Networks, Percolation, and Consumer Demand

: Understanding diffusion processes is key to market strategies as well as innovation and sustainability policies. In promoting new products and technologies, firms and governments need to understand the condi-tionsfavouringsuccessfulspreadoftheseproducts. Weproposeagenericdiffusionmodelbasedonpercolation theory. Our reference is a new product diffusion in a social network through word-of-mouth. Given that consumers differ in their reservation prices, a critical price exists that defines a phase transition from a no-diffusion to a diffusion regime. As consumer surplus is maximised just below a product’s critical price, one can systematically compare the economic efficiency of network structures by investigating their critical price. Networks withlowclusteringwerethemostefficient, becauseclusteringleadstoredundantinformationflowshampering effective product diffusion. We further showed that the more equal a society, the more efficient the diffusion process.


Introduction
. The success of an innovation does not only depend on its intrinsic properties, but also on its di usion process (Griliches ; Mansfield ; Bass ; Davies ). The question of how innovations di use is among the core questions in economics, sociology, marketing, innovation studies, and even physics (Stoneman ; Rogers ; Valente ; Vega-Redondo ). The role of social networks in innovation di usion has become a key question since innovations generally spread through social interactions (Banerjee et al. ). Hence, the structure of social networks in societies may bear important consequences for the rate of technological progress and economic growth (Fogli & Veldkamp ). Our work builds on these questions by analysing the welfare e ects of social networks in new product di usion. The accent on consumers' demand and welfare makes our analysis complementary to studies that focus on market strategies, such as Shao & Hu ( ).
. Though innovations have complex e ects on consumers' welfare, the speed and extent of di usion are arguably the two dimensions of a di usion process that are both universal and fundamental. These aspects are present in empirical marketing studies that focus on the role of contagion and word-of-mouth in the adoption of new products (Garber et al. ; Dover et al. ). If one assumes that any consumer adopting an innovation is improving its welfare, both a faster di usion and widespread di usion automatically imply higher returns to society. The key question becomes how di erent network structures impact the speed and extent of innovation di usion. While the speed of di usion has been extensively studied, the extent of di usion remains understudied. The reason for this lack of attention to the extent of di usion relates to the common assumption that all consumers are willing to adopt any innovation. Following this assumption, di usion will always reach its maximum extent in a connected network. .
In our model, we assume that only a small fraction of consumers is willing to adopt, depending on their preferences. In turn, preferences are expressed by individual reservation prices, and then the share of potential adopters depends inversely on the product price. Such a di usion process can be modelled meaningfully as a .
The article is organised as follows. Section studies the e ects of percolation on innovation di usion. Section looks at percolation in small-world networks. Section considers the time dimension of market di usion. Section addresses alternative demand curves, with a non-uniform distribution of consumers' reservation prices.
In Section we extend our model to percolation in networks with a power law distribution of consumers' connectivity (scale-free networks). Section concludes.

Percolation and Demand
. Let us consider a new product and a network of N potential consumers, where i and j are neighbours if there is a link η i,j connecting them. Links are either existing (η i,j = 1) or absent (η i,j = 0), and do not depend on time.
The di usion process starts exogenously with a small number n << N of initial adopters of the new product (seeds). Information about the innovative product is local (Huang et al. ): consumers who are not among the initial adopters, come to know about the new product only if a neighbour adopts. .
The adoption decision is based on the product's price p, which is defined in the interval [0, 1], and which is assigned before di usion starts. Consumers' preferences are expressed by a reservation price p i ∼ U [0, 1]: only consumers with p i > p are willing-to buy ( Figure  information to their neighbours, while consumers who adopt, pass on information to their neighbours. Hence, the network that truly matters for di usion is the one that results from individual reservation prices. Drawing consumers reservation prices amounts to randomly switching 'o ' nodes and their links ( Figure , right panel). The resulting network of active nodes is called the operational network. Di usion will have a sizeable extent only if a large ('giant') connected component exists in the operational network and at least one seed is part of this giant component. .
What is important to notice is that drawing reservation price has a highly non-linear e ect on di usion. In the example of Figure , we have consumers, and all but one happen to be connected initially (Figure , le panel). An innovation price p = 0.5 means that on average 50% of consumers are willing-to buy, that is consumers. The unwilling-to-buy consumers are "removed", and the resulting network is made of a number of connected components, the largest being formed by only eight consumers. In the best case, when we have an initial adopter who belongs to this component, di usion size will be eight, which is just over half the potential di usion size of willing-to-buy consumers. .
If all consumers are directly linked in a fully connected network (meaning that all consumers have N − 1 neighbours), all consumers are immediately informed once any seed adopts, and di usion will always attain its maximum possible size. Assuming a uniform distribution of consumers' reservation prices, we obtain the standard downward-sloping linear demand curve. 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).

.
It is the local nature of information di usion in social networks that makes the actual adoption curve to depart from the standard linear demand curve. In particular, a sparse network structure introduces a phase transition, leaving many potential adopters uninformed at high prices. We illustrated this point by simulating the percolation model for the case where , consumers form a Poisson random network (Erdös & Rényi ). The le panel of Figure  Values are averages over simulation runs. The standard deviation is largest at the percolation threshold (25%). tion process is initiated by seeds (n = 10). At low prices, the number of adopters follows the linear demand curve reflecting the uniform distribution of reservation prices. This amounts to full di usion of the innovation, meaning that all consumers with a reservation price equal or above the productï£¡s price, actually adopt the product. However, already at values as low as p = 0.4 the di usion size starts to be lower than the number of consumers willing to adopt, and drops down to almost zero above p = 0.7. In these scenarios, di usion is not full, because information of the product's existence does not spread in the network. Put di erently, the operational network is fragmented in many small components of consumers, without any links between these components. As we increase the price, an ever increasing share of consumers that would be willing-to-buy the innovation based on its price, do not actually buy it, because they do never get to know about its existence (Solomon et al. ). .
We can distinguish between two di erent regimes: a di usion regime, where information spreads throughout the whole operational network, and di usion is full or almost-full. A no-di usion regime, where information does not spread, and di usion is very limited leaving many potential buyers uninformed. By lowering the price, the consumers' network undergoes what in physics is called a percolation phase transition (Stau er & Aharony ). A critical threshold value of the price, p c , 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-to-buy (see Figure ). The size of the giant component is highly non-linear, with a sharp increase in size below the critical price p c (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. ( ) for the analysis of several characteristics of random networks. Callaway et al. ( ) apply this formalism to percolation. The case of a Poisson network allows us to evaluate analytically the percolation threshold. In the price space dimension of innovation di usion, and for a network of infinite size, the percolation threshold is given by where k refers to the connectivity of nodes. In the example of Figure the average connectivity is k = 4, so the percolation threshold is p c 0.67. .
The e ects of social networks can be expressed in welfare-theoretical terms. Network ine iciency stems from "lost" consumers, that is, consumers who would have been willing to buy, but who are not informed by any neighbour about the product. That is, for relatively expensive products, a social network may not be able to convey the product's existence to all its members, and consequently not all consumers that would like to buy the product will buy it, because they are not informed. The group of consumers who, at price p, are willing to buy but who do not adopt, we will refer to as lost consumers. By definition, a lost consumer has a reservation price in the range between and p. Hence, given the uniform distribution of reservation prices, the expected loss in consumer surplus of a lost consumer amounts to (1 − p)/2. Accordingly, in the no-di usion regime, the welfare loss amounts to N (1 − p) lost consumers who miss out, on average, on (1 − p)/2 surplus, amounting to Hence, for a given society with N members, the welfare loss due to network ine iciency is solely a function of the price. This implies that, in the following analysis of network ine iciencies, we can proceed by solely focusing on the critical price that separates the di usion regime from the no-di usion regime. That is, we can express the ine iciency of di erent network topologies by the critical price below which full di usion occurs. The lower this critical price, the less e icient is the network in question.

Innovation Di usion in a Small-World Network
. One of the most popular models of social networks is the small-world model introduced by Watts & Strogatz ( ). Several empirical studies have identified small-world properties in real world social and physical networks, with possibly the most notorious one being the six degrees of separation: i.e., it takes on average six steps to reach any individual in the world (Travers & Milgram ). This fact is actually a manifestation of a well defined mathematical property, which is a relatively short average path-length. The small-world 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. Figure shows examples with N = 50 nodes and degree (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 rewired ( Figure , right panel), a procedure that leads to a fully random Poisson network of the type introduced by Erdös & Rényi ( ). In this network the connectivity of nodes follows a Poisson distribution. Also Poisson networks are characterised by a relatively short average path-length with respect to other network structures and in particular with respect to the starting regular network of Figure , le panel. What makes small-world networks interesting is that they have 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. .
The clustering coe icient measures the relative number of triplets out of all possible triplets in a network (Wasserman & Faust ). This fraction is particularly large in the regular network, where each node has two neighbours on each side, while it is almost zero in the Poisson network. By rewiring links at random, the average path-length drops suddenly for small values of the rewiring probability, while the clustering coe icient remains practically unaltered until one rewires a large portion of links. The typical small-world of Watts & Strogatz ( ) Figure : Generation process of small-world networks: example with nodes and degree . Le : regular network (µ = 0). Centre: small-world with µ = 0.1. Right: Poisson network (µ = 1).
is obtained with as few as 1% of links rewired. Its average path-length is almost the same of a Poisson network, but the clustering coe icient is very large and comparable to the original regular network. It is important to notice that while the rewiring process strongly a ects the degree distribution, the average degree remains unchanged, since the numbers of nodes and links are fixed. .
We have simulated percolation in a number of di erent small-world networks, namely for µ = 0.001, µ = 0.01 and µ = 0.1. Figure  di usion size at every price value. The percolation threshold in a small-world is much lower than in a Poisson network, and the non-di usion regime is much larger. Hence, a much lower price is required for innovation to spread in a small-world network compared to a Poisson network. For a typical small-world with µ = 0.01, the price threshold is between 0.2 and 0.3. In the limit case of a regular one-dimensional lattice, the critical price is lowest, between 0.1 and 0.2. The key conclusion here is that small-world networks are highly ine icient compared to Poisson random networks in percolating a new product over a social network of consumers. .
The percolation threshold is defined as the critical value of nodes' activation probability where a phase transition occurs in the size of connected components of active nodes, the so-called percolating cluster. Newman & Watts ( ) show how to evaluate implicitly the percolation threshold of small-world networks of infinite size. In the price space, the critical value p c satisfies the following equation: where µ is the rewiring probability. In Table we report the values obtained for the small-world networks of Figure . The simulation results of Figure would present a sharp discontinuity at these values in the case of Table : Critical price p c and ine iciency in small-world networks with di erent rewiring probability µ. These are theoretical values from Equation , apart from the Poisson network (µ = 1), whose value is from Equation .
an infinite network. At the threshold p c the ine iciency from lost demand reaches its maximum level. Such ine iciency is more severe the lower the rewiring probability µ (Table ). For the typical small-world with µ = 0.01, no less than percent of willing-to-buy consumers are lost, because they never get to know about the existence of the new product. .
The size of the percolating cluster depends on the connectivity distribution of a network. In particular, it depends on the 'accessibility' of a network, that is how many nodes one can reach starting from a given node, within a given distance r (the so-called volume of a 'sphere' of radius r). Due to the redundancy of links in a highly clustered network, the small-world presents relatively low accessibility (one needs to rewire a substantial number of links, starting from a regular network, before reaching the accessibility levels of Poisson networks). .
Concluding, the di usion size in a network depends on the size of the percolating cluster. This size undergoes a transition only at relatively low values of the price. In the price dimension of new products di usion on a consumers network, the critical transition threshold of a small-world is much lower than the threshold for a fully random Poisson network. A given product price can fall into the no-di usion regime of s small-world network, while still being in the di usion regime of the Poisson network. This is the reason behind the relative ine iciency of small-worlds in terms of di usion size. Small-world is very e icient in terms of speed of di usion instead. But this depends on the average path length, as we explain in the following section.
. The two extreme limits represented by the regular network and the Poisson random network can be given a societal interpretation. A regular network reflects a society that is "collectivist", with all nodes having the exact same degree, and with a high clustering coe icient indicative of social cohesion. The fully random network however, corresponds to an "individualistic" society, with some nodes having higher degree than others, and with a low clustering indicative of a lack of cohesion, but short distances. Real social networks may be closer to one or the other limit of the model (which makes the small-world model so relevant), depending of the specific case considered. For instance, some have associated lower income levels to more clustered networks inhibiting innovation di usion (Fogli & Veldkamp ). An alternative and possibly complementary interpretation is to think of networks with high clustering as rural societies where most social ties are established within a village, and networks with low clustering as urbanised societies where clustering is much lower.

Di usion Time
. To some extent, the low level of network e iciency of small-worlds may come as a surprise. Indeed, smallworlds are generally considered very good di usion vehicles, given their short path lengths. However, such an assessment is based on the speed of di usion and not on the extent of di usion. Only in the unlikely case that everyone has a maximum reservation price, the operational network would coincide with the social network itself and full di usion would always be realised. In such instances, the speed of di usion remains the sole relevant performance criterion. .
Following the standard downward-sloping demand curve, we assumed random reservation prices, in particular, uniformly distributed prices. In these instances, we can express network e iciency as the critical price below which full di usion occurs. In this context, as reported by Figure , we observed a relatively low improvement of di usion size when comparing a regular network to a small-world network. The typical small-world with rewiring probability µ = 0.01 (low average path length and high clustering coe icient) has a critical price which is only marginally larger than the regular network. Only for larger values of the rewiring probability, leading to lower levels of clustering, we find networks becoming more e icient. We can therefore claim that whenever di usion is driven by a percolation mechanism, the important factor is a low clustering coe icient, not a short average path-length. .
Here, by di usion time we mean the time the di usion process takes to stop. This is the time required to cover all connected components of the operational network that contain a seed.  Table : Di usion time at p = 0 and at the percolation threshold p = p c for di erent networks with consumers and average degree . µ is the rewiring probability. Values are averages over simulation runs. p = p c is a "worst condition" for di usion time, where only one path connects several nodes in the network. Above the critical price, di usion stops very quickly. On the contrary, p = 0 is a "best condition", where all the network is accessible (apart from unconnected components without a seed). .
The regular network shows the longest di usion time and p = 0 is precisely the percolation threshold. The rewiring mechanism reduces distances in the network, and drives down di usion time both at p = 0 and at the threshold. An important aspect to consider is how the di usion time scales with rewiring. There are two regimes: for low rewiring probabilities, below µ = 0.01, the di usion time scales down fast. Near µ = 0.01, the di usion time is already relatively short, and further reductions from increasing the rewiring probability are negligible. In Figure (le panel) we plot the di usion time values of Table together with the ine iciency measure of Table . The two patterns contrast with each other: the network ine iciency decreases little below µ = 0.01, and drops fast beyond this value. Such di erent scaling patterns mean that di usion size and di usion time are driven by di erent factors. Figure compares percolation results to structural properties of small-worlds networks such as the clustering coe icient and the average distance between nodes (Watts & Strogatz ). The similarity between the two sets of measures is evident. While di usion ine iciency (a negative measure of di usion size) correlates with clustering, di usion time correlates with average path-length. For low values of µ the average path-length decreases linearly as l N = 1 4 − 1 2 µN + O(µ 2 ). When µ = 0 we have the average path-length of a regular network l = N/4, while for large values of µ the approximation above does not hold anymore, and we have l log N instead, as in a Poisson network. .
The di usion time at p = 0 and p = p c are linked to di erent network properties. At p = 0, the di usion time scales with rewiring faster than di usion time at p = p c (Table ). 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 of the percolation process. .
Concluding, while di usion time depends on the average path length of a network, the level of clustering is responsible for the extent of di usion. Small-world network happen to be characterised by a relatively high clustering and low average path length. This is why they are relatively e icient in terms of speed of di usion, but ine icient in terms of di usion size.

Alternative Demand Curves
. So far we have used a uniform distribution of reservation prices p i ∼ 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 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 us consider a Beta distribution of reservation price values, for which the probability density function reads as follows: The factor 1 B(α,β) is a constant, defined by B(α, β) = 1 0 t α−1 (1 − t) β−1 . The parameters α and β control the probability distribution, whose density function can be increasing, decreasing or non-monotonic. Accordingly, the cumulative distribution function F (p) and the resulting demand curve can be convex, concave or S-shaped (Figure ). .
We run batch simulations with di erent demand curves for three di erent network structures, namely the regular one-dimensional lattice, a small-world with rewiring probability µ = 0.01 and a Poisson random network. We first considered four cases with di erent mean value of the individual reservation price. The results reported in Figure clearly show how the potential demand curve (i.e. the distribution of reservation prices) matters for innovation di usion in a network of consumers. For a given network structure, di erent demand curves lead to di erent percolation thresholds. Consider for instance a Poisson network. This one has a percolation  Figure : Percolation with alternative demand curves: Beta distribution of reservation prices p i . Cases with di erent mean. Top: probability density function. Middle: di usion size (averages over simulation runs. The standard deviation is larger at the threshold, and increases as the distribution moves to the right, being 50% for α = 3, β = 1). The dashed line is the demand in a fully connected network, D(p) = N (1 − F (p)), with F the cumulative distribution of Equation . Bottom di usion time (averages over simulation runs. The standard deviation is larger at the threshold, and ranges from 25% to 30%). The networks are a one-dimensional regular network, a small-world with rewiring probability µ = 0.01, and a Poisson network, with nodes and (average) connectivity . threshold p c 0.67 with linear demand (Section , Equation and Figure ). If we use a decreasing probability density function and the resulting convex demand of the example in the le panel of Figure (case α = 1, β = 3), we obtain a much lower di usion 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 ) leads to a larger di usion regime, instead, with a percolation threshold near to p = 0.9. Non-monotone probability distributions (middle panels of Figure ) give a S-shaped potential demand, with a concave and a convex region. When consumers are embedded in a network and innovation di uses as a percolation process, the larger the mode of the distribution, the larger the percolation threshold and the smaller the ine iciency e ect of lost demand due to information transmission (Section ). These considerations also hold for both the regular network and the small-world: i.e., whenever the probability distribution of reservation prices puts more weight on lower values, the percolation threshold decreases, and the di usion regime shrinks. .
The information ine iciency and the loss in demand of a percolation process strongly depend on the reservation price distribution, and for a given distribution they depend on the consumer network structure. For the Poisson network, the more a distribution puts weight on large values of reservation prices, the smaller the ine iciency e ect, and the less demand is lost (Figure , right panel). The more weight on low reservation price values, and the less e icient the di usion process, with a larger loss of demand ( Figure , le panel). For the regular network, the opposite is true. Whenever there is more weight on larger reservation prices, the realised demand increases with a larger threshold, but not as much as for the Poisson network and the di erence with the potential demand is amplified. The reason is as follows: a larger mass of consumers with low reservation price produces many bottlenecks in the network, which reduce information e iciency and lead to a low di usion regime (Section ). A larger mass of consumers with high reservation prices enlarges the connected component of the operational network. This e ect is much stronger the higher the dimensionality of the network, when redundant links are rewired and used to open alternative routes that spread from the seeds of the di u-sion process.
. The percolation threshold gives an absolute measure of di usion, i.e., the "fatter" the potential demand is, the larger the di usion regime. The loss in demand is a relative measure instead, which tells about the e iciency of the network structure. When the potential demand gets "fatter" the Poisson network experiences a smaller loss of demand. However, ine icient network structures such as the regular network or the small-world network are less able to exploit this more favourable distribution of reservation price. If we think of the reservation price as a manifestation of consumers' income, this means that in a more rich society, di usion ine iciency due to the network structure is amplified.
. The di usion time is not a ected by changes in the distribution of reservation prices, apart from the shi of the threshold peak (Figure , bottom panels). Both at p = 0 and p = p c the di usion time does not change much for di erent distributions. The results for the one-dimensional lattice are interesting. The percolation threshold shi s to the right with distributions that have more probability mass at larger values of price, but a peak for the di usion time is missing, that is the di usion time remains long well below the threshold. The reason for this is that in a one-dimensional lattice there are only at most four alternative paths to cover the network, and di usion becomes a linear process. The larger is the distance to cover, the longer the time required, and di usion time becomes proportional to di usion size. .
We have studied the e ect of the dispersion of reservation prices by considering symmetric distributions with mean preserving spread (Figure ). When distribution is less dispersed (and its peak more pronounced), mean- Figure : Percolation with alternative demand curves. Beta distribution of reservation prices. Cases with equal mean (< p i >= 0.5). Top: probability density function. Middle: di usion size (averages over runs. The standard deviation is larger at the threshold, and increases as the distribution moves to the right, being 50% for α = 3, β = 1). The dashed line is the demand in a fully connected network, D(p) = N (1 − F (p)), with F the cumulative distribution of Equation . Bottom di usion time (averages over simulation runs. The standard deviation is larger at the threshold, and ranges from 25% to 30%). The networks are a one-dimensional regular network, a small-world with rewiring probability µ = 0.01, and a Poisson network, with nodes and (average) connectivity .
ing that consumers are more alike, the critical price of di erent network structures converges to the mean value of the distribution (p = 0.5). Put di erently, diverse network structures look more similar for their di usion outcomes. When reservation prices are narrowly distributed, the mean becomes a critical value for the innovation price, above which almost all network nodes are shut down, and above which almost all of them are accessible. In the limit case of a homogeneous reservation price (δ-Dirac distribution peaked at p = 0.5), the demand curve is a step function, the mean reservation price becomes the critical price, and di erent networks present almost the same di usion size. This is exactly the case with the example in the right panels of Figure . Irrespective of the network structure, there is a di usion regime on [0, 0.5], and a no-di usion regime on [0.5, 1]. In this limit the percolation process is able to get the full potential demand for all network structures, with no lost demand from information ine iciency.

.
These results have obvious implications for consumer welfare. First, the convergence of di erent network structure towards a single reservation price results in a net gain of consumers' surplus. Less e icient networks, such as the regular network and the small-world gain in terms of a higher critical price and a smaller loss of consumers with respect to the potential demand. However, even for the Poisson network, the e ect of a smaller critical price is outweighed by a larger demand in the di usion regime, with reduced loss of consumers.

Scale-Free Networks
. When social networks are built on a digital online platform, they o en have a 'hub' structure, which is not well described by small-worlds and Poisson random networks (Amaral et al. ). Few nodes, i.e., the hubs, have many links, while the majority of nodes have only few links. This feature reflects at a more fundamental scale the rich-get-richer formation dynamics of the World-Wide-Web, and the internet (Albert & Barabasi ). Empirical studies of new products adoption have enlightened the importance of social hubs for both the speed and the extent of the di usion process (Goldenberg et al. ). .
A network with a hubs structure is characterised 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. This means that if on average there are nodes with links, we may expect to find nodes with links, nodes with links, and so on. .

The scale-free network model introduced by Barabasi & Albert (
) 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 preferential attachment. Figure shows an example for a scale-free network of nodes, generated with adding two links per every new node. .
Figure compares the di usion size and di usion time in a scale-free network with a small-world network and a Poisson random network, all with average degree k = 4 links. The results in the le panel show that scale-free networks are relatively e icient in terms of di usion size, and roughly match the the demand pattern of consumers arranged in a Poisson network. This is in accordance with the low degree of clustering of both structures. The scale-free network gives a smoother transition between di usion and no-di usion regimes. Its critical price is larger than the Poisson network critical price, meaning that scale-free networks favour di usion when the price is relatively high. The reason is that hubs are useful when the innovation price is relatively high, because whenever a hub adopts the innovation, it passes the information to many neighbours, and very likely find some with reservation price high enough to adopt the innovation. However, when the price is relatively low, this advantage is less important, and the high-dimensionality of a Poisson network is slightly more beneficial to the di usion size. .
The time dimension of di usion is also interesting, and shows that percolation processes in scale-free networks are relatively fast. On average, di usion times in scale-free networks are lower than in Poisson networks and in small-world networks, both at the critical transition threshold and below it ( Figure , right panel). In particular, a low di usion time at the threshold reflects the smoothness of the critical transition of the di usion size. .
Concluding, scale-free networks do not depart much from Poisson networks in terms of their di usion performance. The remarkably di erent network structure remains the small-world one. As long as innovation diffusion occurs following a percolation model, small-worlds happen to be much less e icient than both Poisson and scale-free networks regarding the di usion size. The di usion time is much longer in small-world networks than in Poisson and in scale-free network at relatively low innovation price values, but it becomes smaller at relatively high prices, when we move close to the percolation thresholds of the Poisson and of the scale-free networks.

Conclusions
. A percolation model combines two important factors of innovation di usion, that is, adoption decisions and information spreading. Percolation shows a phase transition from a di usion to a no-di usion regime (phase), for increasing prices. The phase transition indicates the critical price below which di usion is almost complete. We showed that the critical price can be used as a measure of network e iciency, which depends on network topology and the distribution of reservation prices.
. Percolation processes in innovation di usion have two economic implications. First, they highlight an instance of network ine iciency: i.e., a sizeable portion of the demand is not satisfied in the no-di usion phase regime. Second, percolation processes unfold di erently in di erent network topologies. In particular, our results challenge common wisdom according to which small-worlds are favourable for innovation di usion. We showed that whenever di usion works as a percolation process, small-worlds are rather ine icient, as di usion size is driven by low clustering, and not by low average path-length of a network. We further showed that apart from low clustering, a less dispersed distribution of reservation prices across consumers favours di usion, suggesting that not only richer, but also more equal societies, support new product di usion.

.
Our key result on the e ect of clustering on di usion size are in line with empirical evidence on technology di usion (Fogli & Veldkamp ), but against experimental evidence on behaviour di usion (Centola ). Our percolation model o ers a clear benchmark for the adoption mechanism. When innovation adoption is driven by individual preferences only and links between consumers carry only information, clustering has a negative e ect. For behaviour spreading, however, it is more likely that social pressures also play a role, in that individuals become more likely to adopt a behaviour if many neighbours already display the behaviour. An extension of the basic percolation model with a social-pressure function would indeed be a promising line for future research.

.
There are limitations to the present study, which can be possible directions for further research. First, our analysis of the welfare implications of social networks has been limited to the demand side, focusing on consumer surplus that is lost when the consumer remains unaware of a product which they would otherwise be willing to buy. Since, for monopolistic market structures, lower demand translates into lower prices, some of the welfare losses will be compensated by lower equilibrium prices. Campbell ( ) provides such an equilibrium model illustrating the latter e ect, but without considering welfare e ects due to the loss of consumer surplus (and restricted to random networks only). In future research, the percolation theory can be integrated into welfare analysis of markets, where equilibrium prices are derived, rather than treated as a parameter value. Furthermore, a game-theoretical extension can be envisaged following recent work on di usion and imitation (Boyer & Jonard ).
. Second, we assumed in our model that links of consumers networks are uncorrelated with reservation prices. Since the latter probably reflects consumer income, a more realistic description of the di usion process would require us to relax this assumption, and consider some degree of assortativity in network structures, where neighbour consumers are more likely to have similar reservation prices. Assortative networks and income distribution are two key-factors in the study of the implications of percolation for innovation policy.
. Third, in our framework social networks act solely as the medium for information di usion, thus ignoring other possible e ects of social interactions. For example, consumers may more probably become adopters due to local network externalities causing a consumer's willingness to pay to increase with the number of adopters in their neighborhood. Such a mechanism may render the percolation mechanism caused by word-of-mouth relatively less important (Birke & Swann ). Furthermore, a product's price may fall over time due to technological progress and global network externalities such as learning-by-doing. These additional mechanisms will a ect the extent to which a product di use in di erent networks, as shown in previous models (Delre et al. , ). In particular, if an innovation improves over time it may be strategically advantageous to wait for its adoption, and a slower di usion process may enhance social welfare.
. Indeed, the percolation model is first and foremost a theoretical model that applies to many social and natural contexts. It describes di usion dynamics in networks driven by individual traits (here, reservation prices of consumers), rather than an accurate description of various di usion processes that exist in reality. As a null-model based on a single global parameter (here, price), some network topology and some distribution of individual traits, it is a useful benchmark in empirical studies of di usion dynamics. To the extent that empirical data systematically deviate from the predictions of percolation concerning new product di usion, additional mechanisms, such as the ones described above, should be included to better understand the exact nature of di usion in specific technological, social and natural contexts. Center workshops "Socio-Economic complexity" ( ) and "Econophysics and networks across scales" ( ) in Leiden, the UNU-MERIT of Maastricht University, the 5 th Conference on Sustainability Transitions in Utrecht ( ), the Department of Economics of the University of Venice, the ECCS conference in Barcelona, the WEHIA workshop in Reykjavik, the 11 th Workshop on Networks in Utrecht ( ), the 4 th CompleNet Workshop in Berlin ( ), the College of Management of EPFL Lausanne, the LATSIS Symposium and the ABM-CTS conference at ETH, Zurich, and the Amsterdam School of Economics. Funding has been provided by the Netherlands Organisation for Scientific Research (NWO) under the Social Sciences Open Competition Scheme, no. --.

Notes
A related application of percolation theory has been proposed to describe innovation dynamics in a technology space rather than a social network (Silverberg & Verspagen ) The simulation model, called "PercolationPrice", is implemented in NetLogo. The code can be downloaded from the OpenABM database following the link: https://www.openabm.org/model/5955/version/1/view.
The results presented here are obtained from batch simulations experiments, by running the model a number of times for di erent values of the product price. The batch simulation experiment in NetLogo is set with the BehaviorSpace function.
The exact di usion size depends on the number of seeds, but the critical transition threshold does not. Simulations with seeds yield the same patterns. Simulations with less than seeds are less informative, since the variability of results is too large, with a standard deviation near 100%. The reason is that if no seeds fall in the giant connected component, no macroscopic di usion takes place.
A review of studies on the statistical properties of real-world networks is Albert & Barabasi ( ).
The degree distribution of a Poisson random network is p(k) = 1 k e −z z k , 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 ).
Equation is Equation ( ) of Newman & Watts ( ), and holds in the limit µ << 1. In particular it does not hold for the Poisson network (µ = 1). The critical price for the Poisson Network is given by Equation .
The results in Newman & Watts ( ) are obtained with a slightly di erent model, where links are randomly added (not rewired). This modification of the Watts & Strogatz ( ) model is necessary to avoid that one node remains unconnected with positive probability, giving an infinite average path-length. The two models converge as µ → 0.
Strictly speaking, this is the scaling property of the Zipf law. In general, power laws can present a scaling rate di erent from one.
In a scale-free network with k = 2, generated adding one link for a new node, the percolation threshold is p c = 0, and the relationship between size and time of di usion is monotone: a larger di usion size requires a longer di usion time. This is a consequence of the tree-like structure of this scale-free network, where there is only one path from a seed to any node.