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Extremism Propagation in Social Networks with Hubs
Daniel W. Franks*†, Jason Noble**, Peter Kaufmann***, and Sigrid Stagl***
df525@york.ac.uk, dr.Jason.Noble@googlemail.com, [P.Kaufmann | S.Stagl]@sussex.ac.uk
* York Centre for Complex Systems Analysis (YCCSA), Department of Biology &
Department of Computer Science. The University of York, York, YO10 5YW, UK
** School of Electronics and Computer Science, University of Southampton, Southampton,
SO17 1BJ, UK
*** SPRU - Science and Technology Policy Research, University of Sussex, Brighton, BN1
9QE, UK
† Author to whom correspondence should be sent. Tel: +44 (0)1904 328648
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Abstract
One aspect of opinion change that has been of academic interest is the impact of people with
extreme opinions (extremists) on opinion dynamics. Amblard and Deffuant (2004) used an
agent-based model to study the role of small-world social network topologies on general
opinion change in the presence of extremists. They found that opinion convergence to a
single extreme occurs only when the average number of network connections for each
individual is extremely high. Here, we extend the model to examine the effect of positively
skewed degree-distributions, in addition to small-world structures, on the types of opinion
convergence that occur in the presence of extremists. We also examine what happens when
extremist opinions are located on the well-connected nodes (hubs) created by the positively
skewed distribution. We find that a positively skewed network topology encourages opinion
convergence on a single extreme under a wider range of conditions than topologies whose
degree distributions were not skewed. The importance of social position for social influence
is highlighted by the result that, when positive extremists are placed on hubs, all population
convergence is to the positive extreme even when there are twice as many negative
extremists. Thus, our results have shown the importance of considering a positively skewed
degree distribution, and in particular network hubs and social position, when examining
extremist transmission.
Keywords: Social Networks, Scale-Free, Small World, Extremism, Opinion Change
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1. Introduction
People hold opinions about an almost countless number of topics. From religion, politics,
literature and moral values, to entertainment, hair products, and the lives of celebrities, most
people will have an opinion on the matter. These opinions can be shaped by individual
reflection, but are in many cases greatly influenced by social context: if your friends all prefer
Dan Brown to Dostoevsky, then you are likely to do so as well. Dual inheritance theory
(Boyd & Richerson, 1985) tells us that human behaviour cannot be understood without
considering both our genetic and cultural inheritances. But our cultural inheritance is not
absorbed at random or from all possible sources: opinion formation is embedded in social
interaction (Wood, 2000). In other words, what you think depends on who you talk to, and
whose opinions you respect. It follows that understanding the process of social influence and
the social structures that govern human interaction will be central to an understanding of the
dynamics of opinion change. Opinion change across a whole population, of course,
constitutes a change in cultural norms, and thus the dual inheritance program will remain
incomplete without an improved understanding of human social structures and their effects.
Given that opinions influence behaviour, a full account of human behaviour, both adaptive
and maladaptive, likewise depends on a theory of how opinion change is rooted in social
interaction.
One aspect of opinion change that has been of academic interest is the impact of extremists on
opinion dynamics. Extremists are people with marginal opinions that are far away from
ambivalence. The multidisciplinary innovation diffusion literature has produced numerous
case studies where an initially small minority of extremists spread their opinion to a majority
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of the population. Examples include drug use, new cropping techniques, and the introduction
of family planning practices (Rogers 1995). The spread of extreme opinion in Germany in the
1930s, and in Rwanda in the 1990s, stand out as salient and horrific examples. On the other
hand, there are extremists whose opinions (thankfully) never became generally accepted, such
as those of the KuKlux Klan. All minority rights activists, such as those campaigning for
womens' right to vote or those involved in the American civil rights movement, were
considered at some point to be extremists, although their views became mainstream.
Societal structure is important to opinion change, and can be conveniently represented as a
social network. In a social network, nodes represent individuals and connections represent
interactions between those individuals. Many studies have indeed shown that different types
of social interactions, such as scientific collaborations and sexual contacts, can be understood
in terms of networks (Newman, 2003). Evidence has shown that social networks have
distinctive properties (Newman & Park, 2003). Most prominently they are usually
characterized as small-world and scale-free networks. In a small-world network, each node
can be reached by relatively few steps, due to the existence of long-distance connections
bridging different areas of the network. Scale-free networks have a degree distribution that is
positively skewed to the right in a manner similar to a power-law distribution. This results in
some nodes possessing a far greater number of connections than others. However, scale-free
is a theoretical term that needs to be used with care (especially when dealing with networks of
just hundreds of thousands of nodes). Here we refer to our analogous degree distribution in
our relatively small networks as being positively skewed. It may be useful to carefully draw
some appropriate analogies with the properties of scale-free networks. Our own data (taken
from an unrelated study and shown as an example) collected on farmer social networks in six
countries, clearly shows a positively skewed degree distribution (Figure 1). Thus, when
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studying the effects of social network topologies it may be important to capture their small-
world and positively skewed degree distribution structures.
There have been numerous attempts at building models of opinion change. Many models
consider binary opinions (e.g., Latané and Nowak, 1997; Kacpersky and Holyst, 2000).
Because the opinions in these models are only binary representations they do not distinguish
between hard-core extremists and varying levels of tacit supporters and non-supporters.
Axelrod (1986) and Schelling (1978) evaluated opinion change using simple lattices. They
both showed that opinions tend to polarize (i.e., multiple opinions coexist). However, the
opinions were modelled as discrete, and thus extremists were not present. As such, these can
only be used to examine the spread of an opinion to become the minority or majority opinion.
The bounded confidence model (Krause, 2000; Dittmer, 2001) relaxes the assumption of
binary opinions. Instead, opinions are represented as points on an opinion continuum (e.g.,
anywhere between -1 and 1). Interactions become non-linear in that agents influence each
other only if the distance between their opinions is below a threshold. The relative agreement
model is an extension of the bounded confidence model. In the relative agreement model the
level of influence between agents is governed by the distance between opinions and the
certainty of the influencing agent, rather than some predefined level of influence. The model
captures the following ideas: (a) agents with radically different opinions are unlikely to
influence each other; (b) uncertain agents are more susceptible to social influence than agents
that are certain; and; (c) agents that are certain are more influential than uncertain agents. The
assumptions of the relative agreement model have been shown to be supported well by
laboratory experiments (Deffuant et al., 2002), such as those dealing with opinion
radicalization (for one of the first examples see, Moscovici and Lécuyer, 1972). Whereas
these models allow for influence based on persuasion (attractive forces), they do not allow for
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conflicts of opinion to drive opinions further apart (repulsive forces). The meta-contrast
model (Salzarulo, 2006) allows for both attractive and repulsive influence. Thus, the model
allows for the possibility that the final opinions can become more extreme than the most
extreme initial opinion.
Amblard and Deffuant (2004) studied the role of the social network topology on extremist
transmission (see also, Deffuant, 2006). They developed an agent-based model of social
influence using the relative agreement model, and examined the type of opinion convergence
that occurred under various conditions with various small-world topologies (as opposed to a
fully connected network). Extremists were represented as agents with opinions at the extreme
ends of the opinion spectrum (see below for a more technical description). They found that,
for the small-world topology, a critical level of average network degree was needed for the
population to converge on a single extreme opinion. However, the average network degree
needed to be extremely high for this to occur; typically requiring an average degree of around
64 or even higher for lower levels of random connections.
In this paper we further extend the model (Amblard and Deffuant, 2004) to examine the effect
of a positively-skewed-degree-distribution structure on the types of opinion convergence that
occur in the presence of extremists
1
. We also examine what happens when extremist opinions
are located on well-connected nodes.
1
Weisbuch et al. (2005) briefly look at scale-free structures for the relative agreement model, but do not
examine their effect in any depth, and also do not explore social position or hubs.
7
2. The Model
Our agent-based model consisted of one thousand agents embedded in a social network. Each
agent i was characterized by two attributes: an opinion xi and level of uncertainty ui. Social
influence was assumed to take place between all connected individuals in a random order
each time-step. In all cases simulations were run for 2000 time-steps, after which all
simulations found a stable-state.
2.1. Initialization
The population included N individuals of which there were pe.N extremists and (1-pe).N non-
extremists. p+ and p- were the proportions of positive and negative extremists (pe = p+ + p-).
Thus, the population was initialized with p+.N positive extremists and p-.N negative
extremists. When we talk of extremists as being positive and negative we are simply referring
to their position on an abstract opinion continuum (-1 or +1) without any connotations as to
the social value of their opinions (i.e., a negative extremist need not hold a "negative" opinion
in the colloquial sense). Non-extremist opinions were initially drawn from a uniform
distribution between -1 and 1, and the initial uncertainty of non-extremists was set to U.
Extremists were assumed to be those agents with opinions located at the extremes of the
opinion distribution xi (i.e., -1 or 1). They were also assumed to be more confident (i.e., have
a lower value of ui) than non-extremist individuals with an initial uncertainty of ue (where ue <
U). For all simulation runs pe = 0.1 (Deffuant, 2006) and we assumed that equal proportion of
positive and negative extremists in the population (p+ = p- = pe/2).
2.2. The Relative Agreement Model
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The relative agreement model captures social influence between two agents (Deffuant et al.,
2002). In this model, agents’ opinions can be visualized as segments with boundaries defined
as xi - ui and xi + ui. In other words, the centre of the segment is defined by xi and the level of
uncertainty defines the distance of the opinion edges either side of xi. Agent i can influence
agent j, bringing agent j’s opinion closer to agent i’s opinion, if their opinion segments
overlap (i.e., they have at least some common ground). The intensity of the influence depends
on the amount of overlap between the opinion segments; the higher the overlap the higher the
influence. The model also captures the notion that agents that are more certain about their
opinion are more influential. As an agent is influenced, its level of uncertainty is slightly
reduced. A formal statement of a relative agreement between two agents, i and j, follows.
The width of the overlap between opinion segments is given by:
),max(),min( jjiijjiiij uxuxuxuxh
(1)
Thus, the non-overlapping width is:
ijihu2
(2)
The agreement is defined by the overlap minus the non-overlap:
)(2)2( iijijiij uhhuh
(3)
The relative agreement is the agreement divided by the length of agent i’s segment:
9
1
2
)(2
i
ij
i
iij
u
h
u
uh
(4)
If hij ≤ ui, then there is no influence. This corresponds to the idea that agent j’s opinion
segment must intersect agent i’s actual opinion (at the centre of its segment) for agent i to
consider agent j’s opinion. Otherwise, the modifications of agent j’s opinion xj and
uncertainty uj as a result of the interaction with agent i multiplied by the relative agreement:
),(1 ji
i
ij
jj xx
u
h
xx
(5)
),(1 ji
i
ij
jj uu
u
h
uu
(6)
where μ governs the influence intensity for all interactions. A high value of μ results in quick
opinion convergence between neighbouring agents.
2.3. Constructing the Social Network
Networks were constructed in a manner that allowed us to vary the extent to which they
possessed small-world and positively-skewed-degree-distribution characteristics (Noble et al.,
2004). The average number of connections for each agent is denoted k. Each agent’s social
position was represented as a network node, arranged along a one-dimensional ring lattice.
The distance between nodes on this lattice defines the agent’s local neighbourhood; for a
neighbourhood of size n (n = 2k) this would consist of the nodes contained within a distance
10
of k either side of the focal node. There were two axes on which the layouts of our networks
could vary. The first axis of variation allowed us to control the frequency of random long-
range connections. Thus, we used a parameter R that varied the probability with which a
connection would be either local (i.e., from the neighbourhood) or random. A local
connection meant a connection to a randomly chosen node within the local neighbourhood. A
random connection meant a connection to a randomly chosen node from anywhere else on the
network. When R = 1, all links were random, when R = 0 all links were local. Small
intermediate values of R create a network with small-world properties (Watts & Strogatz,
1998).
The second axis of variation allowed us to control the level to which the network degree
distribution was positively skewed (Figure 1). This was done using the parameter P, which
allowed us to vary the level of preferential attachment to well-connected nodes. When P = 0
each node was equally likely to be selected for a connection. As the value of P increased
connections were made in an increasingly preferential way, and well-connected nodes were
more likely to receive a connection than poorly-connected nodes. The value
P
d)(
was
calculated for each eligible node, where d is the node’s degree and σ is a small positive value
(0.1 in this model) that ensures all nodes have a chance of selection. The node selected to
receive a connection was chosen by roulette-wheel selection over these values. Thus, the
parameter P is a preferential exponent, governing the strength of the bias towards connecting
to well-connected nodes (see also, Barabási & Albert, 1999).
[FIGURE 1 HERE]
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To construct the network, nodes were sorted in a random order and given one initial
connection, selected according to the current values of R and P. This procedure helped to
ensure that all nodes were part of the network. Then connections were allocated until K
average connections were reached. The values of R and P were used in the selection of both
source and destination nodes for each connection.
2.4. Convergence Indicator
The model typically results in three types of opinion convergence: ‘central convergence’,
‘both-extremes convergence’, and ‘single convergence’. Central convergence occurs when
extremists have little influence on the general opinion. Both-extremes convergence occurs
when extremists with opposing opinions both influence general opinion enough to cause
opinions to cluster around both extremes. Single extreme convergence occurs when general
opinion converges upon a single extreme opinion. Deffuant et al. (2002) found that the
convergence type exhibited by a simulation run can be conveniently found using an indicator
y. After a population has converged, y is calculated as follows:
22 ppy
(7)
where
p
and
p
are the proportions of initial non-extremists that became extremists to the
positive and negative extreme respectively.
y indicates the type of convergence as follows:
No extreme convergence: y = 0
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Double extreme convergence: y = 0.5
Single extreme convergence: y = 1.0
Intermediate values of y correspond to intermediates of the above situations.
3. Results
Amblard and Deffuant (2004) explored opinion convergence on their small-world network for
variations of k (the average degree) and R (the proportion of non-local connections). We
follow this approach for our study. When varying k and R the following parameters are kept
constant: U = 1.8, pe = 0.1, μ = 0.1. Thus, we are studying opinion formation where non-
extremists begin with arbitrary and uncertain opinions. Lower values of U could stop the
population all converging on extreme opinions, and using an unequal ratio of positive and
negative extremists would aid single extreme convergence. However, these parameters have
been exhaustively studied elsewhere (e.g., Deffuant et al., 2002; Amblard and Deffuant, 2004)
and do not contribute to the argument made in this paper. First, we replicate the work by
Amblard and Deffuant by studying a small-world network. Second, we extend the model to
allow for positively skewed degree distributions to be constructed. Finally, we examine the
effect of placing extremists on well-connected network nodes. In the forthcoming graphs
(Figure 2 to Figure 7) the waves and occasional spikes of the lines are simply noise, and the
graphs would be smoother if we averaged the values over thousands of trials. We were
limited to 50 replications due to the computational time required for the runs.
[FIGURE 2 HERE]
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3.1. Small-World
Small-world networks were constructed by varying R between 0 and 1 (P = 0). Figure 2
shows the type of convergence found for variations of k and R. Notice that for single extreme
convergence to occur regularly (e.g., y > 0.8), the network must have a relatively high number
of random connections, and the average degree must be extremely high (typically k > 60).
Single extreme convergence is always stable as the network tends to saturate in this state.
This result confirms those of Amblard and Deffuant (2004).
[FIGURES 3 & 4 HERE]
3.2. Positively Skewed Degree Distribution and Small-World
Small-world networks were constructed by varying R between 0.2 and 1. When P = 1, we
found that it gives positively skewed degree distributions that qualitatively match our farmer
social networks data. Note that our intention here is to conduct a theoretical study, it is not to
accurately model the farmer data. We simply show both distributions to illustrate that both
network degree distributions are positively skewed, with many individuals connected to few
individuals, and few individuals connected to many individuals. Figure 3 shows the type of
convergence found for variations of k and R. For single extreme convergence to generally
occur, the network does not need an average degree as high as in the case where the network
degree is not skewed, and the proportion of random connections does not need to be as high.
Influence by extremists is higher for almost all data-points in the case with the skewed
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distribution. Figure 4 shows the type of convergence found for variations of k and P.
Convergence to a single extreme occurs readily for high values of k and P.
[FIGURES 5, 6 & 7 HERE]
3.3. Extremists at Network Hubs
Networks with positively skewed degree distributions, like scale-free networks, produce a
relatively limited number of well-connected nodes, or hubs (and random network also
produce hubs, but a limited number of poorly connected nodes). We introduced a new
parameter λ to control the proportion of positive extremists placed at these hubs. Thus, once
the network had been constructed, p+.N.λ positive extremists exchanged their network
position with the agents on the best-connected nodes. Figure 5 shows the type of convergence
found for variations of k and R. The results are similar to the case with the skewed degree
distribution without positive extremists on well-connected nodes, although for values of R <
0.2 a lower average degree is needed for single-extreme convergence.
To examine the effect of placing positive extremists on key nodes on the trajectory of opinion
change, we plotted the proportion of non-extremists that became positive extremists for
networks with positively skewed degree distributions, with and without positive extremists at
hubs (Figure 6). We found that placing positive extremists at network hubs greatly increased
the proportion of non-extremists that become positive extremists. Under some conditions
(roughly k > 22) most of the population (more than 80%) became positive extremists.
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Figure 7 shows that even when negative extremists are twice as prevalent in the population,
opinion convergence to the positive extreme can regularly occur if positive extremists are
placed on network hubs. We reproduced the conditions for Figure 7 without placing positive
extremists at key hubs. The results showed that, under all conditions, the population never
converged on the positive extreme (no graph shown).
4. Discussion
A positively skewed degree distribution increases opinion convergence towards extremes, and
encourages opinion convergence on a single extreme under a wider range of conditions than
topologies that were not skewed in their degree distribution. For single extreme convergence
to occur for the small-world model, Amblard and Deffuant (2004) required an extremely large
number of average connections. Such a large average number of connections is unlikely to
exist in a real-world social network. Constructing a positively skewed degree distribution
reduces the number of connections needed for single-extreme convergence to a more realistic
number of average social network connections. Another effect of the skewed distribution is
that there is little difference between a small-world network with few random connections
(e.g., R = 0.25) and a random network (R = 1).
Why does a positively skewed degree distribution increase extremism transmission in a social
network? We suspect that the creation of some well-connected nodes (hubs), along with the
creation of many poorly connected individuals, is responsible. Extremists located at these key
nodes are connected to more agents, and more parts on the network. Thus, their influence is
more frequent and wider-reaching, making it less likely that certain parts of the network
remain socially isolated from the extremist’s opinion. In the default case, most of the time
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extremists will not be located at hubs. However, extremists take advantage of non-extremists
at hubs, as a means of transmitting their opinion. Non-extremists are initially less confident
about their opinion than extremists. Thus, extremists have more influence over non-
extremists than non-extremists have over extremists (and more influence than non-extremists
have over each other). Once an extremist has spent time influencing a non-extremist at a hub,
the former non-extremist will transmit the opinion throughout the network. Because agents at
hubs are well-connected, it is likely that they are connected to an extremist. In the real world,
extremists might purposely attempt to influence people in high status positions.
Valente (1995) investigated some effects of propagating innovations first to more ‘strategic’
nodes of the network, showing an individual’s network position can affect innovation
diffusion in a social network. Here, we examined the effect of extreme opinions at high social
positions. By placing positive extremists at the well-connected nodes (i.e., nodes to the tail-
end of the degree distribution) we found that when the network converges on a single
extreme, it is always to the positive extreme. This is because when positive extremists are
placed on key nodes they are able to directly influence more agents (as they would have more
direct connections) and have a higher chance of having a random connection that allows them
to influence a different social network neighbourhood. The importance of social position for
social influence was highlighted by the result that, when positive extremists are placed on
hubs, all population convergence is to the positive extreme even when there are twice as many
negative extremists. Thus, the position of an opinion in the social network is more important
than the initial proportion of individuals with the opinion. The network does not have to have
a power-law distribution for this result to stand. However, the network must have multiple
hubs and social networks are known to produce such hubs. In the real world, certain
extremists are likely to fight for high-status network positions in order to preach their
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opinions. For example, people with extreme political opinions might actively attempt to seek
higher social status or widen their social network with the intention of spreading their
opinion.
These findings are of relevance for applications in social science, biology, and computer
science. In social science, one can think of multiple applications where the structure of
networks is of relevance. We have primarily shown how certain social structures can
contribute to changing a population’s values, attitudes and behaviour. Religious extremism is
one of many examples where an extreme opinion exists, and thus has the possibility of
percolating throughout a sub-community such as the one modelled here. We have shown that
such opinions are less likely to pervade the social network if well connected, typically high
status, people are resistant to the opinion. The findings are also useful for studies of
innovation diffusion, and market activities where geographic actor interactions are explicitly
taken into account. In biology, the social network structure can have a large affect on social
learning in animals. For example, in a social learning model that did not explicitly consider
network structure, Noble and Franks (2002) found that who learns from whom affected the
efficiency of different social learning mechanisms. Given our findings, that agents’ social
positions greatly affect their influence, these results might be emphasized. In computer
science, knowing more about social networks should allow us to construct better software for
monitoring, shaping, and exploiting existing human networks. There could also be feedback
into the design of large-scale artificial multi-agent systems, in which the agents necessarily
communicate over a social network.
Our results have shown the importance of considering a positively skewed degree distribution
when examining extremist transmission. Future work will look at the effect of modelling
18
loosely coupled sub-communities, network formation, network growth, and coevolving
network structure with opinions. Such studies will help us to further understand opinion
change and extremism transmission. An interesting future question that this model could be
used to explore is: if most social networks are small-world and have positively skewed degree
distributions, then why do we see the persistence of extreme opinion polarization in
populations? It is clear that social behaviours (and thus human behaviour in general) cannot
be fully understood without considering their underlying network structures. Agent-based
models can help us look for underlying principles that allow us to understand social
behaviours in terms of social networks.
Acknowledgements
We thank two anonymous referees for their comments and suggestions. The model presented
in this paper was developed while D.W.F. was at SPRU (The University of Sussex) working
in the IDARI project, financed under the FP5 Quality of Life and Management of Living
Resources, Key Action 5.
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Figure Captions
Figure 1: The qualitative similarity between degree distributions for (left) a real-world farmer social network in
Estonia (N = 170, K = 7.07); and (right) a simulated social network in our model, with highly preferential
attachment (N = 1000, k = 7.07, R = 0.2, P = 1.0). Note the similarities in the general shape of each graph. They
both show a skewed network degree distribution where many nodes have few connections and few nodes have
many connections.
Figure 2: (left) Contour plot of the mean opinion convergence indicator values, y, for variants of k and R,
averaged over 50 runs. (right) The standard deviation of y. The graph shows the result of varying the network
topology between only local connections, and only random connections; with small-world topologies existing
between the extremes of R. The standard deviation for some mid-range values of y indicates that some trials
may have resulted in single extreme convergence, and some may have reached central convergence.
Figure 3: The mean opinion convergence indicator values, y, averaged over 50 runs for variants of k and R,
when the degree distribution is positively skewed (P = 1). The graph shows the result of varying the network
topology between only local connections, and only random connections; with small-world topologies existing
between the extremes of R. Notice that for single extreme convergence to generally occur, the network does not
need an average degree as high as in the case where the network degree is not skewed. Influence by extremists
is higher for almost all data-points in the case with the positively skewed degree distribution.
Figure 4: The mean opinion convergence indicator values, y, averaged over 50 runs for variants of k and P,
when R = 0.5. The graph (left) shows the result of varying the level of preferential attachment to well-connected
nodes; with positively skewed degree distributions existing for higher values of P. Note that convergence to a
single extreme occurs readily for high values of k and P. The standard deviation is shown on the right.
Figure 5: The mean opinion convergence indicator values, y, averaged over 50 runs, for variants of k and R,
when the degree distribution is positively skewed (P = 1) and positive extremists are placed on the most well-
connected nodes (λ = 1). The graph shows the result of varying the network topology between only local
20
connections, and only random connections; with small-world topologies existing between the extremes of R.
Notice that the graph is a smoothed-out version of the graph in figure 3. Influence by extremists is higher for
almost all data-points in the case with the positively skewed degree distribution. The standard deviation is close
to zero for all data-points.
Figure 6: The mean proportion of initial non-extremists that become an extremist to the positive extent,
averaged over 50 runs. (left) When the degree distribution is positively skewed (P = 1, λ = 0), (right) when the
degree distribution is positively skewed, and positive extremists are placed on the most well-connected nodes (P
= 1, λ = 1). The graph shows the result of varying the network topology between only local connections, and
only random connections; with small-world topologies existing between the extremes of R.
Figure 7: The mean proportion of initial non-extremists that become an extremist to the positive extent,
averaged over 50 runs, when the degree distribution is positively skewed (P = 1) and positive extremists are
placed on the most well-connected nodes (λ = 1). In this case, there were twice as many negative extremists as
positive extremists. The graph shows the result of varying the network topology between only local connections,
and only random connections; with small-world topologies existing between the extremes of R. Note that
despite there being twice as many negative extremists; the population converges to the positive extreme under a
range of conditions.
21
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Figure 1
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Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6
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Figure 7
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31
Daniel Franks Bio
Dan Franks is a research fellow in the York Centre for Complex Systems Analysis, in both
the Department of Biology and the Department of Computer Science. His research centres on
building computational models of ecology and evolution. In particular he is interested in
animal social networks (and how best to sample them), and predator-prey interactions (with a
focus on warning signals, mimicry, and anti-predatory defenses). Address: York Centre for
Complex Systems Analysis, The University of York, York, YO10 5YW, UK. Email:
df525@york.ac.uk
32
Jason Noble Bio
Jason Noble is a research fellow in the Science and Engineering of Natural Systems group,
part of the School of Electronics and Computer Science at the University of Southampton.
His main research interest is in using individual-based simulations to look at the evolution of
social behaviour, particularly communication and social learning. Other interests include
network theory, game theory, philosophy of science, and philosophy of mind. Address:
School of ECS, University of Southampton, SO17 1BJ, UK. Email: jn2@ecs.soton.ac.uk.
