Social Influence, Avalanches and Learning Jean-Benoît Zimmermann (GREQAM) joint work with Alexandre Steyer (PRISM-Paris I) Workshop « Complex Markets » October 2006
Motivations ● New markets formation and growth, innovation success, standardization, ... thus competition (products, standards, technologies ...) ● Rest on individual acceptance of these new products,innovation, standard ● Central question = coherence of individual consumption behaviours, representation and attitudes, decisions
● Dynamic approach of coherence emergence and evolution : a question of diffusion ● Diffusion = dynamic aggregation of individual changing behaviours, beliefs and representations ● Driving forces : – Global signals (equally delivered to all agents) : prices, advertising, legal enforcement, ... – “Local signals” from individual direct interactions the important role of Social Influence .
● Individuals generate their own states, based on the signals they receive from their social environment, and in turn they influence their environment by sending back signals of these states. ● Gabriel Tarde (French Sociologist – 1884) about the fundamental role of social influence in the formation of the value of an invention: “Before becoming a production and exchange of services, society is firstly a production and exchange of needs and a production and exchange of beliefs; this is indispensable”
Bikhchandani, Hiershleifer and Welch (1992) ● “Consider a teenager deciding whether or not to experiment with drugs. A strong motive for experimenting with drugs is the fact that friends are doing so. Conversely, seeing friends reject drugs could help persuade a youth to stay clean” ● “Although the outcome may or may not be socially desirable, a reasoning process that takes into account the decisions of others is entirely rational even if individuals place no value on conformity for its own sake”
1. Diffusion: ● So the diffusion of an innovation, a standard, an opinion or an information should be understood from its dynamic manifestation, testifying to its propagation within a population. ● E.M.Roger (1983) about innovation diffusion “Diffusion is the process by which an innovation is communicated through certain channels over time among the members of a social system ” ● Diffusion takes place from a given number of pioneers (innovators) propagating their influence over the social structure
● Inter-individual influence propagation generates complex dynamics into which the social structure takes a decisive part ● Most of the diffusion models – Either do not integrate any social structure (anybody can influence any other agent with the same probability – epidemic models – or every newcomer can observe all the still established agents – Increasing adoption returns, informational cascades ) – Or consider agents embedded on a networks structure whose links values are drawn randomly (random networks)
● However, in the real world, social networks are issued from long path dependent processes of links formation and evolution that generate peculiar topologies unlikely to be drawn randomly. ● Aim of the paper: – Build a protocol of links evolution driven by the history of past inter-individual interactions – Study the social influence propagation borne by this structure inherited from past. On the contrary to the “informational cascades” of Bickchandani and al. individuals are not likely to observe the decisions of all the preceding indi- viduals but only those of the individuals they are in relation with. Their interactions are “embed- ded” in a social network (Granovetter).
2. A very simplified social system ● Individual's state θ ι ∈ {0,1} ● each individual θ ι =1 is likely to influence k other individuals ● denote p 0 the probability for each of them to be influenced. Switching from 0 to 1. ● Avalanches (Steyer, 1993) are triggered from a given individual (source): social influence spreads step by step along the subgraph (tree) stemming from her.
1 p 0 . i . . k ● on average, i leads k p 0 individuals to switch. ● In turn, each of these latter leads kp 0 other individuals to switch, and so on, in a sort of snowball effect. ● but it is equally possible for the process to stop ( with a probability (1- p 0 ) k at the beginning of the process). ● an avalanche is said of duration t if there are t-1 intermediate individuals between the first and the last switching individual
3. Avalanches size and duration ● The expected number of individuals won over at time t can be written t t ∑ ∑ + τ = τ 1 ( kp ) ( kp ) τ = τ = 1 0 ● if p < 1/ k , the number of influenced individuals tends towards a finite limit and remains confined in a local scale; avalanches are of finite duration ; the distribution of the size of the avalanches is exponential. ● If p > 1/k , the expected number of agents won over is “infinite”; avalanches almost certainly never stop, while the potential market is not totally saturated. ● Let's call these two configurations subcritical and supercritical cases
● The case p=1/k presents very interesting characteristics, such as the appearance of a power law distribution. Like in many empirical phenomena (Sheinkman & Woodford (1994)) ● However, no social or economic mechanism exists to stabilise the system on this particular value. ● In itself, therefore, it has no chance of being observed…
4. Social Learning ● The basic idea here is to build a more social- effective network as issued from past inter- individuals interactions ● Inter-individual links evolution is the result of a “social learning process”, that makes each individual continuously reconsider how her neighbours state or opinion does matter into her proper decision process. ● So the network structure that will drive a diffusion process in the present time, has to be considered as issued from the cumulative path-dependent process of social learning caused by inter- individual interactions in the past.
Homophily (Rogers and Bhowmik, 1971) : ➲ When two individuals have shared the same attitudes in the past, they feel closer and so they are likely to pay more attention to each other. ● So, an agent will socialize more intensively (for instance spare more time or communicate more) with those of her neighbours that have already been sensible to her influence. ● Hence her influence upon these latter tends to increase.
Application – case k=2 ● A given agent i exerts influence on her two neighbours l and m . D enote p l and p m their respective probabilities to actually switch, under its influence p l l i p m m ● Interpretation : - i owns a given amount 1 of socialization potential (e.g. Communication potential, time, affect ...). - She allocates (a, 1-a) towards l and m, - Then she influences them with probabilities p l = f(a) and p m = f(1-a) where f is a concave function p l = a 1/ α (decreasing returns), for instance and p m = (1-a) 1/ α with α > 1
● Homophily : When i has performed a greater influence upon l than m she feels a better proximity to l and she will reallocate her socialization potential in favour to l. (NB the opposite assumption could be imagined as a way to improve the efficiency of influence propagation, hence of diffusion. But this wouldn't correspond to the “social fit”, because individuals utility is nor derived from their “propagation power” but rather from their social satisfaction ) – if l and m make same decision, p l and p m don’t change α → p l α + λ and p m α → p m α - λ – If l adopts but m rejects p l so p l α + p m α = Cte – In the former illustration the allocation (a,1-a) becomes (a+ λ ,1-(a+ λ )) λ
● How does this learning affect the dynamics of the avalanches? The approach : – Starting from an homogenous network as in the static model : each individual is likely to respectively influence k others with a p 0 probability – Trigger are a series of avalanches from a given individual – for each avalanche : ● All individuals' states are firstly put to 0 ● The root individual i 0 is set to 1, influence propagate on the tree issued from i 0 through those of the descendants that consecutively switch to 1 ● Individuals on the tree revise their relationships according to the above principles of social learning – After having achieved this series of learning avalanches, the network is frozen. A diffusion avalanche is then triggered and studied from i 0 on this new tree relational structure.
● Global dynamics of the network, or even of the tree generated from i 0 , is too complex for analytical methods ⇒ 2 alternative methods: – Study the effects of learning on “local” dynamics – Simulations: successive steps learning and diffusion before and after learning
5. Local Analytical Heuristic: ➢ We seek to understand how the evolution of links intensity through relational learning produces transformations in the conditions of diffusion at a “local” level l i m expectation of adoption which governs the “thickness” of avalanches and the probability of stopping which governs the duration of avalanches.
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