User Satisfaction Web Browsing

arXiv:0902.1104v1 [cs.HC] 6 Feb 2009

How happy is your web browsing? A

probabilisti model to des ribe user satisfa tion.

Anirban Banerji Bioinformati s Centre, University of Pune Pune-411007 Maharashtra, India anirbanbioinfo.ernet.in,anirbanabgmail. om

Date : 17

th

De ember, 2008

Abstra t We all go through s ores of web-pages everyday, in sear h of required information. At times we be ome satised with the ontent of some pages, at time we fail to. An obje tive framework that attempts to model the user satisfa tion when he sear hes for some desired pie e of information, is essential for Human-Computer-Intera tion. In the present work, a simple probabilisti model is onstru ted to a hieve pre isely the same. This realisti yet stri tly mathemati al study proposes a marker, the 'satisfa tion retentivity quotient', to model the

omplex realm of user psy hology as he attempts to nd required information and forgets some bits of it here and there, simultaneously.

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We read hundreds of web-pages everyday to nd information of our interest. Startling extent of inter onne tivity [1℄ of the web-do uments have made life easier for us while attempting to retrieve ertain pie e of information from the Internet. We all know how happy we be ome to nd some bits of relevant information and we know how irritated we be ome not to nd the same in s ores of pages. While signi ant progress has been made in many realms of 'Human-Computer Intera tion'(HCI), a fundamental aspe t of the eld, namely an obje tive model that attempts to apture the states of satisfa tion of a user as he traverses through the web in sear h of a parti ular set of information; still eludes the students of HCI and in general, people who use Internet. Although some eorts have been made to quantify ertain aspe ts of 'user satisfa tion' ([2℄ studied it as a fun tion of sear h engines, [3℄ had attempted to des ribe it with respe t to tra demand and apa ity of ea h link in the network, still dierently, [4℄ attempted to des ribe it as a fun tion of information retrieval), a realisti pi ture of the hanging state of satisfa tion level of an Internet user, does not generally emerge from these works. Here, a simple probabilisti model is proposed, whi h tries to model the user psy hology as he wanders around unknown web-sites in sear h of a parti ular set of desired information. An honest attempt is made to des ribe the pro ess as it is, instead of assuming the user to perform unrealisti operations.

Let us attempt to model a situation when a user is sear hing for a hunk of information, of somewhat fuzzy nature (be ause in most of the ases, user himself does not possess ategori al knowledge about what exa tly he is sear hing for, but be omes aware of his pre ise needs when he starts the sear hing operation). Similarly, it is not exa tly ommonpla e for the user to hit-upon a parti ular web-site where the entire bulk of information would be available to

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him in one magi al attempt. Instead, he/she gathers bits and pie es of relevant information from various web-sites. We assume that the required set of information an be broken down to an arbitrarily large number of small bits of information, whi h the user a

umulates and arranges in his mind suitably, to gather the desired knowledge. Let us assume that a ux of tiny bits of pertinent information in the very rst page of any web-site is what aptures the imagination of the user (say, U) and motivates him to read the ontent of the entire site in su ient detail, in the hope of olle ting the maximum ontent of information about the question in his(her, at any rate) mind. We assume further that the satisfa tion level of U is purely a fun tion of the level of information he is re eiving from the web-site (in this work, the design features of the website that do not ontribute to the existing level of information ontent of U, is not onsidered as a reasonable parameter whi h might inuen e the satisfa tion level of U). We assume further that U's oming a ross these tiny bits of pertinent information forms an elementary ow of intensity λ (s aler rate parameter).

It is realisti to assume that the very moment U tou hes upon a pie e of information that he thinks might take him loser to the desired pie e of information he is sear hing for, his satisfa tion level grows. To make the al ulation simpler, we assume that this growth of satisfa tion in U, takes pla e with a onstant s ale of unit magnitude. However, as it generally happens in real-life, very soon U realizes that any typi ally en ountered tiny pie e of hint of an information is not taking him loser to the set of information he wishes to possess, but is aiming at something else that aren't exa tly related to the premise of question he is interested in. His satisfa tion level therefore starts to de ay. To represent the situation reasonably, we assume this de ay fun tion to be having an exponential nature (rather than having a two-state (or some other) hara teristi s), with a

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parameter µ. Thus, while λ is inuen ed predominantly by the ontent of the site; origin of µ is omplex, be ause of its dependen e on various parameters. Nevertheless, those tiny sour es of information satises U to some extent and a

umulative ee t of these a quired information starts to build up in his mind. We assume that the gradual growth of residual satisfa tion of U (attained with the bits of pie es of a quired information from the web-site) over the traversing time t, through the web-page, an be aptured by summing them up. We designate the user-satisfa tion level by X(t). In the present work, a simple mathemati al model is proposed whi h attempts to nd the hara teristi s of this growth of user-satisfa tion upon browsing through a web-site.

At this point we segment our study in two ases. Case 1) :

Let us assume that U tou hes upon the pertinent information at random moments, T1 , T2 , . . . , Ti , . . . , whi h forms an elementary ow of events. The user satisfa tion level at any arbitrarily hosen moment t, due to his intera tion with any parti ular bit of information (say, ith bit of information) en ountered at the moment Ti , is given by :

Si (t) = =

0, (t < Ti )

(1)

e−µ(t−Ti ) , (t > Ti )

(2)

Hen e ompositely we an write, Si (t) = 1(t − Ti )e−µ(t−Ti ) , where 1(t) is a unit fun tion; Ti > 0, t > 0.

Let us now dene a random variable Ω, whi h des ribes the number of su h tiny bits of information that inuen e the satisfa tion level of U. This variable, 4

to be realisti , will be having a Poisson distribution with parameter λt ([5-8℄). Further, to des ribe the real-life situation properly, we represent the usersatisfa tion level X(t) as the sum of random number of random terms :

X(t) =

Ω X

e−µ(t−Ti ) 1(t − Ti )

(3)

i=1

Sin e a Poisson ow of events on any interval (0, t) an be represented, with suf ient a

ura y as a olle tion of points on that interval (des ribed in the 'Supplementary Material', available on request), the oordinate of whi h αi ∈ (0, t) is uniformly distributed on that interval and does not depend on the oordinates of other points. This is natural to expe t even from a non-mathemati al intuitive understanding of the situation also. Be ause the user U omes a ross all these points (representing the exa t instan e of nding a bit of relevant information) during the interval (0, t) and this des ription exhaustively represents the entire event spa e of favorable en ounter for the user during (0, t).

Therefore, eqn -3 an be re-written in the form :

X(t) =

Ω X

e−µ(t−αi )

(4)

i=1

where the random variables αi are independent and uniformly distributed in the interval (0, t).

Sin e the satisfa tion of U is a fun tion of interplay of λ and µ, and pra ti al experien e suggests it to be having a umulative nature, we an attempt to model user satisfa tion as a resultant of ea h event of favorable information gathering and unfavorable de ay. We designate Xi (t) = e−µ(t−αi ) = e−µt eµαi ,

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where Xi (t) represents ea h of these tiny events. Hen e we have :

X(t) =

Ω X

Xi (t) = e−µt

Ω X

eµαi

(5)

i=1

i=1

where Xi (t) are independent similarly distributed random variables, and the random variable Ω does not depend upon the random variables Xi (t) either. Here we note that although X(t) is umulative in nature, essentially it is a sto hasti pro ess.

At this moment, we invoke the known formula regarding mean value and varian e of the sum of a random number of random variables [9℄,[10℄ (if random PΩ variable Z is a sum Z = i=1 Xi , where the random variables Xi are indepen-

dent and have the same distribution with mean value mx and varian e Varx ; the number of terms Ω is an integral random variable whi h does not depend

upon terms of Xi ; has a mean value mΩ and varian e VarΩ ; we know the mean value mz and varian e Varz of the random variable are given by : mz = mx mΩ and V arz = V arx mΩ + m2x V arΩ ). Whereby in the present ase, we have :

mx (t) = mΩ (t)mxi (t)

(6)

V arx (t) = mΩ (t)V arxi (t) + V arΩ (t)m2xi (t)

(7)

and

Sin e the random variable Ω has a Poisson distribution with parameter λt, it follows that mΩ (t) = V arΩ (t) = λt.

To nd mxi (t) :

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mxi (t) = E[Xi (t)] =

1 t


t 0

e−µ(t−x) dx =

1−e−µt . µt

However, it is pragmati to assume that the pro ess of estimation of mxi (t) in user's mind to be less than smooth and therefore to des ribe the situation realisti ally, we need to al ulate some quantity analogous to moment of inertia of mxi (t), if in the mental spa e mxi (t) is des ribed as a line-shaped obje t. Hen e we determine the se ond moment about the origin of the random variable Xi (t) : E[Xi2 (t)] =

1 t


t 0

[e−µ(t−x) ]2 dx =

1−e−2µt . 2µt

Hen e, mx (t) = λ

1 − e−µt µ

(8)

V arx (t) = λt[V arxi (t) + m2xi (t)] = λtE[Xi2 (t)] = λ

1 − e−2µt 2µ

(9)

It is interesting to noti e that as t → ∞, the mean value and varian e of the pro ess X(t) do not depend on time, sin e

limt→∞ mx = mx =

λ µ

(10)

and

limt→∞ V arx (t) = V arx =

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λ 2µ

(11)

This is expe ted purely from an intuitive perspe tive also. After traversing through the web-site(s) for a su iently long time, the user is expe ted to gather a nite amount of desirable information. However, sin e he fails to remember all of it, only a fra tion of the amassed information will be retained by him. Hen e the fra tion

λ µ

an be named as 'satisfa tion retentivity quotient'.

If we represent a su iently a large number by L, then the distribution prole of the se tion of the sto hasti pro ess X(t) for mx =

λ µ

> L, an be

interesting. For this we onsider a nite but su iently large interval (0, t) and assume that for some su iently large number of times Ω, user's intera tion with the desired pie e of information takes pla e on that interval. For su h a situation we see that the pro ess X(t) (eqn -3) is a sum of independent similarly distributed random variables, whi h has an approximately normal distribution (sin e in this ase the onditions of the entral limit theorem are in fa t fullled). Hen eforth, the se tion of the sto hasti pro ess an be onsidered to be having a normal distribution with hara teristi s mx =

λ µ

and V arx =

λ 2µ .

To understand the hara teristi s of user satisfa tion, it is imperative to form an idea about the evolution of it over the time interval of his browsing through web-sit(s). Hen e we pro eed to nd the orrelation fun tion between user satisfa tion proles during dierent instan es of browsing operation. This an be done by onsidering two se tions of the sto hasti pro ess in question, at the moments t and t′ (t′ > t). By virtue of the assumption made, we an assert that the user satisfa tion X(t′ ) at the moment t′ , is equal to the extent of satisfa tion X(t) at the moment t multiplied by the exponent e−µ(t −t) , added with the ′

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satisfa tion Ω(t′ − t), whi h omes into being due to user's oming a ross some interesting bits of information during the time interval (t′ − t). Hen e X(t′ ) is given by :

′

X(t′ ) = [X(t)e−λ(t −t) + Ω(t′ − t)]

(12)

The sto hasti pro esses X(t) and Ω(t′ − t) are evidently independent sin e they are generated due to user's intera tion with desired pie e of information during dierent, non-overlapping time intervals (0, t) and (t, t′ ) respe tively.

˙ ˙ ′ −t), The same an be said about the entered sto hasti pro esses X(t) and Ω(t ˙ ˙ ′ − t) = Ω(t′ − t) − mΩ (t′ − t) as where we dene X(t) = X(t) − mx (t) and Ω(t

entered random fun tions of the aforementioned sto hasti pro esses. Hen e, using eqn -12 we have :

Cx (t, t′ ) = = = =

i h ˙ X(t ˙ ′) E X(t) oi n h −µ(t′ −t) ˙ ˙ ′ − t) ˙ X(t)e + Ω(t E X(t)  2  ′ ˙ E X(t) e−µ(t −t) if (t′ > t)  2  ′ ′ ˙ e−µ(t−t ) if (t′ < t) E X(t )

Thus the orrelation an be ompositely expressed as : i h ′ ′ Cx (t, t′ ) = V arx (min(t, t′ )) 1 − e2αmin(t,t ) e−µ|t −t|

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(13)

Let us onsider the limiting behavior of the sto hasti pro ess when t → ∞, t′ → ∞, but the magnitude of their dieren e τ = t′ − t is nite. In this ase, Cx (τ ) = V arx e−µ|τ | =

λ −µ|τ | . 2α e

Hen e the sto hasti pro ess X(t) representing user satisfa tion pra ti ally attains stationarity in every aspe t when the user spends a long time sear hing for some desired bulk of information, whi h onforms to our experien e. Furthermore, its nature assumes that of a normal distribution when user sear hes for long (in other words, (t → ∞), (t′ → ∞)) and

λ µ

> L.

Of ourse the user an hit upon a web-site where the information regarding all of his interest is kept in one pla e. In su h (unlikely) ase, naturally µ → 0. Here the extent of user satisfa tion will be a Poisson pro ess sin e every new pie e of information that the user will be en ountering will exa tly mat h with the desired set of information he wanted to ollate. Consequently, the de ay in user's interest will o

ur minimally. In su h a ase, the expressions for mx (t), Varx (t) and Cx (t, t′ ) will assume the form :

−µt

limt→∞ mx (t) = limµ→0 λ 1−eµ

−2µt

= limµ→0 V arx (t) = limµ→0 λ 1−e2µ

= λt

i h ′ ′ λ 1 − e2µmin(t,t ) e−µ|t −t| = λ [min(t, t′ )] limµ→0 Cx (t, t′ ) = limµ→0 2µ Case 2) :

In some other real-life ases another situation is frequently en ountered. When

ertain related pie es of information from a web-site onform to user's desired set of information and user omes a ross these related pie es of information

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in a somewhat quantized form. Although this ase is similar to one dis ussed already, there are ertain subtle dieren es. To nd the hara teristi s of user satisfa tion level in this situation, we assume the appli ability of the assumptions made earlier and at the same time assume further that user's oming a ross su h quantum of desired information form an elementary ow with intensity λ. The exa t number of information that onstitute any ith quantum of desired information is assumed to be a random variable Ri , whi h, keeping with the real-life situation, is obviously independent of the number of information that

onstitutes any other quantum. The random variable Ri has a distribution f (R) with hara teristi s of mR and varR .

Just like the ase where user was en ountering the desired information in bits and pie es(eqn 3), here too we an represent the extent of user-satisfa tion by :

X(t) =

Ω X

Ri e−µ(t−αi )

(14)

i=1

where the random variables Ω, Ri and αi are mutually independent.

Keeping with the ase-1 approa h, we designate Xi (t) = Ri e−λ(t−αi ) and then   −µt −2µt E [Xi (t)] = mR 1−eµt and E Xi2 (t) = (V arR + m2R ) 1−e2µt Hen eforth,

mx (t) = λmR

1 − e−µt µ

(15)

and

V arx (t) = λ[(V arR + m2R )

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1 − e−2µt ] 2µ

(16)

Sin e mR > 0 and V arR > 0, eqn -15 will grow faster than eqn -8, similarly eqn -16 will grow faster than eqn -9. This is ompletely in agreement with pra ti al experien es. Sin e user omes a ross the desired bulk of information in a

oherent quantized form, he a quisition of knowledge be omes fast.

This basi s heme of swiftness of knowledge gathering (obviously) doesn't hange when the user browses for a long time and at the limiting ase t → ∞, we have:

limt→∞ mx (t) = mx =

λmR µ

limt→∞ V arx (t) = V arx =

λ(V arR +m2R ) 2µ

Cx (τ ) = V arx e−µ|τ |

Con lusion :

A probabilisti model is proposed here that des ribes the satisfa tion prole of a user when he browses through web-site(s) in sear h of a desired set of information. Sin e the results obtained from theoreti al onsiderations seem to agree pretty mu h with our routine experien es, the reliability of this attempt an be

onsidered trustworthy. The model points to a stationarity in user satisfa tion prole when the browsing operation ontinues for a long time. Most importantly, it suggests a marker, the 'satisfa tion retentivity quotient' that aptures the essen e of the entire pro ess and an help in obje tive des ription of many of the pro esses that the rapidly emerging eld of HCI attempts to model.

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A knowledgment : This work was supported by COE-DBT(Department of

Biote hnology, Govt. of India)S heme. The author would like to thank the present and previous Dire tors of Bioinformati s Centre, University of Pune; Dr. Urmila Kulkarni-Kale and Professor Indira Ghosh, for supporting him to perform this work, although it has got nothing to do with his PhD. resear h.

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