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The Language of Dynamical Networks

The unifying power of the Complexity view is expressed among other in the emergence of a common language which allows the quick, effective and robust / durable communication and cooperation between people with very different backgrounds [10]. One of these unifying tools is the concept of dynamical network.

Indeed, one can think about the 'elementary' objects (belonging to the 'simpler' level) as the nodes of the network and about the 'elementary' interactions between them as the links of the network [11]. The dynamics of the system is then represented by (transitive) operations on the individual links and nodes ((dis)appearance, substitutions, etc.) [12].

The global features of the network correspond to the collective properties of the system that it represents: (quasi-)disconnected network components correspond to (almost-)independent emergent objects; scaling properties of the network correspond to power laws, long-lived (meta-stable) network topological features correspond to (super-)critical slowing down dynamics. In this way, the mere knowledge of the relevant emerging features of the network might be enough to devise methods to expedite by orders of magnitude desired processes [13] (or to delay or stop un-wanted ones). The mathematical tools implementing it are developed presently and include multi-grid and cluster algorithms.

Percolation models

Let us consider a regular arbitrarily large square lattice. Each site i represents a "customer". The basic model considers only one product of fixed quality quantified by a real number q between 0 and 1. The customers have different expectations represented too by numbers p(i) between 0 and 1. The condition that the customer i buys the product is that one of his/ her neighbors bought the product and that p(i) < q. If p(i) < q but no neighbor has bought the product, i is a "potential buyer" because if he only new about the product, the quality of the product q would satisfy its expectations p(i). However, he is not (yet?) an actual buyer, because in the absence with a direct contact with another customer which bought the product, he "doesn't know" enough about the quality of the product and therefore would not buy it at this time.

If one starts a campaign by selling the product to a particular customer j, what will be the outcome amount of sales?

Obviously, this depends on q and on the distribution of p(i).

If one makes the simple assumption that the p(i)'s are independent random numbers distributed uniformly between 0 and 1, then one has a surprise: even if q =0.59, which means it satisfies the expectations of 59% of the customers, its actual sales will be less than 1% !

For people familiar with percolation theory, the reason is obvious: sales take place only within clusters of contiguous sites, if the initial site belongs to a small cluster of potential buyers surrounded by customers with p> 0.59, then the only sales will be within that cluster even if there are "an infinity" of other disconnected clusters "out there". In fact in the case of two dimensional square lattice it is known that if the density of "potential buyers" is less than pc = 0.593... then all the contiguous clusters are finite even if the size of the system is taken to infinity. Consequently, for q < 0.593... not even a finite fraction of the potential market is realized. For q = 0.593 a finite fraction of the lattice becomes actual buyers: the largest cluster is "infinite" (i.e. of the system size).

Obviously the phenomenon does not depend on the network on which the "customers" live. Any lattice which presents a percolation transition would do. I.e. any lattice/ network in which the largest cluster becomes "infinite" when the density of "potential customer" points becomes larger than pc would do. Moreover, the particular way in which the critical density pc of "potential customers" is realized is not important: it can be decided in advance by fixing p(i) for every site, but also it can be (partly) decided randomly at the time that the point is reached by the sales wave. It can also be enforced by having each point spending with some probability some time in the "potential buyer" mode.

The features which do affect the marketing percolation transition are however space and time correlations between the "potential buyer" state.

For instance, we will see that inhibiting the "potential buyer" state for a while after each buy, will lead to periodic waves and spatial correlations. Another extension was considered in the modeling of the race between the HIV and the immune system, where the topologies induced by the possible mutations in the shape spaces of the virus and of the immune cells are different. This has important effects which go beyond the ones that can be deduced on the basis of the knowledge of the static percolation transition.

The marketing percolation transition phenomenon corresponds to the hit-flop phenomenon in the movies industry: some movies make a fortune while others, with no significant quality difference never take-off . However, with appropriate changes the framework can characterize many market phenomena, as well as epidemics, novelty propagation, etc.

FURTHER MODIFICATIONS OF THE SOCIAL PERCOLATION MODEL

In order to make the model more realistic, one has to introduce additional features, but the crucial one is the understanding that the actual sales depend in a dramatic way on the spatial distribution of the customers and on the communication between them. Until recently, the mass of customers was considered uniform and each individual connected to the entire public in an "average" way. Obviously this concept missed the crucial effects mentioned above and many others described below. Also the latest applications take into account the negative influence that dissatisfied customers may have on other potential buyers.

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