Appendix A: Pictures of the networks
Appendix B: Sample Laputa output
We give sample output from Laputa for the “Sherlock Holms network” depicted below running Laputa in the single network mode. Each time step represents a round in the simulation. What happens during a round is determined by the updating rules in Laputa and the value of the parameters, e.g. inquiry chance and initial degree of belief.
Time: 1
Inquirer Mycroft Holmes heard that p from inquirer Sherlock Holmes, lowering his/her expected trust in the source from 0.513 to 0.513.
This raised his/her degree of belief in p from 0.50000 to 0.51299.
Time: 2
(* Nothing happened *)
Time: 3
Inquirer Mrs Hudson received the result that not-p from inquiry, raising his/her expected trust in it from 0.642 to 0.672.
This lowered his/her degree of belief in p from 0.27923 to 0.17761
Time: 4
Inquirer Sherlock Holmes received the result that p from inquiry, raising his/her expected trust in it from 0.371 to 0.452.
Inquirer Sherlock Holmes heard that not-p from inquirer Prof Moriarty, lowering his/her expected trust in the source from 0.656 to 0.573.
This lowered his/her degree of belief in p from 0.91000 to 0.75815
Time: 5
Inquirer Sherlock Holmes received the result that not-p from inquiry, lowering his/her expected trust in it from 0.452 to 0.414.
This raised his/her degree of belief in p from 0.75815 to 0.79160.
Time: 6
Inquirer Sherlock Holmes heard that not-p from inquirer Prof Moriarty, lowering his/her expected trust in the source from 0.573 to 0.525.
This lowered his/her degree of belief in p from 0.79160 to 0.73859
Inquirer Dr Watson heard that not-p from inquirer Mrs Hudson, lowering his/her expected trust in the source from 0.581 to 0.576.
This lowered his/her degree of belief in p from 0.53000 to 0.44843
…
Appendix C: The mathematics behind Laputa
In this appendix we sketch the derivations of the credence and trust update function in Laputa. These derivations make use of a number of idealizations and technical assumptions. The reader may want to consult Olsson (2013) or Vallinder and Olsson (2014) for more details on the intuitive meaning and justification of these assumptions.
The following assumptions are used in the derivation of the credence update function:
Principal Principle (PP):
$$ \begin{aligned} & a < C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right) |S_{\sigma \alpha }^{t} ,p,a < r_{\sigma \alpha } < b} \right) < b \\ & a < C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {\neg p} \right) |S_{\sigma \alpha }^{t} ,\neg p,a < r_{\sigma \alpha } < b} \right) < b \\ \end{aligned} $$
Communication Independence (CI):
$$ \begin{aligned} & C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} ,p,a < r_{\sigma \alpha } < b} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right)} \right) \times C_{\alpha }^{t} \left( p \right) \times C_{\alpha }^{t} (a < r_{\sigma \alpha } < b) \\ & C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} ,\neg p,a < r_{\sigma \alpha } < b} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} (p} \right)) \times C_{\alpha }^{t} \left( {\neg p} \right) \times C_{\alpha }^{t} (a < r_{\sigma \alpha } < b) \\ \end{aligned} $$
Source Independence (SI):
$$ \begin{aligned} & C_{\alpha }^{t} \left( {\mathop {\bigwedge }\limits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right)|p} \right) = \mathop \prod \limits_{{\sigma \in \varSigma_{\alpha }^{t} }} C_{\alpha }^{t} (S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right)|p) \\ & C_{\alpha }^{t} \left( {\mathop {\bigwedge }\limits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right)|\neg p} \right) = \mathop \prod \limits_{{\sigma \in \varSigma_{\alpha }^{t} }} C_{\alpha }^{t} (S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right)|\neg p) \\ \end{aligned} $$
where \( r_{\sigma \alpha } \) is the reliability of the source σ vis-à-vis agent α, \( 0 \le a < b \le 1 \), \( C_{\alpha }^{t} \left( p \right) \) the credence that agent α assigns to p at time t, \( S_{\sigma \alpha }^{t} \left( p \right) \) the proposition that source σ communicates p to agent α at time t, \( m_{\sigma \alpha }^{t} \) is the content of the source σ’s message, and \( \varSigma_{\alpha }^{t} \) is the set of sources that give information to α at t.
Since the trust function, which plays a crucial part in the model, is continuous, the derivation will sometimes need to take a detour through conditional probability densities rather than the conditional probabilities themselves. We will briefly sketch how this can be done here.
We have so far not been specific about the σ–algebra Z that \( C_{\alpha }^{t} \) is defined on. Assume that it is product of several such algebras, the first of which is discrete and generated by atomic events such as p, ¬p, Sβα(p) etc., and the others, which are continuous, are generated by events of the form \( a \le r_{\sigma \alpha } \le b \). Call the first algebra X and the others \( Y_{{\sigma_{0} }} , \ldots , Y_{{\sigma_{n} }} \). It is clear that, as long as time and the number of inquirers are both finite, X will have only finitely many elements. On the other hand, \( Y_{{\sigma_{0} }} , \ldots , Y_{{\sigma_{n} }} \) are certainly infinite. As mentioned, we assume that \( Z = X \times Y_{{\sigma_{0} }} \times \cdots \times Y_{{\sigma_{n} }} \). Given any source σk and time t, we can therefore interpret the part of \( C_{\alpha }^{t} \) defined on the subalgebra \( X \times Y_{{\sigma_{k} }} \) of Z as arising from a joint density function \( \kappa_{\sigma \alpha }^{\tau } \left( {\varphi ;x} \right) \) defined through the equation
$$ C_{\alpha }^{t} \left( {\varphi , a < r_{\sigma \alpha } < b} \right) = \mathop \smallint \limits_{a}^{b} \kappa_{\sigma \alpha }^{t} \left( {\varphi ;x} \right)dx $$
Since we have used the comma to represent conjunction earlier in the paper we use a semicolon here to separate the two variables: the first propositional, and the second real-valued. Like \( \tau \), this distribution’s existence and essential uniqueness are guaranteed by the Radon-Nikodym theorem, and in fact \( \tau_{\sigma \alpha }^{t} \) is the marginal distribution of \( \kappa_{\sigma \alpha }^{t} \) with respect to the reliability variable \( r_{\sigma \alpha } \) in question. Since the conditional distribution of a random variable is the joint distribution divided by the marginal distribution of that variable, this means that we have that
$$ \kappa_{\sigma \alpha }^{t} \left( {\varphi |x} \right) = \frac{{\kappa_{\sigma \alpha }^{t} \left( {\varphi ;x} \right)}}{{ \tau_{\sigma \alpha }^{t} \left( x \right)}} $$
which is what will be used to make sense of what it means to conditionalize on \( r_{\sigma \alpha } \) having a certain value rather than merely being inside an interval. Setting \( r_{\sigma \alpha } = x, a = x - \epsilon \) and \( b = x + \epsilon \) in PP and CI and letting \( \epsilon \to 0 \), we get the versions
$$ Principal\,Principle_{0} \left( {PP_{lim} } \right):\quad \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right) |S_{\sigma \alpha }^{t} ,p;x} \right) = \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {\neg p} \right) |S_{\sigma \alpha }^{t} ,\neg p;x} \right) = x $$
$$ \begin{aligned} & Communication\,Independence_{0} \left( {CI_{lim} } \right):\quad \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} , p;x} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right)} \right) \times C_{\alpha }^{t} \left( p \right) \times \tau_{\sigma \alpha }^{t} \left( x \right) \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} , \neg p;x} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right)} \right) \times C_{\alpha }^{t} \left( {\neg p} \right) \times \tau_{\sigma \alpha }^{t} \left( x \right) \\ \end{aligned} $$
We can now proceed with the actual derivation. By conditionalization, we must have that \( C_{\alpha }^{t} \left( p \right) \) is equal to \( C_{\alpha }^{t} \left( {\mathop {\bigwedge }\nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |p} \right) \). Applying Bayes’ theorem and then SI to this expression gives
$$ \begin{aligned} C_{\alpha }^{t} \left( {p |\mathop {\bigwedge }\nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right)} \right) & = \frac{{C_{\alpha }^{t} \left( p \right) \times C_{\alpha }^{t} \left( {\mathop {\bigwedge }\nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |p} \right)}}{{C_{\alpha }^{t} \left( p \right) \times C_{\alpha }^{t} \left( {\mathop {\bigwedge }\nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |p} \right) + C_{\alpha }^{t} \left( {\neg p} \right) \times C_{\alpha }^{t} \left( {\mathop {\bigwedge }\nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |\neg p} \right)}} \\ & \quad = \frac{{C_{\alpha }^{t} \left( p \right) \times \mathop \prod \nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |p} \right)}}{{C_{\alpha }^{t} \left( p \right) \times \mathop \prod \nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |p} \right) + C_{\alpha }^{t} \left( {\neg p} \right) \times \mathop \prod \nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |\neg p} \right)}} \\ \end{aligned} $$
which gives us the posterior credence in terms of the values \( C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right) |p} \right) \) and \( C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {\neg p} \right) |\neg p} \right) \). Our next task is thus to derive these expressions. Since \( S_{\sigma \alpha }^{t} \left( p \right) \) is equivalent to \( S_{\sigma \alpha }^{t} \left( p \right) \wedge S_{\sigma \alpha }^{t} \), it follows that \( C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right) |p} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} |p} \right) \). Applying first the definition of conditional probability and then the continuous law of total probability, the definition of conditional probability again, and finally CIlim, we get, after some calculations,
$$ \begin{aligned} & C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right) |p} \right) = \frac{1}{{C_{\alpha }^{t} \left( p \right)}} \times C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} ,p} \right) \\ & \quad = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times \mathop \smallint \limits_{0}^{1} \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,p;x} \right) \times \tau_{\sigma \alpha }^{t} \left( r \right) dx \\ \end{aligned} $$
But PPlim ensures that \( \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,p;x} \right) = x \), so we get
$$ C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right) |p} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times \mathop \smallint \limits_{0}^{1} x \times \tau_{\sigma \alpha }^{t} \left( x \right) dx = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times E\left[ {\tau_{\sigma \alpha }^{t} } \right] $$
Parallel derivations give that
$$ C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {\neg p} \right) |\neg p} \right) = C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times E\left[ {\overline{{\tau_{\sigma \alpha }^{t} }} } \right] $$
Now let \( \varSigma_{\alpha }^{t} \left( p \right) \subseteq \varSigma_{\alpha }^{t} \) be the set of sources that give \( \alpha \) the message p at t, and let \( \varSigma_{\alpha }^{t} \left( {\neg p} \right) = \varSigma_{\alpha }^{t} { \setminus }\varSigma_{\alpha }^{t} \left( p \right) \). Plugging the above expressions into our earlier result gives the sought for expression
$$ C_{\alpha }^{t + 1} \left( p \right) = \frac{{C_{\alpha }^{t} \left( p \right) \times \mathop \prod \nolimits_{{\sigma \in \varSigma_{\alpha }^{t} \left( p \right)}} C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |p} \right)}}{{C_{\alpha }^{t} \left( p \right) \times \mathop \prod \nolimits_{{\sigma \in \varSigma_{\alpha }^{t} \left( p \right)}} C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) + C_{\alpha }^{t} \left( {\neg p} \right) \times \mathop \prod \nolimits_{{\sigma \in \varSigma_{\alpha }^{t} }} C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right) |\neg p} \right)}} = \frac{\gamma }{\gamma + \delta } $$
where
$$ \gamma = C_{\alpha }^{t} \left( p \right) \times \mathop \prod \limits_{{\sigma \in \varSigma_{\alpha }^{t} \left( p \right)}} \left( {C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times E\left[ {\tau_{\sigma \alpha }^{t} } \right]} \right) \times \mathop \prod \limits_{{\sigma \in \varSigma_{\alpha }^{t} \left( {\neg p} \right)}} \left( {C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times E\left[ {\overline{{\tau_{\sigma \alpha }^{t} }} } \right]} \right) $$
$$ \delta = C_{\alpha }^{t} \left( {\neg p} \right) \times \mathop \prod \limits_{{\sigma \in \varSigma_{\alpha }^{t} \left( p \right)}} \left( {C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times E\left[ {\overline{{\tau_{\sigma \alpha }^{t} }} } \right]} \right) \times \mathop \prod \limits_{{\sigma \in \varSigma_{\alpha }^{t} \left( {\neg p} \right)}} \left( {C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times E\left[ {\tau_{\sigma \alpha }^{t} } \right]} \right) $$
For the derivation of the trust update expression we assume PP and CI, but not SI. The function we wish to derive is
$$ \tau_{\sigma \alpha }^{t + 1} \left( x \right) = \kappa_{\sigma \alpha }^{t} \left( {x|S_{\sigma \alpha }^{t} \left( {m_{\sigma \alpha }^{t} } \right)} \right) $$
for a source σ of α, and a message \( m_{\sigma \alpha }^{t} \) from that source. Assume that \( m_{\sigma \alpha }^{t} \equiv p \) (the case \( m_{\sigma \alpha }^{t} \equiv \neg p \) is completely symmetrical). Applying the definition of conditional probability, the equivalence \( S_{\sigma \alpha }^{t} \wedge S_{\sigma \alpha }^{t} \left( p \right) \equiv S_{\sigma \alpha }^{t} \left( p \right) \), and the (discrete) law of total probability, we get
$$ \begin{aligned} \kappa_{\sigma \alpha }^{t} (x | S_{\sigma \alpha }^{t} \left( p \right)) & = \frac{{\kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right);x} \right)}}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)} \right)}} = \frac{{\kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} ;x} \right)}}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)} \right)}} \\ & = \frac{{\kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} ,p;x} \right) + \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} , \neg p;x} \right)}}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)} \right)}} \\ & = \frac{{\kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right) | S_{\sigma \alpha }^{t} ,p;x} \right) \times \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} , p;x} \right) + \kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,\neg p;x} \right) \times \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} , \neg p; x} \right)}}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)} \right)}} \\ \end{aligned} $$
Now apply PPlim and CIlim to the factors in both terms of the numerator, and then again the equivalence \( S_{\sigma \alpha }^{t} \wedge S_{\sigma \alpha }^{t} \left( p \right) \equiv S_{\sigma \alpha }^{t} \left( p \right) \):
$$ \begin{aligned} \kappa_{\sigma \alpha }^{t} (x | S_{\sigma \alpha }^{t} \left( p \right)) & = \tau_{\sigma \alpha }^{t} \left( x \right) \times C_{\alpha }^{t} \left( {S_{\sigma \alpha }^{t} } \right) \times \frac{{x \times C_{\alpha }^{t} \left( p \right) + \bar{x} \times C_{\alpha }^{t} \left( {\neg p} \right)}}{{C_{\alpha }^{t} (S_{\sigma \alpha }^{t} \left( p \right))}} \\ & = \tau_{\sigma \alpha }^{t} \left( x \right) \times \frac{{x \times C_{\alpha }^{t} \left( p \right) + \bar{x} \times C_{\alpha }^{t} \left( {\neg p} \right)}}{{C_{\alpha }^{t} (S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} )}} \\ \end{aligned} $$
We can calculate the denominator in this expression by using the definition of conditional probability and expanding twice using the law of total probability (once using the discrete version, and once using the continuous one):
$$ \begin{aligned} C_{\alpha }^{t} (S_{\sigma \alpha }^{t} \left( p \right) | S_{\sigma \alpha }^{t} ) & = \frac{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} } \right)}}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} } \right)}} = \frac{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} , p} \right) + C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right), S_{\sigma \alpha }^{t} \neg p} \right)}}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} } \right)}} \\ & = \frac{1}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} } \right)}} \times \mathop \smallint \limits_{0}^{1} \kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,p;x} \right) \times \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} ,p;x} \right) dx \\ & \quad + \frac{1}{{C_{\alpha }^{t} \left( { S_{\sigma \alpha }^{t} } \right)}} \times \mathop \smallint \limits_{0}^{1} \kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,\neg p;x} \right) \times \kappa_{\sigma \alpha }^{t} \left( {S_{\sigma \alpha }^{t} ,\neg p;x} \right)dx \\ \end{aligned} $$
Let us refer to the last expression as \( \psi \). Applying CIlim, then cancelling, and applying PPlim, we get
$$ \begin{aligned} \psi & = C_{\alpha }^{t} \left( p \right) \times \mathop \smallint \limits_{0}^{1} \kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,p;x} \right) \times \kappa_{\sigma \alpha }^{t} \left( x \right) dx \\ & \quad + C_{\alpha }^{t} \left( p \right) \times \mathop \smallint \limits_{0}^{1} \kappa_{\sigma \alpha }^{t} \left( { S_{\sigma \alpha }^{t} \left( p \right)| S_{\sigma \alpha }^{t} ,\neg p;x} \right) \times \kappa_{\sigma \alpha }^{t} \left( x \right) dx \\ & = C_{\alpha }^{t} \left( p \right) \times \mathop \smallint \limits_{0}^{1} x \times \tau_{\sigma \alpha }^{t} \left( x \right) dx + C_{\alpha }^{t} \left( {\neg p} \right) \times \mathop \smallint \limits_{0}^{1} \bar{x} \times \tau_{\sigma \alpha }^{t} \left( x \right) dx \\ & = C_{\alpha }^{t} \left( p \right) \times E\left[ {\tau_{\sigma \alpha }^{t} } \right] + C_{\alpha }^{t} \left( {\neg p} \right) \times E\left[ {\overline{{\tau_{\sigma \alpha }^{t} }} } \right] \\ \end{aligned} $$
Putting it all together, we finally arrive at the updating rule for trust:
$$ \tau_{\sigma \alpha }^{t + 1} \left( x \right) = \tau_{\sigma \alpha }^{t} \left( x \right) \times \frac{{x \times C_{\alpha }^{t} \left( p \right) + \bar{x} \times C_{\alpha }^{t} \left( {\neg p} \right)}}{{C_{\alpha }^{t} \left( p \right) \times E\left[ {\tau_{\sigma \alpha }^{t} } \right] + C_{\alpha }^{t} \left( {\neg p} \right) \times E\left[ {\overline{{\tau_{\sigma \alpha }^{t} }} } \right]}} $$
Appendix D: Network metrics used
In this appendix we will properly define the network measures we have calculated for the various networks in our study. These measures are all common measures from the network literature [see Easley and Kleinberg (2010), Jackson (2010) or Newman (2010)]. The terminology used in this section will mostly be borrowed from Jackson (2010), but as the measures for the particular networks where calculated using the program Pajek (De Nooy et al. 2011), we will deviate from Jackson (2010) whenever the measures are calculated in a different way in Pajek.
By a network we will understand an undirected graph (N, g), where N = {1, 2, …, n} is the set of nodes also referred to as vertices, individuals, or inquirers. g is a set of pairs (i, j) specifying which links between nodes are present in the network. We will also write ij for (i, j) and if ij ∈ g then we will say that there is a link/edge/tie between i and j. As networks are undirected graphs we will not distinguish between ij and ji (i.e. ij ∈ g will be equivalent to ji ∈ g). In the following (N, g) will refer to a given arbitrary network with N = {1, 2, …, n}. The neighborhood Ni(g) of a node i is the set of nodes that i is linked to, that is \( N_{i} \left( g \right) = \left\{ {j | ij \in g} \right\}. \) The degree of a node i, denoted by di(g), is the total number of nodes i is linked, in other words, \( d_{i} \left( g \right) = \# N_{i} \left( g \right), \) where #A denotes the cardinality of the set A.
Our first measure is the average degree, which says something about how connected each inquirer is on average, and is defined by:
$$ d^{Avg} \left( g \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} d_{i} \left( g \right). $$
A related measure is the density of a network. The density of a network is the fraction of links actually present in the network relative to the number of possible links. For a network with n notes, the number of all possible links is n(n−1)/2 (as the network is undirected). Thus the density is given byFootnote 2:
$$ den\left( g \right) = \frac{\# g}{{n\left( {n - 1} \right)/2}}. $$
Note that
$$ den\left( g \right) = \frac{{d^{Avg} \left( g \right)}}{n - 1} , $$
and thus, average degree and density are highly correlated—even perfectly correlated if the size of the networks is kept fixed. As exhaustively discussed in the introduction, the significance of a network’s density on its potential as an information passing structure, has been widely discussed in the literature.
While average degree and density say something about how connected a network is, there are much more to a network’s structure than captured by these measures. Another structural property one could consider is the distances between nodes (-how many links one has to pass through to reach one node from another), which may say something about how fast things such as diseases or information can spread through a network. It turns out that the distances between nodes are surprisingly small in real networks, which is known as the “Small-world phenomenon” (or the “six degrees of separation” or the “Kevin Bacon effect” (Watts 1999)) and goes back to a famous experiments by Milgram (1967). Let us turn to the formal definitions.
A path from a node i to another node j is a sequence of distinct nodes i1, i2, …, im such that i1 = i, im = j, and ikik+1∈ g for all k ∈ {1, 2, …, m − 1}. In other words, there is a path from i to j if one can reach j from i by following a sequence of distinct links in the network. The length of such a path is the number of links in it (i.e. m − 1). A shortest path from i to j is a path from i to j such that there are no other path from i to j with a shorter length. The distance between two nodes i and j is the length of a shortest path between them (if there is any path between i and j at all) and will be denoted by dist(i, j). We say that two nodes are connected if there is a path (and thereby necessarily a shortest path) between then. A network is connected if every pair of distinct nodes i and j are connected. Traditionally, the average distance of a network is the average distance of any two nodes in it, that is
$$ \frac{1}{{n\left( {n - 1} \right)}}\mathop \sum \limits_{i,j \in N} dist\left( {i,j} \right). $$
However, since there might not always be a path between two nodes i and j (if the network is not connected), the average distance is calculated a little bit different in Pajek. Let,
$$ A\text{ := } \left\{ {\left( {i,j} \right)| i,j \in N, i \ne j, \,{\text{and}}\,dist\left( {i,j} \right)\,{\text{is}}\,{\text{defined}}} \right\}. $$
Then, the average distance, distAvg(g), is calculated in Pajek in the following way:
$$ dist^{Avg} \left( g \right) = \frac{1}{\# A}\mathop \sum \limits_{{\left( {i,j} \right) \in A}} dist\left( {i,j} \right) . $$
The diameter of a network is the length of the longest shortest part in it:
$$ diameter\left( g \right) = max_{i,j \in N} dist\left( {i,j} \right). $$
If the network is not connected, Pajek calculates the above formula as if dist(i j) is 0 for notes i and j that are not connected. Thereby, the diameter is always the maximum of all shortest paths.Footnote 3 Note that the diameter of a network puts an upper bound on the average distance of the network, but the diameter can sometimes be significantly larger than the average distance.
A node’s particular position in a network can be important and centralization measures are all trying to measure this effect. We will consider three different measures if centrality that are widely used in social network analysis. As such these measures are local to nodes of a network, but Pajek also provides global “summations” of the local measures that we will use. The first simple measure is degree centralization, which measures the degree of a node relative to the size of the network, i.e. for a node i the degree centrality of i is defined by:
$$ Ce_{i}^{D} \left( g \right) = \frac{{d_{i} \left( g \right)}}{n - 1} . $$
Pajek provides a measure for the overall degree centrality of a network (“All Degree Centralization”), CeD(g), by the following calculation:
$$ Ce^{D} \left( g \right) = \frac{{\mathop \sum \nolimits_{i \in N} \left( {Ce_{*}^{D} \left( g \right) - Ce_{i}^{D} \left( g \right)} \right)}}{n - 2} , $$
where \( Ce_{*}^{D} \left( g \right) \) is the maximum of the individual degree centralities \( Ce_{i}^{D} \left( g \right) \).
Degree centrality captures some form of importance of centrality of nodes, however, it is often way to simple. For instance, a node with only two links might connect two otherwise separated part of a network and thereby play a central role if information (or other things) has to pass from the one part to the other. Betweenness centrality is a measure, initially defined by Freeman (1977), which is an attempt to capture such potential control over communication. Specifically, betweenness centrality measures how many shortest paths (between other nodes) a given node lies on. Formally, for a node i the betweenness centrality of i is defined by:
$$ Ce_{i}^{B} \left( g \right) = \mathop \sum \limits_{k,j \in N,k \ne j,k \ne i,j \ne i} \frac{{P_{i} \left( {kj} \right)/P\left( {kj} \right)}}{{\left( {n - 1} \right)\left( {n - 2} \right)/2}} , $$
where Pi(kj) is the number of shortest paths between k and j that include i and P(kj) is the total number of shortest paths between k and j. Again, Pajek provides a measure for the overall betweenness centrality of a network (“Betweenness Centralization”), CeB(g), in the following sense:
$$ Ce^{B} \left( g \right) = \frac{{\mathop \sum \nolimits_{i \in N} \left( {Ce_{*}^{B} \left( g \right) - Ce_{i}^{B} \left( g \right)} \right)}}{n - 1} , $$
where \( Ce_{*}^{B} \left( g \right) \) is the maximum of the individual betweenness centralities.
While betweenness centrality may say something about who controls the flow of information in a network, closeness centrality is another measure that may say something about how fast or how far information from a node spreads. Formally, closeness centrality measures how close a node is to all the other nodes of the network. For a node i, the closeness centrality of i is defined as:
$$ Ce_{i}^{C} \left( g \right) = \frac{n - 1}{{\mathop \sum \nolimits_{j \in N,j \ne i} dist\left( {i,j} \right)}} . $$
If i is an isolated node in the network, i.e. di(g) = 0, the convention is that \( Ce_{i}^{C} \left( g \right) \) is 0. If the network is connected, Pajek provides a measure for the overall closeness centrality of a network (“All Closeness Centralization”), CeC(g), by the following calculation:
$$ Ce^{C} \left( g \right) = \frac{{\mathop \sum \nolimits_{i \in N} \left( {Ce_{*}^{C} \left( g \right) - Ce_{i}^{C} \left( g \right)} \right)}}{{\left( {n - 1} \right)\left( {n - 2} \right)/\left( {2n - 3} \right)}} . $$
where \( Ce_{*}^{C} \left( g \right) \) is the maximum of the individual closeness centralities. For more on centrality measures and their use see Freeman (1979) and Borgatti (2005) or any of the textbooks referenced in the beginning of this section.
A final class of measures that we will consider is clustering or transitivity measures. Intuitively such measures say something about how likely it are that any two of my friends are friends themselves. It turns out that in many real social networks clustering is much higher than in most random networks (Watts and Strogatz 1998). Formally, for a given node we can ask how many of its neighbors are connected themselves, which give rise to the following individual clustering measure:
$$ Cl_{i} \left( g \right) = \frac{{\# \left\{ {jk \in g | k \ne j {\text{and}} j,k \in N_{i} \left( g \right)} \right\}}}{{\# \left\{ {jk | k \ne j {\text{and}} j,k \in N_{i} \left( g \right)} \right\}}} = \frac{{\# \left\{ {jk \in g | k \ne j {\text{and}} j,k \in N_{i} \left( g \right)} \right\}}}{{d_{i} \left( g \right)\left( {d_{i} \left( g \right) - 1} \right)/2}} . $$
Taking the average of this measure results in a measure of average clusteringFootnote 4 of an entire network:
$$ Cl^{Avg} \left( g \right) = \mathop \sum \limits_{i = 1}^{n} \frac{{Cl_{i} \left( g \right)}}{n} . $$
This measure of average clustering is referred to in Pajek as “Watts-Strogatz Clustering Coefficient” as it was first proposed by Watts and Strogatz (1998). Note that, commonly, Cli(g) is taken to be 0 if the neighborhood of i only contains one or zero nodes (see Jackson (2010) or Newman (2010)). Conversely, Pajek takes Cli(g) to be plus infinity in this case and does not include the node when calculating the average. In other words, in Pajek:
$$ Cl^{Avg} \left( g \right) = \mathop \sum \limits_{{i \in N, d_{i} \left( g \right) > 1}} \frac{{Cl_{i} \left( g \right)}}{{\# \left\{ {j \left| { d_{j} \left( g \right)} \right\rangle 1} \right\}}} . $$
An alternative way of measuring clustering is by reporting the actual number of “triangles” relative to all possible triangles:
$$ Cl\left( g \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \# \left\{ {jk \in g | k \ne j\,{\text{and}}\,j,k \in N_{i} \left( g \right)} \right\}}}{{\mathop \sum \nolimits_{i = 1}^{n} \# \left\{ {jk | k \ne j\,{\text{and}}\,j,k \in N_{i} \left( g \right)} \right\}}} . $$
This measure is usually referred to as overall clustering, while in Pajek, the measure is referred to as “Clustering Coefficient (Transitivity)”. Note that, overall clustering and average clustering can be different for a particular network.