Optimized reduction of uncertainty in bursty human dynamics
OOptimized reduction of uncertainty in bursty human dynamics
Hang-Hyun Jo, ∗ Eunyoung Moon, and Kimmo Kaski BECS, Aalto University School of Science, P.O. Box 12200, Finland Department of Economics, University of Essex, United Kingdom
Human dynamics is known to be inhomogeneous and bursty but the detailed understanding of therole of human factors in bursty dynamics is still lacking. In order to investigate their role we devisean agent-based model, where an agent in an uncertain situation tries to reduce the uncertaintyby communicating with information providers while having to wait time for responses. Here thewaiting time can be considered as cost. We show that the optimal choice of the waiting timeunder uncertainty gives rise to the bursty dynamics, characterized by the heavy-tailed distributionof optimal waiting time. We find that in all cases the efficiency for communication is relevant to thescaling behavior of the optimal waiting time distribution. On the other hand the cost turns out insome cases to be irrelevant depending on the degree of uncertainty and efficiency.
PACS numbers: 87.23.Ge,89.65.-s,89.90.+n
I. INTRODUCTION
Recently, significant amount of digital data on the be-havior of individuals has enabled us to quantitatively ex-plore the various patterns of human dynamics. One ofthe robust findings is that human dynamics is not ran-dom but correlated, such that the bursts of rapidly oc-curing events alternate with long inactive periods [1–7].The bursty dynamics is typically characterized by theheavy tailed or power-law distributions of waiting timeor inter-event time τ as P ( τ ) ∼ τ − α with α ≈ . ∗ Electronic address: hang-hyun.jo@aalto.fi this model we describe the time frame of how long peo-ple wait for responses from the information providersunder uncertain (risky) situations and get insight intowhich human factors in the daily communication are rel-evant to the bursty dynamics. Here we start with an as-sumption that an agent prefers a fixed amount of payoffover a risky lottery of the same expected payoff, whichis called risk-aversion . A risk-averse agent can reduceuncertainty by communicating for information, which isa time-consuming process. Once an agent requests infor-mation, we consider the time to wait for responses, i.e.the waiting time, as a cost. This reflects a well-knowneconomic perspective that time and information are con-sidered as tradable goods [10, 11]. Based on the trade-offbetween the information gain and the cost of the waitingtime, the agent chooses an optimal waiting time. Notethat we confine our model to risk-averse agents becausethe optimal waiting time for risk-pursuing agents is al-ways zero and the waiting time does not matter for risk-neutral agents unless the information gain is unequal tothe cost of time. By optimizing the waiting time, we de-rive a specific (positive) relationship between the risk andthe optimal waiting time. Then, we show that the opti-mal waiting time distribution follows a power-law, whichserves as theoretical support for the observed bursty dy-namics in e-mail and mobile phone call communications.Our model gives insight into the human communica-tion dynamics, i.e. how people are concerned with thecommunication efficiency and the cost per unit time whiledeciding the communication channel as well as the timeto wait for responses. Since the efficiency and the costper unit time are heterogenous in different communica-tion channels, the optimal channel depends on how ur-gent the situations are and how patient the users are.Furthermore, the efficiency and the cost per unit timealso change the optimal choice of the waiting time. Wefind that if the cost per unit time dominates over theefficiency, people are concerned with both the efficiencyand the cost per unit time under the low risk situation,while they consider only the efficiency under the highrisk situation. This explains the effect of human factors a r X i v : . [ phy s i c s . s o c - ph ] N ov on communication such that in riskier situation peopletend to focus more on reducing the risk without concern-ing the cost, because shortening the waiting time by theefficient communication is preferred to concern the costitself. However, if the cost per unit time is dominated bythe efficiency, the result is opposite. People are concernedwith only the efficiency under the low risk situation andconsider both the efficiency and the cost per unit timeunder the high risk situation, which implies that peopledo not consider the cost per unit time seriously under thelow risk (short waiting time) but do under the high risk(long waiting time). In conclusion the individual agents’attitude towards risk can be one of the explanations forthe pattern of the waiting time in communication.In Section II we describe and analyze the model andexplain the implications of the results. Then we makeconclusions in Section III. II. MODEL
As in [12] we consider the risk-averse utility function, u ( x ) of state X , whose degree of risk-aversion is measuredby A ( x ) = − u (cid:48)(cid:48) ( x ) /u (cid:48) ( x ). For simplicity, we set A ( x ) = 1and then consider a risk-averse agent with utility functiongiven by u ( x t ) = − e − x t + a ( a ≥ , (1)corresponding to a time-dependent state X t . At the be-ginning t = 0, the state X is uniformly distributed from −∞ to ∞ . Then, after waiting time t > X , the agent reduces the uncer-tainty on the state as X t ∼ N (0 , σ /t γ ). That is, thestate becomes more specified. The decreasing speed ofuncertainty is controlled by the parameter γ ≥
0, whichwe call efficiency .As an example let us consider a job applicant who gotoffers from 5 firms, indexed by i ∈ { , , , , } and let X = { w , w , w , w , w } be the set of wages of firms,which is the initial “state” if the applicant is uninformed.He chooses one of them and his utility depends on thewage of the firm chosen such that u ( w i ) = − e − w i + a .Then suppose that the applicant can obtain informationon the unknown wages. If the information he will get attime t = 1 is the exact amount of w , the state becomes X = { w , w , w , w } . At time t = 1, the applicantcan make a clearer decision than he could at time t = 0because the number of unknown wages at t = 1 is 4. Asthe applicant knows the wages one-by-one, his decisiongets improved, and eventually at time t = 5, he will knowthe firm of the highest wage so that the best decisionwithout uncertainty can be made.The time the agent waits for information to reduce theuncertainty can be considered as a cost. Let c ( t ) be thecost of time t . For simplicity, we assume that the cost perunit time is constant but dependent on the uncertaintylevel σ such that c (cid:48) ( t ) = kσ − /β with positive k and β . This assumption describes that given the level of the uncertainty σ , the cost of time is constantly increasingin time t at the rate of uncertainty except for the casewith β = ∞ . The larger value of β corresponds to thelarger cost of the time unit, indicating that the parameter β controls the cost per unit time .Now we define the agent’s expected utility at state X t subtracted by the cost of time as follows:Π( t ) = E [ u ( x t )] − c ( t )= a − (cid:90) ∞−∞ dx (cid:114) t γ πσ exp (cid:18) − x − x t γ σ (cid:19) − c ( t )= a − exp (cid:18) σ t γ (cid:19) − c ( t ) . (2)The agent chooses the optimal waiting time for informa-tion. At t = τ maximizing Eq. (2), the following condi-tion is satisfied: γ σ τ γ +1 exp (cid:18) σ τ γ (cid:19) = c (cid:48) ( τ ) = kσ − /β . (3)From this equation and by means of Lambert function W ( x ), satisfying x = W ( x ) e W ( x ) , we obtain the optimalwaiting time τ as a function of σ , as follows τ ( σ ) = C σ /γ W (cid:16) C σ β − γ ) / [ β ( γ +1)] (cid:17) − /γ , (4)where C ≡ [ γ γ +1) ] /γ and C ≡ γ γ +1) ( kγ ) γ/ ( γ +1) .We first consider the special case with β = γ , i.e. ofbalancing the efficiency and the cost per unit time, lead-ing to τ ( σ ) = τ c σ /γ (5)with τ c ≡ C W ( C ) − /γ . By using the identity of P ( τ ) dτ = P ( σ ) dσ and by assuming the distribution of σ to be P ( σ ) = e − σ , we get the optimal waiting timedistribution P ( τ ) as P ( τ ) = γ τ γ/ c τ γ/ − e − ( τ/τ c ) γ/ (6) ∝ τ − α e − ( τ/τ c ) γ/ , (7)with the power-law exponent α ≡ − γ/ τ c . It turns out that α ≤
1, which requiresthe strong cutoff such as the (stretched) exponential oneas we have assumed.Due to the technical differences among communicationchannels, the optimal waiting time τ can be channel-oriented. For example, one might wait longer for e-mailresponses than for face-to-face responses, and e-mail re-sponses might be faster than post responses. Thus, it isplausible that when people choose communication chan-nels, they might expect different waiting time to differentchannels and choose a suitable channel to the level of un-certainty.This channel preference can be considered in two ways:the efficiency and the cost per unit time. Firstly, theparameter γ can be interpreted as the channel-orientedsensitivity on uncertainty. Based on the empirical obser-vation of α ≈ . γ would be ≈ .
6. This implies that e-mail communication rarely reduces uncertainty as e-mailusers wait time for information. On the other hand, mo-bile phone users can reduce uncertainty more efficiently.In terms of the cost per unit time, the e-mail and theMPC can be compared as follows: c (cid:48) e − mail ( τ ) ≈ < c (cid:48) MPC ( τ ) ≈ . σ τ . exp (cid:18) σ τ . (cid:19) (8)for τ >
0. Since the cost per unit time for e-mails islower than that for MPCs, the empirical observationscan be interpreted such that people are more patientto wait for the e-mail responses than for the MPC re-sponses. Hence e-mail (MPC) is less (more) efficient andless (more) costly communication channel so that e-mailis more suitable than MPC for non-urgent situations andconsequently people would prefer MPC when they needinformation urgently.Since the Eq. (4) cannot be expressed in terms of el-ementary functions, we use the asymptotic expansionsof the Lambert function: W ( x ) ≈ x for x (cid:28) W ( x ) ≈ ln x for x (cid:29)
1. Let us now consider the casewith β > γ , where the cost per unit time is dominantover the efficiency. When σ (cid:28)
1, we obtain the scalingrelation τ ( σ ) ∝ σ δ , δ = 2( β + 1) β ( γ + 1) . (9)On the other hand, for the range of σ (cid:29)
1, the scalingrelation with logarithmic correction is obtained as τ ( σ ) ∝ σ δ [ln( σ/σ )] − /γ , δ = 2 γ , (10)where σ ≡ C − β ( γ +1) / [2( β − γ )]2 . Therefore, we find twoscaling regimes as τ ( σ ) ∼ σ δ for σ < σ × and τ ( σ ) ∼ σ δ for σ > σ × , respectively. Here σ × denotes the crossoveruncertainty and also defines the crossover waiting time τ × ≡ τ ( σ × ). It indicates that the optimal waiting timedistribution shows two scaling regimes with power-lawexponents α = 1 − /δ for τ < τ × and α = 1 − /δ for τ > τ × , respectively. Note that α < α for anypositive β and γ .Intuitively, the optimal behavior of individual agents ischanging according to the risk. Under the low risk situa-tion ( σ < σ × ), the power-law exponent α as a functionof β and γ explains the case where people are concernedwith the efficiency as well as the cost per unit time whenthe risk is not significantly high. On the other hand,under the high risk situation ( σ > σ × ), the power-lawexponent α = 1 − γ/ β .Unlike for the low risk situations, people under the highrisk consider only the efficiency, i.e. it is better to obtain -5 -3 -1 -5 -3 -1 τ k σ k γ /2 (a) 1 2 (log)4/3 4 (log) γ =1,k=10 -4 k=10 -5 k=10 -6 γ =1/2,k=10 -4 k=10 -5 k=10 -6 -20 -10 -4 -2 τ k - / ( γ - ) σ k - γ /[2(3 γ -1)] (b) 16/34 (log) 42 (log) γ =1,k=10 -3 k=10 -4 k=10 -5 γ =1/2,k=10 -3 k=10 -4 k=10 -5 FIG. 1: Numerical solutions of τ ( σ ) for different values ofparameters: (a) β = ∞ and (b) β = 1 /
3. There are twoscaling regimes with different power-law exponents, guidedby the solid lines. The ‘(log)’ following the exponent valuedenotes the fact that the power-law functions with logarithmiccorrection, i.e. Eq. (10) and Eq. (13), are used. information as quickly as possible due to the high costof time. Note that the special condition of β = γ yields δ = δ = 2 /γ , i.e. one scaling regime of optimal waitingtime distributions.We investigate the full range of Eq. (4) by numericallysolving τ ( σ ) for various values of k , β , and γ . The effectof constant k is systemically considered by scaling τ and σ as τ k β/ ( β − γ ) and σk βγ/ [2( β − γ )] , respectively. For eachset of β and γ , the solutions of τ ( σ ) for different valuesof k collapse to the single scaling function f : τ ( σ ) = k − β/ ( β − γ ) f ( σk βγ/ [2( β − γ )] ) , (11)where the scaling function f has two scaling regimes as f ( x ) ∼ (cid:26) x δ if x < x × , x δ if x > x × . (12)In the case with β = ∞ , implying c (cid:48) ( τ ) = k , we obtainthe numerical solutions for γ = 1 and 1 / k =10 − , 10 − , and 10 − . As shown in Fig. 1(a), for eachvalue of γ , the curves for different values of k collapse toa single curve with two scaling regimes.Next, we consider the case where the cost per unit timeis dominated by the efficiency, i.e. β < γ . We find twoscaling regimes of τ ( σ ) as following: τ ( σ ) ∝ (cid:40) σ δ [ln( σ /σ )] − /γ , δ = γ if σ < σ × , σ δ , δ = β +1) β ( γ +1) if σ > σ × , (13)where σ = C β ( γ +1) / [2( γ − β )]2 . The power-law exponentfor the low risk regime, α , turns out to depend only on γ , while the power-law exponent for the high risk regime, α , is a function of both β and γ . In other words, whenthe cost per unit time is dominated by the efficiency,people are concerned with only the efficiency under thelow risk situation. This is because under the low risk,implying short waiting times, the cost of waiting timeis still negligible compared to efficiency. On the otherhand, people under the high risk are concerned with theefficiency as well as on the cost per unit time because thecost of time is not negligible any more due to the longwaiting time.For the full range of τ ( σ ), we have numerically solvedthe Eq. (4). In case with β = 1 /
3, implying c (cid:48) ( τ ) = kσ − / , we obtain the numerical solutions for γ = 1 and1 / k = 10 − , 10 − , and 10 − . As shown inFig. 1(b), for each value of γ , the curves for differentvalues of k collapse to a single curve of Eq. (11) with twoscaling regimes. III. CONCLUSION
In order to investigate the role of human factors inthe bursty dynamics of individuals, we have studied anagent-based model. Our model assumed an intrinsic hu-man behavior, such as a trade-off between informationgain to avoid uncertainty and the cost of time to wait for information. With the agent’s optimal choice for thewaiting time, the empirical waiting time distributions indaily communication process have been explained.The results we obtained by analytical derivation showtwo scaling regimes for the waiting time distributionswith different power-law exponents, denoted by α . Thisindicates that people have different attitudes on valuingthe efficiency and the cost per unit time, controlled by γ and β , respectively. While α is generally expected tobe a function of both β and γ , the value of α turns outto be independent of β in two cases: a) When β > γ and under high risk situation, people are not concernedwith the cost per unit time because it is better to obtaininformation as quickly as possible due to the high cost ofwaiting time. b) When β < γ and under low risk situa-tion, the cost of waiting time becomes negligible so thatpeople do not have to be concerned with the cost of timeunless the waiting time is significantly long.We showed that the scaling behavior of optimal wait-ing time distributions reveals which human factors in thedaily communication are considered as relevant to scal-ing. We also expect that further studies to identify intrin-sic properties of social agents as well as extension of thismodel to interacting agents on complex networks wouldbe helpful in figuring out the origin of bursts in humandynamics even in more details. Acknowledgments
Financial support by Aalto University postdoctoralprogram (HJ), from EU’s FP7 FET-Open to ICTeCol-lective Project No. 238597, and by the Academy of Fin-land, the Finnish Center of Excellence program 2006-2011, Project No. 129670 (KK) are gratefully acknowl-edged. [1] J.-P. Eckmann, E. Moses, and D. Sergi, Proceedings ofthe National Academy of Sciences of the United Statesof America , 14333 (2004).[2] A.-L. Barab´asi, Nature , 207 (2005).[3] U. Harder and M. Paczuski, Physica A: Statistical Me-chanics and its Applications , 329 (2006).[4] B. Gon¸calves and J. J. Ramasco, Physical Review E ,026123 (2008).[5] T. Zhou, H. A. T. Kiet, B. J. Kim, B. H. Wang, andP. Holme, EPL (Europhysics Letters) , 28002 (2008).[6] F. Radicchi, Physical Review E , 026118 (2009).[7] M. Karsai, M. Kivel¨a, R. K. Pan, K. Kaski, J. Kert´esz,A.-L. Barab´asi, and J. Saram¨aki, Physical Review E , 025102 (2011).[8] A. V´azquez, J. G. Oliveira, Z. Dezs¨o, K.-I. Goh, I. Kon-dor, and A.-L. Barab´asi, Physical Review E , 036127(2006).[9] R. D. Malmgren, D. B. Stouffer, A. E. Motter, andL. A. N. Amaral, Proceedings of the National Academyof Sciences , 18153 (2008).[10] D. Nichols, E. Smolensky, and T. N. Tideman, TheAmerican Economic Review , 312 (1971).[11] Y. Barzel, Journal of Law and Economics , 73 (1974).[12] J. W. Pratt, Econometrica32