic message ϕ has been trusted in a given derivation D. We denote such measure by \(\phantom {\dot {i}\!}\mid \!{Trust}(\phi ^{V})\!\mid _{D}\).
Theorem 1
∣ Trust(ϕ ^{ V }) ∣_{ D }=∣ Verify(ϕ ^{ V }) ∣_{ D }, for all v _{ i }∈V.
Proof
By induction on the length of D, provided that verify_high and verify_sceptic are the only rules that introduce a formula \(\phantom {\dot {i}\!}Verify(\phi ^{v_{i}})\) which is the premise of a trust rule. □
This computable method allows us to offer a simple resolution for the case in which consistency fails and one wants to decide on the basis of the more trusted formula on the derivation tree.
Definition 5
(Conflict Resolution by Trust Majority) Given a derivation D _{1} terminating in \(\Gamma ^{v_{i}}\vdash Write(\phi ^{v_{i}})\) and a derivation D _{2} terminating in \(\Gamma ^{v_{j}}\vdash Write(\neg \phi ^{v_{j}})\), a new step holds which takes as premises \(\Gamma ^{k}\vdash Read(\phi ^{v_{i}})\) and \(\Gamma ^{k}\vdash Read(\neg \phi ^{v_{j}})\) respectively, and concludes \(\Gamma ^{v_{k}} \vdash \phi ^{v_{k}}\) if and only if \(\mid \!Trust(\phi ^{V})\!\mid _{D_{1}} > \mid \!Trust(\neg \phi ^{V})\!\mid _{D_{2}}\).
This suggests that at any stage of branch merging, the most popular (trusted) content is preserved, hence enforcing a network effect.
A different resolution strategy can be enforced by computing the number of times an atomic message ϕ has been distrusted in a given derivation D. We denote such measure by ∣ DisTrust(ϕ ^{ V }) ∣_{ D }.
Theorem 2
∣ DisTrust(ϕ ^{ V }) ∣_{ D }=∣ ¬Verify(ϕ ^{ V }) ∣_{ D }, for all v _{ i }∈V.
Proof
By induction on the length of D, provided that unverified_contra and unverified_lazy are the only rules that introduce a formula \(\phantom {\dot {i}\!}\neg Verify(\phi ^{v_{i}})\) which is the premise of a distrust rule. □
Definition 6
(Conflict Resolution by Distrust Majority) Given a derivation D _{1} terminating in \(\Gamma ^{v_{i}}\vdash Write(\phi ^{v_{i}})\) and a derivation D _{2} terminating in \(\Gamma ^{v_{j}}\vdash Write(\neg \phi ^{v_{j}})\), a new step holds which takes as premises \(\Gamma ^{k}\vdash Read(\phi ^{v_{i}})\) and \(\Gamma ^{k}\vdash Read(\neg \phi ^{v_{j}})\) respectively, and concludes \(\Gamma ^{v_{k}} \vdash \phi ^{v_{k}}\) if and only if \(\mid \! DisTrust(\phi ^{V})\!\mid _{D_{1}} < \mid \! DisTrust(\neg \phi ^{V})\!\mid _{D_{2}}\).
From Definition 4, the following holds:
Lemma 1
For each (un)SecureND^{ sim } derivation D with a valid formula Γ ^{ V }⊩ϕ ^{ V }, there is a graph G that is unanimously labelled by ϕ.
Proof
The proof requires to construct a graph G with a node for each distinct v _{ i }∈V occurring in D and an edge for each judgement instantiating one or more rules with two distinct nodes on each side of the derivability sign. Starting from the node occurring at the highest position of D validating ϕ, by application of one or more sequences of rules the conclusion in such branch of D represents a new node in G labelled by ϕ. If all branches of D terminate with a formula validating ϕ, as by assumption and according to Definition 4, then all nodes in G will be labelled by ϕ. □
The construction of such a graph G for experimental purposes is the aim of “Related work” section. Notice, moreover, the following structural properties on SecureND ^{ sim } derivations.
Lemma 2
For a derivation D of (un)SecureND^{ sim }, the value of ∣ Trust(ϕ ^{ V }) ∣_{ D } is directly proportional to the number of verify_high rule applications and the number of distinct sceptic( v _{ i } )∈V occurring as labels in the premises of the derivation.
Proof
By structural induction on D, selecting the appropriate step as indicated by Theorem 1. □
Lemma 3
For a derivation D of (un)SecureND^{ sim }, the value of ∣ ¬Trust(ϕ ^{ V }) ∣_{ D } is directly proportional to the number of unverified_contra rule applications and the number of distinct lazy( v _{ i } )∈V occurring as labels in the premises of the derivation.
Proof
By structural induction on D, selecting the appropriate step as indicated by Theorem 2. □
Lemma 4

the number of instances of the verify_high rule applications.

the number of instances of the verify_sceptic rule applications, for each v _{ i }∈V.

the number of instances of the unverified_contra rule applications.

the number of instances of the unverified_lazy rule applications, for each v _{ i }∈V.
where \(\phantom {\dot {i}\!}\phi ^{v_{i}}\) occurs in the first premise.
Proof
This follows directly by Lemmas 2 and 3, and for the graph analysis by Lemma 1; the more verification operations and the more sceptic agents, the higher the convergence towards validity; the more distrust operations on the same formula, and the more the lazy agents, the lower the convergence. □
We offer in the following sections an agent based simulation which implements the set of rules described in (un)SecureND^{ sim } and proceed with an experimental analysis of its conditions and results.
Design and implementation
In this section we illustrate the design and implementation of a NetLogo model (Wilensky 1999) based on (un)SecureND^{ sim } to investigate properties related to knowledge distribution depending on the epistemic attitude of the seeding agents and on the network topology. NetLogo is a wellknown, widely used modelling platform for complex systems of interacting agents.
We start with basic definitions and analysis of the topologies of interest.
Definition 7
(Graph) A network is an undirected graph G=(V,E), with a set V={v _{ i },…,v _{ n }} of vertices representing our agents and a set E={e _{(i,j)},…,e _{(n,m)}} of edges, representing transmissions among them.
Definition 8

v _{ i }(p) denotes a vertex labelled by an atomic formulas and expresses an agent i knowing p;

v _{ j }(¬p) denotes a vertex labelled by the negation of an atomic formula and expresses agent j knowing ¬p;

v _{ k }() denoted a vertex with no label and expresses an agent k who does not hold any knowledge yet.
An edge between two labelled nodes is denoted by e(v _{ i }(p),v _{ j }()) and denotes a transmission channel from agent i to agent j such that the former can transmit p over to the latter. The case e(v _{ i }(p),v _{ j }(¬p)), i.e., where an edge is constructed between two agents holding contradictory information, is admissible and it requires a resolution procedure. This is generalised below to the case where one agent who does not hold any knowledge yet receives contradictory information from two distinct agents. To this aim, a nonstandard notation with three nodes e(v _{ i }(p),v _{ j }(),v _{ k }(¬p)) is used in the following to abbreviate the presence of two edges e(v _{ i }(p),v _{ j }()) and e(v _{ k }(¬p),v _{ j }()) between three nodes, one holding p, one holding ¬p and one with no information. This is another case where a node with an empty label requires a resolution procedure to choose between labels p and ¬p.

In a linear network, each vertex has an edge to the next vertex higher in the ranking; by transitivity, this order is also total. See Fig. 2 b.

In a random network, for as long as new nodes are introduced, edges are created making sure that for each vertex at least one edge with another vertex is established; the ranking is here assigned by nodes labelled at the beginning (the seeding node) and never overwritten, the order is partial. See Fig. 3 a.

Scalefree networks (Milgram 1967; Watts and Strogatz 1998) use the BarabasiAlbert method to establish edges. Initialised by m=3 nodes, each node with 0 neighbours is asked to create an edge with a vertex in the network; for each new vertex v _{ j } without neighbours, v _{ j } is connected to up to n<m existing vertices with a probability p(v _{ j }) defined by the following expression:where \(k_{v_{j}}\) is the number of neighbours of agent v _{ j } and the sum is made over all preexisting nodes v _{ i }. Newly added nodes tend to prefer nodes that already have a higher number of links. The ranking in this case is given to each node by a simple function \(\frac {1}{\mid edges\mid }\). Scalefree world networks are characterised by the clustering coefficient with a degree distribution that follows a powerlaw. See Fig. 3 b.$$\mathbf{p}({v_{j}}) = \frac{k_{v_{j}}}{\sum_{v_{i}} k_{v_{i}}} $$
The maximum number of vertices in our graphs is set at 300. The scalefree network model can be assumed to be representative of real social networks cases: the assumption is that the degree distribution (as encoded in the network topology) is the main factor to be investigated by the model. It is known that many global network properties (like resilience to attack, short distance between any pair of nodes etc.) depend on network structure as defined much more than network size, so it is a reasonable starting point for our model. As most interesting realworld networks follow the scalefree model, the analysis of real world networks concerning trust and information spreading can be sufficiently modelled by this network type.
 1.
overly lazy network: in this type of network, the proportion of sceptic nodes is set at 20%, with their confirmation rate at 5%, the latter expressing the proportion of such agents that will after all ask for verification;
 2.
balanced network: in this type of network, the proportion of sceptic nodes is set at 50% and their confirmation rate at 95%, to account for a 5% of random sceptic agents who decide not to ask for verification after all;
 3.
overly sceptic network: in this last type of network, the proportion of sceptic nodes is set at 80%, their confirmation rate at 100%, hence verification is always implemented.

The first one, see Fig. 7, takes into account the number of links with nodes labelled by p, the number of links with nodes labelled by ¬p and sums them to the respective overall rankings, obtaining values Score P and Score¬P. This implementation sensibly refines the pure majority by counting of the formal system in “The logic (un)SecureND^{ sim } ” section, by adding the ranking of the agents involved as a parameter of the related score. For each pair of edges from nodes with contradictory information p,¬p to an unlabelled node, if the value of scoreP is higher than the value of score¬P, the new node is labelled by p, by ¬p otherwise. We assume here a context in which agents refer to a popularity criterion in order to choose which of two contradictory pieces of information to preserve.

The second version, see Fig. 8, analyses the number of distrusted links appended to each neighbour with each contradictory piece of information and it selects the new label from the least distrusted one, proceeding by random choice when an equal number of distrusted links is detected. We assume here a context in which agents refer to a popularity criterion in order to choose which of two contradictory pieces of information not to preserve. It then executes the subroutine Distrust on the selected link.
The observerproperty of trustworthiness (i.e., the total number of trusted links) and distrustfulness (i.e., the total number of distrusted links) are a known property of the network at any given time and used to perform conflict resolution. By the procedure clearP, trusted and distrusted links obtained by a first message passing operation are preserved in subsequent executions of the procedure Transmission over the same graph to analyse their effect on epistemic costs. An objective of the experimental analysis in “Experimental results” section is to compare results of the two resolution subroutines to determine the effects of distinct conflict resolution strategies based on trust and distrust.
Experimental results
Experiments are run over the four distinct types of networks. Scalefree networks better represent the topology of complex graphs as they occur, for example, in social networks. On the other hand, linear networks are more common in hierarchical structures that can be encountered in conditions of access control. The experiments have been executed on a machine with 7.7 GB of memory running 64bit Ubuntu 15.10. We have collected data from several scalefree networks of fixed dimensions between 10 and 300 nodes. The seeding of contradictory information is done by associating an atom p to a lazy node, and its negation to a sceptic one (although this association can be altered at will). The code and result of the experiments are available at https://github.com/gprimiero/securendsim.

The knowledge plot, i.e., the final labelled graph is never consistent with the previous execution.

There is no systematic distribution of consensus across the 30 runs.

There is no systematic relation between the resulting knowledge plot and the costs of the transmission.

There is no systematic relation between the knowledge plot and the ranking of the seeding nodes (nodes labelled at the beginning).
This indicates that memoryless networks do not offer a reliable experimental setting to investigate issues of consensus and epistemic costs of trusted graphs.
The second set of experiments has been performed again on several networks of different size, but ensuring at each run of the main algorithm that the trust graph obtained by a previous run is preserved. This is obtained by executing at the end of each execution a procedure clearP, which eliminates all labels from the graph, but preserves ranking and trusted links. In this way we can better average on the number of trusted and distrusted links which are created and destroyed at each execution.
Under these experimental conditions, we analyse consensus, costs, ranking of the seeding nodes and time complexity in networks with trust and distrust.
Consensus
Here and in the following we will call a graph that satisfies consensus a unanimous graph.

Total networks reach consensus most often.

Scalefree networks always perform better than linear or random ones in terms of number of runs that reach consensus.

The data for random networks is not overly reliable, as full labelling might not be reached (increasingly often in the number of nodes). For this reason, a timeout is set at 1000 steps. A step indicates here one message passing operation. The proportion of runs that timeout is given in Fig. 9 b, showing a nonstrictly linear increase. Accordingly, the number of runs that reach consensus is bound to decrease.
These experimental results on consensus support empirically the properties of SecureND^{sim} derivations provided in “Structural properties on derivations” section, Lemma 4. To observe this, consider that total networks are graphs in which the number of edges between nodes is maximal, corresponding to derivations with the maximal number of branches, one for each pair of agents (v _{ i },v _{ j }) appearing respectively in the premises and in the conclusion. Similarly, overly sceptic networks are graphs corresponding to derivations where more instances of the verify_sceptic rule are used. In both cases, the number of executions resulting in consensus are maximal, as stated by the first item in Lemma 4. On the other hand, linear networks are graphs corresponding to derivations where the number of agents for which the ranking can be transitively established is maximal, and overly lazy networks are graphs corresponding to derivations where more agents implement the unverified_lazy rule.
Epistemic costs
The second type of experimental analyses concerns epistemic costs. With this term we refer to the computational expenses required to perform verification and distrust operations: these correspond in the calculus to instances of rules verify_high and verify_sceptic for trust and rules unverified_contra and unverified_lazy for distrust; in the algorithm they correspond to Verify and Distrust procedures. The effect of these procedures in the network is to generate trusted and distrusted links respectively. Given that there are proportionally more nodes than links and a message might pass more than once over a given node (through several senders), the values for costs are expected to be higher than those for links. Moreover, given the conditions for Verify are more than those for Distrust, the values of trust can be expected to be higher than those for distrust. The aim is to asses these values in the different topologies, to evaluate the proportion between trust and distrust costs and to use them as parameters to evaluate these actions with respect to consensus and complexity.

Random networks are by far the most epistemically expensive.

Linear networks are slightly less expensive than scalefree ones, by a very small margin. If one is balancing costs against information diffusion, scalefree should always be preferred to linear networks, as the associated costs do not diverge much (in general, scalefree networks are more resilient than linear ones).

Given the previous observation on consensus and timeouts, it is obvious that random networks are the worse performing ones.
We now compare in more detail average trust costs in scalefree networks in all configurations (balanced, overly lazy and overly sceptic). The results are plotted in Fig. 12 b. By definition, a network with a higher number of sceptic nodes and confirmation requests will have higher trust costs. For small networks up to 40 nodes the costs are within a small range between 9 and 52; the difference increases significantly between larger overly lazy and overly sceptic networks. Cost difference remains comparably restricted for balanced and overly lazy networks (with a minimum difference of 15 average points at 50 nodes). This suggests that if one is trying to balance trust costs against consensus, large lazy networks should be preferred over balanced ones, as in the latter case the number of runs with uniform labelling tends to drop, while the costs still increase.
The comparison between tables shows that the average number of trusted and distrusted links grows in parallel, while the related costs decrease in a similar vein across the different topologies. Nonetheless, this proportion is not linear. Trust propagates at a much higher rate than distrust in these balanced networks, and there is a small difference between scalefree and linear networks, where the former presents more distrust than trust cost when compared to the latter. These observations suggest that trust is a more frequent and more relevant property in information transmission than the latter in general, and that linear networks are less affected than scalefree ones by distrust propagation.
Let us briefly compare these experimental results with theoretical properties of SecureND^{sim} derivations from “Structural properties on derivations” section. Lemma 2 states that, given a fixed number of sceptic agents in a derivation, the resulting value of trust instances, defined as epistemic costs, is only due to the applications of the verify_high rule. The applications of the rule in question map directly to the number of order relations satisfied by agents in the derivation, and hence to the number of agents that are higher in the order than the agent appearing in the conclusion. Our experimental results show that this cost value is higher in random networks than in graphs with a linear order, where the latter correspond to derivations such that ∀v _{ i },v _{ j }∈V.(v _{ i }<v _{ j })∨(v _{ i }>v _{ j }). In the latter ones, a higher number of transitively valid relation (due to the totality of the graph) means fewer instances of the verify_high rule are applied. For the case of distrust costs, Lemma 3 states that, given a fixed number of lazy agents, the value of distrust instances is only due to the applications of the unverified_contra rule. The explanation above, mapping order relation to topologies, holds in this latter case as well.
Rankings
 1.
whether a strictly higher ranking for one of the seeding nodes implies a greater chance to obtain a unanimous graph labelled by the same formula;
 2.
which type of scalefree networks (between overly sceptic, balanced and overly lazy) has the higher probability to reach a unanimous graph from a seed with higher ranking.

There is no strict correlation between a highly ranked seed and the labelling of the network: the number of cases where the consensus is reached and the label is the same as the one from the higher ranked seed, is relatively small (min \(\frac {1}{7}\), max \(\frac {8}{26}\)).

An overly sceptic scalefree network offers the highest probability to obtain a unanimous graph labelled with the input of the higher ranked node among the seeds; the comparison between the lazy and the balanced network sees the former obtain better results in general, and the latter only for significantly large networks.

Contradictory information transmission from differently ranked nodes tends to be more expensive than from equally ranked nodes in balanced and overly lazy networks: here the costs are induced by a less stable labelling for the information transmitted by higher ranked nodes.

In lazy networks, the higher costs of differently ranked seeds tend to collapse for maximally large networks, where the costs are less than the corresponding seeding with equally ranked nodes.

Contradictory information transmission from equally ranked nodes tends to be more expensive than from differently ranked nodes in overly sceptic networks: this can be a symptom of the greater overall epistemic balance of the sources spreading information, combined with the more common attitude of agents to require confirmation.
Distrust and epistemic attitude
Time complexity
The final analysis concerns the time complexity of our algorithm. Its running time efficiency is computed as a function relating the size of the network (in terms of the number of nodes) with the number of steps required for termination. Recall that each step in the simulation expresses one message transmission or epistemic operation of verification or distrust. We wish to know whether and in which way the network topology affects this relation.

Linear networks are the most computationally expensive in terms of time it takes for the procedure to terminate.

Scalefree are at most as expensive as random ones.

Total networks have a linear increase of the computational complexity in the number of nodes and require the shortest time to terminate.
The difference between total networks (the cheapest one) and linear networks(the most expensive one) is over 150 steps. The algorithm has complexity O(n), see Fig. 18 b for best fit of a linear function to the data for scalefree and random networks.
Conclusions
We have offered an agentbased modelling of contradictory information transmission across a network. Agents are heterogeneously qualified as either sceptic or lazy, and they are ranked. This model simulates some typical real case scenarios, like those of social networks or (rolebased) access control systems. We consider in particular networks with memory, where the result of a given transmission in terms of trusted edges is preserved at the next transmission and the new labelling can therefore be compared. Our algorithm associates costs to confirmation processes. We identify trust as a property of communications (rather than as a relation), when such confirmation is performed. We focus on contradiction resolution by a trustworthiness metric, computed by the popularity of the information in the reachable network and the ranking of the associated node. We further compare this resolution strategy with a another metric based on distrust, where the least trusted content is rejected.
Our results suggest that a sceptic approach is favourable when maximisation of consensus is the goal; a lazy approach should be pursued when minimisation of costs is the goal. We have also suggested that ranking of initial nodes is only of little relevance to consensus reaching, while a rigidly structured network (linear) is the most expensive in this respect. Finally, in the comparison between trust and distrust, it clearly results that the former is a better mean to information propagation than the latter. Moreover, we have highlighted how the presence of contradictory information is by itself the cause of distrust generation, independently from the initial attitude of the agents.
While the simulated model allows for change of such attitude, by determining confirmation and rejection rates of sceptic and lazy agents, in future extensions, we plan to allow agents to reject information explicitly and in turn allow relabelling. This could imply a onetime refutation on the transmission channel or a permanent one in networks with memory. The effects of such operations on the various types of networks is unknown and would offer an important opening in the analysis of negative concepts for computational trust in multiagent systems. The dynamic of agents can be further extended by allowing change of their epistemic status (sceptic vs. lazy) after a sufficient number of (un)successful interactions (i.e., not by some prefixed rate).
The present analysis also misses a finergrained analysis of structural conditions under which certain results (e.g. higher epistemic costs and consensus) are obtained. A more systematic analysis would allow to prune isomorphic networks (e.g. in view of the initial edge structure and ranking). Currently, applications of an extension of this model are being explored in the context of swarm robotics, see for example (Paudel and Clark 2016).
Declarations
Acknowledgements
The authors wish to thank the reviewers of Complex Networks V for feedback on the initial steps of this research. In particular, the first author wishes to thank participants to the Workshop for useful observations following the presentation. This work was partially supported by the Engineering and Physical Sciences Research Council [grants EP/K033905/1 and EP/K033921/1].
Authors’ contributions
All authors have contributed equally to this research. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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