The Effect of Well-informed Minorities and Meritocratic Learning in

Social Networks

Marwa Shekfeh and Ali A. Minai

Department of Electrical Engineering and Computer Science, University of Cincinnati, Cincinnati, OH 45221-0030, U.S.A.

Keywords:

Associative Learning, Social Learning, Cognitive Agents, Misinformation.

Abstract:

A signiﬁcant amount of information acquisition in human groups occurs through social learning, i.e., individ-

uals learning through communication with their peers. Since people communicate what they know and their

information is not completely accurate, such peer-to-peer learning can lead to the spread of both knowledge

and misinformation over social networks. How much of each occurs depends on many factors, including the

quality of knowledge in the group as a whole, its initial distribution over the network, and the learning styles

of individuals. The number of conﬁgurations in which these factors can occur is inﬁnite, but multi-agent net-

work models provide a promising way to explore plausible scenarios. In this paper, we use such a model to

consider the joint effect of two factors: 1) The proportion of initially well-informed and ill-informed agents

in the population; and 2) The choice of each group to learn in one of two plausible ways. The simulations

reported ﬁnd that both factors have a large effect.

1 INTRODUCTION

The spread of misinformation in social networks has

recently been a topic of major interest because of

the increasingly important role social media is play-

ing in politics and policy (Del Vicario et al., 2016;

Allcott and Gentzkow, 2017; Shu et al., 2017; Wal-

drop, 2017; Vosoughi et al., 2018; Oliveira and Chan,

2018; Tambuscio et al., 2018). However, the social

propagation of misinformation has been part of hu-

man society since time immemorial, and methods to

counter it have become part of the social mores and

codes in virtually all societies. The challenge being

faced at the present time arises from the sudden, ex-

ponential, and non-geometric ampliﬁcation of social

networking with the advent of the Internet. It has,

therefore, become very important to understand the

factors that contribute to the spread of misinforma-

tion or can mitigate such spread. Of course, this is

an extremely complex issue that can be addressed at

many different levels using a variety of approaches.

In this paper, we describe a simple, abstract multi-

agent model called MANILA (Multi-Agent Network

for the Implicit Learning of Associations) to explore

the implicit social propagation of false conceptual as-

sociations. We apply this model to look at the effect of

two factors on the spread of such misinformation: 1)

The presence of an extremely well-informed minority

in the population; and 2) The preference of individu-

als to attend to peers based on their perceived like-

mindedness versus their reputation for being well-

informed. The model considers both the quality and

quantity of information, and attempts to capture the

implicit nature of social learning as well as some of

its cognitive complexities.

2 MOTIVATION

Human knowledge is necessarily imperfect, and the

ubiquity of social learning makes it inevitable that

false information would spread to some degree in hu-

man populations (Buntain and Golbeck, 2017; Men-

doza et al., 2010; Castillo et al., 2013). However, not

all individuals are equally well- or ill-informed, and

it is interesting to consider how the presence of ex-

ceptionally well-informed individuals in a population

inﬂuences the quality of knowledge in the larger, less

well-informed sub-population. Here, one can con-

sider varying degrees of being well- or ill-informed as

well as a varying presence of each class in the overall

population. Exploring this entire space of possibili-

ties is practically impossible even in a computational

model, but a few canonical cases can be considered.

One of these is when the population is divided into

those who only have accurate information and those

104

Shekfeh, M. and Minai, A.

The Effect of Well-informed Minorities and Meritocratic Learning in Social Networks.

DOI: 10.5220/0008980201040115

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 1, pages 104-115

ISBN: 978-989-758-395-7; ISSN: 2184-433X

whose information is tainted to some plausible degree

by inaccuracies. The proportion of each group in the

population can then be varied systematically, as is the

case in the present study.

A second interesting factor is which peers the

agents in each sub-population learn from. Again,

there are many possibilities, but the MANILA model

includes three pure strategies for accepting informa-

tion received from a peer: 1) Based on the strength

of the social connection; 2) Based on perceived like-

mindedness; and 3) Based on the empirically ob-

served reputation of the peer for having accurate in-

formation. Well-deﬁned mixtures of these strate-

gies are also possible. In this paper, we consider

the canonical situation where well-informed agents

prefer to learn from high-quality peers whereas the

less well-informed agents prefer to learn from like-

minded ones. We then consider whether the mixture

of some quality preference in the latter population can

have a signiﬁcant effect.

3 BACKGROUND

The two main features of MANILA are the spread

of information in the social network and the im-

plicit learning of (true and false) associations that re-

sults from it. This section relates these features of

MANILA to prior work.

The diffusion of information in social networks

has been studied empirically for a long time, result-

ing in several models (Granovetter, 1978; Liggett,

1985; Kempe et al., 2003). Many other models have

also been developed for the diffusion and spreading

of ideas, innovations, information, and disease on

social networks (Adar and Adamic, 2005; Leskovec

et al., 2006; Leskovec et al., 2007; Watts and Dodds,

2007; Liben-Nowell and Kleinberg, 2008; Golden-

berg et al., 2008). The effect of model structure on

the spread of information (and misinformation) has

also been studied (Weng et al., 2013). Lamberson

(Lamberson, 2010) proposed the term “cognitive ad-

vantage” as a factor in how one should study idea

propagation, and criticized previous models that took

into consideration only the underlying structure of the

social networks without looking at the cognitive and

psychological proﬁle of agents diffusing information

or ideas through the social network. MANILA also

incorporates the cognitive preferences and reputation

of agents in the model.

Though learning and adaptation are not part of all

multi-agent models (MAS), in most cases the phe-

nomenon of interest does require the inclusion of

adaptation – often in the form of reinforcement learn-

ing (Sutton and Barto, 1998), where agents learn to

improve their choices based on positive or negative

feedback from the environment or a critic. Learning

in MAS is a natural extension of classic reinforcement

learning, but adds a crucial social component, with

learning depending not only on rewards, but also on

communication, attention and information diffusion

(Weiss (ed.), 1999). In almost all cases, however, re-

inforcement learning is explicit, with each action or

sequence of actions eliciting a direct, visible reward

from a critic. MANILA, in contrast, uses a type of

reinforcement learning that differs from the classic

paradigm because it uses reward only implicitly and

indirectly.

Bayesian models of learning in social networks

have been studied by several researchers (Gale and

Shachar, 2003; Rosenberg et al., 2009; Acemoglu

et al., 2011; Lobel and Sadler, 2012; Mueller-Frank,

2013). These models are complex, and focus mainly

on proving the convergence of beliefs to the truth.

However, most of them do not model the dynamics

of learning and information diffusion, or the asso-

ciative nature of knowledge. In contrast, MANILA

takes into consideration the representation of knowl-

edge as epistemic networks, the communication of

this knowledge, and the ﬂow of ideas over a social

network with a speciﬁc structure – albeit in a simple,

idealized way.

Implicit learning (Reber, 1967; Seger, 1994)

refers to learning that occurs incidentally and with-

out awareness of learning. Most work on this – in-

cluding computational models (Dienes, 1992; Mathis

and Mozer, 1994) – focuses on individuals without

reference to social factors (Berry, 1997; Dienes and

Berry, 1997). In MANILA, however, the term refers

to the acquisition or loss of conceptual associations

by agents as a side-effect of their communication with

each other, and incorporates both social and cognitive

factors.

4 MODEL DESCRIPTION

4.1 Overview

MANILA is a system with cognitive and generative

agents who receive and learn associative informa-

tion implicitly from interaction with their peers in a

social network. The social network is assumed to

have a small-world (SW) architecture as proposed by

Watts and Strogatz (Watts and Strogatz, 1998). The

knowledge of each agent, i, is represented as an epis-

temic network (EN), E

, whose nodes represent con-

cepts and edges indicate associations between con-

The Effect of Well-informed Minorities and Meritocratic Learning in Social Networks

105

cepts. Therefore, This system has a network-of-

networks structure with two levels of networks: The

social network connecting the agents, each of whom

has an epistemic network. To mimic natural dis-

course, agents are assumed to communicate not in

terms of concepts, but ideas, which are combinations

of concepts, as postulated in most models of ideation

(Campbell, 1960; Mednick, 1962; Brown et al., 1998;

Paulus, 2002; Fauconnier and Turner, 2003; Simon-

ton, 2003; Simonton, 2010). Formally, an idea is de-

ﬁned as a 0.5-quasi-clique (Brunato et al., 2008) of

6 to 10 concept nodes in the EN. Concepts and ideas

are deﬁned purely in abstract terms to perform sys-

tematic simulations, but a lexical network based on a

text corpus could be used as well.

Agents in MANILA generate and express ideas,

i.e., combinations of concepts, based not on their ex-

pectation of reward – which they have no model to

calculate – but on their subjective assessment of the

idea’s coherence within their own mind, which they

use as a surrogate for its value. This models the nat-

ural situation in human expression where individuals

express ideas based on their own convictions rather

than on a calculation of external reward, with the tacit

assumption that those convictions, in fact, represent

real value or truth.

Similarly, when an agent receives an idea from a

peer, it assimilates that idea into its own mind based

not on some explicit reward that the idea has visibly

generated, but based on its esteem and regard for the

peer from whom it came. This esteem, in turn, can be

based on several factors, including those that have no

bearing on the veracity of the idea or the agent.

The Oracle

Since the focus of this work is on false associations,

there needs to be a criterion of true associations, and

a mechanism by which the correctness or incorrect-

ness of expressed ideas can be perceived in the so-

cial network. In keeping with the abstract nature of

the model, we assume that there is a ﬁxed large ideal

epistemic network (IEN) of concepts and true associa-

tions known only to an Oracle, which thus represents

the ground truth. Agents’ initial ENs represent par-

tial, noisy samples from the IEN, reﬂecting the fact

that an individual agent’s information is typically in-

complete and possibly inaccurate. Whenever an agent

expresses an idea, it is evaluated by the Oracle and

elicits a reward to the agent based on the idea’s con-

sistency with the Oracle’s IEN. Thus, the Oracle plays

the role of the critic in classic reinforcement learning.

However, in this case, the reward represents only the

implicit beneﬁt and reputation an agent acquires for

being right, and is not publicly visible or linked ex-

plicitly to a speciﬁc idea in the generating agent’s per-

ception. Agents accumulate the rewards they receive

with a decay factor, and the cumulative reward of an

agent at a given time is visible to other agents as an

indication of the agent’s merit status. Thus, over time,

every agent can see which of their peers are more (or

less) well-informed in an aggregate sense.

In a real-world situation, the role of the Oracle

would be played by fact-checkers, news reporters, ex-

perts, and the general social consensus on what is or

is not acceptable as fact. In MANILA, the Oracle pro-

vides a purely abstract but ﬁxed (and therefore objec-

tive) reference against which the correctness of asser-

tions can be evaluated.

The Social Network

The social network in MANILA is deﬁned by a sym-

metric small-world adjacency matrix C = [c

i j

], where

i j

∈ {0, 1} indicates whether there is a social connec-

tion from A

to A

Epistemic Network Initialization

The initial EN for agent A

is constructed in three

steps:

1. A set of ideas is sampled randomly from the Ora-

cle’s idea repertoire.

2. The selected ideas are superposed to create an

epistemic network with N edges, all of which are

true by construction (because they came from the

IEN).

3. Next, qN true edges are removed and the same

number of false edges are added, giving the ini-

tial EN for A

as a partial and noisy version of

the IEN, with q ∈ [0, 1] controlling the degree of

misinformation (incorrect associations) in the ﬁ-

nal EN.

4.2 Bayesian Model for Learning

Association Weights

For each Agent, A

, if there is an edge e

i j

= 1 between

concept nodes i and j in its EN, it has a weight w

i j

representing the agent’s conﬁdence in the association

between the concepts represented by nodes i and j.

The weights are initially chosen independently from

a uniform distribution between 0 and 1, and change

based on an agent’s learning process using a Bayesian

formulation.

Idea Generation and Expression

Each agent potentially generates an idea at each time

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

106

Figure 1: Mean Agent Knowledge after 1000 epochs for

wise agents (upper blue curve) and normal agents (lower

red curve) for various percentages of wise agents.

Figure 2: Mean Agent Wisdom after 1000 epochs for wise

agents (upper blue curve) and normal agents (lower red

curve) for various percentages of wise agents.

step by sampling its EN via an attentional search pro-

cess. The edges in the generated idea are binary rather

than weighted:The agent is asserting which associ-

ations exist or do not exist in the idea, and not in-

dicating how strongly it believes each association in

its own mind. However, the association weights are

taken into account by the agent in the process of gen-

erating ideas. Idea generation and expression occurs

as follows:

1. Attentional Focus: This step identiﬁes the con-

cepts and associations that agent A

currently has

in mind. A seed set of nodes is identiﬁed in the

EN of the source agent A

. This is then used to

obtain the biased set of nodes via spreading acti-

vation. After removal of low-degree nodes, this

gives the sub-network, E

(t), which is designated

the current attended epistemic network (AEN) of

the agent at time t, and represents the internal at-

tentional focus of the agent (Iyer et al., 2009; Iyer

et al., 2010).

2. Generation of Potential Ideas: This step gener-

ates the set G

(t) of all possible potential ideas,

(t), within the AEN by applying the quasi-

clique search algorithm, with the number of nodes

per idea restricted to be between 6 and 10.

3. Evaluation of Potential Ideas: Once agent A

has generated the set of potential ideas, G

(t) =

(t)}, it evaluates the coherence, q

(t), of each

idea based on the density of its internal connec-

tivity and weight of its edges in its AEN. For an

idea g

with nodes N

and edges E

, the weighted

coherence is given by:

∏

e∈E

(1)

where w

is the weight of the edge e and n = |E

Thus, more densely connected ideas with higher

weights are considered more coherent. Coherence

is the agent’s subjective assessment of the quality

of an idea.

Coherence values below a threshold φ are set to 0.

The result is a coherence vector Z

(t) = {z(g

(t))}

for the set of ideas in G

(t). If no idea meets

threshold φ, the coherence vector is null, and the

agent remains silent in that time-step.

4. Idea Selection and Expression: One of the ideas

from G

(t) is chosen for expression by using a

roulette wheel algorithm based on the coherence

of the ideas, i.e., the probability of choosing idea

(t) for expression is:

comm

(t)) =

(t)

∑

u∈G

(t)

(2)

Once A

decides to express the idea, it broadcasts

it to all its immediate neighbors in the social net-

work.

The seed set is also continually updated by the nat-

ural dynamics of the system based on recently gener-

ated ideas and what the agents hears from others (see

(Shekfeh, 2017) for details.)

Idea Evaluation and Reward

Ideas expressed by agents are evaluated by the Oracle

based on the number of true and false associations in

it. If an idea g is expressed by agent A

at time t, it

The Effect of Well-informed Minorities and Meritocratic Learning in Social Networks

107

generates a reward using the function R that is com-

puted by the Oracle as follows:

R(χ(g)) =



0.5[1 − (χ(g))

1−µ

] i f χ(g) ≤ 0

0.5[1 + (χ(g))

1−µ

] i f χ(g) > 0

(3)

where µ is the reward selectivity parameter and χ(g)

represents the Oracle’s quality evaluation function for

ideas. Given an idea g, its quality is calculated as :

χ(g) =

true

− e

f alse

total

(4)

where e

total

is the number of edges in g, e

true

is the

number of edges in g that are also present in the IEN,

and e

f alse

is the number of edges present in the g but

not in the IEN.

The merit of agent k is then computed as follows:

(t + 1) = (1 − σ)ξ

(t) + σR(χ(g)) (5)

where σ ∈ (0, 1) is the merit adaptation rate parame-

ter and is set to a small value. Thus, the more accurate

the ideas expressed by an agent, the more merit it ac-

quires over time.

Idea Reception and Assimilation

Idea Reception: When a receiving agent A

receives

an idea from source agent A

, it can choose to ig-

nore it or assimilate it into its own EN, thus learning

that idea. The idea is assimilated with a probability

based on the receiving agent’s general receptivity to

ideas and its speciﬁc attentiveness towards the send-

ing agent A

. The attentiveness of A

towards A

de-

pends on three possible factors: 1) The social weight,

, between the agents; 2) The empirical epistemic

afﬁnity, ψ

, that A

has inferred with A

(see below);

and 3) The merit differential, ξ

− ξ

, of A

and A

The attentiveness of agent A

towards agent A

is rep-

resented by the esteem of A

for A

= c

(ψ

) + c

(ϕ

) + c

(ξ

− ξ

) (6)

where f

( ), f

( ), and f

( ) are monotonically in-

creasing sigmoid functions with range 0 to 1, and

, c

and c

, are parameters representing the so-

cial, epistemic, and perceptual components of the

attentiveness function, respectively. These are de-

ﬁned on the simplex c

+ c

= 1, so that the tu-

ple (c

, c

) deﬁnes the learning style of agent A

There are three pure learning styles:

1. Social Learning Style (1, 0, 0), where the agent

learns preferentially from those to whom it has

strong social connections. The social selectivity

function determines how esteem depends on so-

cial weight with the source agent. It is deﬁned as:

(ϕ) =



0.5[1 − (1 − 2ϕ)

1−α

] i f ϕ ≤ 0.5

0.5[1 + (2ϕ − 1)

1−α

] i f ϕ > 0.5

(7)

where α is the social selectivity parameter. If

α = 0 , f

has linear dependence on social weight.

As α increases towards 1, f

(ϕ(r, s)) approaches

a threshold function at ϕ(r, s) = 0.5, so the agent

accepts ideas only from peers with social connec-

tion ϕ(r, s) > 0.5.

2. Like-Minded Learning Style (0, 1, 0), where the

agent learns preferentially from agents that have

previously expressed ideas similar to its own. This

function determines how esteem depends on epis-

temic afﬁnity that the receiving agent perceives

with the source agent. It is deﬁned as:

(ψ) =



0.5[1 − (1 − 2ψ)

1−β

] i f ψ ≤ 0.5

0.5[1 + (2ψ − 1)

1−β

] i f ψ > 0.5

(8)

where β is the epistemic selectivity parameter. If

β = 0, f

becomes a linear function of epistemic

afﬁnity. As β increases towards 1, f

(ψ(r, s)) ap-

proaches a threshold function at ψ(r, s) = 0.5, so

the agent accepts ideas only from peers with epis-

temic afﬁnity ψ(r, s) > 0.5 .

3. Meritocratic Learning Style (0, 0, 1), where the

agent learns preferentially from those who have

higher accumulated merit than itself. The f

func-

tion determines how esteem depends on the dif-

ference between the merit status of the receiving

agent and the source agent. It is deﬁned as:

(∆) =



0.5[1 − |∆|

1−γ

] i f ∆ ≤ 0

0.5[1 + |∆|

1−γ

] i f ∆ > 0

(9)

where the differential status, ∆ = ξ

−ξ

is the dif-

ference in current merit status between the receiv-

ing agent and the source agent, and γ is the merit

status selectivity parameter. If γ = 0, f

is a linear

function of reward status. As γ increases towards

1, f

(∆) approaches a threshold function at ∆ = 0,

so ideas from an agent with higher status are al-

ways accepted and those from agents with lower

status are not.

The probability of agent A

accepting an idea from

agent A

is:

(r, s) = κ

(10)

where κ

∈ [0, 1] is the general receptivity of

agent r to ideas from others. We use κ

= 1 for all

agents.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

108

Idea Assimilation: Once the receiving agent A

has

decided to accept an idea, g, from the source agent A

it must be assimilated into A

’s EN. The process of

assimilating a new idea involves updating the weights

of edges using a Bayesian model. The process for

updating the edge between concept nodes i and j in

the received idea comprises the following steps:

• If i and j are connected (disconnected) in both

g and E

, they remain connected (disconnected)

in E

. However, in the connected case, the edge

weight is adjusted according the weight adjust-

ment rule which is described in the following sec-

tion.

• If i and j are connected in g but not in E

, A

con-

nects them with a small weight, w

, with proba-

bility p

add

= Λ

– the esteem for agent A

• If i and j are not connected in g but are connected

in E

, the weight in E

is decreased by an amount

∆w

i j

which is also described in the Learning Rule

section

Thus, the result is to make E

more consistent with

the received idea g. Both addition of associations and

weight adjustment of edges is possible, which is cru-

cial for improving initially noisy ENs.

The Learning Rule

The association weights in the ENs of individual

agents change based on the ideas they assimilate from

their peers, which increase or decrease the agents’

conﬁdence in each received association. A single

weight update equation is as follows:

i j

(t + 1) =

¯e

i j

(1 − λ)w

i j

(t) + λ

i j

(t)

(1 − w

i j

(t)) + ρ

i j

(t)

+ (1 − ¯e

i j

)

(1 − λ)w

i j

(t) + λ

i j

(t)

(1 − w

i j

(t)) + w

i j

(t)

where: w

i j

(t) is the belief at time t of agent A

that the

edge is True; ¯e

i j

≡ the state of the edge in the message

sent by agent A

; e

i j

≡ the state of the edge in the EN

of receiving agent A

: e

i j

= 1 means the edge exists,

else 0; λ is the learning rate; and ρ

is the odds ratio

of A

believing an association received from A

. The

last quantity is computed from Λ

, the esteem that

Agent A

has for Agent A

, as follows:

= 1 − log(1 −Λ

) (11)

Thus, an esteem of Λ

= 0 implies ρ

= 1, i.e., Agent

thinks that Agent A

is just as likely to be wrong in

its assertion of e

∗

i j

= 1 as it is to be right, it will learn

nothing from Agent s and has no esteem for it.

Of course, it is necessary that 0 < λ < 1. For the

complete derivation of this equation, see (Shekfeh,

2017).

Two points should be noted here. First, that the

esteem of A

for A

determines both whether the for-

mer accepts an idea from the latter and the degree to

which the idea changes its own beliefs. Second, the

learning happening here is implicit in that informa-

tion elements (conceptual associations) are learned as

a side-effect of assimilating ideas rather than individ-

ually, and that the agents are not trying explicitly to

become wiser or more knowledgeable: It simply oc-

curs as an implicit consequence of communication,

attention, and inﬂuence among agents.

Activation and Forgetting

Not all knowledge in a person’s mind is equally ro-

bust: Things must be brought to mind somewhat reg-

ularly, or may be forgotten entirely. This effect is

modeled in the system by processes of activation and

forgetting.

An association (edge) in an agent’s EN is acti-

vated when it receives the agent’s attention in the pro-

cess of searching for ideas to express, or is updated

via a received idea, that is:

1. If the two concepts occur in an idea received from

another agent.

2. If the agent thinks of the two concepts as part of

an idea it generates.

If the association is not activated at a time step, its

strength changes as follows:

i j

(t + 1) =



(1 − ε

−

i j

(t) i f w

i j

(t) ≤ θ

i j

(t) else

(12)

where ε

−

is a small memory decay rate parameter.

Thus, if the absolute strength of the association

between concepts i and j falls below θ

, it begins to

be forgotten and requires activation in order to sur-

vive. Through this process, both correct and incor-

rect associations can, in principle, disappear from the

agent’s mind – and from the entire population – over

time. Recently activated concepts also provide the

seed for idea search in the agent’s mind.

Epistemic Afﬁnity Update

The epistemic afﬁnity, ψ

, that agent A

perceives for

agent A

is based on the ideas it has seen from A

and quantiﬁes the degree to which A

says things that

agrees with. Initially, ψ

is set to 0.5 , i.e., neutral

The Effect of Well-informed Minorities and Meritocratic Learning in Social Networks

109

afﬁnity. Every time A

receives an idea, g, from A

, A

calculates the raw similarity, S

(g), of the idea with

its own EN as:

(g) =

present

− e

absent

total

(13)

where e

present

is the number of edges in g that are also

present in EN

, e

absent

is the number of edges in g

that are not present in EN

, and e

total

is the number of

edges in g . Thus, if all the associations in g are al-

ready found in the receiving agent’s EN, the raw sim-

ilarity is +1, and if none are, the raw similarity is -1.

A normalized familiarity value is then calculated as:

Γ(S) =



0.5[1 − (S)

1−ν

] i f S ≤ 0

0.5[1 + (S)

1−ν

] i f S > 0

(14)

where ν

is a small value ∈ (0, 1). A comparison with

Eqns. (3) and (4) shows that the familiarity of idea

g for agent A

is a subjective version of the Oracle’s

objective reward metric: They represent an estimate

of the perceived truth of the idea by the agent and the

Oracle, respectively.

Then the epistemic afﬁnity of agent A

for A

updated as:

(t + 1) = ψ

(t) + η[Γ(S) − ψ

(t)] (15)

where η ∈ (0, 1) is a small adaptation rate parameter.

The n × n matrix ψ(t) = [ψ

(t)] is deﬁned as the

epistemic afﬁnity matrix.

4.3 Metrics of Agent Information

Agents are evaluated in terms of their information via

two metrics:

1. Knowledge: This measures the quantity of cor-

rect information that the agent has in its EN com-

pared to the Oracle’s IEN, with a penalty for in-

correct knowledge. The performance of agent A

in time step t is computed as:

(t) =

true

− e

f alse

total

(16)

where e

total

is the number of edges in IEN, e

true

is the number of true edges in EN

, and e

f alse

the number of false edges in EN

. Thus, if an

agent knows all the true associations but no false

associations, its knowledge is 1. If it has as many

false associations as true ones, its knowledge level

is 0, and when the number of false associations

exceed true ones, knowledge becomes negative.

This metric is similar in spirit to the standard re-

call metric, but allows for negative values.

2. Wisdom: This measures the quality of an agent’s

information regardless of its quantity. For an

agent A

, this is measured as:

(t) =

true

+ e

f alse

(17)

Thus, it is identical to the standard precision met-

ric.

5 SIMULATIONS AND RESULTS

The simulations reported in this study use an undi-

rected small-world social network with n = 500

agents and average degree of 20. All agents begin

with ENs of approximately the same size, generated

as described below. A fraction n

of agents in the

network are initialized as wise agents, i.e., q = 0 and

all the associations in their initial EN are true. The

remaining fraction 1 − n

of agents are initialized as

normal agents, with 5% false edges in their initial EN

(q = 0.05). All the wise agents are assumed to always

learn in the pure meritocratic style, while the learn-

ing styles of the normal agents are varied as described

below.

5.1 Scenarios and Parameters

The simulations explore the effect of two factors on

implicit learning in the network: 1) The fraction of

wise agents, n

; and 2) The learning style of the nor-

mal agents. The following 9 values of n

are sim-

ulated: n

= 0 (baseline case with no wise agents),

0.02, 0.05, 0.1, 0.2, 0.3, and 0.4. For each of these

cases, four different learning style scenarios are sim-

ulated for the normal agents:

• Case I: All normal agents learn in the pure like-

minded (0, 1, 0) style.

• Case II: 5% of normal agents learn in the pure

meritocratic (0, 0, 1) style and the remaining 95%

in the pure like-minded (0, 1, 0) style.

• Case III: All normal agents have a mixed (0, 0.95,

0.5) learning style: Mostly like-minded with a

small degree of meritocratic learning.

• Case IV: All normal agents learn in the purely

meritocratic (0,0,1) style – as do the wise agents.

Thus, in all there are 36 different simulation sce-

narios, with results averaged over 10 independent tri-

als for each one. The knowledge and wisdom of each

agent is tracked over time in each run. The struc-

ture of the social network is ﬁxed across all simu-

lations, but the positions of wise and normal agents

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110

(a) 0% Wise Agents (b) 20% Wise Agents (c) 40% Wise Agents

(d) 0% Wise Agents (e) 20% Wise Agents (f) 40% Wise Agents

(g) 0% Wise Agents (h) 20% Wise Agents (i) 40% Wise Agents

(j) 0% Wise Agents (k) 20% Wise Agents (l) 40% Wise Agents

Figure 3: Distribution of Knowledge in the normal agent population over time for Case I (top row); Case II (second row);

Case III (third row); and Case IV (bottom row). In each image, the leftmost column shows the color-coded histogram of initial

Knowledge distribution, and subsequent columns show the same at the end of successive a 50-epoch intervals.

(and learning style assignments in Case II) are varied

randomly over trials. A total of 1, 000 epochs are run

in each trial, where an epoch comprises giving each

agent the opportunity to generate and express an idea.

5.2 System Initialization

The initial ENs are set up at the start of each simu-

lation trial by randomly selecting a set of true ideas

from the ideas pool gathered a priori by parsing the

The Effect of Well-informed Minorities and Meritocratic Learning in Social Networks

111

(a) 0% Wise Agents (b) 20% Wise Agents (c) 40% Wise Agents

(d) 0% Wise Agents (e) 20% Wise Agents (f) 40% Wise Agents

(g) 0% Wise Agents (h) 20% Wise Agents (i) 40% Wise Agents

(j) 0% Wise Agents (k) 20% Wise Agents (l) 40% Wise Agents

Figure 4: Distribution of Wisdom in the normal agent population over time for Case I (top row); Case II (second row); Case

III (third row); and Case IV (bottom row). In each image, the leftmost column shows the color-coded histogram of initial

Wisdom distribution, and subsequent columns show the same at the end of successive a 50-epoch intervals.

IEN and collecting all possible ideas in it. The indi-

vidual EN of a normal agent is then parsed again to

remove some true edges and add as many false ones

to reach the noise level q as deﬁned in the agent pa-

rameters. Wise agents do not have any noise, so all

their associations are true. Then, the initial status of

each agent is computed according to the wisdom for-

mula. So the wisdom of normal agents is initialized as

the number of correct ideas after adding noise divided

by the original number of correct ideas when the EN

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112

is created. Accordingly, the merit status of all wise

agents is initialized to 1.

Finally, the initial epistemic similarity is com-

puted between all pairs of agents by calculating the

overlap between their initial ENs.

5.3 Results

Figures 1 and 2 plot the ﬁnal levels of mean knowl-

edge and wisdom attained by each agent group as

a function of n

and the learning cases for normal

agents. The following observations can be made:

• In the absence of wise agents (left-most data point

in each graph), by far the best strategy for nor-

mal agents is to learn from better informed peers

(lower right graph in each ﬁgure.)

• Introducing even a small fraction of wise individ-

uals into the population improves the learning of

the whole community.

• Increasing the fraction of the wise minority in

the population improves the learning of the nor-

mal population dramatically, except when the nor-

mal agents adopt a meritocratic learning strategy,

in which case the presence of more wise agents

makes only a modest difference. This suggests

that the effect of more wise agents is not mainly

due to the greater available volume of correct

knowledge they bring, but due to the greater num-

ber of connections to wise peers available to nor-

mal agents.

• of the two non-pure learning cases for the normal

majority (Cases II and III), it is better for all nor-

mal agents to have a small meritocratic element in

their learning rather than having a small fraction

of normal agents learn purely meritocratically.

• Interestingly, the wise minority shows exactly the

same pattern of improvement in their ﬁnal knowl-

edge level with higher n

as is seen for the nor-

mal majority – albeit with an increment reﬂecting

their edge in initial knowledge due to the absence

of false associations. This can be explained by

the fact that, since the wise agents follow a pure

meritocratic strategy, they learn mostly from their

wise peers and hardly ever from the normal major-

ity. As such, increasing n

increases the pool of

peers from whom each wise agent can learn, lead-

ing to faster learning. And since the normal agents

do not pay much attention to the merit status of

their peers (except in Case IV), the learning pat-

tern for the wise minority simply passes through

to the normal majority.

• The general pattern of knowledge and wisdom is

similar for the normal majority across all scenar-

ios. Of course, the wise minority always retains its

perfect wisdom because wise agents learn with a

purely meritocratic strategy, and thus hardly ever

accept an idea from less well-informed peers. It

will be interesting to see how wise agents fare if

some of them adopt a different learning style.

Figures 3 and 4 show the time evolution of the dis-

tribution of knowlegde and wisdom among the nor-

mal agents when n

is 0, 0.2, and 0.4. The most no-

table feature of all these plots is that normal agents all

learn together: There are no stragglers or any bifur-

cation. The ﬁgures also show that the added beneﬁt

from a larger wise minority start showing up early.

In the beginning, the distribution remains quite stable

around the initial distribution for both knowledge and

wisdom. But it then reaches a breakout point where

learning suddenly takes off. Interestingly, the delay

before breakout depends greatly not only on the size

of the wise minority (horizontal comparisons) but also

on the learning style of the normal majority (vertical

comparisons). This effect is stronger with knowledge

than with wisdom. The adoption of a purely meri-

tocratic learning style by the normal agents leads to

a qualitatively different learning pattern, with break-

out occurring almost immediately, a very linear rise in

performance, and much higher ﬁnal level of both wis-

dom and knowledge. The latter effect is very small if

the wise minority is larger.

Many other things can be analyzed based on the

simulations (e.g., the spread or disappearance of indi-

vidual true/false edges in the social network), but this

will be discussed in future reports.

6 CONCLUSIONS

In this paper, a multi-agent network of generative cog-

nitive agents was used to look at the effect of having

a well-informed wise minority in a larger population

of normal agents that carry a certain amount of false

information. It was found that the size of the minor-

ity has a direct effect on the implicit learning perfor-

mance of normal agents. The study also found that

the performance depends very strongly on the learn-

ing style adopted by the normal agents. In particular,

this is seen in the transition of learning from an initial

ﬂat phase to a breakout with rapid learning.

The MANILA model is versatile and powerful

enough to be useful for exploring many other issues

related to the spread and learning of information in

human networks, including the spread of false infor-

mation – a critical problem at this time. Such issues

will be addressed in future studies.

The Effect of Well-informed Minorities and Meritocratic Learning in Social Networks

113

ACKNOWLEDGEMENTS

This work was partly supported by National Science

Foundation INSPIRE grant BCS-1247971 to Ali Mi-

nai.

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