Innovation diffusion in networks:
the microeconomics of percolation∗
Paolo Zeppinia,b,c Koen Frenkena
November 10, 2013
a
b
School of Innovation Sciences, Eindhoven University of Technology, The Netherlands
CeNDEF, Amsterdam School of Economics, University of Amsterdam, The Netherlands
c
Istituto di Economia, Scuola Superiore Sant’Anna, Pisa, Italy
Key words: connectivity, demand, market efficiency, phase transition, small-worlds.
JEL classification: C63; D42; O33.
Abstract
We implement a diffusion model for an innovative product in a market characterized by a structure of social relationships. Diffusion is described as a percolation process, based on the economic hypothesis of heterogeneous reservation prices.
There is a threshold value for the innovation price that defines a phase transition
from a diffusion to a no-diffusion regime. This has strong implications for market
demand. We study percolation in Small-worlds networks, and our results challenge
the common hypothesis that these structures facilitate diffusion. We show that a
percolation mechanisms benefits from low clustering rather than low average path
length. Connectivity scaling behaviour is the driving factor of diffusion. Hence,
social structures with low clustering (individualistic society) are most beneficial for
innovation diffusion. Moreover, percolation experiments with non-uniform distributions of reservation prices show that not just “richer” societies, but also more equal
societies favour diffusion in networks of consumers.
1
Introduction
The success of an innovative technology and the market penetration of a new product
largely depend on the diffusion process. Seminal works on innovation diffusion (Griliches,
1957; Mansfield, 1961; Bass, 1969; Davies, 1979) have identified a number of factors that
∗
We are grateful to Luis Izquierdo and Fernando Vega-Redondo for their help and comments. We acknowledge feedback from the following conferences: ECCS 2013 in Barcelona, WEHIA 2013 in Reykjavik,
11th Workshop on Networks in Utrecht, 4th CompleNet Workshop in Berlin, LATSIS 2012 Symposium
and ABMCTS 2012 conference at ETH Zurich, and seminars participants at the College of Management
of EPFL Lausanne, and the Economic Colloquia of the University of Amsterdam.
drive and sometimes delay the establishment of new technologies, as consumer heterogeneity, imperfect information, sunk costs of adoption. Yet, the role of market network
structure in the diffusion of innovations is a rather recent research topic (Banerjee et al.,
2012), and can play a fundamental role in the success or failure of innovations. This is
particularly relevant to the problem of more efficient innovations that do not diffuse in
face of less desirable incumbent technologies.
In this article we study innovation diffusion in networks with a percolation model. Percolation models has been successfully applied to social dynamics (Solomon et al., 2000)
and epidemiology (Davis et al., 2008). We claim that percolation is a very ‘economic’
model in that it combines the contagion mechanism of information diffusion (Bass, 1969)
with the economic approach of rank models (Stoneman, 2002). The diffusion process for an
innovative product strongly affects the demand function whenever consumers are embedded in a network, with a phase transition separating a “diffusion” from a “non-diffusion”
regime. Campbell (2013) is a first economics study of the implications of percolative
diffusion on market demand. With our model in part we complement Campbell’s study
with the market efficiency aspect of percolation, and then we focus on the role played by
the network structure, the role of the role of the distribution of consumers’ reservation
prices (the demand shape) and the time dimension of diffusion beside the diffusion size.
The principal type of network structure in our study is the Small-world network (Watts
and Strogatz, 1998). The common wisdom is that Small-worlds favour diffusion processes - in particular innovation diffusion - thanks to “long distance” short-cuts (REFERENCES...). We claim that this is not true in general, and very much depend on the
“infection” or adoption mechanism of the diffusion process. In particular, for a percolative
diffusion process Small-worlds perform well in terms of diffusion time, but very poorly
in terms of diffusion size. The main point for understanding percolation in small-worlds
is how relevant properties such as the average “network accessibility” and the average
“path-length” scale with the density of short-cuts. We find that accessibility, which is the
“opposite” of clustering, is the main driver of diffusion size, while the average path-length
is the main factor behind diffusion time.
In percolation models the usual assumption is a uniform distribution of consumers’
reservation prices, which gives a linear market demand. We show that non-uniform distributions strongly affect the percolation process. In particular, a decreasing probability
distribution gives a smaller diffusion phase. Moreover, the “lost demand” due to percolation is larger for this such distribution with respect to a uniform distribution. This result
is empirically relevant, since reservation prices are likely to be correlated with consumers’
2
income, which is known has a negative power law distribution (REFERENCES...). A
percolative diffusion process is favoured not only by “richer consumers”, but also by a
more equal wealth distribution.
The rest of the article is organized as follows. Section 2 studies the effects of percolation
on market demand. Section 3 looks at percolation in Small-world networks. Section
4 analyses the structural factors of diffusion. Section 5 addresses non-linear demand
curves, with a non-uniform distribution of consumers’ reservation prices. Section 6 studies
percolation in scale-free networks. Section 7 concludes.
2
Percolation and market demand
Let’s consider an innovative product and a network of N potential consumers, where i
and j are friends if there is a link ηi,j connecting them. Links are either existing (ηi,j = 1)
or absent (ηi,j = 0), and do not depend on time. The diffusion process starts with a small
number n << N of initial adopters (seeds). Information about the innovative product is
local: consumers that are not initial adopters know about the innovation only if a neighbour adopts. The adoption decision is based on the innovation price p, that we assume to
be defined in the interval [0, 1], and that is assigned before diffusion starts. Consumers’
preferences are expressed by a reservation price pi ∼ U [0, 1]: only consumer with pi > p
are willing-to buy, and can adopt the innovation (Fig. 1, central panel). A consumer not
Figure 1: Percolation in a network of consumers considering whether to buy an innovative product with price
p ∈ [0, 1]. Left: original network. Centre: after drawing reservation prices pi ∼ U [0, 1], only consumers with pi > p
are willing-to-buy (white nodes), while the others are not (filled nodes). Right: consumers that are not willing to buy
are removed from the networks, and their links canceled. The resulting giant connected component is enlightened
with the dashed line.
adopting does not pass on information to her neighbours. Adoption decisions and information diffusion influence each other. As a consequence, the connections network that
matters for diffusion is the one that results from individual reservation prices. Drawing
3
consumers reservation prices amounts to randomly switch ‘off’ nodes (Fig. 1, right panel).
We call the resulting network of active nodes the operational network. Diffusion will have
a sizeable extent only if a giant connected component exists in the operational network.
What is important to notice is that drawing reservation price has a highly non-linear effect on diffusion. In the example of Fig. 1 we have 30 consumers, and all but one happen
to be connected initially (Fig. 1, left panel). An innovation price p = 0.5 means that on
average 50% of consumers are willing-to buy, that is 15 consumers. But unwilling-to-buy
consumers are “removed”, and the resulting network is made of a number of connected
components, the largest being formed by only 7 consumers. In the best case, when we
have an initial adopters who belongs to this component, diffusion size will be only seven,
which is less then half the potential diffusion size.
A uniform distribution of consumers’ reservation prices would result in a linear demand
curve if all were informed about the innovation, or if they would be some how fully
connected. When p is the price of the innovative product, the probability that a consumer
is willing-to-buy is q = 1 − p, and the expected number of adopters is N (1 − p). This
is the classical downward sloping linear demand curve of textbook microeconomics. The
local effects of information diffusion based on a percolation mechanism makes the demand
depart from this linear shape. In particular, the social network structure of information
flow introduces a phase transition, which strongly affects the demand at high prices. We
illustrate this point by simulating the percolation model for the case where consumers
form a Poisson random network (Erdös and Rényi, 1960). The left panel of Fig. 2
reports the final number of adopters for different values of the innovation price. The
percolation process is initiated by 10 seeds (early adopters).1 At low prices, the number
of adopters follows the linear behaviour dictated by a uniform distribution of reservation
prices. This amounts to full diffusion of the innovation, meaning that all consumers with
a reservation price equal or above the actual price adopt the product. But already at
values as low as p = 0.4 the diffusion size starts to be lower, and drops down to almost
zero above p = 0.7. In this scenario innovation diffusion is not full, because information
does not spread efficiently in the network. As the price grows larger, an ever increasing
number of consumers that would be willing-to-buy the innovation based on its price do
not actually buy it, because they do not get to know about its existence. There are
1
In all simulations presented in this article we maintain this setting: a network of 10000 nodes with
10 seeds. The diffusion size depends on the number of seeds, but not the critical transition threshold.
We have run simulations with 100 seeds, and obtained very similar patterns. With less than 10 seeds the
variability of results is too large, with a standard deviation that reaches 100%.
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Figure 2: Percolation in a Poisson random network with N = 10000 nodes (consumers), average connectivity 4,
and 10 seeds (early adopters). Left: number of adopters as a function of the product price. Left: adoption values
reported as a demand curve. Values are averages over 50 simulation runs. The standard deviation is largest at the
percolation threshold (25%).
two different regimes then: a diffusion regime, where information spreads efficiently and
diffusion is full. A no-diffusion regime, where information does not spread, and diffusion
is not full. By increasing the innovation price the consumers’ network undergoes what in
Physics literature is called a percolation phase transition (Stauffer and Aharony, 1994).
A threshold value of the price separates the two regimes (phases). This is the value that
marks a fundamental change in the structure of the system, namely the value below which
we see the appearance of a giant connected component of consumers that are willing-tobuy (see Fig. 1), a giant-connected-component in the operational network defined by the
price p. The size of connected components in a percolation process is highly non-linear,
with a sharp change at the percolation threshold.
The percolation threshold is a mathematical property of a network, which can be
computed, at least numerically. A powerful approach is based on Generating Functions,
introduced by Newman et al. (2001) for the analysis of several characteristics of random
networks. Callaway et al. (2000) apply this formalism to percolation. The case of a
Poisson network allows to evaluate analytically the percolation threshold. In the price
space dimension of innovation diffusion, the percolation threshold is
pth =
hki − 2
,
hki − 1
(1)
where the random number k refers to the connectivity of nodes. In the example of Fig. 2
the average connectivity is hki = 4, so the percolation threshold is pc ' 0.67.
Percolation with its phase transition has important effects on the microeconomics of
innovation diffusion. The demand function that results from percolation can be compared
5
to the benchmark case of a linear demand function (Fig. 2, right panel). There are two
important effects. The first effect is that in general the market price is lower with respect
to a linear demand. The second effect is an instance of market inefficiency, that is not all
potential consumers actually buy the innovation. We will first explain the price reduction
effect, and then consider the market efficiency effect.
The type of market organization influences the effects of percolation on demand. In a
scenario of perfect competition there is almost no price reduction effect when the market
clears at a price below the threshold, where actual demand and potential (linear) demand
coincide. In the non-diffusion regime, when marginal production costs are relatively high
and the market price is above the threshold, also perfect competition may present a
lower price than a linear demand would have, because the actual demand resulting from
percolation departs from the potential demand (Fig. 2, left panel).
The effect of percolation is stronger in a monopoly scenario. A monopolist can charge
a higher price than in perfect competition. Microeconomics theory predicts a price markup that is inversely proportional to the elasticity of demand (Tirole, 1988). In the left
panel of Fig. 2 a phase transition makes demand more elastic below the threshold price,
which translates in a smaller mark-up (Campbell, 2013). The monopolist finds it harder
to increase the price because of the lower demand in the non diffusion regime. The exact
amount of price reduction from percolation depends on the production costs structure.
However, even for very low marginal production costs the price mark-up of a monopolist
is limited from above due to the phase transition of demand. Of course, the more the monopolist attempts to enter the non-diffusion regime, the larger will be the price reduction
with respect to the mark-up that can be charged with the potential (linear) demand.
The second effect of percolation is an instance of market inefficiency, a market failure
due to inefficient information diffusion, which counters the positive effect of a lower price
on consumers’ welfare. The market inefficiency comes from “missing” buyers, that is
consumers with a relatively high reservation price who do not buy the innovative product
because they are not informed. The social network structure is not able to convey full
information about the product, and not all consumers that would buy the product are in
the possibility to buy it.
The analysis above can be resumed in terms of consumers welfare. A lower price increases consumer welfare, while a lower demand reduces it. Without making assumptions
on the market structure from the supply side, we can always assume that market is in a
Walrasian equilibrium, and consider the price where supply equals demand. If such price
is below the threshold price, the effect of percolation on welfare is quite simple, because
6
the actual demand almost coincides with the potential demand: welfare is reduced by
percolation, because there is only the effect of market inefficiency, while the price level
is unaffected (Fig. 3, left panel). In the non-diffusion regime the effect of percolation on
1
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Figure 3: Intuitive picture of market equilibrium in a network of consumers where an innovation diffuses according
to a percolation process. The line with diamonds is the simulated demand for a Poisson random network. The
dashed line corresponds to the potential linear demand. Left: market equilibrium in the diffusion regime. Right:
market equilibrium in the non-diffusion regime.
consumers welfare is ambiguous, instead (Fig. 3, right panel). Here the lower price of
a reduced demand adds to consumer welfare, but this is lowered by market inefficiency.
Above the threshold there are ‘missing’ buyers which are not reached by information
about the innovative product, and do not count for consumer welfare.
Concluding, the market inefficiency in percolative diffusion is the prevailing effect at
low prices, in the percolation regime, when the consumer welfare is always lower than
in the case of perfect information. At high prices, the market inefficiency effect and the
price effect are comparable, but in the best case they almost offset each other. A scenario
where a percolation regime is preferable in terms of consumers welfare seems unlikely.
3
Innovation diffusion in small-worlds
One of the most popular models of social networks is the Small-worlds model introduced
by Watts and Strogatz (1998). Several empirical studies have identified small-world properties in real world social and physical networks,2 with possibly the most notorious one
being the six degrees of separation: it takes on average six steps to reach any individual
in the world. This fact is actually a manifestation of a well defined mathematical prop2
A review of studies on the statistical properties of real world networks is Albert and Barabasi (2002).
7
erty, which is a relatively short average path-length. The small-worlds network model is
constructed starting with a regular one-dimensional lattice, and introducing a rewiring
probability µ based on which any link can be re-wired. Fig. 4 shows examples with
N = 50 nodes and degree 4 (the total number of links is 4 × 50/2 = 100). In the middle
panel there is a small-world network where eleven links have been rewired (the rewiring
probability was µ = 0.1). The limit case µ = 1 is a network where all links have been
Figure 4: Small-worlds construction process: example with 50 nodes and degree 4. Left: regular lattice (µ = 0).
Centre: Small-world with µ = 0.1. Right: Poisson network (µ = 1).
rewired (Fig. 4, right panel), a procedure that leads to a fully random Poisson network
of the type introduced by Erdös and Rényi (1960). In this network the connectivity of
nodes follows a Poisson distribution. Also Poisson networks are characterized by a relatively short average path-length with respect to other network structures and in particular
with respect to the starting regular lattice of Fig. 4, left panel. What makes the smallworld network so interesting is that it has a short average path-length comparable with
the one of a Poisson network while preserving another character of the original lattice,
which is a high level of clustering. In Fig. 5 we report the simulation results of Watts
and Strogatz (1998).
1
C(p) / C(0)
0.8
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0
0.0001
L(p) / L(0)
0.001
0.01
0.1
1
p
Figure 5: Clustering coefficient C(p) and average path-length L(p) as a function of the rewiring probability (horizontal
axis) in Small-World networks (Watts and Strogatz, 1998).
8
The clustering coefficient measures the relative number of triplets out of all possible
triplets in a network (Wasserman and Faust, 1994). We may notice how this fraction is
particularly large in the starting regular lattice where each node has two neighbours on
each side, while it is almost zero in the Poisson network. What makes the small-world
network model so interesting is that by rewiring links at random, the average path-length
drops suddenly already for small values of the rewiring probability, while the clustering
coefficient remains practically unaltered until one rewires a large portion of links. This
is evident in Fig. 5, where the typical small-world setting results the one obtained with
as few as 1% of links rewired, with an average path-length which is almost the same of
a Poisson network, but a clustering coefficient which is still very large and comparable
to the original lattice. It is important to notice that while the rewiring process strongly
affects the degree distribution, the average degree remains unchanged, since the numbers
of nodes and links are fixed. In the examples of Fig. 4 the average degree is 4.3
The two limits represented by the regular lattice and the Poisson network have a
clear social interpretation: the first one describes a “collectivist” society, where clustering
is high but distances between individuals are long. The fully random network instead
corresponds to an “individualistic” society, with low clustering but short distances. Real
world social networks are likely to score somewhere in between, and this is what makes
the small-world model so relevant. Real social networks may be closer to one or the other
limit of the model, depending of the specific case considered. For instance, less developed
countries are considered to be characterized by a collectivist society, while individualistic
societies are typical of industrialized countries (Fogli and Veldkamp, 2012).
We have simulated percolation in a number of different small-world networks. Fig. 6
reports the results in terms of final number of adopters as average values over 20 different
simulation runs. We observe that a Poisson network represents the best scenario, with
largest diffusion size at every price value, as expected. The percolation threshold in a
small-world is much lower than in a Poisson network, and the non-diffusion regime is
much larger: a lower price is required for innovation to spread. For a typical small-world
with 1% of short-cut links, the price threshold is between 0.2 and 0.3. In the limit case
of a regular one-dimensional lattice, the threshold is between 0.1 and 0.2.
The results of Fig. 6 are in agreement with theory. The percolation threshold is defined
as the critical value of nodes’ activation probability where a phase transition occurs in the
3
The degree distribution of a Poisson random network is p(k) =
1 −z k
z ,
ke
where k is the degree, and
the parameter z is the average degree. Each possible link has a probability q such that, given the total
number of nodes N , the average degree z = qN is constant (Vega-Redondo, 2007).
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Figure 6: Percolation in different small-world networks: final diffusion size (number of adopters) with 10 initial
adopters. Values are averages over 20 runs. Standard deviation is larger at the threshold, and largest for the Poisson
network (about 40%). All networks have 10000 nodes (consumers) and an average degree 4.
size of connected components of active nodes, the so-called percolating cluster. In Newman
and Watts (1999) it is shown how to evaluate implicitly the percolation threshold of smallworld networks.4 After adjusting this equation for our theoretical setting, we find that a
price percolation threshold pth in a small-world network must satisfy the equation:
p2th = 4µ(1 − pth ),
(2)
where µ is the rewiring probability. The theoretical price percolation threshold can be
evaluated just by specifying the rewiring probability µ. In Table 1 we report the values
obtained for the small-world networks of Fig. 6. The simulation results of Fig. 6 would
µ = 0 µ = 0.001 µ = 0.01 µ = 0.1 µ = 1
pth
0
0.06
0.18
0.46
0.83
Table 1: Price percolation threshold in small-world networks.
present a sharp discontinuity at these values in the case of an infinite network. Notice
also how the theoretical value obtained for the Poisson network (µ = 1) slightly differs
from the value computed in Section 2, which is the correct one (pth = 0.67). The results
in Newman and Watts (1999) are obtained with a slightly different model, where links
4
Equation (30) of Newman and Watts (1999) is defined on a probability space expressing the proba-
bility that a node is active. Our analysis is defined on a price space p ∈ [0, 1], where a node is active with
probability 1 − p. In our case the connectivity range of links for each node in the original regular lattice
is equal to 2, meaning 2 neighbours on either sides.
10
are randomly added and not rewired. This modification of the Watts and Strogatz (1998)
model is necessary to avoid analytical problems that originates from a strictly positive
probability that one node can remain unconnected, giving an infinite average path-length.
However, the two models converge as µ → 0, and in our analysis we will consider only
small values of µ, that is the typical range of small-worlds (Figure 5).
In terms of innovation market demand, small-world show a higher elasticity than poisson networks, and consequently the effects explained in Section 2 are even stronger, such
as a lower market price together with the reduction of demand due to market inefficiency.
The market inefficiency effect can be computed directly by measuring the loss in consumer
surplus. The latter is the difference between the maximum price that consumers are willing to pay, the reservation price, and the price they do actually pay. Consequently the
consumer surplus is the definite integral of the demand, and can be evaluated as the area
below the demand curve and above the market price. We have computed numerically this
definite integral by adding together the contributions of each single price value in the simulations, which amount to a numerical interpolation of the demand curve.5 Fig. 7 reports
this measure of consumer surplus as a function of the price. The market inefficiency effect
Consumer Surplus
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Figure 7: Consumer surplus as a function of the price in different network structures. Values are the numerical
integration of simulated demand curves reported in Fig. 6.
5
The definite integral of the demand is computed as follows. In the simulation of the model we have
considered twenty-one price values pi ∈ [0, 1] equally spaced by ∆p = 0.05. For each price value pi we
have a simulated value of the demand (diffusion size) n∗i . The contribution of each price value pi to the
integral is the addition of the area n∗i × ∆p and the area 21 (n∗i+1 − n∗i ) × ∆p. This approach to the definite
integration of the demand amounts to interpolate the demand itself with a piece-wise curve that is made
connecting all values (n∗i , n∗i+1 ).
11
measured by the loss in consumer surplus is considerably strong for small-world networks.
The reduction in consumer surplus experienced by these networks is even larger than the
reduction in demand (Fig. 6), because for a given price the surplus measures the overall
loss above such price, which in a sense amounts to an amplification of the effect on demand. In particular, the typical small world, with rewiring probability µ = 0.01 presents
a negligible value of the consumer surplus in the non-diffusion regime (at prices p > 0.2)
and a strong reduction in the diffusion regime.
The lost consumer surplus measured from the data of Fig. 7 is incomplete, since it only
includes the market efficiency effect, and not the price reduction effect. Given a demand
curve, market equilibrium occurs at different prices for different supply schemes. Fig. 8
intuitively compares the market equilibrium in a Poisson random network and in a smallworld with rewiring probability µ = 0.01. In the diffusion regime (left panel) it is evident
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Figure 8: Intuitive picture of market equilibrium for demand resulting from percolation in a Poisson random network
and in a small-world with rewiring probability µ = 0.01. Left: market equilibrium in the diffusion regime. Right:
market equilibrium in the non-diffusion regime.
that a Poisson random network outperforms the small-world in terms of consumer surplus:
the market inefficiency effect is by far more important than the price reduction effect. In
the non-diffusion regime, when supply is such that market equilibrium realizes above the
threshold price, the two effects are comparable, as we have seen in Section 2. In principle,
the consumer welfare can be larger in a small-world than in a Poisson random network.
In this case the lower information efficiency that characterizes innovation diffusion in a
small-world actually ‘protects’ consumers, avoiding them from being exploited for instance
by a greedy monopolist. However, the price reduction effect can be at most comparable
to the efficiency effect, but never substantially larger. Small-world structures are not an
efficient environment for innovation diffusion then, at least when a percolation mechanism
based on individual preferences is at work. In the next section we explain why this is so.
12
4
Scaling properties of percolation in small-worlds
A common hypothesis states that diffusion processes work well in small-worlds, and this
performance is considered a virtue of long-distance links (REFERENCES...). These are
short-cuts that connect far away regions in the network. The intuition is that shortcuts strongly reduce the average path length. Our study challenges the generality of
this hypothesis, showing how the effect of network characteristics on diffusion strongly
depends on the diffusion mechanism. In this section we show that whenever diffusion is
the outcome of combined information diffusion and adoption decision, the main factor
is not the average path-length, but clustering, so that lower clustering leads to larger
diffusion sizes. The reason behind this result is a dual property of clustering, which is the
scaling law of connectivity. Whenever clustering is low, the network happens to “spread”
relatively more, meaning that more neighbours are added at increasing distances of a
given node. This turns out to be the main driver of diffusion as long as a mechanism
based on individual adoption preferences is involved, and not just information contagion
or social pressure.
In percolation results reported by Figure 6 we observe a relatively low improvement
of diffusion size from rewiring. In particular, the typical Small-world with rewiring probability µ = 1% (Figure 5) has a percolation threshold only marginally larger than the
regular lattice. We claim that whenever diffusion is driven by a percolation mechanism,
the important factor is clustering, not average path-length, and clustering is detrimental.
In Fig. 9 we report the percolation diffusion size as a function of the rewiring probability
µ for an innovation price p = 0.5, which is well in the no-diffusion regime for the regular lattice (Fig. 6). There is a phase transition also in this space, occurring nearly at
µ = 0.1, that is exactly where the clustering coefficient drops down (Fig. 5). Results do
not show any feature at lower values of µ, where the average path-length undergoes its
phase transition.
The message from this analysis is the following: the average path-length is not the
crucial factor for percolative diffusion. Clustering is way more important, with a negative
effect. Low clustering turns into a topological “spreading” of the network, where the
number of neighbours increases with distance. In a random network the number zr of
neighbours at distance r is given by (Newman et al., 2001)
z2
zr =
z1
r−1
z1 .
(3)
Connectivity spreads whenever the second order neighbours are in larger number than
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Figure 9: Diffusion size in small-world networks as a function of the rewiring probability µ. The innovation price is
p = 0.5, and we start with 10 initial adopters (seeds). Values are averages over 20 simulation runs. The standard
deviation is largest at the threshold, being about 50% for µ = 0.1, 20% in the no-diffusion regime, and much smaller
in the diffusion regime.
direct neighbours, z2 > z1 . In general the number of second order neighbours depends
on the variance of the connectivity k, z2 = hk 2 i − hki (Vega-Redondo, 2007). Then
connectivity spreads in a network whenever hk 2 i > hki. For Poisson networks hk 2 i = hki2 ,
so this condition simplifies to hki > 1. Considering that in a Poisson network
z2
z1
= hki,
and the expected number of neighbours at distance r is hkir . In our model of innovation
diffusion we set the average connectivity to hki = 4, and we obtain for the following value
for the expected number of neighbours at distance r in a Poisson network:
zrP oisson = 4r .
(4)
This is to be compared with the regular one-dimensional lattice. Here the connectivity is
constant, k = 4, and also at distance r we have:
zrCircle = 4.
(5)
Small-world networks present non-trivial scaling properties for connectivity, and different
regimes within the bounds represented by Eq. (4) and Eq. (5). Newman and Watts
(1999) find two regimes.6 For a given density of rewired links µ, when two nodes are
close to each other on the circular reference system, their average distance scales linearly
with their relative coordinate and with the network size. When they are far apart and
for large networks, their average distance scales logarithmically. These two regimes are
6
As already pointed out in Section 3, we use analytical results obtained for the slightly different
small-world model where links are randomly added and not-rewired. The reason is purely technical,
since rewiring can lead to an infinite average path-length with positive probability, and for low values of
rewiring probability µ (the range of interest) the two models coincide.
14
separated by a characteristic distance ξ =
1
,
2µ
which is defined as the typical distance on
the circular reference system between the two ends of a short-cut (rewired link).7 Below
ξ the network presents characteristics close to the regular lattice, and above which it is
close to the Poisson network. The average number of neighbours at distance r is given by
the ‘surface’ A(r), which is defined as the number of nodes at distance r from any given
node. Adapting the computation of Newman and Watts (1999) to our model setting, we
find that in a small-world this number is given by A(r) = 4e8µr . The exponential scaling
of the surface holds when the distance is comparable with the characteristic distance ξ.
At short distances r << ξ, the surface scales linearly. Resuming, we can express the
number of neighbours at distance r in a small-world as follows:


4(1 + 4r/ξ) if r << ξ,
SW
zr =

4e4r/ξ
if r >> ξ.
(6)
When µ << 1, as in a typical small-world, the regime r << ξ is much more important. For
instance, with µ = 1% the characteristic distance is ξ = 50. This means that neighbours
of orders up to r = 50 scale only linearly, and very slowly (since the proportionality
coefficient is µ). This regime is the one that counts when diffusion works as percolation.
Small values of r are crucial, because the surface A(r) at short distances is more affected
by inactive nodes (those for which pi < p) than at long distances. Inactive nodes are
uniformly distributed, and while at long distances information can easily find a way
through the surface, at short distances it may stops easily after a few steps. Only when
the rewiring probability is large the surface enlarges fast enough to overcome this hurdle,
because the characteristic distance of the small-world is much reduced. For instance when
µ = 0.5 the surface starts to scale up exponentially already after ξ = 2 steps.
In order to understand the effect of the rewiring mechanism on percolation, let’s
consider a regular lattice with connectivity four, and see how the number of neighbours
at a given distance r (the surface of radius r) changes for a node with a rewired link.
Before rewiring, the node has four r-order neighbour. If one link is rewired, one of these
four neighbours is lost, and four new r-order neighbours are found. In Figure 10 we give
the example with r = 2. Assuming that in r steps we do not find another rewired link,
the surface of radius r increases from four to seven. In case of a mechanism of random
links addition instead of rewiring, the surface increases by a factor 2, from four to eight.
7
For the expression of the characteristic distance ξ and for the expression of the surface A(r), we
adapt equation (9) of Newman and Watts (1999) to our model setting, where the original lattice has a
connectivity range equal to 2, that is two neighbours on either sides.
15
Figure 10: The surface of radius r = 2 for a node with a rewired link. The surface is equal to 4 nodes before rewiring,
and equal to 7 after rewiring.
However, in both cases only the surfaces of the (few) nodes with rewired links increase,
. The surface of other nodes is unaffected.
which are µ kN
2
In general, in a small-world with initial connectivity k, the surface of radius r for a
node with a rewired link changes from k to 2k − 1 (or to 2k in case of link addition).
The relevant factor for percolation in a random network is its average accessibility, that is
the accessibility averaged across nodes. In a small-world with link rewiring (or addition)
probability µ the average surface of the sphere with radius r centered in any node is
(1 − µ)k + µ(2k − 1). The average volume, or average accessibility, is simply r[(1 −
µ)k + µ(2k − 1)]. Notice that in what we have written there is the assumption that we
do not find another rewired link within r steps. This assumption holds provided that
µ << 1 and that r << ξ. Table 2 summarizes these considerations both in case of
link rewiring and link addition. Considering that a small-world is obtained with very
low rewiring probability µ, the average number of neighbours at a distance r << ξ is k,
and the average volume is rk. From this analysis we derive the two following messages:
first, in terms of accessibility a small-world is not different from the initial one-dimensional
regular lattice, unless the rewiring probability µ is large. Second, the two different random
mechanism of small-world creation do not differ in terms of accessibility of a small-world,
provided that µ remains low. To conclude, since percolation is primarily driven by the
average accessibility of the network, and that accessibility of the first few steps is primarily
important, due to shut-down nodes, the diffusion size in a small-world is not substantially
larger than in the original one-dimensional lattice. Short-cuts do not add substantially to
16
rewiring
adding
r-surface for “normal” nodes
k
k
r-surface for “affected” nodes
2k − 1
2k
average r-surface for any node
(1 − µ)k + µ(2k − 1)
(1 − µ)k + 2µk
average r-volume for any node r[(1 − µ)k + µ(2k − 1)] r[(1 − µ)k + 2µk]
Table 2: Number of neighbours at distance r (surface) and within distance r for nodes interested (affected) or non
interested (normal) by the random rewiring or addition mechanism in a small-world with connectivity k and link
rewiring or addition probability µ (assuming r << ξ =
1
).
2µ
the diffusion size, which remains limited by the low dimensionality of the original lattice.
The average path-length l presents a completely different scaling, instead. When the
rewiring probability µ is low it goes down linearly as
l
N
= 14 − 21 µN + O(µ2 ). When µ = 0
we have the average path-length of a circle l = N/4, while for large values of the rewiring
probability µ the approximation above does not hold anymore, and we have l ' log N
instead, as in a Poisson network. This means that l goes down at sustained rate already
for very small values of µ, reaching the scaling of a Poisson network at relatively low
values of the rewiring probability.
A complementary measure of network connectivity is the average accessibility of a
network. The accessibility of a node is defined as the total number of nodes that can
be reached with a given number r of steps, that is the volume V (r) of the sphere with
radius r. A realization of a random network is characterized by a distribution of different
accessibility values across nodes. In order to get an idea of the accessibility in smallworlds, let us consider the average accessibility of one network realization. The left panel
of Fig. 11 shows the dependence of average accessibility on the rewiring probability µ at
different distance values r. We notice the striking match with the simulation results for
the diffusion size in Fig. 9, with the most meaningful measure being the accessibility of
nodes within distance r = 5. This good match demonstrates that connectivity scaling
behaviour, that is the accessibility of a network, is the true determinant factor of diffusion
as percolation. The right panel of Fig. 11 shows how average accessibility depends on
the distance r, for different small-world networks. Here we see that accessibility scales
exponentially in a Poisson network, in agreement with Eq. (4). Notice that at distance
values r > 5 this is not true anymore, due to the finiteness of the networks considered
(10000 nodes). In small world networks accessibility scales linearly, at least for short
distances. All this is in agreement with the two scaling regimes explained by Eq. (6). For
17
800
1000
r=1
circle
r=2
SW 0.001
600
r=3
SW 0.01
500
r=4
400
r=5
average accessibility
average accessibility
700
300
200
SW 0.1
100
Poisson
50
10
40
30
20
100
0
0,001
10
0
0
1
0,01
µ
0,1
0
1
1
2
3
r
4
1
2
5
3
4
5
6
7
6
8
7
Figure 11: Average nodes accessibility of small-worlds (10000 nodes, average connectivity 4). Left: average nodes
accessibility as a function of the rewiring probability µ (horizontal axis, logarithmic scale), for different distance
values r. Right: average nodes accessibility (logarithmic scale) as a function of distance r, for different small-world
networks (the onset graph reports the same data in linear scale only for µ = 0, 0.001, 0.01 ).
a Poisson network (µ = 1) the characteristic distance is ξ = 1/2, so that all (discrete)
distance values r are larger and the scaling is always exponential. In a small-world with
µ = 0.1 we have ξ = 5, and already after the first steps we exit the linear scaling behaviour.
For a small-world with µ = 0.01 the characteristic distance is ξ = 50, so we have linear
scaling as long as r << ξ/4 = 12.5. This is what we see in the onset of the left panel in
Fig. 11 for distance values up to r = 4, while already at r = 5 the scaling slightly departs
from a perfect linear behaviour. In the case of a small-world with rewiring probability
µ = 0.001, we have ξ = 500, so that for the values considered we are always in a regime
where r << ξ/4 = 125, and scaling is perfectly linear, as predicted by Eq. (6).
Usually a low average path-length and connectivity spreading are associated Albert
and Barabasi (2002). But in small-world networks this is not the case. In such networks
topological spreading is much more correlated to clustering, so as to have a number
of neighbours that spreads largely only when clustering is low, while high clustering
corresponds to moderate connectivity spreading. The reason is that links that are used
to form the triplets of clusters are not used to ‘spread’ to longer distances in the network.
In different words, many links in a small-world with high clustering are redundant, and
this redundancy hampers the percolation process. When diffusion works as percolation,
a consumer does not need to know the information about the innovative product more
than one time, and consequently clustering is detrimental, while its opposite, connectivity
spreading, is the true important factor of diffusion.
This result is in line with empirical evidence on technology diffusion Fogli and Veldkamp (2012), and against results on behaviour diffusion in online social networks exper-
18
iments Centola (2010). This suggests that a percolation model may be better suited for
technological innovation diffusion than for behaviour diffusion.
A final remark: random network may not present full diffusion at zero price, and for
positive but very low price values regular lattices show larger diffusion sizes. The reason
is that random networks may have unconnected component, and unless there is an early
adopter in those components, consumers will never buy.
So far we have been concerned with the diffusion size of the innovative product in
the consumers network. Another important aspect of innovation diffusion is the diffusion
time. In what follows with diffusion time we mean the time it takes for the diffusion
process to stop. This is the time required to cover all connected components of the
operational network that contain a seed. Figure 12 reports the diffusion time for different
innovation prices obtained for the simulations already analysed in Figure 6. The results
700
600
circle
SW 0.001
time
500
SW 0.01
400
SW 0.1
300
random
200
100
0
0
0.1
0.2
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
Figure 12: Diffusion time (steps) for percolation in small-world networks with 10000 nodes (consumers) and average
degree 4. Values are averages over 20 runs. The standard deviation is larger at the percolation threshold (30%).
present the typical pattern of second order phase transitions, with a time peak at the
percolation threshold. Because of this, the diffusion time is an unambiguous way to
locate the percolation phase transition.
In order to compare the percolation diffusion time in different networks we can look
at two price values, p = 0 and the percolation threshold p = pth . The latter is a kind
of “worst condition” for the diffusion time, being the maximum time it would take to
complete the adoption process for the network considered. On the contrary, p = 0 is a
“best condition”, where all the network is accessible (apart from unconnected components
without a seed). Price values above the percolation threshold do not count much, since
the diffusion time is low due a small diffusion size.
19
The diffusion time at p = 0 and at the percolation threshold from the simulations of
Figure 12 are reported in the following table and in Figure 13 The diffusion time of a
diffusion time
µ = 0 µ = 0.001 µ = 0.01 µ = 0.1 µ = 1
p=0
693
328
73
16
9
p = pth
693
401
165
65
46
Table 3: Diffusion time at p = 0 and at the percolation threshold p = pth for different networks with 10000 consumers
and average degree 4. µ is the rewiring probability. Values are averages over 20 simulation runs.
800
700
value at p = 0
600
value at p = p_th
time
500
400
300
200
100
0
0
0.001
0.01
µ
0.1
1
Figure 13: Diffusion time (steps) for percolation in different small-world networks. Values are averages over 20 runs.
Here there are 10000 nodes (consumers) and all networks have an average degree 4.
regular lattice is the highest, as expected, and p = 0 is exactly the percolation threshold.
The rewiring mechanism reduces distances in the network, and drives down the diffusion
time both at p = 0 and at the threshold. The important aspect to consider here is how the
diffusion time scales down with rewiring in both cases. There are clearly two regimes: for
small rewiring probabilities, below µ = 0.01, the diffusion time scales down relatively fast.
Above µ = 0.01, the fall in diffusion time slows down considerably, and for instance there
is no much improvement passing from a small-world with µ = 0.1 to a Poisson network.
If we look at the diffusion size in Figure 9 we find an exactly opposite scaling behaviour:
there we have considerable changes above µ = 0.01 and little changes below. This means
that diffusion size and diffusion time are driven by different network factors. The former
depends on the accessibility of the network, as we have seen in Section 4. The diffusion
time is linked to the size of the network, instead. The scaling behaviour of the average
path length in small-worlds reported by Watts and Strogatz (1998) (Figure 5) is exactly
the same scaling of the diffusion time. The rewiring mechanism has a strong effect on the
20
distances between nodes in the networks, but much smaller effect on the accessibility of
nodes. Consequently, small-world structures perform very well in terms of diffusion time,
but much worse in terms of diffusion size.
The diffusion time at p = 0 and p = pth are linked to different measures of distance in
a network. At p = 0 the diffusion time scales down with rewiring faster than the diffusion
time at p = pth (Figure 13). If we compare a Poisson network to a small-world network
as the one with µ = 0.01, the diffusion time at p = 0 is more than seven times larger in
the latter, but at the percolation threshold it is only 3.5 times larger. The diffusion at
p = 0 depends on the average path-length of the network, while the diffusion time at the
percolation threshold depends on the diameter of the network (the two nodes at maximum
distance). At p = 0 the operational network coincides with the full network, and many
alternative paths are available to reach any node starting from the seeds (early adopters)
of the percolation process. The diffusion time depends on the shortest one, and the lower
the average path length, the shortest the diffusion time. At the percolation threshold a
different scenario realizes. The connected operational component is relatively large, but
very often only one path connects distant parts of this network. The innovation has to
cover long distances compared to scenarios with a price below the threshold, even if the
connected operational component is smaller. The diameter gives a measure of diffusion
time in this case, since it is the distance between the two nodes most far away.
In Figure 14 we report the diameter and the average path-length for different smallworlds with 1000 nodes.8 The diameter and the diffusion time at p = pth are nicely
300
average path-length
diameter
250
200
150
100
50
0
0
0.001
0.01
µ
0.1
1
Figure 14: Average path-length and diameter for one single realization of five different small-worlds with 1000
consumers and average degree 4.
8
We used only 1000 nodes due to computing power limitations. The absolute values of hLi and d are
of course different than in 10000 nodes networks, but the scaling properties are preserved.
21
correlated above µ = 0.01. Below this value, the number of short-cuts becomes critically
low, and the percolation mechanism starts to have a large impact on the accessibility of
the network. This results in short-lived diffusion processes which are not driven by the
diameter anymore.
5
Non-linear demand
So far we have used a uniform distribution of reservation prices pi ∼ U [0, 1] across consumers (nodes). Given the innovation price p ∈ [0, 1], the uniform distribution gives a
linear “prior” or “potential” demand D(p) = 1 − p. In case of full information the fraction
1 − p of consumers adopt the innovation on average.
In general, for a distribution f [0, 1] of reservation price values, the demand with full
information is D(p) = N × P rob(adoption), where
P rob(adoption) = P rob[pi > p]
(7)
= 1 − P rob[pi < p]
Z p
f (x)dx = 1 − F (p).
= 1−
0
F (p) is the cumulative distribution function, so that D(p) = N [1 − F (p)]. In this section
we study percolation with non-uniform distributions of reservation prices. In particular,
we want to understand how the percolation mechanism depends on the potential demand.
Let’s consider a Beta distribution of price values, for which the probability density
function reads as follows:
1
pα−1 (1 − p)β−1 , p ∈ [0, 1].
(8)
B(α, β)
R1
is a constant, defined by B(α, β) = 0 tα−1 (1 − t)β−1 . The parameters
f (p; α, β) =
The factor
1
B(α,β)
α and β control the probability distribution, whose density function can be increasing,
decreasing or monotonic. Accordingly the cumulative distribution function F (p) and the
resulting demand curve can be convex, concave or S-shaped.
We run batch simulations with four different demand curves for three different network structures, namely the regular one-dimensional lattice, a Small-world with rewiring
probability µ = 0.01 and a Poisson random network. As before, we set a connectivity
equal to 4 (exact value for the regular lattice, average value for the random network structures), and consider 10000 consumers. As before, we run 20 repetitions of the percolation
process in each setting defined by an innovation price p ∈ [0, 1], with steps ∆p = 0.05.
22
10000
10000
circle
10000
circle
circle
9000
SW 0.01
9000
SW 0.01
8000
Poisson
8000
Poisson
8000
Poisson
8000
Poisson
7000
No-network
7000
No-network
7000
No-network
6000
α=2
β=2
6000
α=3
β=1
n-of-adopters
No-network
6000
α=1
β=3
5000
5000
4000
6000
α=3
β=2
5000
4000
3000
n-of-adopters
SW 0.01
n-of-adopters
9000
n-of-adopters
SW 0.01
7000
5000
4000
3000
4000
3000
3000
2000
2000
2000
2000
1000
1000
1000
1000
0
0
0
0.1
0.2
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
3
3
α=1
β=3
2.5
0.5
price
0.6
0.7
0.8
0.9
1
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
0
1.5
α=3
β=2
1.5
1
1
1
0.5
0.5
0.6
0.8
1
0
0
0.2
0.4
p
0.6
0.8
1
p
0
0
0.4
0.5
price
0.6
0.7
0.8
0.9
1
α=3
β=1
1.5
0.5
0.4
0.3
2
1
0.2
0.2
2.5
0.5
0
0
0.1
3
2
f(p)
f(p)
1.5
0.2
2.5
2
2
0.1
3
α=2
β=2
2.5
0
0
f(p)
0
f(p)
10000
circle
9000
0.2
0.4
0.6
p
0.8
1
0
0
0.2
0.4
0.6
0.8
1
p
Figure 15: Percolation with non-linear demand. We use a Beta distribution of reservation prices, with probability
density function reported in the bottom panels. The results are averages over 20 simulation runs. The standard
deviation is larger at the threshold, and increases as the distribution mass moves to the right (about 54% for α = 3,
β = 1). There are 10000 consumers arranged in a one-dimensional lattice, a small-world (rewiring probability
µ = 0.01), and a Poisson network, with (average) connectivity 4. The dashed line in the top panels is the demand
curve without network effects, D(p) = 1 − F (p).
The results are reported in Fig. 15 These data should be compared with the simulations
results in Figure 6, which refers to the case of a linear demand obtained with a uniform
distribution of reservation prices.
The potential demand curve matters for innovation diffusion in a network of consumers,
because percolation is not independent on the distribution of reservation prices. For a
given network structure, different demand curves lead to different percolation threshold.
Let’s consider for instance a Poisson network. This one has a percolation threshold
pth ' 0.67 with linear demand (Section 2, Eq. 1 and Figure 2). If we use a decreasing
probability density function and the resulting convex demand of the example in the left
panel of Figure 15 (case α = 1, β = 3), we obtain a much smaller diffusion regime,
with a percolation threshold between p = 0.3 and 0.4. An increasing distribution of
reservation prices, with a concave demand (right panels of Figure 15) leads to a larger
diffusion regime, instead, with a percolation threshold near to p = 0.9. Non-monotone
probability distributions (middle panels of Figure 15) give a S-shaped potential demand,
with a concave and a convex region. When consumers are embedded in a network and
innovation diffuses as a percolation process, the larger the mode of the distribution, the
larger the percolation threshold and the smaller the inefficiency effect of lost demand due
to information transmission (Section 2).
The considerations above hold also for the the regular lattice and the small-world:
23
whenever the probability distribution of reservation prices puts more weight on lower
values, the percolation threshold decreases, and the diffusion regime shrinks. But our
results show more. The lost demand due to information inefficiency is not the same with
different distributions: the more a distribution puts weight on large values of reservation
prices, the smaller the inefficiency effect, and the less demand is lost. On the contrary, the
more weight of the distribution on low reservation price values, the more important the
role of the network structure, and the less efficient the diffusion process on a network, with
a larger portion of lost demand. The intuition is that a larger mass of consumers with
high reservation prices leads to a larger connected component for the operational network
(Section 2), while a distribution with larger mass of consumers with a low reservation price
places a relatively large number of bottlenecks in the network which reduce information
efficiency and lead to a relatively low diffusion regime.
Concluding, the percolation threshold gives an absolute measure of diffusion efficiency:
the “fatter” the potential demand is, the larger the diffusion regime, as expected. The
loss in demand is a relative measure of efficiency instead, according to which we see that
“fatter” demand curves present smaller losses, and inefficient network structures such as
the circle or the small-world suffer less with respect to the full information scenario. All
this means that not only “richer” societies favour diffusion in networks of consumers, but
also “more equal societies” (VERIFY...).
We have measured the diffusion time of percolation with non-linear demand curves.
Figure 16 contains the results for the same examples of Figure 12. While the effect of
circle
600
SW 0.01
700
Poisson
α=1
β=3
300
400
time
400
α=2
β=2
300
300
200
100
100
0
0
0
0.2
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
circle
600
SW 0.01
α=3
β=2
400
100
0.1
700
SW 0.01
500
200
0
circle
Poisson
600
500
time
time
800
SW 0.01
600
Poisson
500
circle
700
0.1
0.2
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
400
α=3
β=1
300
200
200
0
Poisson
500
time
700
100
0
0
0.1
0.2
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
price
0.6
0.7
0.8
0.9
1
Figure 16: Diffusion time for percolation experiments with a Beta distribution of reservation prices. The results
are averages over 20 simulation runs (standard deviation is larger at the threshold, and ranges from 25% to 30%).
The networks are a one-dimensional regular lattice, a small-world with rewiring probability µ = 0.01, and a Poisson
network, all with (average) connectivity 4 and 10000 nodes.
the distribution of reservation prices on the percolation threshold is strong, it does not
seem to be so on the percolation diffusion time. Both at p = 0 and p = pth the diffusion
time does not change much for different distributions. The results for the one-dimensional
lattice are interesting. The percolation threshold shifts to the right with distributions that
have more probability mass at larger values of price, but a peak for the diffusion time
24
is missing: the diffusion time remains long also well below the threshold. The reason is
that in a one-dimensional lattice there are only at most four alternative paths to cover
the network, and diffusion becomes a linear process. The larger is the distance to cover,
the longer the time required, and diffusion time becomes proportional to diffusion size.
6
Scale-free networks
Real social networks often present a “hub” structure, which is by no means captured by
Small-worlds and Poisson random networks. Few nodes, the hubs, have many links, while
the majority of nodes only have few links. This network structure is characterized by a
power law distribution of the degree. A power law distribution is also called “scale-free”:
for any value of the degree, the probability of occurrence of nodes with such degree “scales”
down with the degree at the same rate. In a double logarithmic scale, this distribution
is linear (Figure 17, left panel). This means that if on average there are 10 nodes with
1000 links, we may expect to find 100 nodes with 100 links, 1000 nodes with 10 links, and
so on. Many socio-economic systems present a scale free network structure, such as the
World-Wide-Web, the internet, science collaboration networks, and many others (Albert
and Barabasi, 2002).
Figure 17: Scale-free network simulated in t = 500 steps, adding m = 1 links for each new node, starting with N0 = 2
nodes. Left: degree distribution p(k) in a linear scale (up) and in a log-log scale (down). Right: network graph.
The scale-free network model introduced by Barabasi and Albert (1999) is essentially
an algorithm to generate a graph with a power law degree distribution. The basic idea is
a self-reinforcement mechanism of link creation, which builds on two factors, growth and
25
preferential attachment. At each time step a new node added to the network, and linked
to existing nodes with a probability which is proportional to their degree (an instance
of rich-get-richer positive feedback). The model has some variants, as for instance the
possibility to add more than one link for a new node. In this case the network can have
a triadic structure, while a tree-like structure is generated if only one link is introduced
for each new node (Figure 17, right panel). In general, if we grow a network with this
algorithm, starting with N0 nodes and adding m new links for each new node until we
have N = mt + N0 nodes, the final number of links is as follows:
Nlinks = mt −
(m − N0 )(m − N0 + 1)
,
2
(9)
which equals approximately mt links as soon as t >> m. The connectivity k of such
network is distributed with a probability density function p(k) ∼ k −γ where γ = 2.9 ± 0.1.
The striking property of such scale-free networks is that p(k) does not depend on time
t and on size N = mt + N0 : for any starting number of nodes N0 and any rate of links
addition m, at each time step t the connectivity distribution has the same shape γ. The
mean of such distribution is time invariant, but it does depend on the initial number of
nodes N0 (although it is independent on m):
hki =
γ−1
kmin ,
γ−2
(10)
where kmin is the lower bound of the distribution support. If we add m new links at each
time step, such lower bound is exactly kmin = m. In a scale-free network generated with
m = 1 then we have hki ' 2, while for m = 2 we have hki ' 4.
In this section we study how percolation works on scale-free networks. Fig. 18 compares the diffusion size and the diffusion time in two scale-free networks with a small-world
network and a Poisson random network. We consider scale-free networks with average
degree 2 (generated with m = 1) and 4 (generated with m = 2). The results in the left
panel show that scale-free networks are relatively efficient in terms of diffusion size, and
present a smooth transition from the non diffusion to the diffusion phase. The scale freenetwork with average degree 4 has a critical transition threshold at a higher price than
the Poisson network, meaning that it favours diffusion when the price is relatively high.
Below the threshold the Poisson network catches-up and overcomes the scale free-network.
The reason is that when the innovation price is relatively high hubs are useful, because
whenever a hub adopts the innovation, it can “test” many neighbours and very likely
find some with reservation price high enough to adopt the innovation. This effect is even
more striking when comparing a scale-free network of average degree 2 to a small-world
26
SW 0.01
10000
Poisson
160
SF 1
SF 2
8000
linear demand
7000
SF 1
SF 2
140
120
6000
time
n-of-adopters
SW 0.01
180
Poisson
9000
100
5000
80
4000
60
3000
40
2000
20
1000
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
price
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
price
Figure 18: Percolation in scale-free networks built with one new link at each time step (SF 1) and two new links
(SF 2), in a Small-world with rewiring probability 0.01 and a Poisson network, both with average connectivity 4.
All networks have 10000 nodes, and percolation experiments start with 10 seeds (early adopters). Values reported
are averages over 20 simulation runs.
of average degree 4: below p = 0.6 percolation in the scale-free network takes off, with
sizeable diffusion size values where the small-world scores negligible results. The critical
transition in the small-world is sharper, and for low innovation prices diffusion is larger
than in the scale-free network. These observations show a trade-off in the effect of hubs
compared to the small-world structure: hubs favour diffusion when this is more difficult
(high prices), but fails to reach a relatively large diffusion size when it diffusion occurs
more easily (low prices). In the latter regime, hubs turn out to be not important.
The time domain of percolation in scale-free networks is also interesting. The diffusion
time for the two scale-free networks considered is relatively very low, both at the threshold
p = pth and at p = 0. In the scale-free network SF 2 (hki = 4) the diffusion time is lower
than in the Poisson network also for innovation price values where the diffusion size is
larger, like p = 0.7. The time peak at the percolation threshold is relatively low, an aspect
that mirrors the smooth phase transition in the size space. For the scale-free network SF
1 (hki = 2) there is not even a time peak, and it is difficult to speak of phase transition
at all. For this network, the relationship between size and time of diffusion in percolation
is monotone, meaning that larger diffusion size require longer diffusion time. This is
a consequence of the tree-like structure of this particular scale-free network (Figure 17,
right panel), where there is only one route from the seeds to any node, since only one
path connects any two nodes.
27
7
Conclusions
A percolation model combines two important factors of economic diffusion, which are
information diffusion and adoption decisions. Percolation shows a phase transition from
a diffusion to a no-diffusion regime (phase), with a threshold price that depends primarily
on the network structure, and then on the distribution of reservation prices. The percolation critical transition has two economic implications. First, it is an instance of market
inefficiency: a sizeable portion of the demand is not satisfied in the no-diffusion phase
regime. Second, the demand curve is more elastic in the no-diffusion phase, and the market equilibrium price is lower. Market inefficiency is the prevailing effect at low prices, in
the percolation regime, when consumer welfare is always lower than with full information.
At high prices, the market inefficiency effect and the price effect are comparable, but in
the best case they almost offset each other.
We have studied with particular attention the case of small-worlds. Our results challenge the common hypothesis according to which small-worlds are a favourable environment for innovation diffusion. We show that whenever diffusion works as a percolation
process, small-worlds are rather inefficient. The main factor of diffusion size in percolation is not the average path-length of the network, but network accessibility. Whenever
clustering is low, the network connectivity happens to “spread” relatively more, meaning
that more neighbours are added at increasing distances of a given node. Small-worlds
are characterized by relatively low average path-length and relatively high level of clustering, and consequently are incapable to give large values of diffusion size. In particular,
there is a characteristic distance in a small-world network, below which accessibility scales
linearly, and above which it scales exponentially. Since for percolation the first steps of
adoption starting from a seed (early adopter) are mostly important, the linear scaling
behaviour at short distances is the relevant one, and it happens to represent a bottle-neck
for innovation diffusion. The diffusion time of percolation is a completely different story.
For diffusion time the average path-length and the diameter of a network are the relevant
factor, and then small-worlds result particularly efficient in this respect.
Our results on the effect of clustering for diffusion size are in line with empirical
evidence on technology diffusion (Fogli and Veldkamp, 2012) and against experimental
evidence on behaviour diffusion (Centola, 2010). For technology diffusion highly clustered collectivist societies present lower innovation diffusion rates, compared to individualist societies. On the other hand, high clustering favours behaviour diffusion, due to
social reinforcement. Our percolation model offers a clear benchmark for the adoption
28
mechanism: when innovation adoption is driven by individual preferences only, and links
between consumers only carry information, clustering has a negative effect.
The potential demand curve matters for innovation diffusion in a network of consumers,
because percolation is not independent on the distribution of reservation prices. For a
given network structure, different demand curves lead to different percolation threshold.
When consumers are embedded in a network and innovation diffuses as a percolation
process, the larger the mode of the distribution, the larger the percolation threshold and
the smaller the inefficiency effect of lost demand due to information transmission. The
percolation threshold gives an absolute measure of diffusion efficiency, according to which
a larger potential demand leads to larger diffusion regimes. The loss in demand due to
networks effects is a relative measure of efficiency instead: when the potential demand is
larger, inefficient network structures such as the circle or the small-world suffer less with
respect to the full information scenario. All this means that not only “richer” societies
favour diffusion in networks of consumers, but also “more equal societies”.
We conclude with some ideas for future research. The percolation model can accommodate the diffusion of multiple innovations. Goyal and Kearns (2012) study the
strategic implications of diffusion for two competing innovative firms. An interesting
question is whether competition affects diffusion, and how the percolation critical transition affects competition. Another possibility is to introduce local reinforcement and/or
learning curves. Local reinforcement can affect reservation prices, accounting for imitation, peer-effect, and social pressure in general, while learning curve are an endogenous
way of modelling the fall in the innovation price due to global adoption. Percolation
represents a simple benchmark of diffusion dynamics. A useful study would be to test innovation diffusion in a controlled web experiment (Centola, 2010), to see if the percolation
mechanism is grounded on the evidence of human behaviour.9
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