Information propagation and collective consensus in blogosphere: a game-theoretical approach

0.31 MB·English

by  Lianghuan Liu

epl draft
Information propagation and collective consensus in blogosphere:
7a game-theoretical approach
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L.-H. Liu, F. Fu and L. Wang(a)

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aCenter for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
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PACS 89.75.Hc–Networks and genealogical trees
] PACS 87.23.Ge–Dynamics of social systems
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p PACS 02.50.Le–Decision theory and game theory
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c Abstract.-Inthispaper,westudytheinformationpropagationinanempiricalbloggingnetwork
o by game-theoretical approach. The blogging network has small-world property and is scale-free.
s Individuals in the blogosphere coordinate their decisions according to their idiosyncratic prefer-
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s encesandthechoicesoftheirneighbors. Wefindthatcorrespondingtodifferentinitialconditions
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andweights,theequilibriumfrequencyofdiscussionshasatransitionfromhightolowasaresult
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s ofthecommon interestinthetopicsspecifiedbypayoffmatrices. Furthermore,underrecommen-
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dation, namely, individuals in blogging networks refer to additional bloggers’ resources besides
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theirnearest neighborspreferentially accordingtothepopularityoftheblogs,thewholeblogging
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networkultrafastlyevolvesintoconsensusstate(absorbingstate). Ourresultsreflectthedynamic
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pattern of information propagation in blogging networks.

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1 Introduction. – Blog, which is short for “web log”, Much previous research investigating the phenomenon
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hasgaineditsgroundbyonlinecommunityasanewmech- of information propagation on networks has been done
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anism for communication in recent years [1]. It is often a by adopting classic Susceptible-Infected-Removed (SIR)
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/personal journal maintained on the Web, which is eas- model in epidemiology [2–5]. Thinking about a rumor
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ily and frequently updated by the blogger. In the past spreading on social networks: first, an ignorant (I) ac-
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ifew years,blogs are the fastest growing part of the World quires the information from her/his neighbors and be-
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yWide Web (WWW). Advanced social technologies lead comes a spreader (S). Finally, she/he loses interest about
hto change of the ways of people’s thinking and commu- the information and no longer spreads it again, and be-
pnicating. In the virtual space of blogs, which is usually comes a stifler (R). Accordingly, SIR models information
:
vreferred to as blogosphere, bloggers present their percep- propagationin which the stiflers arenever againsuscepti-
ition, life experience, ideas, etc, on their blogs, which is ble to the information — just like conferring lifetime im-
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instantly accessible and open to readersaroundthe world munity. Yet, SIRS models the situation in which a stifler
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awhocancommentonthepostsofotherbloggers. Thiscre- eventuallybecomessusceptibleagain. Forinstance,inblo-
ates the eden of free minds and ideas to trigger sparks of gosphere,theSIRSmodelcanbeinterpretedasfollows[6]:
inspiration. Interestingly, the dynamic hierarchy of links abloggerwhohasnotyetwrittenaboutatopicisexposed
and recommendations generated by blogs creates power- to the topic by reading the blog of a friend. She/he de-
ful collaborative filtering effect to the tremendous infor- cides to write about the topic, becoming infected. The
mation flow travelling through the blogosphere everyday. topic may then spread to readers of her/his blog. Later,
Therefore, the blogosphere provides an extraordinary on- she/he may revisit the topic from a different perspective
line laboratoryto analyze how trends, ideas and informa- and write about it again. Thus the life cycle of the topics
tion travel through social communities. Further investi- is analogous to the diseases’.
gationinto the blogosphere will help understand not only
the dynamic pattern of information propagation in such In the realm of sociology, extensive study of the diffu-
ecosystem, but also the collective behavior taking place sion of innovation in social networks has been conducted
on such social networks. by examining the power of world of mouth in innovations
diffusion [6–8]. Generally speaking, there are two funda-
(a)E-mail:[email protected] mental models describing the process by which nodes in
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L.-H. Liu et al.
networks adopt new ideas or innovations: threshold mod- that the dependency of C(k) on k is nontrivial, and thus
elsandcascademodels(seeref.[6]andreferencestherein). exhibitssomedegreeofhierarchyinthenetwork. Besides,
In this paper, we empirically investigate the dynamic the average clustering coefficient of the undirected blog-
pattern of information propagation in blogosphere from ging network is 0.46. The average shortest path length
game-theoretical perspective. Individuals in the blogo- hli=4.28. Consequently,ourbloggingnetworkhassmall-
sphere coordinate their decisions according to their id- world property and is scale-free. A detailed study of the
iosyncratic preferences and the choices of their neighbors. structure of the blogging network is presented in ref. [9].
Assume that individuals have two choices, A and B. The Let us introduce the game-theoretical model by which
payoffofanindividualchoosingAorBiscomposedoftwo we study the information propagation on the empirical
components: an individual and a social component. The blogging network. The social network can be represented
individualpartcomesfromone’spreferenceirrespectiveof bya directedgraphGconsisting ofavertex setV andan
others’choices in the network while the social component edgesetE. Eachvertexirepresentsabloggeri(herblogs
of payoff results from the mutual (reciprocal) choices of represents herself) in the blogosphere. A directed edge
individual neighbors. As a result of such a coordination e(i,j) from i to j indicates that j’s actions influence i’s
game, one adapts her/his strategy by imitation at each actions. DenoteΓi astheneighborsetofverticestowhich
time step, that is, follows the more successful strategy node i’s outgoing edges connect. At each time step, each
measured by the payoff. Analogous to replicator dynam- individualhastwochoices: AandBcorrespondingto“not
ics, after generations and generations, the system arrives todiscussthe topic(No)” and“towritesomething onthe
atequilibrium state. The dynamic behaviour ofsuch pro- topic (Yes)” respectively. Let xi represent individual i’s
cessofinformationpropagationisaffectedsignificantlyby state (A or B). For convenience, the states are denoted
the blogging network structure. Thus we study the dy- by the two-dimensional unit vectors,
namics of information propagation empirically on a blog-
1 0
gingnetworkwhichis collectedby ourWWW robot. The A= and B = (1)
(cid:18) 0 (cid:19) (cid:18) 1 (cid:19)
dynamicpatternofinformationpropagationasa resultof
The individual’s choice depends upon the payoff result-
the common interest in the topics specified by different
ingfromone’sidiosyncraticpreferenceandsocialinfluence
payoff matrices is also observed.
(trend). Therefore,the payoffofanindividualchoosingA
The remainder of this paper is organized as follows.
orB iscomposedoftwo components: anindividualanda
Sec. II deals with the blogging network data and explains
socialcomponent. Theindividualpartf (x )ofthepayoff
the modelwe adoptto study the informationpropagation i i
resultsfromtheintrinsicpreferenceforAorBirrespective
inblogosphere,andSec.IIIgivesouttheresultsandmakes
of others. The social component of the payoff is depen-
some explanations. Conclusions are made in Sec. IV.
dent on the externalities created by the choices of one’s
The blogging network and the model. – Since neighbors Γi. The social payoff is supposed to take the
the global blogosphere has more than 20 million blogs, form j∈ΓixTi Mxj,wherethesumissummedoveralli’s
we focused in our preliminary investigation on its sub- outgoPing linked neighbors Γi. The payoff matrix M for
community — the Chinese blogs hosted by Sina. We the two strategies A and B (the choices A and B can be
viewedthissub-blogosphereasacloseworld,i.e.,thelinks interpreted as) is:
outgoing of the Sina blog community were omitted. We A B
obtained a directed and connected blogging network con- A a b (2)
sisting of 7520 blogs’ sites which was crawled down by B c d
our WWW robot. In fig. 1 and fig. 2, we report the wherea>candd>b. Thisisacoordinationgamewhere
in- and out-degree distributions of the directed blogging individuals should choose an identical action, whatever it
network. It is found that both in- and out-degree dis- is, to receive high payoff. Hence matching the partner’s
tributions obey power-law forms where P(kin) ∼ ki−nγin choice is better off than not matching (a > c and d >
with γin = 2.13 ± 0.66, P(kout) ∼ ko−uγtout with γout = b). For simplicity, and without loss of the feature of the
2.28±0.096. The average degree of our blogging network coordination game, we set b = c = 0 and d = 1−a with
hkini = hkouti = 8.42. We noticed that about 18.4% of 0<a <1. Thus the rescaled payoff matrix is tuned by a
the blogs have no outgoing links to other blogs, but the single parameter a. The payoff P of individual i is:
i
in-degreeofeachvertexinthebloggingnetworkisatleast
P =(1−w)f (x )+w xTMx (3)
one since our blogging network was crawled along the di- i i i i j
rected links. The fraction of reciprocal edges (symmetric jX∈Γi
links) is about 31%. The degree-dependent clustering co- wherethe weightw ∈(0,1)indicatesthe balancebetween
efficient C(k) is averaged over the clustering coefficient individual and social payoff. Here we use the strategy
of all degree k nodes. In fig. 3, we can see that for the update rule similar to imitation. In any one time step,
undirected blogging network, it is hard to declare a clear individual i adopts the choice A with probability propor-
power law in our case. Nevertheless, the nonflat cluster- tionaltothetotalpayoffofher/himandher/hisneighbors
ing coefficient distributions shown in the figure suggests choosing A:
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Information propagationand collective consensus etc

In-degree Distribution 0.1 Out-degree Distribution
0.1
)(Pkin0.01 slope= 2.13 )(PKout0.01
1E-3 slope= 2.28
1E-3
1E-4
1 10 100 1 10 100
kin Kout
Fig. 1: (Color online) Thein-degreedistribution P(kin) obeys Fig. 2: (Color online) Theout-degreedistribution P(kout) fol-
apower-lawP(kin)∼ki−nγin withγin =2.13±0.66. Theline’s lows a power-law P(kout) ∼ ko−uγtout with γout = 2.28±0.096.
slope is −2.13 for comparison with the distribution. Theslopeofthestraightlineis−2.28forcomparison withthe
distribution.
P
W = j∈SiA j (4) over 100 generations after a transient time of 5000 gener-
xi←A Pj∈{i∪Γi}Pj ations. The evolution of the frequency of discussions as a
P function of a and w has been computed corresponding to
where SA = {k|k ∈ {i∪Γ } and x = A}. Otherwise,
i i k different initial conditions. Furthermore, each data point
individual i adopts B with probability 1−W . This
xi←A results from averaging over 10 runs for the same parame-
update rule is in essential “following the crowd” in which
ters.
theindividualsareinfluencedbytheirneighborsandlearn
We present the results of equilibrium frequency of dis-
from local payoff information of their nearest neighbors.
cussionsasafunctionofthe parameterspace(a,w)corre-
Within this imitation circumstance, the individual tends
sponding to different initial conditions in fig. 4. The den-
to keep up with the social trend based upon the payoff
sityofdiscussionsisindicatedbythecolorbar. Itisfound
information gathered from local neighbors.
thattheinitialfractionofdiscussersaffectstheequilibrium
Results and discussions. – We consider the infor- resultsquantitatively. Infig.4,fora),b),c)panelsrespec-
mation propagation on the blogging network when some tively,thereis acleartransitionfromhightolowforfixed
bloggers are initially selected at random as seeds for dis- w whenaisincreasedfrom0to1. Asaforementioned,the
cussing some certain specified topic in their blogs. All payoff matrix element a indicates the common interest in
the bloggers are assumed to be identical in their interests the topic travelling in the blogosphere, that is, when a is
and preferences, thus the individual part of payoff func- near zero, bloggers in the blogosphere show high interest
tion f (x ) is identical for all i. For simplicity, we set in the topic and would like to write something about the
i i
f(A) = 0.4 and f(B) = 0.5 in our simulations (the same topic; while a is near one, it means that people lose in-
magnitude as 0<a<1). In this situation, bloggers pref- terest in the topic, and reluctant to discuss. Besides, the
erentially discuss the topic in their blogs, hence we can weight w also plays a role in equilibrium results. When
examine the world-of-mouth effect in blogosphere empir- w approaches to zero, namely, individuals neglect the so-
ically. In addition, all individuals are influenced by their cial influence andonly depend upon individual preference
outgoing linked neighbors. Individual i’s social payoff is to discuss or not. While w tends to one, individuals are
summed over her/his all outgoing edges in which she/he completely influenced by their friends regardless of their
comparesher/his choice with her/his friends’ and obtains own idiosyncratic preferences. Otherwise, for intermedi-
payoffaccording to the payoff matrix of eq. (2). The syn- atew, i.e.,the choicesarebalancedbetweentheirindivid-
chronousupdatingruleisadaptedhere. Ateachtimestep, ual preference and social influence, the “self-organized”
each blogger updates her/his decision whether to discuss bloggersperformina collectivewaythatwithout acenter
ornotaccordingtoeq.(4). Allbloggersintheblogosphere control most of the individuals in the blogosphere change
are assumed to coordinate their choices to their friends conformablyfromfrequentlydiscussingthetopictolosing
(whoseblogstheoutgoinglinksareconnectedto),because interestinthetopicasaincreasesfromzerotoone. More-
conformity with friends in choices leads to the solid basis over, the critical value a of a at which the frequency of
c
to communicate and enjoy the fun of the topics. Equilib- discussionstransitsfromhightolowisaffectedbytheini-
rium frequencies of discussions were obtained by average tial fraction of discussers. It is observed that for interme-
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L.-H. Liu et al.

1
Clustering Coefficient 0.8
frequency of disccussions
0.1 0.6
C(k) fd
0.4
0.01
0.2
0.0
1 10 100 0.0 0.2 0.4 0.6 0.8 1.0
k a
Fig. 3: (Color online) The plot of degree-dependent clustering Fig.5: (Coloronline)Thefrequencyofdiscussionsasafunction
coefficientC(k)versusdegreekinundirectedbloggingnetwork. of a corresponding to w=0.66 and initial condition 21%.
munity is sustained at high value, even though there are
diateweightw,thecriticalvaluesofa arearound0.1,0.3,
c oftensmallfluctuationsaroundtheequilibriumstate. And
and0.5 correspondingto the initial fractionsof discussers
yet,whenallindividualshavelowinterestindiscussingthe
1%, 21%,and 51%respectively. Thus althoughthe initial
topic(whenaisnearone),theyarereluctanttomentionit
conditioninfluencestheequilibriumresult,differentinitial
in their blogs. For instance, the discussion of “influenza”,
conditions do not change the equilibrium results qualita-
israreatnon-influenzaseason,butburstsoutininfluenza
tively. Inotherwords,forfixedweightwandcertaininitial
season. Consequently, being consistent with the dynamic
condition,the density ofdiscussions has a cleartransition
pattern of information propagation in real world, our re-
from high to low when a increases from 0 to 1. We show
sults demonstrate that the frequency of discussions has a
the frequency of discussionsf vs a for w=0.66and21%
d transition from high to low due to the common interest
ofinitialdiscussionsinfig.5. Thefrequencyofdiscussions
specified by the payoff matrix for different weights w and
decreases from around 84% to 4% with increasing a. The
initial conditions.
transitionhappens arounda =0.3. Interestingly,wefind
c Inordertoinvestigatetheroleofrecommendationtoin-
thatthereareabout18.4%bloggershavenooutgoinglinks
formationpropagationinblogosphere,weconsideramod-
atall. Hence,their stateskeepinvariantbecausetheir de-
ified model based upon the above one. In blogging com-
cisions are not affected by neighbors. Accordingly, the
munity, the system often recommends some recent posts
whole blogosphere can never evolve into absorbing states
onitsmainpage. Thustherecommendedpostsarethein-
inwhichallindividualsmakethesamechoiceAorB.The
formation resources which are noticeable for the bloggers
typical evolution of frequency of discussions with respect
to acquire. In addition, when bloggers surf in the blogo-
to time correspondingto different a with w =0.66,initial
sphere,theprobabilityablogbeingvisitedisproportional
condition 21% is shown in fig. 6. With a = 0.11, the bl-
to its in-degree. Therefore, for simplicity, we assume that
ogosphere quickly evolves into the truncated equilibrium
besidestheneighborstowhichtheblogoutgoinglinkscon-
state where the frequency of discussion often drops down
nect, each blogger refers to additional K blogs which are
andrecoversto previouslevelaftera while. Nearthe crit-
chosen with probability proportional to their in-degrees,
icalvalueofa witha=0.31,thefrequencyofdiscussions
c i.e., the probability p that individual i chooses j’s blog
ij
is decreased at first. Yet, it strives to achieve the high
(j *Γi) as information reference is,
level very soon and is retained small fluctuations around
the equilibrium state. When a is increased to 0.41, the p =kin/ kin (5)
ij j l
frequency of discussions descends quickly, and then oscil- Xl
lates around the equilibrium state. With a = 0.81, the
Sinceeachbloggerindependently choosesK blogsaccord-
frequency of discussions fluctuates with some “spikes”—
ing to the probability proportional to in-degree, the cho-
occasionally, it suddenly erupted from 4% to 8%. There-
sen K blogs of each blogger might be different. All the
fore,theseresultsshowninfig.6cantosomeextentreflect
individualsareinfluencedbyboththeirneighborsandthe
thedynamicpatternofinformationpropagationinthereal
additionalK blogs. Let A denote the K blogs individual
i
blogosphere. When most bloggers show great enthusiasm
i chooses. The payoff P of individual i becomes,
i
in the topic (when a is near zero), they extensively dis-
cuss the topic in their blogs. For example, the fraction of P =(1−w)f (x )+w xTMx (6)
i i i i j
the bloggers talking about “Microsoft” in computer com- j∈{XΓi∪Ai}
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Information propagationand collective consensus etc
Fig. 4: (Color online) The frequency of discussions as a function of the parameter space (a,w). Panel a), b), c) correspond to
theinitial conditions 1%, 21%, 51% respectively.

1.0
(a) (b)
0.8
0.8
0.6 0.6
fd

a=0.11 a=0.31
0.4
0.4
0.2
0.2
0.0
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000

0.22 (c) (d)
0.20 0.2
a=0.41
0.18 a=0.81
0.16
fd 0.14

0.12
0.1
0.10
0.08
0.06
0.04
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
t t
Fig. 6: (Color online) Panel a), b), c) and d) show the evolution of discussions corresponding to a = 0.11,0.31,41,0.81
respectively. The weight w is fixedas 0.66 and 21% of thebloggers are initially selected as seeds.
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L.-H. Liu et al.

0.25
1.0 (a) (b) 1.0
0.20 frequency of discussions
0.8
0.8
0.15
fd 0.6 0. 10 fd0.6
0.4
0.4 0.05
0.2
0.2 0.00
0.0
0 50 100 150 200 0 50 100 150 200
t t 0.0 0.2 0.4 0.6 0.8 1.0
a
Fig. 7: (Color online) The evolution of discussions when indi-
vidualschooseadditionalKblogsproportionaltothein-degree Fig.8: (Coloronline)Thefrequencyofdiscussionsasafunction
as references. Left panel a) shows the case with a=0.11, and of a corresponding to w = 0.51,K = 10 and initial condition
b) with a = 0.81. The weight w is 0.51, K = 10, and 21% of 21%.
thebloggers are initially selected as seeds.
blogosphere. Moreover, under the circumstance of rec-
And the according update rule is, ommendation, the recommended blogs based on their in-
degrees act as leaders influencing other bloggers. Hence,
W = j∈SiAPj (7) thewholeblogosphereevolvesintoabsorbingstateswhere
xi←A P P all bloggers achieve the consensus of choices. Based upon
j∈{i∪Γi∪Ai} j localinformationΓ andlimitedglobalinformationA ,in-
P i i
where SA ={k|k∈{i∪Γ ∪A } and x =A}. dividual i finally collectively synchronizes her choice with
i i i k
Thecorrespondingresultsareshowninfigs.7,8. Fig.7 others. Therefore, our results may help understand the
shows the evolution of frequency of discussions with K = collective behaviours of bloggers in the blogosphere.
10, w = 0.51 and initial condition 21%. With a = 0.11,
the whole blogosphere ultrafastly evolves into absorbing ∗∗∗
state where all bloggers discuss the topic in their blogs
(see fig. 7(a)). While for a = 0.81, all individuals choose The authors are partly supported by National
nottomentionthetopicatall(seefig.7(b)). Bycontrast, Natural Science Foundation of China under Grant
without the recommendation, the whole blogosphere can Nos.10372002and60528007,National973Programunder
never evolve into collective consensus (see figs. 4, 5, 6 for Grant No.2002CB312200, National 863 Program under
comparison). In fig. 8, we can see that the frequency of Grant No.2006AA04Z258 and 11-5 project under Grant
discussionstransits fromone to zeroas a increasesfrom0 No.A2120061303.
to1. As aresult,under recommendation,the blogosphere
quickly attains the consensus state in which all bloggers
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sitionfromhightolowresultingfromthecommoninterest
specifiedbythepayoffmatrix. Tosomeextent,ourresults
reflectthe dynamic patternof informationpropagationin
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