More spreaders but less influence: An analysis of indirect influenceon Twitter

Xin Shuaia, Ying Dingb, Jerome Busemeyerc, Shanshan Chenc, Yuyin Sunb, Jie Tangd

aSchool of Informatics and Computing, Indiana University Bloomington, IN, USAbSchool of Library and Information Science, IN, USA

cDept. of Psychological and Brain Science, Indiana University Bloomington, IN, USAdDept. of Computer Science and Technology, Tsinghua University, Beijing, China

Abstract

Social influence in social networks has been extensively researched. Most studies have focusedon direct influence, while another interesting question can be raised as whether indirect influenceexists between two users who’re not directly connected in the network and what affects such influ-ence. In addition, the theory of complex contagion tells us that more spreaders will enhance the in-direct influence between two users. Our observation of intensity of indirect influence, propagatedby n parallel spreaders and quantified by retweeting probability in two Twitter social networks,shows that complex contagion is validated globally but is violated locally. In other words, theretweeting probability increases non-monotonically with some local drops. A quantum cognitionbased probabilistic model is proposed to account for these local drops.

Keywords: Twitter, Complex Contagion, Quantum Cognition

1. Introduction

Thanks to the fast development of Web2.0,many online social networks have emerged,where the observation of information dif-fusion, or social influence, in large-scaledata becomes possible. Social influence hasbeen studied by many researchers, includingthe validation of influence (Anagnostopouloset al., 2008; Crandall et al., 2008; Gomez Ro-driguez et al., 2010), the propagation of in-fluence among multiple types of social me-dia (Gruhl et al., 2004; Cha et al., 2009; Honget al., 2011), the maximization of influencespread in the whole network (Kempe et al.,2003; Chen et al., 2009), and the probabilis-tic modeling of direct influence (Xiang et al.,

2010; Tang et al., 2009; De Choudhury et al.,2007). However, most relevant studies fo-cused on direct influence, while another rele-vant question regarding social influence maybe raised as to whether a user can exert indi-rect influence on his/her friends’ friends andwhat affects such influence. Normally, multi-ple intermediate persons called spreaders areinvolved in the indirect communication be-tween two persons, i.e., the sender and the re-ceiver. Those spreaders may have a combi-national effect on the indirect influence prop-agated from the sender to the receiver.

Some have studied the problem of serial in-direct influence as shown in Figure 1(a), wherethe node A exerts indirect influence on the

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node B through the n serial intermediate nodes,C1 from Cn. For example, Fowler & Christakis(2008) found that happiness spreads amongpeople as far as 3 hops. Liu et al. (2010) triedto quantify the indirect influence of n-degreefriends at the topical level. However, very fewstudies have focused on another type of indi-rect influence, which we call parallel indirectinfluence, as shown in Figure 1(b). C1, ...,Cn

nodes are located in parallel between node Aand node B to propagate influence from theformer to the latter.

A C1 Cn B

n nodes

direct influence

indirect influence

(a) Serial indirect influence

A B

C1

Cn

direct influence

indirect influence

n no

des

(b) Parallel indirect influence

Figure 1: Two types of indirect influence

A concept closely related to parallel indirectinfluence is complex contagion. Unlike sim-ple contagion, which can spread in social net-works after just one contact with a single in-fected neighbor like a disease, complex conta-gion is a phenomenon where multiple sourcesof exposure to a new idea are required be-fore an individual adopts the idea (Centola &Macy, 2005). That is to say, repeated expo-sures of an individual to an idea recommendedby his/her multiple neighbors positively affectthe probability he/she will eventually followthat idea. Romero et al. (2011) studied the

spread of hashtags in Twitter and quantifiedthe probability of a user adopting a new hash-tag as the function of the number of his/herneighbors who have already adopted it. Theyfound that the spread of political hashtags val-idates the complex contagion, where the adop-tion probability increases monotonically as thenumber of neighbors who had already adoptedthe same hashtags increases, until a plateau isfinally reached. By contrast, for idiom hash-tags, complex contagion does not take effect,and the adoption probability decays rapidlywhen more neighbors have adopted the samehashtags.

The problem we are studying is similarto Romero et al. (2011), but we focus on mes-sage spread behavior and indirect influence onTwitter. A concrete example of this is shown inFigure 2, where Alice sends out original mes-

Figure 2: Typical information spread in a social network

sages, Charlie and Carol further spread Alice’smessages (i.e., by retweeting) and Bob finallyreceives them. After that, Bob may choose tofurther spread Alice’s messages to others, justlike his two neighbors Charlie and Carol havedone, or not. Here, the intent of Bob to fur-ther spread Alice’s messages would reflect theintensity of the indirect influence of Alice onBob, which can be measured as the probabil-ity that Bob will further spread Alice’s mes-sages, given that Charlie and Carol have al-ready spread these messages. If complex con-

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tagion takes effect, the influence intensity willbe higher when both Charlie and Carol spreadAlice’s messages than when either or none ofthe two spread them.

In this paper, we examine the intensity ofindirect influence as the function of the num-ber of parallel spreaders between two users onTwitter who don’t have direct following rela-tions. We found that complex contagion is ob-served globally but is violated locally. Espe-cially, when the number of spreaders increasesfrom one to two, there’s an obvious drop inthe intensity. The newly emerging field ofquantum cognition is applied to interpret thelocal drops in terms of interference effect onthe process of decision-making. Recently, anarticle from NewScientist indicated how hu-mans may actually think in a “quantum” man-ner (Buchanan, 2011). Research from cogni-tive science has also provided some initial ev-idence of quantum-like cognitive interferencein human decision-making (Khennikov, 2010;Busemeyer et al., 2009). These cognitive ex-periments showed that the classical law of to-tal probability was violated. Instead, quan-tum probability (Gudder, 1988) was appliedto explain the experimental results. In addi-tion, quantum cognition has been employed tofurther advance the theory of information re-trieval (IR) (Piwowarski et al., 2010; Zucconet al., 2009; Zhang et al., 2010).

Our main contributions are:

• Examined the change of parallel indirectinfluence between the sender and the re-ceiver, quantified by retweeting proba-bility, with the number of spreaders andfound that such probability increases non-monotonically with some local drops;

• Verified the existence of complex con-tagion in the indirect influence on theglobal scale;

• Proposed a probabilistic model based onquantum cognition to explain local dropsin retweeting probability.

This paper is organized as follows: Section 2defines the problem. Section 3 shows the ex-perimental results. Section 4 proposes quan-tum cognition model. Section 5 lists relatedliterature and Section 6 concludes the study.

2. Problem Definition

Twitter users send and read messages calledtweets, which contain no more than 140 char-acters. One user can read another user’s mes-sages by following them. In addition, oneuser’s message can be re-sent by his followersvia retweeting. A retweeting message startswith the identifier “RT @username”. Such fol-lowing/follower relationships connect Twitterusers and form the social network where in-formation flows through retweeting. Given acollection of tweets C = {t}, V represens allTwitter users while E = {(u, v) |u, v ∈ V} rep-resents all following relations where u followsv. We provide several formal definitions as fol-lows:

• DEFINITION 1. [Following Triple] ∀tstarting with “RT @y: RT @x” postedby z, we build a following triple Txyz =

(x, y, z), x, y, z ∈ V and claim that (z, y) ∈E and (y, x) ∈ E. We also define C(Txyz)as the total count of tweets that belongs toTxyz and C(v), v ∈ V as the total count oftweets v posted.

• DEFINITION 2. [Spreaders] ∀a, b ∈ V ,we define spreaders between a and b asS ab = {y|Tayb , NULL, y ∈ V}.

• DEFINITION 3. [N-spreader RetweetingPattern] ∀a, b ∈ V we define a retweetingpattern Pab = {Tayb|y ∈ S ab} and |S ab| =

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n. Consequently, we define a n-spreaderretweeting pattern as Pn = {Pab||S ab| = n},and Pab is an instance of Pn.

• DEFINITION 4. [Retweeting Probabil-ity] ∀Pab , ∅, we define the probabilityof b retweeting from a as Pr(b|a; S ab) =∑

y∈S abC(Tayb)/C(a). Consequently, we

define the retweeting probability of n-spreader retweeting pattern as Pr(n) =∑

Pab∈PnPr{b|a; S ab}/|Pn|.

• DEFINITION 5. [Indirect Influence]∀x, z ∈ V ∩ Pxz , ∅, we think x exerts in-direct influence on z. Pr(n) indicates theaverage intensity of indirect influence inn-spreader retweeting pattern.

Starting from 1-spreader retweeting patternto illustrate how information spreads in Twit-ter (Figure 3(a)), where B follows C and Cfollows A. We assume that B does not di-rectly follow A so it can only read A′s mes-sages through C. A posts a tweet T and C readsand retweets it by adding the sign RT @A.Such information flow is represented by solidarrays because it is actually observed. Then Breads T and decides whether to further retweetit by adding a sign RT @C. The informa-tion flow from B is represented by a dashedarray because it is not an actual but a poten-tial flow. We can measure the intent of B tofurther propagate A′s messages transferred byC using the retweeting probability Pr(B|A; C).For instance, A posts a total of 100 tweets and20 of them are retweeted by C. Then B fur-ther retweets 5 of the 20 tweets, which makesPr(B|A; C)= 5/100 = 0.05.

In Figure 3(b), two spreaders C1 and C2 areinvolved in transferring messages from A to B.Assume that A posts 100 tweets, C1 retweets15 of them and C2 retweets 30 of them. ThenB retweets 5 of C′1s 15 tweets and 10 of C′2s 30tweets. As a result, Pr(B|A; C1,C2)= (5 + 10)

A C BT RT @A: T RT @C: RT @A: T

(a) 1-spreader

A

C1

C2

B

T1RT @A: T1

RT @A: T2

RT @C1: RT @A: Ti

RT @C2: RT @A: Tj

T2

(b) 2-spreader

A

C1

C2 B

RT @A: T1

RT @A: T2RT @C1: RT @A: Ti

RT @C2: RT @A: Tj

Cn

RT @A: T

3

RT @Cn: RT @A: Tn

T1

Tn

T2

(c) n-spreader

Figure 3: Retweeting patterns with different number ofspreaders

/100 = 0.15. The general n-spreader retweet-ing pattern is shown in Figure 3(c), and ourresearch question can be formulated as: Givenn spreaders, how does the curve Pr(n) changewith n?

According to complex contagion, Pr(n),which represents the intensity of indirect in-fluence, increases monotonically with n, sincemore spreaders involved in transferring thesender’s messages leads to higher probabil-ity of the receiver to further propagate thesender’s messages. We examined the proba-bility in the real Twitter data to see whetherthis is true.

3. Results

In this section, we show the results ofretweeting probability in two different Twit-

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ter datasets and testify the existence of com-plex contagion. One dataset consists of tweetsfrom the public timeline that can be seen as awhole social network, while the other consistsof tweets from a certain user’s ego network.

3.1. Dataset1The first dataset1 contains 467 million

tweets from 20 million Twitter users fromJune to December 2009 , which covers 20%-30% of total public tweets during this period.Figure 4 shows the distribution of n-spreaderretweeting pattern. As the number of spread-ers increases, the corresponding number ofinstances drops, indicating that the situationwhere too many spreaders are involved is in-frequent. The maximum number of spreaderswe found is 29.

0 5 10 15 20 25 30No. of Spreaders

102

103

104

105

106

107

log(N

o.

of

Inst

ance

s)

Figure 4: Distribution of the number of instances in alln-spreader retweeting patterns in dataset1

We plotted the curve Pr(n) for n = 1 to 10in Figure 5. The reason that we didn’t showthe part of the curve for n > 10 is that the rel-atively rare occurrence of patterns with a largenumber of spreaders makes the result subjectto random disturbance and yields unreliableobservations. From Figure 5 we can see that

1http://snap.stanford.edu/data/twitter7.html

drop

Figure 5: The curve of Pr(n) in dataset1

0 5 10 15 20 25No. of Spreaders

101

102

103

104

105

log(N

o.

of

Inst

ance

s)

Figure 6: Distribution of the number of instances in alln-spreader retweeting patterns in dataset2

drop

Figure 7: The curve of Pr(n) in dataset2

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the global trend of Pr(n) is increasing as n in-creases. That is to say, overall, the intensity ofindirect influence tends to become higher, or atleast persists, as more spreaders are included,which validates the phenomenon of complexcontagion in the global level. However, thereare two drops spotted in Pr(n), i.e. from n=1 to 2 and 8 to 9. To testify the statisti-cal significance of the two decreased probabil-ity values we observed, we used t-test (Gos-set, 1970) of difference between two means2

since retweeting probability is a type of meanvalue according to DEFINITION 4. Specifi-cally, we tested two hypothesis: Pr(1) > Pr(2)and Pr(8) > Pr(9). Two indices, t-score and p-value were calculated, with higher t-score andlower p-value indicating more statistically sig-nificant difference. Generally, we consider thedifference (decrease) as statistically significantif p-value is less than 0.05; otherwise, the dif-ference (decrease) is very likely to be causedby random noise. The results are shown inTable 1: We can see both p-values and t-

decrease from 1 to 2 from 8 to 9t-score 88.44 8.38

p-value 0.00 0.00

Table 1: T-test results of decrease of retweeting proba-bility in dataset1

scores show that the two drops of retweetingprobability are statistically significant, espe-cially when one spreader becomes two spread-ers. This implies that the decrease is indeedcaused by some other reason rather than ran-dom noise. In other words, the indirect in-fluence decays occasionally as the number ofspreaders increases, wherein the complex con-tagion is violated locally.

2http://www.rossmanchance.com/applets/TOSCalculations/TOSCalculations.html

3.2. Dataset2The second dataset comes from Knowledge

Engineering Lab at Tsinghua University andcovers the time span from August to Decem-ber in 2009. The process of tweets crawl-ing started from @yanglicai, a popular Twitteruser in the Chinese community, then extendedto all contacts of @yanglicai identified fromhis replying and retweeting messages. Subse-quently, tweets from the contacts of his con-tacts are crawled as well. Finally, 192,999tweets from 8254 users were obtained, and25.5% of all crawled tweets were retweetingmessages. If dataset1 represents the whole net-work, we view dataset2 as an ego network with@yanglicai located at the center with all otherusers having direct/indirect interactions with@yanglicai.

decrease from 1 to 2 from 4 to 5 from 7 to 8t-score 5.15 1.51 5.11

p-value 0.00 0.06 0.00

Table 2: T-test results of decrease of retweeting proba-bility in dataset2

We did the same investigation for the sec-ond dataset as the first dataset. Figure 6 showsthe distribution of n-spreader retweeting pat-terns in this ego-network. Again, the numberof instances decreases as n increases. We plot-ted the curve of Pr(n) in Figure 7. Similar todataset1, several drops are spotted, i.e. fromn =1 to 2, 4 to 5 and 8 to 9. We also did thesame t-test with results shown in Table 2. Thep-value of the test for the decreased probabil-ity occurring when n changes from 4 to 5 is alittle larger than 0.05, implying that such de-crease is only marginally trustable and is pos-sibly caused by random noise. However, theother two drops still show high statistical sig-nificance because both p-values are close to 0.In summary, the phenomenon that retweetingprobability increases non-monotonically with

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some local drops is observed in both the wholenetwork (dataset1) and certain ego-network(dataset2), verifying that complex contagion isglobally validated but locally violated.

4. Quantum Cognition

Although the effect of complex contagionon the indirect influence is observed on aglobal scale, the local scale decreasing in in-fluence needs further interpretation. Actually,a reason for this may be related to human psy-chology and cognition, a topic too complex toexplain. The emerging field of quantum cog-nition, however, might be able to provide apotential interpretation for the decreased in-fluence phenomenon. This field applies theformalism of quantum theory to model cog-nitive phenomena such as memory, judgmentand decision making. Notably, in the pro-cess of decision making where a decision de-pends on multiple factors, quantum cognitionassumes that these factors are not indepen-dent but have quantum-like interference effectson the final decision in a manner similar tothe explanation for results from double-slit ex-periments (Khennikov, 2010; Pothos & Buse-meyer, 2009; Yukalov & Sornette, 2011).

4.1. A Physical Metaphor

From the perspective of quantum cogni-tion, the information spread is compared hereto the physical wave quantum phenomenon.In the wave theory, the phase and magni-tude of a wave at a certain point is the linearsuperposition of all waves from all sources.This superposition can result in constructive(when the wave phases align) or destructive(when the waves are out of phase) interfer-ence. Such quantum interference was first ob-served in the well-known double-slit exper-iment Young (1804), where quantum wavesthat pass through two slits interfered with each

other and generated a pattern of bright anddark bands on the screen. (Figure 8(a))

Detector

Screen

Light Source S1

S2

(a) Double-slit experiment in quantum physics

Alice

Charlie

Carol

Bob

(b) Double-slit phenomenon in information spread

Figure 8: Double-slit phenomenon in both physics andinformation

We use our example in Figure 2 to illustratethe physical metaphor for information spreadin Figure 8(b). Alice’s sending out an initialmessage to Charlie and Carol creates an infor-mation ‘wave’ in the form of a primary wave-front. Then Charlie and Carol forward Alice’smessages by creating two secondary wave-fronts. Finally, the two waves arriving at Bobmay have constructive/destructive interferenceeffects on his intent to further propagate the in-formation flow. If constructive interference oc-curs, the intensity of Bob’s intent to propagateAlice’s messages will increase, which is verysimilar to the effect of complex contagion. Ifdestructive interference occurs, the intensity ofBob’s intent will decrease, which goes against

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complex contagion.We give an intuitive illustration to show

how Bob’s intent decreases due to a destruc-tive interference effect. Assume that initiallyonly Charlie spreads Alice’s messages whileCarol does not. Bob receives Alice’s mes-sages through Charlie and becomes interestedin these messages and further spreads them,because Bob has obtained relevant and freshinformation. Later on, Carol also begins tospread Alice’s messages but they largely over-lap with those already spread by Charlie. Bobtherefore becomes less interested in Alice’smessages because he’s overwhelmed with re-dundant information. Thus Bob’s intent to fur-ther spread Alice’s messages decreases, and sodoes the indirect influence of Alice on Bob.Here, the interference between two spreadersleads to destructive effects on the indirect in-fluence from the sender to the receiver, whichmay explain the local decrease spotted in Fig-ure 5. If a third spreader joins in and providesthe receiver with new and relevant informationabout the sender, the indirect influence mayrise again.

4.2. Cognition ModelAlthough an intuitive interpretation of lo-

cally decreased influence is given in terms ofquantum cognition, formal mathematical mod-eling is needed. We propose a cognitive modelcalled a q-attention model. This model origi-nates from Batchelder & Riefer (1990), whosework presents a family of processing modelsfor the source-monitoring paradigm in humancognition but is especially tailored to our so-cial influence research in Twitter. Note thatmost concepts and notations used in our quan-tum modeling come from the theory of quan-tum probability (Gudder, 1988).

The q-attention model is designed for thesituation where a group of Twitter users S n ={C j

}, j = 1, 2, ..., n, follow user A and are also

followed by user B. S n receive tweets from Aand also retweet them to B. In addition, weassume that there exists an upper limit on thetotal number of users B can pay attention toamong all B′s followees, due to human’s cog-nitive processing capacity. We set the numberas N ≤ n . In order to formulate the proba-bility Pr(B|A; S n) as a function of n, we pro-pose two versions of q-attention model: one isbased on classical probability theory, and theother is based on quantum probability theory.

A

CN+1

CN

Cn

B

C1

C2

q1

q2

qN

Pr(B | A;Sn)

(a) Classical q-attention model

A

CN+1

CN

Cn

B

C1

C2B | A;S

n

!1

!2

!N

(b) Quantum q-attention model

Figure 9: Q-attention model

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4.2.1. Classical q-attention modelWe model the cognitive process of B

retweeting from A′s messages through C j ∈ S n

by dividing it into two phases: First, B paysattention to C′js messages; Second, B readsthose messages retweeted by C j from A anddecides whether to further retweet them (Fig-ure 9(a)). In the first phase, we define q j as theprobability that B pays attention to C j and re-quire

∑Nq=1 q j = 1 . In the second phase, we

have already defined Pr(B|A; C j) to measureB′s retweeting intent through C j. Finally, theprobability that B retweets from A through C j

can be computed by the classical law of totalprobability:

Pr(B|A; S n) =∑

C j∈S n

q j · Pr(B|A; C j) (1)

We replace Pr(B|A; S n) with Pr(n) to repre-sent the general case and two implications canbe drawn:

• The total effect of all spreaders on thereceiver is just the linear superpositionof an individual effect of each spreader,which is represented by an independentsingle term without interfering with otherspreaders’ effects;

• Pr(n) increases monotonically as n in-creases, indicating that adding spreadersonly increases the final intent of B to fur-ther retweet A′s messages.

Actually, the classical q-attention modelmatches the sense of complex contagion. In-spired by the description of influence curve ofpolitical hashtag in Romero et al. (2011), wecan represent Pr(n) as a monotone increasingfunction. In the simplest case, we can assumethat Pr(n) is just a linear increasing functionof n from 1 until N . When n ≥ N, Pr(n) ≡Pr(N), because the upper limit of human’s

information processing probability is reachedand the intensity of influence persists at a rel-atively stable level. To summarize, Pr(n) canbe approximately represented by the followingpiecewise defined function:{

Pr(n) = Pr(1) +Pr(N)−Pr(1)

N−1 (n − 1), n = 2, ...,N − 1Pr(N + k) = P(N), k ∈ N+

(2)

4.2.2. Quantum q-attention modelTo take into account the cognitive interfer-

ence effect that is ignored in the classical q-attention model, we now formulate a quantumversion of the q-attention model. According toquantum probability theory Gudder (1988), aprobability p of an event is not primitive butderived from something more primitive calleda probability amplitude ϕ (a complex num-ber) — the probability is the square of mag-nitude of the amplitude, i.e. p = |ϕ|2 . Asis shown in Figure 9(b), the structure of quan-tum q-attention model is almost the same asthe classical version, except for the followingthree differences:

• The classical probability of paying atten-tion to C j, previously denoted by q j , isreplaced by a probability amplitude de-noted by ϕ j . B is assumed to be in asuperposition state, denoted by:

|S n〉 =

N∑j=1

ϕ j ·∣∣∣C j

⟩(3)

where ϕ j represents the “potential” of Bto consider the tweets from C j, but the po-tential is represented by a complex num-ber. The probability of paying attentionto the tweets from C j is obtained from thesquared magnitude q j = |ϕ j|

2 and againwe require

∑Nj=1 |ϕ j|

2 = 1. Overall, Equa-tion 3 represents B′s cognitive state of se-lecting spreaders C j to pay attention to ina quantum manner;

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• The classical probability that B retweetsfrom A through C j, previously denotedby Pr(B|A; C j), is replaced by the prob-ability amplitude

⟨B|A; C j

⟩, which repre-

sents the B’s “potential” to retweet butsuch potential is represented by a com-plex number. The probability of retweet-ing is obtained from the squared magni-tude Pr(B; A,C j) = |

⟨B; A,C j

⟩|2;

• The quantum probability obeys the lawof total amplitude rather than the lawof total probability. The probabilitythat B retweets from A passed along byC j ∈ S n is represented by Pr(B; A, S n) =

| 〈B; A, S n〉 |2. To determine the amplitude

〈B; A, S n〉, we replace the sum of proba-bilities shown in Equation 1 with the sumof amplitudes given below:

〈B|A; S n〉 =∑

C j∈S n

ϕ j ·⟨B|A; C j

⟩(4)

To make it clear, we show some examplesof how quantum q-attention model naturallyencompass interference effects. First, we con-sider n = 1 in which case only one spreaderC1 exists. Then 〈B|A; S 1〉 = |ϕ1 · 〈B|A; C1〉 |

2 =

|ϕ1|2 · | 〈B|A; C1〉 |

2 = q1 · Pr(B|A; C1). Appar-ently, the quantum version is exactly the sameas the classical version because no interferenceoccurs.

Next we consider n = 2 in which case S 2 ={S 1, S 2

}. From Equation 4 we have:

〈B|A; S 2〉 = |ϕ1 · 〈B|A; C1〉 + ϕ2 · 〈B|A; C2〉|2

= |ϕ1 · 〈B|A; C1〉|2 + |ϕ2 · 〈B|A; C2〉|

2

+(ϕ∗1 · ϕ2

)· 〈B|A; C1〉

∗ · 〈B|A; C2〉

+(ϕ1 · ϕ

∗2)· 〈B|A; C1〉 · 〈B|A; C2〉

∗.

The first two terms corresponds to the sameprobability that we obtain from the classical

model, while the last terms form a conjugatepair: (

ϕ∗1 · ϕ2)· 〈B|A; C1〉

∗ · 〈B|A; C2〉 =

|ϕ1 · ϕ2 · 〈B|A; C1〉 · 〈B|A; C2〉| ·

(cos (θ12) + i · sin (θ12))(ϕ1 · ϕ

∗2)· 〈B|A; C1〉 · 〈B|A; C2〉

∗ =

|ϕ1 · ϕ2 · 〈B|A; C1〉 · 〈B|A; C2〉| ·

(cos (θ12) − i · sin (θ12))

where θ12 is the phase difference of thetwo complex number ϕ1 · 〈B|A; C1〉 and ϕ2 ·

〈B|A; C2〉, which is dependent on the relationbetween C1 and C2. The sum of the conjugatepair produces a real number that we call an in-terference term:

I12 = 2 · |ϕ1 · ϕ2 · 〈B|A; C1〉 · 〈B|A; C2〉 | · cos(θ12)

Note that the cosine term can be positive(producing constructive interference), nega-tive (producing destructive interference) orzero (no interference). Notably, if the cosinevalue is sufficiently negative, then the prob-ability of B retweeting from A through twospreaders C1 and C2 can be smaller than theprobability given only one spreader C1, i.e.|ϕ2 · 〈B|A; C2〉 |

2 + I12 < 0.Finally we consider the case n = 3 and S 3 ={

S 1, S 2, S 3

}. Similarly, we obtain:

〈B|A; S 3〉 = |ϕ1 · 〈B|A; C1〉|2 + |ϕ2 · 〈B|A; C2〉|

2

+ |ϕ3 · 〈B|A; C3〉|2 + I12 + I13 + I23,

with the interference term defined as:

Ii j = 2 · |ϕi · ϕ j · 〈B|A; Ci〉 ·⟨B|A; C j

⟩| · cos(θi j)

(5)Again, each interference term can be positiveor negative. If they are all large negative num-bers, then we may then find a decrease in theprobability for three spreaders compared withone or two spreaders. There is also a possi-bility that all the interference terms will cancel

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out each other in which case the combinationalinterference would be equal to none. In otherwords, the quantum version in this case is re-duced to the classical version.

4.2.3. Comparison between two q-attentionmodels

We now systematically compare the clas-sical and quantum q-attention models andmathematically interpret the locally decreasedinfluence based on the quantum q-attentionmodel. The architecture of the two models isactually the same yet the main conceptual dif-ference between the two lies in the represen-tation of probability. Based on above equa-tions, we can obtain the mathematical relationbetween the two models as:

Prq(n) = Prc(n) +

n−1∑i=1

n∑j=i+1

Ii j (6)

where Prq(n) represents the average result ofthe quantum model while Prc(n) represents theaverage result of the classical model. Theonly difference between the two models is thesum of interference terms, which representsthe interference effect. If we consider the deltavalue, we obtain:

4Prc(n) = Prc(n)−Prc(n−1) = qn ·Pr(B|A; Cn)(7)

4Prq(n) = Prq(n) − Prq(n − 1)

= qn · Pr(B|A; Cn) +

n−1∑j=1

I jn(8)

From Equation 7, we can easily ob-tain 4Prc(n) > 0, i.e., Prc(n) increasesmonotonously with n. In other words, B′sintent to retweet A′s messages always be-comes stronger as more spreader C j areinvolved in relaying A′s messages. However,the situation becomes more complex inEquation 8, due to the interference items. It

becomes very difficult to predict the changingtendency of B′s retweeting intent. Especially,if

∑n−1j=1 Ii j > −|qn · Pr(B|A; Cn)|, we have

4Prq(n) < 0. In other words, sometimes thedestructive effect will become so strong that itdecreases B′s intent.

Compared with the classical q-attentionmodel, the quantum q-attention model fullycaptures the trend of indirect influence shownin Figures 5 and 7. When constructive cog-nition interference occurs or interference can-cel out (i.e., the classical model), the influ-ence increases and complex contagion takeseffect; when destructive cognition interferenceoccurs, the influence may decrease and com-plex contagion is violated. In addition, theconstructive/null effect occurs more frequentlythan the destructive effect, corresponding tothe global increase of indirect influence withoccasional local drops. Although our proposedquantum cognition model shows promise, it isa purely theoretical hypothesis based on cur-rent research in the field and needs further em-pirical verification.

5. Discussion

Equation 6 shows that the core part ofquantum q-attention model is the interferenceterms, which can be modeled as a linear com-bination of different cos(θi j) terms. Then theremain problem is how to evaluate the capabil-ity of quantum model (i.e. those θi j) in char-acterizing the dynamic of indirect influenceregarding retweeting behavior in Twitter. Itis difficult to solve because those interferenceterms are affected by complex human psy-chological or other factors, which needs fur-ther investigation and experimentation. More-over, the quantum cognition model is morelike a conceptual and theoretical prototypi-cal model, rather than traditional probabilisticmodels, like Markov random filed or Bayesian

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network, which already have mature frame-works for parameters learning and predictionevaluation. Therefore, the complete evalua-tion of the quantum model is beyond our workat the moment and we will leave it for futurework.

In spite of the difficulty of model evalua-tion illustrated above, we still attempt to jus-tify the mathematical capability of the quan-tum q-attention model in interpreting the co-existence of global increase and local dropsfound in the empirical data, by answering twoquestions: First, if we have already observedchange of the retweeting probability with thenumber of spreaders in one empirical dataset,can we find the valid mathematical solutions ofcos(θi j)? Second, can we apply some simplerule to generate simulated/fake cos(θi j) terms,which can lead to global increased but locallydropped retweeting probabilities against dif-ferent different number of spreaders?

To answer the first question, we rely on thesimple classic linear model described in Equa-tion 2 and 1. In addition, we assume thatthe empirical values of retweeting probabil-ities represent the output of quantum modeland the difference between the quantum andclassical model are fully captured by interfer-ence terms, as is shown in Equation 6. Thenby solving the linear equations composed ofEquation 1,2,6, we can obtain θi j for differentvalues of n. There’re two points that worth at-tention. First, if |cos(θi j)| > 1, then it is aninvalid solution and the quantum q-attentionmodel fails. Second, most of the time, the so-lutions are not unique and multiple combina-tive solutions of cos(θi j) can be found.

We take dataset1 as the example to illus-trate how to find those cos(θi j). We set N ashalf of the total number of retweeting patternsfound in dataset1, i.e. N=14. When n = 2,we solve the linear equations above and obtaincos(θ12) = −0.524, which is a valid value since

−1 ≤ cos(θ12) ≤ 1. It also indicates that theinterference between C1 and C2 is destructive,which leads to a local drop. When n = 3, wesolve the linear equations and use the alreadyknown value of cos(θ12). We obtain

44.4 · cos(θ13) + 7.8 · cos(θ12) = 7.9· (9)

Obviously, cos(θ13) and cos(θ12) have morethan one solutions and they may have oppo-site signs. In other words, when C3 is added,the interference between C1 and C3 and the in-terference between C2 and C3 may behave indifferent directions, leading to the rise of theglobal indirect influence again. To be general,those cosine terms satisfy the following equa-tion when n = k > 2:k−1∑

i

Di f f (i)+2(αcos(θ1k)+k∑

j=2

βcos(θ jk)) = Di f f (k)

(10)where Di f f (i) = Prq(i) − Prc(i), α =√

Pr(1) · β and β =Pr(N)−Pr(1)

N−1 . We also try tofind values of cos(θi j) for other n values andthe solutions are all valid, which indicates themathematical plausibility of proposed quan-tum model. However, It is worth note that thecos(θi j) items calculated from one dataset donot apply to other dataset, because differentsets of users have different interactions.

To answer the second question, we run asimple simulation to generate those cos(θi j)without any prior knowledge by assuming thatcos(θi j) is just uniformly distributed from [-0.5, 0.5] and can be randomly sampled. Herewe examine three types of retweeting prob-abilities: classical model value from Equa-tion 2, emprical value from the dataset1, andsimulated quantum model value from Equa-tion 10, 6 and randomly sampled cos(θi j). Weplot two groups of values by using differentrandom number generator seeds. The sim-ulated curves in both Figure 10(a) and Fig-ure 10(b) show the trend of global increase

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plus local drop, which matches our motiva-tion of building quantum q-attention model.However, the change of retweeting probabil-ity in Figure 10(b) shows large deviation fromthe empirical curve, implying that cos(θi j) isnot randomly distributed but follows some rulethat needs further examination in order to fitthe real data.

1 2 3 4 5 6 7 8 9 10No. of Spreaders

0.020

0.022

0.024

0.026

0.028

0.030

0.032

0.034

0.036

Avera

ge R

etw

eeti

ng P

robabili

ty

classicalsimulatedempirical

(a) simulation with random seed 1

1 2 3 4 5 6 7 8 9 10No. of Spreaders

0.020

0.025

0.030

0.035

0.040

0.045

Avera

ge R

etw

eeti

ng P

robabili

ty

classicalsimulatedempirical

(b) simulation with random seed 2

Figure 10: Simulation results of quantum model usingrandomly sampled interference items

Although we demonstrate the mathematicalplausibility of quantum model above, our workjust initiates a new direction of applying quan-tum cognition to social influence study, which

is not seen in any other current related re-searches yet. We will leave the study of com-plete parameter estimation and model evalua-tion of quantum q-attention models for futurework, possibly by combining existing machinelearning techniques and psychological experi-ments.

6. Related Work

6.1. Information and Influence PropagationThe origin of information propagation in

social networks comes from social influence,which occurs when an individual’s thoughts,feelings or actions are affected by other peo-ple. Information propagation characterizes theway that a node in social networks can spreadan information meme to its neighbor nodesvia exerting social influence on them. Theexistence of social influence has been vali-dated by many researchers (Anagnostopouloset al., 2008; Crandall et al., 2008; Gomez Ro-driguez et al., 2010). Different types of in-fluence across various fields has been stud-ied, including social media like Blog (Gruhlet al., 2004), Flickr (Cha et al., 2009) and Twit-ter (Hong et al., 2011; Yang et al., 2010), aswell as academia (Ding, 2011; Ding & Cronin,2011; Yan et al., 2011). The maximization ofinfluence spread in the whole network was in-vestigated (Kempe et al., 2003; Chen et al.,2009) as well. In addition, different meth-ods of probabilistic modeling of social influ-ence were also proposed. Tang et al. (2009)used Topical Affinity Propagation to modelthe topic-level social influence on large so-cial network; De Choudhury et al. (2007)found three contextual factors that affect theinfluence propagation between two friends inMySpace; Xiang et al. (2010) developed anunsupervised model to estimate relationshipstrength from interaction activity and user sim-ilarity. Although many interesting phenomena

13

have been observed from the above researches,most of the time only direct communicationbetween two adjacent persons in a social net-work is considered in modeling and analyz-ing the information propagation. Normally,multiple intermediate persons called spreadersare involved in the indirect communication be-tween two persons, i.e., the sender and the re-ceiver. Those spreaders may not be indepen-dent but have combinational interference onthe information flow from the sender to the re-ceiver. Some have studied the problem of se-rial indirect influence where indirect influenceis propagated through n serial nodes. Fowler &Christakis (2008) found that happiness spreadsamong people as far as 3 hops. Liu et al. (2010)tried to quantify the indirect influence of n-degree friends at the topical level. However,very few studies have focused on the parallelindirect influence, where indirect influence ispropagated through n parallel nodes, the focusof our paper.

6.2. Quantum cognitionA recent article from NewScientist pointed

out the strong connection between quantumtheory and human thought processes, wherehuman cognition is found to follow fuzzy logicmore than classical logic (Buchanan, 2011).Cognitive science researches have also iden-tified the existence of quantum-like cognitiveinterference in human decision-making (Buse-meyer et al., 2009; Khennikov, 2010), wherethe experiments showed that the classic lawof total probability was violated. Instead,quantum probability (Gudder, 1988) was in-troduced to explain the experimental results.Quantum cognition has been introduced inthe IR field with the hope to build moreuser-centric methods in searching and rank-ing. Arafat & Rijsbergen (2007) applied QT toaddress some fundamental issues in searches,which investigated and modeled user cogni-

tion and their interactions during the searchprocess. Hou & Song (2009) proposed an ex-tended vector space model (EVSM) to modelcontext-sensitive high-order information. Zuc-con et al. (2009) suggestted ranking the docu-ment relevance using QT as judgment of rele-vance that is not independent from other doc-uments, and the interference of other docu-ments play an important role in judging the rel-evance. They proposed a novel quantum prob-ability ranking principle (QPRP) to model asituation, where a document relevance assess-ment is influenced by other documents. Zhanget al. (2010) proposed the use of quantum fi-nite automation (QFA) to represent the transi-tion of the measurement states (the relevancedegrees of the document judged by users) anddynamically model the cognitive interferenceof users when they are influenced by other re-lated documents. Although quantum cognitionhas been extensively applied in IR, our papertakes the first lead in applying quantum cogni-tion to analyze the indirect social influence insocial networks.

6.3. Complex ContagionComplex contagion (Centola & Macy,

2005) refers to the phenomenon where mul-tiple sources of exposure to a new idea arerequired before an individual adopts the idea.Simple contagion, however, can spread in so-cial networks with just one contact with a sin-gle infected neighbor, as a disease may func-tion. Centola (2010) found that individuals aremore likely to acquire new health practiceswhile living in networks with dense clustersof connections, that is, when in close contactwith people they already know well. For in-stance, people are more likely to participateregularly in the health forum if they had morehealth buddies who registered for it. Romeroet al. (2011) studied the spread of hashtagsin Twitter and quantified the probability of a

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user adopting a new hashtag as the function ofthe number of his/her neighbors who have al-ready adopted it. They found that the spreadof political hashtags validates the complexcontagion, where the adoption probability in-creases monotonically as the number of neigh-bors who have already adopted the same hash-tags increases, until finally reaching a plateau.By contrast, for idiom hashtags, complex con-tagion does not take effect and the adoptionprobability decays rapidly when more neigh-bors have adopted the same hashtags.

7. Conclusions

In this paper, we investigated the propaga-tion of parallel indirect influence on Twitterwith a focus on how the intensity of influencechanges with the number of spreaders. Weconstructed two social networks formed by thefollowing relations (confirmed by RT in mes-sages) in Twitter, quantified the intensity of in-direct influence with the retweeting probabil-ity, and plotted the curve of retweeting prob-ability with the number of spreads. We foundthat the phenomenon of complex contagion isvalidated globally since the overall trend ofthe retweeting probability is gradual increas-ing against the number of spreaders. How-ever, the probability decreases locally as well.We applied quantum cognition theory in an at-tempt to interpret the local anomaly.

Although some interesting findings aboutindirect influence on Twitter are observed inthis study, a great deal of future work remainsto be done on the topic. First, the phenomenonof locally decreased influence needs furtherstudy. Quantum cognition itself may not besufficient to explain it. Some theories frompsychology, sociology, and probability willhelp address these complexities. Second, weplan to compare the indirect influence curvesfor different topics. We think that different pat-

terns in retweeting behaviors exist for differ-ent topics on Twitter. It will be very meaning-ful to compare them and see whether complexcontagion works or not in different contexts.Third, we hope to testify our quantum cogni-tion model on other social networks includingFacebook. For instance, “like” is a commonaction in Facebook. We can model the proba-bility of a use who “like” some item on Face-book as the number of his/her social neighborswho already “liked” the same item, and seewhether complex contagion with local dropsreemerges. Finally, our current work is moredescriptive and observational than mathemat-ical formulations. We hope to build a proba-bilistic model to represent and predict indirectinfluence in the future.

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