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Opulence. Maybe there's something in the air of Paris that inspires it.<br><br>


'''DeGroot learning''' refers to a rule-of-thumb type of social learning process. The idea was stated in its general form by the American statistician [[Morris DeGroot|Morris H. DeGroot]];<ref name=DeGroot74/> antecedents were articulated by John R. P. French<ref name=French56/> and Frank Harary.<ref name=Harary59/> The model has been used in [[physics]], [[computer science]] and most widely in the theory of [[Social network analysis|social networks]].<ref name = Jackson2008/>
There's the proliferation of palais (petit or grand) for designers to show their collections in, the galleries filled with florid, lurid art, and the haute couture, of course, opulence cubed and then wrapped in duchesse satin for good measure. They all do something to designers, no matter how minimally minded.<br><br>It's certainly done something to Marc Jacobs. Compare and contrast the first Louis Vuitton collection he presented oh-so-quietly 16 years ago with the ball-busting extravaganza unveiled last Wednesday, part �Turn Back Time� video, part Paul Verhoeven's Showgirls<br><br>
It was an indulgent farewell to a house that has helped redefine the meaning of luxury goods in the 21st century. Indulgent for Jacobs, but also for the audience, a funereal feast for the eyes, Jacobs' all-black swansong. The Stephen Jones headdresses were inspired by Ziegfeld Follies and Cher's 1986 Oscars outfit<br>
Says it all<br>
Jacobs declared that the collection was obsessed with �pure adornment�, reasoning that �connecting with something on a superficial level is as honest as connecting with it on an intellectual level�. That's an interesting [http://search.huffingtonpost.com/search?q=conceit&s_it=header_form_v1 conceit]. Contemporary fashion, like contemporary art, tends to be overthoug<br><br>


== Setup and the learning process ==
If not by designers, then certainly by critics. After all, we have to justify our presence there. And the Air Miles we rack <br>.
Take a society of <math> n </math> agents where everybody has an opinion on a subject, represented by a vector of probabilities <math> p(0) = (p_1(0), \dots, p_n(0) ) </math>. Agents obtain no new information based on which they can update their opinions but they communicate with other agents. Links between agents (who knows whom) and the weight they put on each other's opinions is represented by a trust matrix <math> T </math> where <math> T_{ij} </math> is the weight that agent <math> i </math> puts on agent <math> j </math>'s opinion. The trust matrix is thus in a one-to-one relationship with a [[Weighted graph|weighted]], [[directed graph]] where there is an edge between <math> i </math> and <math> j </math> if and only if <math> T_{ij} > 0 </math>. The trust matrix is [[Stochastic matrix|stochastic]], its rows consists of nonnegative real numbers, with each row summing to 1.
Marco Zanini also showed his final collection for the house of Rochas. He's moving to Schiaparelli. Like Jacobs, he was obsessed with surface, with dumb, straightforward beauty. His was sugary sweet, in pastels that made your teeth ache, dedicated to Tennessee Williams' The Glass Menager<br>.
Banish thoughts of tormented southern belles. Zanini focused on the crystalline beauty of said ornaments, bonding devor� velvet to organza and freckling iridescent fabrics with Swarovski g<br><br>


Formally, the beliefs are updated in each period as
Maria Grazia Chiuri and Pier Paolo Piccioli took a precious approach, too. Their Valentino collection was inspired by the Rome Opera, apparently. In actual fact, it was all about the heavily embellished surfaces of lace and tulle. Lawn shirts seemed included purely as foil for all that decoration, a palette clean<br><br>


:<math>
It would have made it easier to enjoy Hedi Slimane's latest Saint Laurent collection if you could take it at face value. Look! Sequinned lips! Flames! Lurex pop-socks! Removed from the heritage of Saint Laurent, it had a pop, pap appeal. If you stopped looking for a hidden depth, the soul-searching that Yves Saint Laurent made an intrinsic part of his fashion, Slimane ticked bo<br><br>
p(t) = T p(t-1)
</math>


so the <math> t </math> th period opinions are related to the initial opinions by
Girls who want to look like that will love to dress in t<br>s.
Karl Lagerfeld has always been about surface. His toying with the hallmarks of Chanel - tweeds, camellias, pearls, chains, those two-tone shoes - has always been about ironic appropriation, post-modern reinterpretation. It's the fashion equivalent of Jeff Ko<br><br>


:<math>
He showed his latest Chanel collection in an art gallery. At least, it was on the surface. It was all fake, only the clothes were real. And they were pure Chanel, the art-house backdrop just that. I kept thinking of something Dinos Chapman once said to me: �I think that the art world and the high-end fashion world� are the same peop<br>.�
p(t) = T^t p(0)
</math>


== Convergence of beliefs and consensus ==
People wanted to buy the Chanel works of art as surely as they wanted to buy the Chanel clothes. They both became post-modern commod<br>ies.
That's looking below the surface, though, behind the opulence of Chanel's specially woven, artfully unravelling tweeds that resembled rag-rugs, the canvas bags with a 2.55 fa�ade painted on the front, the graffitied art-student backpacks. They were just great, covetable fashion, brilliant pro<br><br>s.


An important question is whether beliefs converge to a limit and to each other in the long run.  
That brings us, inevitably, to Phoebe Philo. She's known for great products - photography is banned in the C�line showroom for fear of rampant copying. And rightfully so. Still, that oft-imitated C�line hallmark is an ascetic aesthetic. Philo is the last designer one imagines inclined to opu<br><br>e.
As the trust matrix is [[Stochastic matrix|stochastic]], standard results in [[Markov chain]] theory can be used to state conditions under which the limit


:<math>
But her riotously messy spring C�line collection felt fresh and energetic, pleated skirts bouncing below an elongated torso, fringe swaying on latticed leather [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bags Outlet], mashed-up metal formed into enamelled bracelets and <br>els.
p(\infty) = \lim_{t \to \infty} p(t) = \lim_{t \to \infty} T^t p(0)
Couture, of course, breeds opulence. Riccardo Tisci halted his made-to-measure line this year, but the handicraft of couture infected his spring collection, from the crystal-encrusted masks, to feather-embroidered bodices, to a series of sequinned, sinuous multi-pleat evening <br><br>s.
</math>
exists for any initial beliefs <math> p(0) \in [0, 1]^n </math>. The following cases are treated in Golub and Jackson
<ref name = GolJack2010/> (2010).


=== Strongly connected case ===
Those were old-school opulent. For modern couture, and a contemporary opulence, fashion turns to Raf Simons. His last Dior haute-couture collection met mixed reviews, but has inspired many a designer. The throbbing mood of Africa that beat through the collections, albeit slightly hackneyed, can be traced to S<br>ons.
This time, he pushed his aesthetic further still. It felt a collection in flux, sitting halfway between Dior past and Simons future, a cross-pollination. The inspiration was flowers, a theme at the very root of Dior. But the best summary was the least ostentatious: Simons' shirtdresses, twisted takes on the white cotton coats of workers corkscrewing around the body in a fascinating surface of complex <br><br>s.


If the social network graph (represented by the trust matrix) is [[strongly connected]], convergence of beliefs is equivalent to each of the followings
So, French fashion is all about surface. Where does that leave Rei Kawakubo and Junya Watanabe, or Hussein Chalayan? The visual is just a fragment of what they offer, thinking clothes for the thinking woman. Jacobs may be in love with �beauty for beauty's sake�, but this trio of talents begs for somethin<br>more.
* the graph represented by <math> T </math> is [[Aperiodic graph|aperiodic]]
They all had a stellar, cerebral season: Watanabe creating an idiosyncratic ode to the spaghetti Western, Chalayan a paean to the windswept beach, while Kawakubo presented 23 non-outfits that challenged our perceptions of what fashion actually represents. She stated that she had no new ideas, so didn't create c<br><br>es.
* there is a unique left [[Eigenvalues and eigenvectors|eigenvector]] <math> s </math> of <math> T </math> corresponding to [[Eigenvalues and eigenvectors|eigenvalue]] 1 whose
entries sum to 1 such that, for every <math> p \in [0, 1]^n </math>, <math> \left( \lim_{t \to \infty} T^t p \right)_i = s \cdot p </math>


for every <math> i \in \{1, \dots, n \} </math> where <math> \cdot </math> denotes the [[dot product]].
If only other designers could follow her lead and thin out th<br>herd.
The surface of this deep-thinking threesome's clothing was universally impressive, but it's what lies beneath that really interests. Opulent intelligence? Their clothes beg dissection and discussion. To intellectualise a fashion show isn't automatically to over- intellectual<br><br>it.


=== General case ===
Ultimately, that's what fashion is about. Surface is all well and good. But you have to get someone inside the damn clothes for it to all make<br><br>se.


It is not necessary to have a [[strongly connected]] social network to have convergent beliefs, however,
It's ideological, as well as physical. At least, it is when it's really great.
the equality of limiting beliefs does not hold in general.
 
We say that a group of agents <math> C \subseteq \{1, \dots, n \} </math> is ''closed'' if for any <math> i \in C </math>, <math> T_{ij} > 0 </math> only if <math> j \in C </math> . Beliefs are convergent if and only if every set of nodes (representing individuals) that is strongly connected and closed is also [[Aperiodic graph|aperiodic]].
 
=== Consensus ===
 
A group <math> C </math> of individuals is said to reach a ''consensus'' if <math> p_i(\infty)= p_j(\infty) </math> for any <math> i, j \in C </math>. This means that, as a result of the learning process, in the limit they have the same belief on the subject.
 
With a [[strongly connected]] and [[Aperiodic graph|aperiodic]] network the whole group reaches a consensus.
In general, any strongly connected and closed group <math> C </math> of individuals reaches a consensus for every initial vector of beliefs if and only if it is aperiodic. If, for example, there are two groups satisfying these assumptions, they reach a consensus inside the groups but there is not necessarily a consensus at the society level.
 
== Social influence ==
 
Take a [[strongly connected]] and [[Aperiodic graph|aperiodic]] social network. In this case, the common limiting beliefs are determined by the initial beliefs  through
 
:<math>
p(\infty) = s \cdot p(0)
</math>
 
where <math> s </math> is the unique unit length [[Eigenvalues and eigenvectors|left eigenvector]] of <math> T </math> corresponding to the [[Eigenvalues and eigenvectors|eigenvalue]] 1. The vector <math> s </math> shows the weights that agents put on each other's initial beliefs in the consensus limit. Thus, the higher is <math> s_i </math>, the more ''influence'' individual <math> i </math> has on the consensus belief.
 
The eigenvector property <math> s = s T </math> implies that
:<math> s_i = \sum_{j=1}^n T_{ji} s_j </math>
 
This means that the influence of <math> i </math> is a weighted average of those agents' influence <math> s_j </math> who pay attention to <math> i </math>, with weights of their level of trust. Hence influential agents are characterized by being trusted by other individuals with high influence.
 
== Examples ==
These examples appear in Jackson <ref name=Jackson2008 /> (2008).
 
=== Convergence of beliefs ===
[[File:De Groot learning convergent example.png|200px|right|thumb|A society with convergent beliefs]]
Consider a three-individual society with the following trust matrix:
 
:<math>
T =
\begin{pmatrix}
0 & 1/2 & 1/2 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{pmatrix}
</math>
 
Hence the first person weights the beliefs of the other two with equally, while
the second listens only to the first, the third only to the second individual.
For this social trust structure, the limit exists and equals
 
:<math>
\lim_{t \to \infty} T^t p(0) = \left(\lim_{t \to \infty} T^t\right) p(0) = \begin{pmatrix}
2/5 & 2/5 & 1/5 \\
2/5 & 2/5 & 1/5 \\
2/5 & 2/5 & 1/5 \\
\end{pmatrix} p(0)
</math>
 
so the influence vector is <math> s = \left( 2/5, 2/5, 1/5 \right) </math> and the consensus belief is
<math> 2/5 p_1(0) + 2/5 p_2(0) + 1/5 p_3(0) </math>. In words, independently of the initial beliefs,
individuals reach a consensus where the initial belief of the first and the second person has twice as
high influence than the third one's.
 
=== Non-convergent beliefs ===
[[File:De groot learning nonconvergent beliefs.png|200px|right|thumb|A society with non-convergent beliefs]]
If we change the previous example such that the third person also listens exclusively to the first
one, we have the following trust matrix:
 
:<math>
T =
\begin{pmatrix}
0 & 1/2 & 1/2 \\
1 & 0 & 0 \\
1 & 0 & 0 \\
\end{pmatrix}
</math>
 
In this case for any <math> k \geq 1 </math> we have
 
:<math>
T^{2k - 1} =
\begin{pmatrix}
0 & 1/2 & 1/2 \\
1 & 0 & 0 \\
1 & 0 & 0 \\
\end{pmatrix}
</math>
 
and
:<math>
T^{2k} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1/2 & 1/2 \\
0 & 1/2 & 1/2 \\
\end{pmatrix}
</math>
 
so <math> \lim_{t \to \infty} T^t </math> does not exist and
beliefs do not converge in the limit. Intuitively, 1 is updating based on 2 and 3's beliefs while
2 and 3 update solely based on 1's belief so they interchange their beliefs in each period.
 
== Asymptotic properties in large societies: wisdom ==
 
It is possible to examine the outcome of the DeGroot learning process in large societies,
that is, in the <math> n \to \infty </math> limit.
 
Let the subject on which people have opinions be a "true state" <math> \mu \in [0, 1] </math>. Assume that individuals
have [[Independence (probability theory)|independent]] noisy signals <math> p_i^{(0)}(n) </math> of <math> \mu </math>
(now superscript refers to time, the argument to the size of the society).
Assume that for all <math> n </math> the trust matrix <math> T(n) </math> is such that the
limiting beliefs <math> p_i^{(\infty)}(n) </math> exists independently from the initial beliefs.
Then the sequence of societies <math> \left( T(n) \right)_{n = 1}^{\infty} </math> is called ''wise'' if
 
:<math>
\max_{i \leq n} | p_i^{(\infty)} - \mu | \xrightarrow{\ p\ } 0
</math>
 
where <math> \xrightarrow{\ p\ } </math> denotes [[Convergence of random variables|convergence in probability]].
This means that if the society grows without bound, over time they will have a common and accurate
belief on the uncertain subject.
 
A necessary and sufficient condition for wisdom
can be given with the help of [[#Social influence|influence vectors]]. A sequence of societies is wise if and only
if
:<math>
\lim_{n \to \infty} \max_{i \leq n} s_i(n) = 0
</math>
that is, the society is wise precisely when even the most influential individual's influence vanishes in the
large society limit. For further characterization and examples see Golub and Jackson<ref name=GolJack2010 /> (2010).
 
== References ==
<references>
* <ref name = DeGroot74>DeGroot, Morris H. 1974. “[http://www.jstor.org/stable/10.2307/2285509 Reaching a Consensus.]” ''Journal of the American Statistical Association'', 69(345): 118–21.</ref>
 
* <ref name = French56>French, John R. P. 1956. “A Formal Theory of Social Power” ''Psychological Review'', 63: 181–94.</ref>
 
* <ref name = GolJack2010>Golub, Benjamin & Matthew O. Jackson 2010. "[http://www.aeaweb.org/articles.php?doi=10.1257/mic.2.1.112 Na&iuml;ve Learning in Social Networks and the Wisdom of Crowds,]" American Economic Journal: Microeconomics, American Economic Association, vol. 2(1), pages 112-49, February.</ref>
 
* <ref name=Jackson2008>Jackson, Matthew O. 2008. ''Social and Economic Networks.'' Princeton University Press.</ref>
 
* <ref name = Harary59>Harary, Frank. 1959. “A Criterion for Unanimity in French's Theory of Social Power” in Dorwin Cartwright (ed.), ''Studies in Social Power'', Ann Arbor, MI: Institute for Social Research.</ref>
 
</references>
 
[[Category:Social learning theory]]

Latest revision as of 20:51, 29 March 2014

Opulence. Maybe there's something in the air of Paris that inspires it.

There's the proliferation of palais (petit or grand) for designers to show their collections in, the galleries filled with florid, lurid art, and the haute couture, of course, opulence cubed and then wrapped in duchesse satin for good measure. They all do something to designers, no matter how minimally minded.

It's certainly done something to Marc Jacobs. Compare and contrast the first Louis Vuitton collection he presented oh-so-quietly 16 years ago with the ball-busting extravaganza unveiled last Wednesday, part �Turn Back Time� video, part Paul Verhoeven's Showgirls

It was an indulgent farewell to a house that has helped redefine the meaning of luxury goods in the 21st century. Indulgent for Jacobs, but also for the audience, a funereal feast for the eyes, Jacobs' all-black swansong. The Stephen Jones headdresses were inspired by Ziegfeld Follies and Cher's 1986 Oscars outfit
Says it all
Jacobs declared that the collection was obsessed with �pure adornment�, reasoning that �connecting with something on a superficial level is as honest as connecting with it on an intellectual level�. That's an interesting conceit. Contemporary fashion, like contemporary art, tends to be overthoug

If not by designers, then certainly by critics. After all, we have to justify our presence there. And the Air Miles we rack
. Marco Zanini also showed his final collection for the house of Rochas. He's moving to Schiaparelli. Like Jacobs, he was obsessed with surface, with dumb, straightforward beauty. His was sugary sweet, in pastels that made your teeth ache, dedicated to Tennessee Williams' The Glass Menager
. Banish thoughts of tormented southern belles. Zanini focused on the crystalline beauty of said ornaments, bonding devor� velvet to organza and freckling iridescent fabrics with Swarovski g

Maria Grazia Chiuri and Pier Paolo Piccioli took a precious approach, too. Their Valentino collection was inspired by the Rome Opera, apparently. In actual fact, it was all about the heavily embellished surfaces of lace and tulle. Lawn shirts seemed included purely as foil for all that decoration, a palette clean

It would have made it easier to enjoy Hedi Slimane's latest Saint Laurent collection if you could take it at face value. Look! Sequinned lips! Flames! Lurex pop-socks! Removed from the heritage of Saint Laurent, it had a pop, pap appeal. If you stopped looking for a hidden depth, the soul-searching that Yves Saint Laurent made an intrinsic part of his fashion, Slimane ticked bo

Girls who want to look like that will love to dress in t
s. Karl Lagerfeld has always been about surface. His toying with the hallmarks of Chanel - tweeds, camellias, pearls, chains, those two-tone shoes - has always been about ironic appropriation, post-modern reinterpretation. It's the fashion equivalent of Jeff Ko

He showed his latest Chanel collection in an art gallery. At least, it was on the surface. It was all fake, only the clothes were real. And they were pure Chanel, the art-house backdrop just that. I kept thinking of something Dinos Chapman once said to me: �I think that the art world and the high-end fashion world� are the same peop
.�

People wanted to buy the Chanel works of art as surely as they wanted to buy the Chanel clothes. They both became post-modern commod
ies. That's looking below the surface, though, behind the opulence of Chanel's specially woven, artfully unravelling tweeds that resembled rag-rugs, the canvas bags with a 2.55 fa�ade painted on the front, the graffitied art-student backpacks. They were just great, covetable fashion, brilliant pro

s.

That brings us, inevitably, to Phoebe Philo. She's known for great products - photography is banned in the C�line showroom for fear of rampant copying. And rightfully so. Still, that oft-imitated C�line hallmark is an ascetic aesthetic. Philo is the last designer one imagines inclined to opu

e.

But her riotously messy spring C�line collection felt fresh and energetic, pleated skirts bouncing below an elongated torso, fringe swaying on latticed leather Celine Bags Outlet, mashed-up metal formed into enamelled bracelets and
els. Couture, of course, breeds opulence. Riccardo Tisci halted his made-to-measure line this year, but the handicraft of couture infected his spring collection, from the crystal-encrusted masks, to feather-embroidered bodices, to a series of sequinned, sinuous multi-pleat evening

s.

Those were old-school opulent. For modern couture, and a contemporary opulence, fashion turns to Raf Simons. His last Dior haute-couture collection met mixed reviews, but has inspired many a designer. The throbbing mood of Africa that beat through the collections, albeit slightly hackneyed, can be traced to S
ons. This time, he pushed his aesthetic further still. It felt a collection in flux, sitting halfway between Dior past and Simons future, a cross-pollination. The inspiration was flowers, a theme at the very root of Dior. But the best summary was the least ostentatious: Simons' shirtdresses, twisted takes on the white cotton coats of workers corkscrewing around the body in a fascinating surface of complex

s.

So, French fashion is all about surface. Where does that leave Rei Kawakubo and Junya Watanabe, or Hussein Chalayan? The visual is just a fragment of what they offer, thinking clothes for the thinking woman. Jacobs may be in love with �beauty for beauty's sake�, but this trio of talents begs for somethin
more. They all had a stellar, cerebral season: Watanabe creating an idiosyncratic ode to the spaghetti Western, Chalayan a paean to the windswept beach, while Kawakubo presented 23 non-outfits that challenged our perceptions of what fashion actually represents. She stated that she had no new ideas, so didn't create c

es.

If only other designers could follow her lead and thin out th
herd. The surface of this deep-thinking threesome's clothing was universally impressive, but it's what lies beneath that really interests. Opulent intelligence? Their clothes beg dissection and discussion. To intellectualise a fashion show isn't automatically to over- intellectual

it.

Ultimately, that's what fashion is about. Surface is all well and good. But you have to get someone inside the damn clothes for it to all make

se.

It's ideological, as well as physical. At least, it is when it's really great.

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