The curious absence of turnout cascades: a re-assessment of turnout in a small

world

1. Introduction

A number of scholars have incorporated group based interests and social influence into

theories of political participation and turnout (Uhlaner 1989; Morton 1991; Knack

1992; Rosenstone and Hansen 1993; McClurg 2003; Abrams, Iversen et al. 2011;

Sinclair 2012; Campbell 2013; Schmitt-Beck and Lup 2013). What is common to group-

based models is that voters’ interests are in some way dependent on their membership

of groups or networks. The effect of network membership may derive from a number of

sources including enhanced rational incentives (e.g. relational benefits and bloc voting)

or the fulfilment of shared social norms and the avoidance of associated social

sanctions. Network influence is central to the concept of two-step mobilization

introduced by Lazarsfeld (1954) and integral to the process of indirect mobilization

(Rosenstone and Hansen 1993). By this account, arguably the most crucial role of social

networks is though indirect mobilization whereby the direct effects of party based

mobilization are magnified by the effect of second-order interpersonal mobilization.

This occurs where turnout is encouraged by friends, families, neighbours and the wider

social networks of voters. Inter-personal influence can lead to enhanced interest (or

disinterest) in politics (Huckfeldt and Sprague 1995). Equally, mobilization can occur as

a result of individuals being asked to vote either by friends and acquaintances

(Fieldhouse and Cutts 2012). In other words, in social theories of voting whether a

citizen votes is a function of whether others in their social network vote. The idea of the

conditional voter gives rise to a prediction that voting should spread though social

networks leading to voting ‘cascades’ and bandwagon effects (Fowler 2005; Rolfe

2012).

Building on the idea of network influence, the obvious implication is that social

influence will cause multiplier effects (Gerber and Rogers 2009). There is evidence that

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there are genuine causal effects. For example, using a GOTV field experiment Nickerson

(2008) estimates that 60% of the average treatment effect is passed on to the spouse of

the primary recipient (see also Sinclair, 2012). In an important and conceptually

ground-breaking article, James Fowler (2005) estimated that direct mobilization of one

additional voter might be expected to lead to a total increase in turnout of up to four

votes due to indirect network influences, referring to this phenomena as turnout

‘cascades’. Fowler’s article makes an important and widely cited argument about the

potential power of indirect mobilization. He uses the analogy of turnout cascades,

suggesting exponential spreading of direct mobilization effects via interpersonal

influence. Using agent based simulation he demonstrates that even a modest degree of

contagion can cause a chain reaction, substantially raising aggregate voter turnout.

Cascade models, however, present a puzzle. If social influence is as strong as evidence

suggests, and participation is conditional on the participation of social referent, then

moderate levels of direct, candidate or party-mobilization should lead to geometric

increases in turnout. Although Fowler's analysis only implies that a single mobilised

agent leads to four additional votes, it suggests that if there were several mobilised

agents, the effects would be additive. Although, in principle, this need not be the case, in

Fowler’s model the effects of mobilised agents are indeed approximately additive. That

is, if there are 2% of additional unconditional voters, the additional number of votes will

be approximately equal to 8% of the electorate. We know that although local turnout is

responsive to local campaigns, the effects are rather more modest than the cascade

model would imply. At least there is no observational evidence of an effect of this size.

There are a number of reasons the magnitude of turnout cascades may be overstated.

Using a complex agent based model Anonymous (forthcoming) find that every additional

vote generated by a party based mobilization, might generate between 0.1 – 0.2

additional votes (i.e. a multiplier of 1.1 or 1.2). They suggest four possible reasons why

the effect is relatively small. First, regardless of network structure, first-order

mobilization will only change turnout behaviour of citizens close to the voting threshold

(see Arceneaux and Nickerson 2009). Second, and related but additionally to this,

insofar as networks tend to be made up of like-minded citizens, higher-order effects will

peter out when they meet groups of electors who are already fairly certain to vote (or

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not to vote). In other words, the more homogeneous networks are (with respect to the

propensity to vote), the fewer opportunities there are for electors to change the minds

of others. Third, whilst some voters may be influenced by their networks, exogenous

factors may confound the contagion effect: for example, an elector may be persuaded it

is important to vote, but may lack a candidate they are willing to support, or their

participation may be interrupted by circumstances (they were ill or too busy to vote).

Fourth, and most importantly for the argument presented here, the assumptions of the

simple cascade model may be overly generous. In Fowler’s small world simulations just

one agent is constrained to unconditionally vote (or not vote), and cannot be influenced

by other agents. In this article we focus on the operationalization of that model and the

assumptions that underpin it, including this assumption of a single unconditional voter

or ‘zealot’. We do not challenge the assumptions about the underlying levels of influence

or the structure of networks as, although these are important, we do not think they are

unreasonable. Rather, the reason the model may over-estimate the size of turnout

cascades is in the assumption underlying the crucial experimental manipulation.

2. Fowler’s Small World Model

Fowler’s small world model of turnout cascades (hereafter FSWM) is an agent based

simulation of voting whereby electors belong to social networks and the turnout

intention of each agent can be altered by interaction with other agents. In short, agents

probabilistically copy the turnout behaviour of neighbouring agents with whom they

interact.

A defining characteristic of a ‘small world’ model is the network structure. In a small-

world, regardless of the population size and the average network size (or average

degree), people are linked to others through a relatively small number of intermediaries

(i.e. the average path length is small). The structure of the network in Fowler’s model is

built using the Watts-Stogatz model (Watts and Strogatz 1998) and is designed such

that the average degree (average number of discussants), the average path length, the

level of clustering (or the probability that two of one’s discussants also talk to one

another) and the number of interactions are all consistent with data derived from real

(observational) political network data. The small world network structure is

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theoretically derived, but the key parameters in the model are based on analysis of the

ISLES data (Huckfeldt and Sprague 2000) and also a meta-analysis of previously

published research. The imitation rate is not directly empirically grounded but is

backwardly engineered from correlations in turnout within networks, again taken from

ISLES data.

In order to experimentally measure the size of the cascade, a series of simulations are

performed. Each simulation is based on the idea that one voter may be ‘flipped’ to the

“voting state” by direct mobilization. The simulation is repeated many times and the

average total turnout is compared with the equivalent simulation with the voter flipped

to the “non-voting state”. The essence of the turnout cascade is the second-order effects

or, in other words, the extent of indirect mobilization (Rosenstone and Hansen 1993).

Based on 10,000 trials, the average increase in turnout is about 4. This represents a very

substantial multiplier which, if correct, should have important implications for political

campaigning and for how we understand turnout.

Each simulation is based on the idea that one voter may be ‘flipped’ by direct

mobilization and may not be flipped back by social influence. At the same time all other

agents are subject to social influence and are neither unconditional voters nor

unconditional non-voters. Why the experimental subject (the one voter at the centre of

the ‘experiment’) should be unique is not clear as it is neither a plausible nor a

necessary assumption. Drawing on wide ranging experimental evidence from social

psychology and behavioural economics Rolfe (2012) estimates that 10-20% of

experimental subjects (and by inference electors) are unconditional co-operators, and

by extension, unconditional voters. Although these are approximations, we can be

reasonably sure the true number of conditional voters in the real world is not 100%

(minus one). Rather there will be some distribution of conditional and unconditional

voters and non-voters (which we refer to generically as zealots). We hypothesise that

the size of the cascade effect will be constrained by the proportion of zealots in the

electorate. In this paper we demonstrate this, replicating the Fowler model

mathematically and using simulations. In addition, we track the cascade at the micro-

level and quantify the fraction of the additional votes that are due to direct influence of

the mobilised agent, and the fraction due to higher order effects.

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FSMW also has important implications about the possible effect of network

characteristics on indirect mobilisation. Fowler finds that the size of the turnout cascade

rises quickly as the path length drops, and that clustering coefficient has a positive

effect in the size of the turnout cascades. Here, we investigates the extent to which the

induced turnout really depends on network structure, particularly whether

characteristics other than the average number of discussants for each agent (which is

itself closely related to network clustering and path length) have a noticeable effect.

In addition, the model allows for the estimation of the net favourable change in turnout

(additional votes to the agent's preferred candidate) an agent can induce. Fowler finds

that the net favourable change in turnout induced by an agent displays a non-

monotonic, curvilinear, relation with the local clustering coefficient of that agent, and

argues that this finding has important implications for the literature on social capital.

He proposes the following explanation: as networks become more clustered this

initially has a positive effect on the size of the cascade as the opportunities for the initial

message to be passed on increases; however, the argument goes, as networks become

very clustered the opportunities for second order effects become saturated as everyone

within a group talks to each other, but are spending less time talking to people in the

rest of the network. Fowler says, “when these relationships within a group become too

dense, civic engagement actually declines because people are less connected to the rest

of society” (p.287). Here we test out this explanation by measuring the total turnout

induced by an agent as a function of her local clustering coefficient. We propose the

counter-explanation that the effect is just due to the fact that, in the model, agents with

higher local levels of clustering tend to be connected to agents with similar opinion to

theirs.

3. Analysis

Fowler´s model (Fowler 2005) considers a population of N agents with an average

degree (average number of ties per agent) equal to k. The dynamics of the model can be

described as follows:

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At time t = 0 each citizen, i, is assigned a turnout behaviour at random such that

si = 1 (vote) or si = 0 (abstain). In the initial round at time t =1, Nk interactions

take place one at a time in which a tie (i', j') is chosen at random and with

probability q citizen i' imitates the turnout behaviour of citizen j': si' =sj'. The

round is then repeated D times until the end of time, t = D. In this way, each

citizen to interact with each of his or her neighbours D times on average. In

period t = D+1 an election is held in which citizens in the voting state, si = 1, vote.

This process is known as the “Voter Model” in the probability (Liggett 2005; Cox and

Griffeath 1986) and statistical physics (Dornic et al. 2001; Sood and Redner 2005)

literature. More precisely, Fowler's definition corresponds to the link-update voter

model, in which at each interaction a link is chosen at random (as opposed to choose an

agent at random and then one of its neighbours). The fact that imitation happens only

with probability q amounts just to a change in the time-scale, so that D iterations of

Fowler's formulation correspond to q D iterations of the voter model, in which q=1 1.

Because in this model social influence can take place through multiple, recurrent, paths,

it is not straightforward how to measure the turnout cascade induced by a given agent.

A first possibility is to define the size of the turnout cascade as the number of agents

who vote when the agent in question (the ego) is initially in the voting state, minus the

number of agents who vote when the ego is initially in the non-voting state. This

definition differs from Fowler's in that we allow the ego to be subjected to social

influence, just like any other agent in the system. In other words, the target agent is a

conditional voter. It is known that in the link-update voter model the average state of

the system is conserved (Suchecki, Eguíluz and San Miguel 2004). This implies that the

expected value of the turnout cascade (as we have just defined it) is exactly equal to one

(see the technical appendix for a mathematical proof of this result valid for any

network). That is, an initial extra voter leads (on average) to just one extra vote, which

means that there is no multiplier effect.

1 Actually, D iterations in Fowler's formulation correspond to Bin(DN,q)/N iterations in the (link update) Voter Model, where Bin(A,b) is a binomial random variable with parameters A and b, and N is the total number of agents. Note that E[Bin(DN,q)/N]=qD, Var(Bin(DN,q)/N)=Dq(1-q)/N, and that this variance tends to zero if N is large.

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3.1 Introducing zealots

Fowler uses a different definition of turnout cascade, in which the state of the ego is

fixed over all the time evolution of the process. That is, the ego cannot be influenced by

any other agent, while all the other agents are susceptible to social influence. In this

way, Fowler's definition quantifies the maximum possible effect a given agent could

have, if she keeps committed to her initial option during all the course of the process.

This quantity, however, introduces an apparent conceptual difficulty. If we consider the

plausible situation in which the ego remains with a fixed state, being immune to social

influence, why is there not any other agent in the population who is also immune to

social influence? To address this conceptual inconsistency (which we have just shown is

crucial to the multiplier or cascade effect) we allow a given fraction of agents to be

“zealots”, whose voting (or non-voting) state is also fixed and cannot be changed

through social influence, while they can still convince other agents to adopt their state.

More generally, one could consider that the “susceptibility” to social influence varies

from agent to agent. To keep things simple, here we assume that there are only two

types of agents: those with a given non-zero susceptibility (all with an equal

susceptibility defined by the rate of influence in the model) and those immune to social

influence (the zealots). As we shall see, the presence of even a small fraction of zealots

can have a major impact and greatly limit the size of turnout cascades an agent can

induce.

The voter model with one or several agents with a fixed state has been analysed

theoretically in the literature (Mobilia 2003; Mobilia, Petersen and Redner 2007). In this

case the average state is not conserved, and an exact explicit solution valid for an

arbitrary network is not known. Still, one can perform some approximations that allow

to derive a very simple expression for the expected size of the turnout cascade (see the

technical appendix for details):

E [ ΔT ]=1+1− z0− z1 −1/ N

z0+z1+1/ N(1 − e− q k ( z0+ z1+1 /N ) D ) , [1 ]

where E[.] denotes the expected value, ΔT is the turnout cascade, z0 and z1 are

(respectively) the fraction of unconditional non-voters and unconditional voters, N is

the total number of agents and D is the number of iterations of the process. We see that

in the case with no zealots (other than the ego), z0=z1=0, the turnout cascade tends to N,

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that is, if only one agent has a fixed state, eventually all the population will adopt this

agent's state. The time (number of interaction rounds) to reach this state, however, is of

the order of N/(qk) which can be exceedingly long for large populations. If some fraction

of zealots are present, the maximum size of the turnout avalanche is limited to

1z0+z1+1/ N (interestingly, this is independent of all the other parameters of the model)

and the time to reach this maximum value is now of the order of 1

qk ( z0+ z1+1/ N ) , which

is much smaller if the total fraction of zealots, z=z0+z1, is much greater than 1/N (i.e.

there is a non-vanishing proportion of zealots). In figure 1 we show the average turnout

obtained simulating the model, in the case of no zealots (other than the ego) and in the

case of 20% of zealots, for a collection of social networks and parameter values similar

to the one employed by Fowler. The figure exemplifies how turnout cascades can be

greatly diminished when zealots are present.

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Fig. 1 Expected total change in turnout with no zealots (light crosses) and 20% zealots

(dark dots). The maximum expected change in turnout predicted by expression [1] is

displayed as a solid line. N=1000. Results averaged over 1000 realisations to estimate

expected values.

We would like now to quantify how much of the extra turnout induced by the ego is due

to direct interaction with the ego and how much of it is a higher order effect, only

indirectly due to interactions with the ego. We define the size of the first-order effect as

the number of agents whose turnout decision is set due to direct interaction with the

ego, i.e. the number of agents whose last coping event was with the ego. In the appendix

we show that the expected value of the size of the first-order effect is given by:

E [ ΔT 1 ]= 1− z−1 /N1−1 /N ⟨ kego

k i

(1− e−qk i D ) ⟩ , [ 2 ]

where⟨ . ⟩denotes average over the network. This is an exact expression valid for any

form of the social network. In figure 2 we plot the different order effects obtained by

simulating the model, together with the results of the simple theoretical expressions, for

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different parameters of the model. Note that despite the strong approximation used to

derive expression [1], which ignores all the network characteristics except the average

degree, the agreement with the simulations is rather good, particularly when the

proportion of zealots is not too small. This already suggests that network characteristics

other than the average degree have a minor impact in the turnout cascades. Expressions

[1] and [2] allow us to gain insight into the behaviour of the model further than that

gained through simulations. They show that the influence of the zealots on the cascade

size depends only on the total proportion of zealots, z=z0+z1, regardless the relative

proportion of unconditional voters and unconditional non-voters. It also shows that the

cascade size depends on the imitation probability, q, the average number of

acquaintances, k, and the number of iterations of the process, D, only through the

product of these three parameters, qkD, and that there is a maximum value of the

cascade size that depends on the proportion of zealots alone.

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Fig. 2 Size of the turnout cascade induced directly (squares) and indirectly (circles) by

the ego. Left panel corresponds to 20% proportion of zealots while right panel

corresponds to no zealots (note the different scale in the y-axis). The symbols are the

results of numerical simulations and the lines come from expressions [1] and [2].

Parameter values are N=10000, k=20, C=0.61, q=0.05. The dynamics was averaged over

3000 realisations to estimate the expected values.

With the parameter values that Fowler considers the most likely, q=0.05, D=20, k=4, one

obtains a total expected change in turnout (including the ego himself) of 4 when there

are no zealots (other than the ego) and of 3.2, 2.2, 1.6 and 1.2 when there is a

proportion of 20%, 40%, 60% and 80% of zealots, respectively. The contribution to the

expected turnout change due to first-order effect is 1 in the case of no zealots and 0.8,

0.6, 0.4, 0.2 in the case of 20%, 40%, 60% 80% of zealots, respectively.

3.2 The effect of clustering

Perhaps the most surprising result from figure 1 is the apparent increase in cascade size

with the clustering coefficient of the underlying network, particularly since the

clustering coefficient, C, does not appear in expression [1]. To better understand this

feature, we have to consider what is actually plotted in the figure. As Fowler, we have

run the model varying independently the average number of discussions per

acquaintance, D, the imitation probability, q, the average number of acquaintances, k,

and the rewiring parameter in the Watts-Strogatz network model that, together with k,

determines the clustering coefficient, C. For each combination of parameter values, we

run the model and compute the turnout induced by a particular agent. Since this is a

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stochastic quantity that varies between runs even for fixed parameter values, we

average it over 1000 runs to estimate its expected value. In each panel of figure 1 we

plot the results for each combination of parameter values but using a different variable

for the X-axis. This means, for example, that for a given value of the average number of

discussions in the top left panel, each point corresponds to different values of k, q and C.

This way of presenting the data is a bit misleading if we try to interpret the role of the

clustering coefficient, C. The problem is that, in the Watts-Strogatz network used, the

clustering coefficient is correlated with the average number of acquaintances, k. When

the rewiring is zero, the clustering is maximum and the relation between C and k is:

Cmax=3 (k /2 −1 )2 ( k− 1 )

.With a non-zero rewiring, the clustering decreases, but the positive

dependence of the clustering with k approximately continues to hold. Because of this,

points in panel 3 of figure 4 with low C also have low k and the dependence of the

turnout change with k causes the apparent increase with the clustering coefficient

observed. In figure 3 we plot, in the case of no zealots, the change in turnout as a

function of the clustering coefficient, but distinguishing points corresponding to

networks with different values of k. The figure shows that, while networks with larger

values of k can lead to larger turnout changes, for a constant k the clustering coefficient

has no noticeably effect on the turnout change. Indeed, linear regressions of E[ T] Δ on C

for fixed values of k yield dependences that are not statistically significant for any value

of k. A similar result is obtained if some proportion of zealots is present.

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Fig. 3 Expected total change in turnout as a function of clustering coefficient of the

underlying network. Different symbols correspond to networks with different average

degree, k, as indicated. Parameter values as in Fig. 1.

Fowler also finds that there is a power-law relationship between turnout induced and

path length of the network, with the size of the turnout cascade rising quickly as the

average path length drops. However, the average path length in the network is also

correlated with the average degree. Higher average degree implies more connections

and so smaller distances between nodes. If we plot the turnout change as a function of

the average path length, we observe a clear negative dependence, and indicated by

Fowler. However, this is again due to the dependence of the average path length on the

average degree, as figure 4 shows. Indeed, while a regression of the change in turnout in

the average path length yields a strong negative dependence:

ln ( E [ ΔT ])=2.20± 0.02 − (0.60 ± 0.01 ) ln (⟨ L⟩ )(standard errors in parenthesis), regressions

for fixed values of the average degree, k, yield dependencies that are never statistically

significant. While we have used a logarithmic scale to explore the power-law

dependence proposed by Fowler, similar results are obtained if a linear scale is used.

When some proportion of zealots is present, equivalent results are obtained.

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Fig. 4 Expected total change in turnout as a function of the average path length of the

underlying network, in a logarithmic scale. Different symbols correspond to networks

with different average degree, k, as indicated. Parameter values as in Fig. 1.

We arrive to the following conclusion: once the average number of discussants is

controlled for, the effect of clustering and path length practically disappears. This

implies that these network characteristics have a much smaller effect in the model than

previously thought, and singles out average degree as the most important network

characteristic.

So far we have focussed on the extra turnout induced by a given agent, without

considering whether this extra turnout is actually to the electoral advantage of the

agent, as defined by the increase in turnout amongst supporters of the agents preferred

candidate as compared to that amongst supporters of the rival candidates. A relevant

question concerns the size of the net favourable turnout change induced by an agent. In

Fowler's model, each agent has a preference for one of two possible options. This

preference is set at the beginning and it is held fixed. While the social influence process

affects the voting intention (voting versus abstaining), it does not affect the option the

agent would vote for if she actually votes. An extra assumption is that the preference of

connected agents is correlated. We can, then, ask what is the net favourable turnout

induce by an agent, that following Fowler, is defined as the net outcome of the ego's

preferred option (votes for this option minus votes for the other option) when she is in

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the voting state, minus the net outcome of the ego's preference when she is in the non-

voting state.

Since an agent is more likely to influence the turnout decision of agents she is linked to,

whose preferences are correlated with hers, the expected net favourable turnout

induced by an agent will be positive. The higher the correlation on the preferences of

linked agents, the more the net favourable turnout change will approach the total

turnout change, as obtained by Fowler.

We can also investigate whether agents whose contacts are connected to one another

tend to induce a larger net favourable turnout change. In the left panel of figure 5 we

plot the expected net favourable turnout induced by an agent as a function of her local

clustering coefficient (fraction of the agent's acquaintances that are connected to one

another). To eliminate the possible dependence of the number of acquaintances of the

agent, we plot separately agents with a different number of acquaintances. We also

include a fit to a second-order polynomial, as proposed by Fowler. A clear positive

influence of the local clustering coefficient, c, is observed for small values of c, while the

effect tends to saturate or even become negative for larger values of c. The effect of the

local clustering in the total turnout induced is portrayed in the right panel of figure 5,

which shows a weak negative dependence. The fact that the net positive turnout change

initially increases with the clustering coefficient while the total turnout change

decreases, suggests that the first effect is due to the fact that agents with large clustering

coefficient tend to be connect to agents with similar preference to themselves. This is

confirmed in figure 6, where we plot the fraction of acquaintances with the same

preference as a function of the local clustering coefficient.

We conclude that the non-monotonic dependence of the net favourable turnout induced

with the local clustering coefficient is a result of the way in which preferences are

assigned in the model, rather than a network effect. The fact that the total turnout

change monotonically decreases with the clustering of the agent, completely rules out

Fowler's explanation:

“Clustering increases the number of paths available to influence other people

in the network... At the extreme, however, individuals in groups that are very

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highly clustered may have several paths of influence within the group but

they will also have fewer connections to the rest of the social network...”.

(Fowler, 2005, p.282).

We also point out that the dependency with the clustering is very weak, with the

second-order polynomial fits proposed by Fowler yielding r^2 values of the order of 10 -

4 for the total change and 10-5 for the net positive change (i.e. clustering explains only

0.01% and 0.001% of the variances in turnout observed, when fitting individual turnout

cascades rather than average values). This implies that a very large number of

simulations need to be averaged to observe this dependency, and leads to the very noisy

data observed in figure 5.

Fig. 5 Expected net favourable (left panel) and total (right panel) turnout change as a

function of the local clustering coefficient of the treatment agent. The error bars

correspond to one standard deviation and the solid lines the OLS fit to second order

polynomials. Parameter values are N=5000, 104 realisations, 2*105 networks considered,

⟨ k=4 ⟩ ,rewiring probability 0.05 per link(C≃0.43 ) ,correlation coefficient among opinions

of linked agents

0.68.

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Fig. 6 Probability that an agent's neighbour has the same preference as the agent, as a

function of the local clustering coefficient of the agent. Error bars correspond one

standard deviation, obtained averaging over 1.6*10^7 networks. Other parameter values

as in Fig. 5.

4. Conclusions

Fowler’s small world model of turnout has important implications for our

understanding of direct and indirect mobilisation, and on the cost effectiveness of Get-

Out-The-Vote drives (Green and Gerber, 2004). Its message is simple and powerful: if

you can mobilise one elector to vote, there are likely to be spill-over effects due to

processes of social conformity and imitation (Huckfeldt and Sprague, 1995; Sinclair,

2012). In the FSWM one additional voter leads to a total increase in turnout of up to

four votes due to indirect network influences, leading to the concept of turnout

‘cascades’. However, empirical studies of campaign effects in general and GOTV spill-

over effects more specifically (Nickerson 2008; Sinclair, 2012) suggest more modest

effects. Using agent based simulation and mathematical modelling of the FSWM we have

demonstrated a number of important reasons why the estimates of the cascade effects

may be misleading.

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First, we show that as the number of unconditional voters (or non-voters) increases, the

size of the cascade is reduced. While the model with no zealots replicated Fowler’s

estimate of four additional votes for the introduction of one unconditional voter, this

was reduced to 3.2 when 20% of electors were zealots and to 1.8 when 50% were

zealots. Although the reduction is reasonably modest – and the multiplier effect still

quite high – this does demonstrate that the size of the effect is constrained by the

proportion of electors which are subject to influence. In fact the size of the cascade is

dependent on two factors: the product of the imitation rate, the average number of

acquaintances and the number of iterations (or time elapsed) on the one hand, and the

proportion of zealots on the other.

This highlights a conceptual shortcoming of the FSWM. If all other electors are

conditional voters, why should the target voter be a zealot at all? In a GOTV scenario,

any one directly mobilised must by definition be a conditional voter and should

therefore be subject to influence by other citizens. When there are no zealots, if we

initialise the target agent as a voter, but allow her to be influenced by others (that is she

is a conditional voter) then the average size of the turnout cascade (taking into account

the target agent himself) is 1.00. If the number of zealots among the rest of the

population is greater than zero, the average size of the turnout cascade is, actually,

smaller than unity. In other words, there is no multiplier when the one additional voter

can be influenced by the rest of the electorate. It seems only reasonable that if the

experimental subject was persuadable by a political party (as his hypothetical

conversion through direct mobilization implies) then she could potentially be

influenced by other voters not to vote. While voter malleability may be non-symmetrical

– it may well be that conditional voters may be easier to persuade to vote than to

abstain (Rolfe 2012) – it is certainly plausible, indeed probable, that voters may be

influenced not to vote just as they may be persuaded to turn-out (Partheymüller and

Schmitt-Beck, 2013). In the symmetrical case analysed here, the indirect mobilisation

effect completely vanishes. In the general case, as long as mobilised agents can be

persuaded to abstain by social influence (even if with a smaller probability than the

general population), the multiplier effect will be smaller than the one obtained by

Fowler. A more systematic study of the effect of non-symmetrical malleability is left to

future studies.

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A second important finding derived from the FSWM is that average path length and

clustering coefficient of the underlying social network have a relatively large impact on

the expected size of the turnout cascades. However, our analysis shows that this is not

the case since, once the average degree is controlled for, the dependence of the size of

the cascade on the average path length or on the clustering coefficient disappears. This

singles out average degree as the most important network characteristic, and suggests

that other network properties can have a smaller importance in promoting social

influence than previously thought.

The third important finding of the FWSM, and the one with the most important real-

world implications, relates to the net-favourable effect of the turnout cascade on votes

received by rival candidates. Fowler finds a curvilinear relation between the local

clustering coefficient of an agent and the size of the net favourable turnout change that

is generated. He interprets this as a network effect, suggesting that as networks become

more clustered this initially has a positive effect on the size of the cascade, as the

opportunities for the initial message to be passed on increases; however as the network

becomes very clustered the opportunities for second order effects become saturated, as

everyone within a group talks to each other, but are spending less time talking to people

in the rest of the network. We, however, rule out this explanation by showing that the

total turnout induced by an agent actually decreases with the local clustering of this

agent, and argue that the effect in net favourable turnout is just due to the fact that, in

the model, agents with larger clustering tend to be connected to agents with similar

opinion to theirs.

Although we cast doubt on the likely magnitude of turnout cascades and on the specifics

of how they depend on network structure, we acknowledge the importance of the

underlying message in Fowler's (2005) article. It is absolutely crucial to recognise the

social dimension of voting in order to appreciate the true extent of the power of voter

mobilization (Sinclair, 2012) and how the dynamics of turnout are the product of a

complex system characterised by social influence and feedback effects (Franklin, 2004).

However, it is also important to remember that while highly abstract and simple agent

based models are excellent tools for elaborate thought experiments, over simplification

19

can lead to misleading results (Edmonds and Moss, 2005). As shown here, a number of

apparently interesting insights, for example about the impact of network structure on

indirect mobilisation, may be artefacts of more mundane assumptions, in this case

about the size of networks and how open to influence people are. The curious absence

of turnout cascades in the real word might therefore be explained by two observations.

First, voters in a real and complex world are subject to a huge number of cross-

pressures that are not full captured in the small world model of turnout (Anon,

forthcoming). Second, notwithstanding this, people may follow different decision rules:

some may be conditional voters; some may be unconditional voters or abstainers (Rolfe,

2012). If we start from the premise that ‘one-extra-voter’ can be influenced just like

everyone else, then the turnout cascade in the FSWM melts away to nothing.

20

5. Technical appendix

We provide here a derivation of the key mathematical results used in the main text.

We start by defining a set of variables{si }i=1N that correspond to the state of the

population, with si=1 if citizen i has the intention to vote and si=0 if she has the intention

to abstain. We initially consider the process in which all the citizens, including the ego,

are subjected to social influence. According to the definition of the process given in the

main text,S={s i }i=1N is a Markov chain that evolves according to:

Pr ( S , τ n+1 )=∑i=1

N

Pr (Si' , τn ) wi (S i

' )+Pr ( S , τn )[1−∑i

N

wi (S ) ] , [ 3 ]

where Pr stands for probability, τn labels the time point after iteration n,w i ( S )is the

probability that citizen i changes her intention (by coping one on her neighbours) in a

given iteration when the state of the population is S, andSi'is the state that differs from S

only in the intention of citizen i, formally:

S i'={s j

' ∨s j' =s j ∀ j ≠i ;si

'=(1− si ) },

w i ( S )=qk i

Nk ∑l=1

N Ai , l

k i[ si (1− sl )+(1− si ) sl ] , [4 ]

with ki the degree (number of acquaintances) of citizen i, and Ai,j the connectivity matrix

of the network (Ai,j=1 if there is a link between i and j, Ai,j=0 otherwise). In [4] qki/Nk is

the probability that citizen i copies the intention of one of her acquaintances, Ai,l/ki is the

probability that i copies citizen l, and si(1-sl)+(1-si)sl is the probability that the voting

intention of i actually changes in that event.

Multiplying [3] by sj and summing over all possible values of S, we obtain the evolution

equation of the expected value of sj:

E [s j , τn+1 ]=∑S

s j Pr (S , τn+1 )=∑S

s j Pr ( S , τ n)+∑S

s j∑i

[wi (S i' ) Pr (S i

' , τ n)− w i ( S ) Pr ( S , τn ) ]

Noting that for any function of S, f(S),if∑S

s j f (Si' )=∑

Ss j f (S ) i≠j and

∑S

s j f (S j' )=∑

S(1− s j ) f ( S ) ,we obtain:

E [s j , τn+1 ]=E [ s j , τn ]+∑S

( 1− 2 s j ) w j ( S ) Pr ( S ) .

Taking into account that(1 −2 s j ) [ s j (1 − s l )+ (1− s j ) sl ]=s l− s j ,(since sj2=sj), we obtain:

E [s j , τn+1 ]=E [ s j , τn ]+ qNk ∑l

A j ,l {E [sl , τn ]− E [s j , τn ] } [ 5 ]

21

Using that∑j ,l

A j ,l f ( sl )=∑i

k i f (s i ) ,∑j ,l

A j , l f ( s j )=∑j

k j f (s j ) ,which holds for undirected

networks (as the ones considered in FSWM), we see that [5] leads to:

E [∑js j , τ n+1]=E [∑j

s j , τ n],that is, when there are no zealots, the expected number of citizens voting is constant

under the dynamics proposed by Fowler (Suchecki et al. 2004).

When some number of zealots is present, the sums over i in [3] should only run over

non-zealot citizens (since zealots have a constant turnout intention) and the expected

number of voters is not conserved any more. Still, we can derive the evolution of the

expected number of voters using the following approximation. We first substitute Ai,l in

[4] by the probability of i and l being connected in a random network, with the same

degree sequence, Ai , l≈k i k l

kN.Since the considered network is of the Watts-Strogatz type

which has low degree heterogeneity, we can further assume that all the nodes have the

same degree,k i ≈ k(for other types of networks with higher degree heterogeneity, other

methods exists, see for example (Gleeson 2013)). With these approximations, the

evolution equation of the expected intention of citizen i is:

E [si , τn+1 ]=E [si , τn ]+ qkN [∑l

E [ sl , τn ]N

− E [s i , τn ] ]By summing the previous expression over all non-zealot citizens, we obtain the

evolution equation for the expected number of non-zealot voters:

V (τn+1 )=V ( τn )+ qkN

(1− z )V z − zV (τ n) , [ 6 ]

withV (τ ) ≡ E [ ∑i∈N nz

si , τ ] ,V z ≡ ∑α∈N z

sα , z≡N zealots

N,and Nz and Nnz denoting the set of all

zealot, and non-zealot citizens, respectively. The solution of the recurrence relation [6]

is:

V (τn )=V (τ0 )(1− qkzN )

n

+(1 − z )V z

z [1−(1− qkzN )

n]We can now compute the expected size of the turnout cascade, that is, the difference

between the expected number of voters when the ego is an unconditional voter (one

extra zealot in the 1 state) and the expected number of voters when the ego is an

unconditional non-voter (one extra zealot in the 0 state):

22

E [ ΔT ]=V z ( z1+1/ N , z0 )+V ( τ DN , z1+1/ N , z0 ) −V z ( z1, z0+1/N )− V ( τ DN , z1 , z0+1/ N ) .

After n=ND iterations, using that for large N(1− qkzN )

DN

≈ e− qkzD ,we finally obtain:

E [ ΔT ]=1+ 1− z−1 /Nz+1/ N

(1− e−qk ( z+1 /N ) D ) ,

formula [1] of the main text.

To compute the first order influence of the ego, that is, the number of citizens whose

preference was set by direct interaction with the ego, we note that the probability, p,

that citizen i has her preference set by interaction with the ego, provided that she is a

non-zealot acquaintance of the ego, evolves as:

p (τn+1 )=p (τ n )(1−qk i

Nk i− 1

k i)+[1 − p (τn ) ]

qk i

N1k i

, [ 7 ]

whereqk i

Nis the probability that citizen i copies the intention of one of her

acquaintances, and1k i

,k i− 1

k iare the probabilities that i chooses to copy the ego, and an

acquaintance different to the ego, respectively. The solution of [7] at n=ND with initial

condition p (t 0 )=0is 1k i [1−(1−

qk i

N )ND ]≈ 1

k i(1−e− qki D ) ,the equality becoming exact in the

large N limit. The expected number of non-zealot acquaintances of the ego is

(1− NzN − 1 )kego .Since, for large N, the ego's acquaintances evolve independently

(regarding whether the ego was the last citizen to influence them), we finally obtain:

E [ ΔT 1 ]=1− z−1 /N1−1 /N ⟨ kego

k i

(1− e−qk i D ) ⟩ ,

formula [2] of main the text. For networks with low degree heterogeneity, like the

Watts-Strogatz model, the formula is approximately equal to the simpler expression:

E [ ΔT 1 ]=1− z−1 /N1−1 /N

(1− e− qkD ) .

6. Funding

23

This work was supported by the Engineering and Physical Sciences Research Council

[grant number EP/H02171X].

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