Aprendizaje y Redes de Información · 2020. 10. 14. · da a la opini on del agente j en la...

Imitacion e Influencia SocialPageRank

Aprendizaje y Redes de Informacion

Alvaro J. Riascos Villegas

Octubre de 2020

Universidad de los Andes y Quantil | Matematicas Aplicadas Aprendizaje en Redes y Redes de Informacion


Contenido

1 Imitacion e Influencia Social

2 PageRank



Modelo de DeGroot (1974)

Este es un modelo de transmision de informacion, formacionde opinion y formacion de consensos.

Fijemos una red donde los nodos son agentes.

Cada agente i tiene una opinion (creencia) inicial sobre alguntema que representamos con pi (0) ∈ [0, 1]

La interaccion entre estas opiniones se representan por unamatriz n× n,T donde Tij representa el peso que el agente i leda a la opinion del agente j en la formacion de la opinion de ipara el siguiente periodo (T es estocastica por filas: suma porfilas es uno).

Las opiniones evolucionan en el tiempo de acuerdo a:

p(t + 1) = Tp(t)


Ejemplo

Supongamos que tenemos 3 agentes y T es:

1/3 1/3 1/31/2 1/2 0

0 1/4 3/4

Para incorporar lıderes de opinion se puede fijar para un lideri , Tii = 1,Tij = 0, j 6= i y para algun k, Tki 6= 0.

Ejemplo

Si Tij > 0 entonces se hace un enlace dirigido de i a j .8.3. IMITATIONAND SOCIAL INFLUENCEMODELS: THEDEGROOTMODEL295

1

2

3

1/3

1/3

1/2 1/41/3

3/4

1/2

Figure 8.3. An Updating Process

Then, as agents update again, beliefs become

p(2) = Tp(1) =

0B@ 1=3 1=3 1=3

1=2 1=2 0

0 1=4 3=4

1CA0B@ 1=3

1=2

0

1CA =

0B@ 5=18

5=12

1=8

1CA :

Iterating this process leads to beliefs that converge:

p(t) = Tp(t� 1) = T tp(0)!

0B@ 3=11

3=11

3=11

1CA :

The way in which we can calculate the limit belief is discussed in detail below.

This process has the following motivation, discussed by DeMarzo, Vayanos and

Zwiebel [183]. Agents are connected by a (possibly directed) network indicating whose

information they get to observe over time. At time t = 0, each agent sees a noisy

signal pi(0) = �+ ei where ei 2 IR is a noise term. An agent i then hears the opinionshis or her neighbors, and assigns precision �ij to the signal of agent j. If agents are

Bayesian and the noise terms are normally distributed with zero mean, then agent i

Ejemplo

En este ejemplo en el que T es:

1/3 1/3 1/31/2 1/2 0

0 1/4 3/4

Ejemplo: convergencia

Decimos que T es convergente si no importan las creenciasiniciales, las creencias en el lımite convergen.

8.3. IMITATIONAND SOCIAL INFLUENCEMODELS: THEDEGROOTMODEL297

1

2

3

1/2

1/2

1 1

Figure 8.3.2. A Society with a Convergent Updating Process

A social in�uence matrix T is convergent if limt Ttp exists for all initial vectors of

beliefs p.

This is illustrated in the following example.

Example 8.3.2 Convergence

There are three individuals and an updating matrix described by

T =

0B@ 0 1=2 1=2

1 0 0

0 1 0

1CA :

This updating process is pictured in Figure 8.3.2.

Here, one can check that

T 2 =

0B@ 1=2 1=2 0

0 1=2 1=2

1 0 0

1CA ; T 3 =

0B@ 1=2 1=4 1=4

1=2 1=2 0

0 1=2 1=2

1CA ; T 4 =

0B@ 1=4 1=2 1=4

1=2 1=4 1=4

1=2 1=2 0

1CA ; : : : :

and

T t !

0B@ 2=5 2=5 1=5

2=5 2=5 1=5

2=5 2=5 1=5

1CA :

Ejemplo: convergencia

En este caso:

0 1/2 1/21 0 00 0 0

No es difıcil mostrar que:

T∞ =

2/5 2/5 1/52/5 2/5 1/52/5 2/5 1/5

y

p1(∞) = p2(∞) = p3(∞)) = (2/3, 2/3, 1/3) · (p1(0), p2(0), p3(0))

Ejemplo: No hay convergencia

Sea T :

0 1/2 1/21 0 01 0 0

es facil mostrar que T = T 2 = T 4....

Ahora considere p(0) = (1, 0, 0).

Aperiodicidad e irreducibilidad

El problema con este ejemplo es que que tiene un ciclo(dirigido) y todos los ciclos son longuitud que es un multiplode dos. Por eso se dice que es un grafo periodico.

Un grafo es aperiodico si el mayor divisor comun de todos losciclos es uno.

Veanse los ejemplos anteriores de convergencia.


Un grafo es fuertemente conexo o irreducible si cada par depuntos se pueden conectar con enlaces dirigidos.

Un sunconjunto de nodos es cerrado si ningun enlace sale deese subconjunto hacia un nodo que esta por fuera (i.e., unagente en el subconjunto no lo influencia nadie de afuera delsubconjunto).

Aperiodicidad e irreducibilidad: Ejemplo8.3. IMITATIONAND SOCIAL INFLUENCEMODELS: THEDEGROOTMODEL301

3

5

4

1/4

1/211/2

3/4

1

2

6

1/4

1/41

1/3

1/4

2/3

1/4

Figure 8.3.2. Closed Sets of Nodes

3

5

4

1/4

1/211/2

3/4

1

2

6

1/4

1/41

1/3

1/4

2/3

1/4

Figure 8.3.2. The Only Closed and Strongly Connected Sets of Nodes

Existen muchos conjuntos de nodos cerrados.

Los unicos cerrados y fuertemente conexos son {1} y {3, 4, 5}(vease la siguiente figura).

Aperiodicidad e irreducibilidad: Ejemplo

8.3. IMITATIONAND SOCIAL INFLUENCEMODELS: THEDEGROOTMODEL301

3

5

4

1/4

1/211/2

3/4

1

2

6

1/4

1/41

1/3

1/4

2/3

1/4

Figure 8.3.2. Closed Sets of Nodes

3

5

4

1/4

1/211/2

3/4

1

2

6

1/4

1/41

1/3

1/4

2/3

1/4

Figure 8.3.2. The Only Closed and Strongly Connected Sets of Nodes


Theorem

T es convergente si y solo si todo conjunto de nodos que esfuertemente conexo y cerrado, es aperiodico.

Una condicion suficiente para ser aperiodico es que Tii > 0para algun i (no es necesaria, vease segundo ejemplo).

Consenso

Un grupo de agentes llega a un consenso si sus creencias en ellımite son iguales para todos los agentes del grupo.

Proposicion: Cualquier grupo de individuos fuertementeconexo y cerrado llega a un consenso si y solo si es aperiodico.

Veanse ejemplos anteriores.


Contenido

1 Imitacion e Influencia Social

2 PageRank



PageRank

PageRank es una funcion que le asigna un numero a cadapagina en la Web (o subconjunto de paginas).

Interpretamos la Web como un grafo dirigido de paginas quetiene enlaces a otras:

166 CHAPTER 5. LINK ANALYSIS

by modifications that are necessary for dealing with some real-world problemsconcerning the structure of the Web.

Think of the Web as a directed graph, where pages are the nodes, and thereis an arc from page p1 to page p2 if there are one or more links from p1 to p2.Figure 5.1 is an example of a tiny version of the Web, where there are only fourpages. Page A has links to each of the other three pages; page B has links toA and D only; page C has a link only to A, and page D has links to B and Conly.

BA

C D

Figure 5.1: A hypothetical example of the Web

Suppose a random surfer starts at page A in Fig. 5.1. There are links to B,C, and D, so this surfer will next be at each of those pages with probability1/3, and has zero probability of being at A. A random surfer at B has, at thenext step, probability 1/2 of being at A, 1/2 of being at D, and 0 of being atB or C.

In general, we can define the transition matrix of the Web to describe whathappens to random surfers after one step. This matrix M has n rows andcolumns, if there are n pages. The element mij in row i and column j has value1/k if page j has k arcs out, and one of them is to page i. Otherwise, mij = 0.

Example 5.1 : The transition matrix for the Web of Fig. 5.1 is

M =

0 1/2 1 01/3 0 0 1/21/3 0 0 1/21/3 1/2 0 0

In this matrix, the order of the pages is the natural one, A, B, C, and D. Thus,the first column expresses the fact, already discussed, that a surfer at A has a1/3 probability of next being at each of the other pages. The second columnexpresses the fact that a surfer at B has a 1/2 probability of being next at Aand the same of being at D. The third column says a surfer at C is certain tobe at A next. The last column says a surfer at D has a 1/2 probability of beingnext at B and the same at C. 2


PageRank

El modelo de un surfista que camina aleatoriamente en la web(random surfer) se puede representar con una matriz detransicion M (el surfista puede comenzar en cualquier nodo yse mueve segun los enlaces, cada uno con la mismaprobabilidad):

166 CHAPTER 5. LINK ANALYSIS

by modifications that are necessary for dealing with some real-world problemsconcerning the structure of the Web.

Think of the Web as a directed graph, where pages are the nodes, and thereis an arc from page p1 to page p2 if there are one or more links from p1 to p2.Figure 5.1 is an example of a tiny version of the Web, where there are only fourpages. Page A has links to each of the other three pages; page B has links toA and D only; page C has a link only to A, and page D has links to B and Conly.

BA

C D

Figure 5.1: A hypothetical example of the Web

Suppose a random surfer starts at page A in Fig. 5.1. There are links to B,C, and D, so this surfer will next be at each of those pages with probability1/3, and has zero probability of being at A. A random surfer at B has, at thenext step, probability 1/2 of being at A, 1/2 of being at D, and 0 of being atB or C.

In general, we can define the transition matrix of the Web to describe whathappens to random surfers after one step. This matrix M has n rows andcolumns, if there are n pages. The element mij in row i and column j has value1/k if page j has k arcs out, and one of them is to page i. Otherwise, mij = 0.

Example 5.1 : The transition matrix for the Web of Fig. 5.1 is

M =

0 1/2 1 01/3 0 0 1/21/3 0 0 1/21/3 1/2 0 0

In this matrix, the order of the pages is the natural one, A, B, C, and D. Thus,the first column expresses the fact, already discussed, that a surfer at A has a1/3 probability of next being at each of the other pages. The second columnexpresses the fact that a surfer at B has a 1/2 probability of being next at Aand the same of being at D. The third column says a surfer at C is certain tobe at A next. The last column says a surfer at D has a 1/2 probability of beingnext at B and the same at C. 2

Obsevese que todas las entradas de M son positivas y suscolumnas suman 1 (i.e., matriz estocastica por columnas).

La probabilidad de que un un surfista que comienza en algunode los nodos con una distribucion de probabilidad v (i.e, vectorcolumna) se mueva al nodo i el siguiente periodo es: (Mv)i .

PageRank

M define una cadena de Markow.

Un estado estacionario de la cadena de Markov es unadistribucion de probabilidad sobre los nodos v tal que:v = Mv .

Si un surfista comienza con probabiliad v en cada nodo, elsiguiente periodo estara por la misma probabilidad en cadanodo.

PageRank

Este vector muestra cuales son los nodos mas probables endonde estarı un surfista que comienza y recorre el grafo deforma aleatoria. Para ver esto, usamos un poco de teoıa decadenas de Markov.

Para calcular el estado estacionario iteramos sucesivamente laecuacion vn+1 = Mvn comenzando desde cualquierdistribuicion v0.

Recordemos las condiciones para que haya convergencia.


Un grafo es fuertemente conexo o irreducible si cada par depuntos se pueden conectar con enlaces dirigidos.

Un sunconjunto de nodos es cerrado si ningun enlace sale deese subconjunto hacia un nodo que este por fuera (i.e., unagente en el subconjunto no tiene posibilidades de visitarnodos por fuera del subconjunto).

Theorem

T es convergente sı y solo sı todo conjunto de nodos que esfuertemente conexo y cerrado, es aperiodico.

Una condicion suficiente para ser irreducible es que Tii > 0para algun i (no es necesaria).

PageRank: Nodos muertos

En la practica cuando 50− 80 iteraciones son suficientes paraconverger con una precision muy alta.

El ejemplo anterior satisface las condiciones del teorema. Ladistribucion estacionaria a la que converge es: (39 ,

29 ,

29 ,

29).

Ejercicio: resolver el sistema lineal y hacer la iteracion.

El siguiente ejemplo tiene un nodo muerto (la matriz no esestocastica: tiene una columna de ceros).

Observese que no aplica el teorema (la matriz no esestocastica.5.1. PAGERANK 171

BA

C D

Figure 5.3: C is now a dead end

Note that it is substochastic, but not stochastic, because the sum of the thirdcolumn, for C, is 0, not 1. Here is the sequence of vectors that result by startingwith the vector with each component 1/4, and repeatedly multiplying the vectorby M :

1/41/41/41/4

,

3/245/245/245/24

,

5/487/487/487/48

,

21/28831/28831/28831/288

, . . . ,

0000

As we see, the probability of a surfer being anywhere goes to 0, as the numberof steps increase. 2

There are two approaches to dealing with dead ends.

1. We can drop the dead ends from the graph, and also drop their incomingarcs. Doing so may create more dead ends, which also have to be dropped,recursively. However, eventually we wind up with a strongly-connectedcomponent, none of whose nodes are dead ends. In terms of Fig. 5.2,recursive deletion of dead ends will remove parts of the out-component,tendrils, and tubes, but leave the SCC and the in-component, as well asparts of any small isolated components.4

2. We can modify the process by which random surfers are assumed to moveabout the Web. This method, which we refer to as “taxation,” also solvesthe problem of spider traps, so we shall defer it to Section 5.1.5.

If we use the first approach, recursive deletion of dead ends, then we solve theremaining graph G by whatever means are appropriate, including the taxationmethod if there might be spider traps inG. Then, we restore the graph, but keep

4You might suppose that the entire out-component and all the tendrils will be removed, butremember that they can have within them smaller strongly connected components, includingspider traps, which cannot be deleted.

Si se itera el estado estacionario converge a un vector deceros: es decir, cuando un surfista llega a un nodo muerto, enel siguiente periodo desaparece!

PageRank: Spider traps

El siguiente ejemplo satisface las condiciones del teorema peroarroja un resultado anti intuitivo (e.g., spider trap)

Si se itera, el estado estacionario se va concentrar en el estadoC . 174 CHAPTER 5. LINK ANALYSIS

BA

C D

Figure 5.6: A graph with a one-node spider trap

get

1/41/41/41/4

,

3/245/24

11/245/24

,

5/487/48

29/487/48

,

21/28831/288

205/28831/288

, . . . ,

0010

As predicted, all the PageRank is at C, since once there a random surfer cannever leave. 2

To avoid the problem illustrated by Example 5.5, we modify the calculationof PageRank by allowing each random surfer a small probability of teleportingto a random page, rather than following an out-link from their current page.The iterative step, where we compute a new vector estimate of PageRanks v′

from the current PageRank estimate v and the transition matrix M is

v′ = βMv + (1− β)e/n

where β is a chosen constant, usually in the range 0.8 to 0.9, e is a vector of all1’s with the appropriate number of components, and n is the number of nodesin the Web graph. The term βMv represents the case where, with probabilityβ, the random surfer decides to follow an out-link from their present page. Theterm (1− β)e/n is a vector each of whose components has value (1− β)/n andrepresents the introduction, with probability 1− β, of a new random surfer ata random page.

Note that if the graph has no dead ends, then the probability of introducing anew random surfer is exactly equal to the probability that the random surfer willdecide not to follow a link from their current page. In this case, it is reasonableto visualize the surfer as deciding either to follow a link or teleport to a randompage. However, if there are dead ends, then there is a third possibility, whichis that the surfer goes nowhere. Since the term (1− β)e/n does not depend onthe sum of the components of the vector v, there will always be some fraction

PageRank: Componetes fuentes

Es un conjunto de nodos que no reciben ninguna visita ytienen por lo menos un enlace que apunta hacia afuera delconjunto.

Intutivamente, una caminata aleatoria en una red con unconjunto de nodos de este tipo tendria a asignarlesprobabilidad cero de ser visitados. Probablemente una subestimacion de la importancia de los nodos.

PageRank: tributacion y teletransporte

Suponemos que en cada etapa de la caminata, el surfista eligecon probabilida β si utiliza la estructura del grafo paracontinuar navegando o con probabilidad (1− β) salta deforma aleatoria a cualquier otro estado todos con la mismaprobabilidad.

vn+1 = βMvn + (1− β)1

n

donde β es un numero cercano a 1.


Observese que modificacion evita las spider traps y los nodosmuertos.

Sin embargo, cuando hay nodos muertos el vector al que secoverge no es distribucion de probabilidad sobre todos losnodos, pero es distinto de cero.


Consideremos de nuevo el ejemplo del spider trap.174 CHAPTER 5. LINK ANALYSIS

BA

C D

Figure 5.6: A graph with a one-node spider trap

get

1/41/41/41/4

,

3/245/24

11/245/24

,

5/487/48

29/487/48

,

21/28831/288

205/28831/288

, . . . ,

0010

As predicted, all the PageRank is at C, since once there a random surfer cannever leave. 2

To avoid the problem illustrated by Example 5.5, we modify the calculationof PageRank by allowing each random surfer a small probability of teleportingto a random page, rather than following an out-link from their current page.The iterative step, where we compute a new vector estimate of PageRanks v′

from the current PageRank estimate v and the transition matrix M is

v′ = βMv + (1− β)e/n

where β is a chosen constant, usually in the range 0.8 to 0.9, e is a vector of all1’s with the appropriate number of components, and n is the number of nodesin the Web graph. The term βMv represents the case where, with probabilityβ, the random surfer decides to follow an out-link from their present page. Theterm (1− β)e/n is a vector each of whose components has value (1− β)/n andrepresents the introduction, with probability 1− β, of a new random surfer ata random page.

Note that if the graph has no dead ends, then the probability of introducing anew random surfer is exactly equal to the probability that the random surfer willdecide not to follow a link from their current page. In this case, it is reasonableto visualize the surfer as deciding either to follow a link or teleport to a randompage. However, if there are dead ends, then there is a third possibility, whichis that the surfer goes nowhere. Since the term (1− β)e/n does not depend onthe sum of the components of the vector v, there will always be some fraction

Si se itera la ecuacion de PageRank modificada con β = 0,8,se converge a ( 15

148 ,19148 ,

95148 ,

19148)

El nodo C logra un porcentaje importante de visitas delsurfista pero bastantes menos que sin ser teletransportado.

PageRank en la practica

Lo primero que hace un buscador es, dados unos terminos debusqueda primero elige un conjunto de paginas que contenganesos terminos y sobre este conjunto de palabras se hacePageRank.

Tıpicamente el score de PageRank tiene un peso importantepero se consideran tambien otros criterios (lugar y frecuenciadonde aparecen los terminos en cada pagina, etc.).

PageRank y Centralidad en Grafos

Considere la siguiente definicion de centralidad (Prestigio deKatz) de un nodo i , p(i):

p(i) =∑

j

gijp(j)

deg(j)(1)

En un grafo dirigido podrıa tomarse el grado de enlacessalientes.

Sea gij =gij

deg(j)

Entonces gij define una matriz estocastica:∑

i gij = 1

Luego esta definicion de centralidad es equivalente alproblema de PageRank.

Aprendizaje y Redes de Información · 2020. 10. 14. · da a la opini on del agente j en la...

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