[PPT] - Social Influence, Avalanches and Learning Jean-Benot Zimmermann PowerPoint Presentation

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Social Influence, Avalanches and Learning

Jean-Benoît Zimmermann (GREQAM) joint work with Alexandre Steyer (PRISM-Paris I) Workshop « Complex Markets » October 2006

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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 Motivations

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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.

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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”

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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”

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So the diffusion of an innovation, a standard, an
pinion 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

ver the social structure
1. Diffusion:

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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)

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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

bserve 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).

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2. A very simplified social system
Individual's state θι∈ {0,1}
each individual θι=1 is likely to influence k other

individuals

denote p0 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.

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1 p0 . i . . k

on average, i leads k p0 individuals to switch.
In turn, each of these latter leads kp0 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- p0)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

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3. Avalanches size and duration
The expected number of individuals won over at time t can

be written

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

f 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

∑ ∑

= =

= +

t t

kp kp

1

) ( ) ( 1

τ τ τ τ

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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
bserved…

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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.

4. Social Learning

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➲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.

Homophily (Rogers and Bhowmik, 1971) :

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Application – case k=2

A given agent i exerts influence on her two neighbours l

and m . Denote pl and pm their respective probabilities to actually switch, under its influence

pl l i pm 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

pl = f(a) and pm = f(1-a) where f is a concave function (decreasing returns), for instance pl = a1/α and pm = (1-a)1/α with α > 1

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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, pl and pm don’t change – If l adopts but m rejects pl

α→ pl α + λ and pm α →pm α - λ

so pl

α + pm α = Cte

– In the former illustration the allocation (a,1-a) becomes

(a+λ,1-(a+λ)) λ

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How does this learning affect the dynamics
f 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 p0 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 i0 is set to 1, influence

propagate on the tree issued from i0 through those

f 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 i0 on this new tree relational structure.

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Global dynamics of the network, or even of the

tree generated from i0 , 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

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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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Proposition 1 : If α > 1, the local

expectation of adoption (at the scale of on individual agent's influence on his direct neighbours) is decreasing subsequently to each step of learning.

Proposition 2: If α < 1 / (1−p0) , the local

probability of the avalanche stopping (at the scale of an individual agent's influence on his direct neighbours) is decreasing subsequently to each step of learning.

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Agents influenced in t=0 Probability

l

p1

m

p1 l p0 (1 - p0)

α α

λ

/ 1

) (

+

p

α α

λ

/ 1

) (

−

p l and m p0² p0 p0 m p0 (1 - p0)

α α

λ

/ 1

) (

−

p

α α

λ

/ 1

) (

+

p none (1 – p0)² p0 p0 Table 2: probabilities of influence after one step of learning

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Agents influenced in t Probability

l t

p

1

+

m t

p

1

+

l

) 1 (

m t l t

p p

−

α α α α

λ λ

/ 1 / 1

) ) 1 ( ( ) (

+ + = +

n p pl

t

α α α α

λ λ

/ 1 / 1

) ) 1 ( ( ) (

+ − = −

n p p m

t

l and m

m t l t p

p

l t

p

m t

p

m

m t l t p

p ) 1 ( −

α α α α

λ λ

/ 1 / 1

) ) 1 ( ( ) (

− + = −

n p pl

t

α α α α

λ λ

/ 1 / 1

) ) 1 ( ( ) (

− − = +

n p p m

t

none

) 1 )( 1 (

m t l t

p p

− −

l t

p

m t

p Table 3: probabilities of influence after t+1 steps of learning

More generally after a given duration t of learning there exists

Ν ∈

n

such that

α α

λ / 1

) ( n p p

l t + =

and

α α

λ / 1

) ( n p p

m t − =

r conversely by inverting l

and m.

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When 1 < α < 1 / (1−p0)

Avalanches evolution through learning is characterized by a decreasing thickness but a growing duration.

Diffusion becomes less and less efficient on

an instantaneous local level, but it lasts longer and longer.

This leads to a situation very similar to the

unrealistic kp=1 case of our first homogeneous network model.

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6. Numerical simulations
Aim: compare diffusion on a tree before and after

learning

Starting from an homogeneous network (every agent

can influence two others with a probability p0)

Triggering a series of diffusion avalanches on the tree

proceeding from a given individual before and after learning

At each step all the indivduals are put to 0 and the root

individual to 1

Measuring the size and duration distribution of the

avalanches (diffusion effect restricted from one single pioneer)

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6.1. Before learning: Exponential distribution of avalanches duration 6.2. Social learning

150 learning avalanches
At each step, links are reevaluated

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On this new structure of tree : power law distribution of avalanches duration

Here we obtain a probability roughly inversely proportional to the cube of the duration

More precisely with an exponent slightly greater

than 3, meaning finite moments of first and second

rder.

Inverse cumulative distribution of avalanche durations after learning

0,0001 0,001 0,01 0,1 1 5 10 15 20 DURATIONS Ln(Prob of longer Durat

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Size of the avalanches (i.e. the total number of

influenced agents generated by an avalanche) is

proportional to a power of their duration. Here we obtain a power of 1.18 : thus the network exerts an amplification effect.

By combination this leads to a power law

distribution of the size of the avalanches with an exponent = 2.7 ∈ ]2,3[ ⇒ infinite variance

(but finite expected value)

The relation between size and duration of avalanches (after learning).

1 10 100 1 10 100

Ln(duration) Ln(size

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So the network structure has been deeply

transformed trough social learning process.

Avalanches have finite durations on the contrary to

the exponential growth invading the whole network with no limits than the space of the network itself

We are therefore not in the case of the

supercritical network

Impossible to calculate a standard deviation of the

size of the avalanches.

We are not either in the case of a subcritical

network

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We are in an intermediary situation:

– Like in supercritical state, the size of

avalanches is unlimited (finite first order and infinite second order moments)

– Like in the subcritical state, their duration is

limited (finite first and second order moments)

In this new state arising naturally out of social

learning, the noise generated by avalanches into the diffusion curve is infinitely variable and doesn't therefore satisfy the conditions of application of the central limit theorem.

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7. Consequences (as a way of conclusion)
Predictive models of market size and growth when social

interactions play an important role

– Trend to underestimate the market size and growth (ex:

mobile phone)

– Diffusion dynamics on such networks structure: noise itself

generates new avalanches

– Need for econometric methods casting off the distribution of

the residual error or integrating specific distribution of it

Marketing approaches (viral marketing, ethnic and groups based

marketing ...)

– Limits of the classical epidemic models – Importance of the opinion leaders

Financial markets : bubble formation
Rumours spreading, Radical vote
...