Punctuated equilibrium dynamics in human communications
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Punctuated equilibrium dynamics in humancommunications
Dan Peng a , Xiao-Pu Han b , Zong-Wen Wei a , Bing-Hong Wang a,c,d a Department of Modern Physics, University of Science and Technology of China, Hefei230026, China b Alibaba Research Center for Complexity Sciences, Hangzhou Normal University,Hangzhou 311121, China c College of Physics and Electronic Information Engineering, Wenzhou University,Wenzhou 325035, China d The Research Center for Complex System Science, University of Shanghai for Scienceand Technology, Shanghai, 200093 China
Abstract
A minimal model based on individual interactions is proposed to study thenon-Poisson statistical properties of human behavior: individuals in the sys-tem interact with their neighbors, the probability of an individual acting cor-relates to its activity, and all individuals involved in action will change theiractivities randomly. The model creates rich non-Poisson spatial-temporalproperties in the activities of individuals, in agreement with the patterns ofhuman communication behaviors. Our findings provide insight into varioushuman activities, embracing a range of realistic social interacting systems,particularly, intriguing bimodal phenomenons. This model bridges priorityqueues and punctuated equilibrium, and our modeling and analysis is likelyto shed light on non-Poisson phenomena in many complex systems.
Keywords:
Punctuated equilibrium, Non-Poisson properties, Power-lawdistribution, Social networks, Human dynamics89.75.Fb, 05.40.Fb, 89.75.Da
1. Introduction
In the recent decade, along with the fast development of online socialservices, the understanding and predicting of human behaviors has attractedmuch attention of researchers. One of the remarkable features of statistical
Preprint submitted to Elsevier October 21, 2014 atterns of human behavior is the wide-spread non-Poisson properties [1, 2,3, 4, 5, 6, 7, 8, 9, 10], which usually shows heavy-tail distribution on temporalstatistics or spatial patterns and sharply differ from the Poissonian picture intraditional understanding [11, 12]. Several mechanisms based on separatedindividuals have been proposed to explain the origin of bursts and heavytails, including priority-queuing processes [1, 13, 14, 5], Poisson processesmodulated by circadian and weekly cycles [15, 16, 17], adaptive interests[5, 18], preferential linking [5], and memory effects [19].However, one major concern of behavior types in human dynamics is theinteraction and communication behavior between individuals. Such humanactions have a strong impact on resource allocation, circulation of informa-tion, even evolution of social network structure, therefore it have been thefocus of research. In real life, everyone is influenced by the surroundingsocial environment, for instance, the interval between sending two consec-utive E-mails is influenced by the actions of this individual and the othercommunicating partners. The researchers achieved a breakthrough first inthe simplest model of two interacting bodies [20]. And a minimal modelof interacting priority queues to discuss bimodal phenomenon observed inShort Message correspondence is proposed [21]: inter-event time distributionwas neither completely Poisson nor power law but a bimodal combination ofthem. Notably, this bimodal phenomenon was also observed in inter-eventtime distribution of the calling activity of mobile phone users [22], two con-secutive transactions made by a stock broker [13] and successive transactionsof experimental futures exchange [23] etc . Indeed, some purported power-lawdistributions in complex systems may not be power laws at all. In fact, strictpower-law distribution is rarely observed in empirical studies albeit burstsand heavy tails are widespread. In addition to the widespread bimodal distri-bution, human activity patterns may be power-law distribution followed bydistinct cutoff [24, 10], multimodal distribution of power-law with differentscaling exponent [25, 26, 27], or in some instances it is more consistent withMandelbrot distribution [28], and so on. Human activity patterns exhibitsuch a wealth of statistical properties. How to quantitatively understand hu-man dynamics, dose there exist an universal fundamental governing humandynamics and individual? The understanding about these questions need tobe studied deeply.In this paper, a simple model incorporating solely individual interactionbased on network is proposed to explain and develop human dynamics, es-pecially the origin of bursts and heavy tails. Our model links punctuated2quilibrium models and queueing models, and reproduces varieties of distri-butions on spatial-temporal patterns observed in empirical researches, e.g. ,exponential distribution, power-law distribution, and bimodal properties.
2. The model
In real life, a certain type of activity of an individual is influenced by theactions of this agent and the other communicating partners. For example,one person may reply in no time after receiving a message, or even send onemore to someone else; without receiving any message, however, this personmay not send out any message in a long time. That is, most of the time,people’s social behavior is activated by others. Whereas, it is conceivablethat the interaction make individual passive, for instance, an ongoing topicbetween interacting individuals terminates, or one party is reluctant to keepup interacting and so on. Hence we consider that social interactions make anindividual more active, or more inactive. Yet an activated individual mightkeep reticent or contact to more than one neighbor. On the other hand, an in-dividual might act at will without environmental stimulation. Incorporatingall these considerations, the schemes of our model are as follows: N nodes (individuals) are arranged on a network. We define a series ofmassage sendings that are activated by a common source node as a “burst”.Each node may be in two different status: affected by a burst (A), or un-affected status (U). Initially ( t = 0), each node (for node i , say) is in Ustatus, and its activity value is set as a random number a i , equally distributedbetween 0 and 1. At each processing step t , the evolution of the system obeysthe following rules:i) with probability p , the node with the highest activity is chosen to sendmessages to its n neighbors; or with probability 1 − p , an arbitrary node ischosen to be the sender, denoted as S t . Here n is the number of messagessent in one processing step and is not larger than the degree of the sender;ii) If the sender is in status U, namely, the current message sending is outof the previous burst, so all the other nodes turn to U status.iii) If the sender has received message in previous time steps, with prob-ability q , the sender sends a message back to the node which sent the lastmassage to the sender. Or with probability 1 − q , it sends a message to arandomly selected neighbor of sender S t , and sending other n − t + 1 and repeat the above procedures.This model has three free parameters for a given network: n , p and q .The interaction come about when messages are sent ( n ≥
1) by the mostactive individual ( p = 1) and all individuals involved in the action mutatetheir activities. When p <
1, it incorporates Poisson initiation of messages.If the network is regular (for example, a Ring) and n equals the degree ofnodes, this simple model is same as Bak-Sneppen model. If n = 0, the directinteraction between individuals is ignored completely, this case is equivalentto Barab´asi queueing model. Thus we give an association between priority-queue models and criticality phenomena.
3. Simulation Results
In the first instance, we concentrate on the dynamics on a Ring thatevery individual has two neighbors. It has several modes for the sending ofmessages. The first one is that the sender send no message to its neighbor,namely, n = 0, and the interaction between individuals is eliminated. Thiscase is equivalent to Barab´asi queueing model, in which the node with highestactivity is updated with probability p . The second mode is that the sendersend messages to all neighbors, namely, n = k . For Ring ( n = 2), when p = 1,the model is same as Bak-Sneppen model and exhibits long-range correlationand punctuated equilibrium (see Fig. 1(a) and (b)). Decreasing the value of p , along with the growth of randomness, the long-term correlation weakenand the stable value of global average activity h a i increases (see Fig. 1(a)).When p = 0, sender is completely randomly-chosen, and the system is in adisordered state with h a i = 0 . < n < k (for Ring, theonly case is n = 1), which is impacted by the selection of receivers (decidedby q ). In the extreme case q = 1, massage sendings usually are restricted toone pair of individuals, as shown in Fig. 1(c).In the following we meticulously investigate statistical patterns on a Ringfor different p and q . For simplicity of discussion, we fix N = 10 and n = 1.Here all distributions in the following discussions are counted in stable period.4 < a > t p = 0 p = 0.9 p = 0.99 p = 1 (a) i (c) s t ep t i m e i (b) Figure 1: (Color online) (a) The evolution of the global average activity h a i for different p , and the model runs on a Ring with parameters N = 10 , n = 1, q = 0. (b) and (c)Spatial-temporal patterns of the model running on a Ring in the stable period under theparameters setting N = 100, p = 0 . n = 1 and different q . (b) q = 0, the systememerges long-range correlation similar to the result of BS model. (c) q = 1, messageusually propagates between a pair of nodes. P ( τ ), here the inter-event time τ is defined as the time interval between two consecutive massagesendings of a node. For simplicity of discussion about p , we fix q value,in particular q = 0. As p = 0, at each step, sender is selected randomly, P ( τ ) = N (1 − N ) τ − . In the case p = 1 (BS model), the system evolvesinto a typical status of self-organization criticality and P ( τ ) obeys power-law distribution with exponent − .
56 (Fig. 2(a)). With the decrease of p ,the long-range correlations in each avalanche range is partially broken bythe randomness of sender selection, an exponential tail on P ( τ ) emerges andstretch P ( τ ) into bimodal form (Fig. 2(a)), corresponding to long intervalsof uncorrelated message sendings. And when p = 0, the system is completelyPoissonian-like (Fig. 2(a)).By contrast, the impact of q on P ( τ ) is not so strong. As shown in Fig.2(b), when p is very close to 1, as q increases from 0 to 1, a small Poissonian-like region appears in the head of P ( τ ), and the tail keeps power-law-likeform.The waiting time τ w is the time interval between the message sending andthe last receiving of a node. The results of P ( τ w ) are very similar to P ( τ ),as shown in Fig. 2 (c) and (d).Meanwhile, the spatial distribution P ( d ) of the distance d between thesenders of two consecutive time steps, is deeply impacted by both p and q .Fig. 2(e) shows the correlated range (the power-law region of P ( d )) enlargesfrom zero to the global range as p increases from 0 to 1. And localizedinteractions ( q >
0) weaken the long-range correlation (see Fig.2(f)).Moreover, the spatial-temporal patterns of bursts also show similar scal-ing properties. The burst duration time t b that is defined as the total num-ber of time steps in a burst period, is expected to be 1 as p = 0, and followspower-law-like distribution while p value very closes to 1. This power-law-likedistribution of burst sizes was observed in the distribution of Orkut session[29] and the spread of news and opinions [30]. It is interesting that P ( t b ) isstrongly sensitive to the reduction of p , seemingly follows exponential distri-bution (see Fig. 3(a)), implying that scaling temporal pattern results frompriority-activity mechanism of acting in system. In contrast, the setting of q almost can not impact on P ( t b ) (Fig. 3(b)). Ref. [10] reported variousdistribution patterns of burst sizes illustrated in Figs. 3(a) and (b), such asexponential, power law and even a distribution similar to P ( t b ) at p = 0 . p → q →
0, the distribution P ( N b ) obeys6 -15 -13 -11 -9 -7 -5 -3 -1 -15 -13 -11 -9 -7 -5 -3 -1 -8 -7 -6 -5 -4 -3 -2 -1 Slope = -1.56 (a) q = 0 p =0 p = 0.9 p = 0.99 p = 1 P () (b) p = 0.9999 q = 0 q = 0.9 q = 0.99 q = 1 Slope = -1.56Slope = -2.00 (c) q = 0 p =0 p = 0.9 p = 0.99 p = 1 P ( w ) w Slope = -1.90 (d) p = 0.9999 q = 0 q = 0.9 q = 0.99 q = 1 w Slope = -3.12 (e) q = 0 p =0 p = 0.9 p = 0.99 p = 1 P ( d ) d Slope = -3.16 (f) p = 0.9999 q = 0 q = 0.9 q = 0.99 q = 1 d Figure 2: (Color online) The distribution P ( τ ) for (a) different p with q = 0 and (b)different q with p = 0 . P ( τ w ) for (c) different p with q = 0 and (d) different q with p = 0 . P ( d ) for (e) different p with q = 0 and (f) different q with p = 0 . N = 10 and n = 1. The black dash lines denote the power laws with the corresponding exponents. -9 -8 -7 -6 -5 -4 -3 -2 -1 -7 -6 -5 -4 -3 -2 -1 q = 0 p = 0 p = 0.9 p = 0.99 p = 0.9999 P ( t b ) t b (a) Slope = -1.14 (b) p = 0.9999 q = 0 q = 0.9 q = 0.99 q = 1 t b Slope = -1.00 (c) q = 0 p = 0 p = 0.9 p = 0.99 p = 0.9999 P ( N b ) N b (d) p = 0.9999 q = 0 q = 0.9 q = 0.99 q = 1 N b -5 -4 -3 -2 -1 N b P ( N b ) Figure 3: (Color online) The distribution P ( t b ) for (a) different p with q = 0 and (b)different q with p = 0 . P ( N b ) for (c) different p with q = 0 and(d) different q with p = 0 . N = 10 and n = 1.The dash lines show the fitting curves. And in panel (c) and (d), the fitting function is P ( N b ) = 45 N − . b exp [ − (0 . x + 4 . -12 -10 -8 -6 -4 -2 -13 -11 -9 -7 -5 -3 -1 -8 -7 -6 -5 -4 -3 -2 -1 (a) p = 1, q = 0 r = 0 (Regular Network) r = 0.01 r = 0.2 ER Network P () Slope = -1.56 Slope = -1.50Slope = -2.00 (b) p = 1, q = 0 r = 0 (Regular Network) r = 0.01 r = 0.2 ER Network P ( w ) w (c) p = 0, q = 0 r = 0 (Regular Network) r = 0.01 r = 0.2 ER Network P ( w ) w Figure 4: (Color online) Effect of topological randomness. The model runs on WS networksfor different rewriting probability r and ER random network. (a) the inter-event timedistribution P ( τ ), parameter settings p = 1, q = 0; (b) the waiting time distribution p ( τ w ), parameter settings p = 1, q = 0; (c) the waiting time distribution p ( τ w ), parametersettings p = 0, q = 0. Other parameters are set on N = 10 , n = 1. Dashed lines showthe fitting slopes of curves. power law with exponential cutoff, here N b is the affected range of burst andis defined as the number of nodes involved in the burst. In contrast with P ( t b ), besides p , P ( N b ) also shows sensibility on q : large reply probability q will break the scaling property on P ( N b ) (see Fig. 3(d)). When p = 1, thereusually exists only one burst. Topology of media usually deeply affect the dynamics of system. In thissection, we run the model on different modeling networks to investigate thetopological effect on the process.The above discussions are based on the dynamics on regular networks.With the increase in the probability r of randomly rewriting edges, the aver-age distance of the network rapidly reduces, and regular network changes tobe small-world network. So, we firstly run the model on three types of media:Nearest-neighboring regular network, Watts-Strogatz (WS) small-world net-work with different rewriting probability r , and Erd os-R´enyi (ER) randomnetwork. All the networks have the same size ( N = 10 ) and the same av-erage degree ( h k i = 4), and the model runs on the typical parameter settingleading to scaling property: n = 1, p = 1, q = 0.Fig. 4(a) shows the differences on the inter-event time distribution P ( τ )for these networks. Topological randomness mainly affects P ( τ ) in its tail: it9 -12 -10 -8 -6 -4 -2 -12 -10 -8 -6 -4 -2 (a) p = 1, q = 0n = 1, = 2.0n = 1, = 2.7n = 1, = 2.9n i = k i , = 2.0n i = k i , = 2.7n i = k i , = 2.9 P () (b) p = 1, q = 0n = 1, = 2.0n = 1, = 2.7n = 1, = 2.9n i = k i , = 2.0n i = k i , = 2.7n i = k i , = 2.9 P ( w ) w Slope = -1.56
Figure 5: (Color online) Effect of heterogeneity of network degree. The model runs onrandom scale-free networks with different power law exponent α of degree distribution. (a)the inter-event time distribution P ( τ ) for different α and different n , parameter settings;(b) the corresponding waiting time distribution P ( τ w ). Other parameters are set as p = 1, q = 0, N = 10 . Dashed lines show the fitting slopes of curves. drives the second peak rising on the tail of P ( τ ) but almost has no effect onthe power-law head, implying another possible origin of the bimodal propertywould be the small diameter of network, because the long-range correlationin the critically is limited by the diameter of the network. In contrast, theimpact of the randomness on the waiting time distribution is weak. P ( τ w ) onsmall-world networks and ER networks generally keeps power-law-like formand only has a slightly change on the power law exponents (see Fig. 4(b)).In addition, a noticeable case is that, when p = 0 and q = 0, topologicalrandomness has contributions on the emergence of heterogeneous waitingtime distribution. As shown in Fig. 4(c), the tail of P ( τ w ) on ER randomnetwork decays much slower than the one on other networks and obviouslydeviates the exponential form. In this case, the punctuated equilibrium doesnot exist, and thus this effect would be purely related to topological impacts.Furthermore, the simulation results on different random scale-free net-works are compared to investigate the impact of heterogeneous topology.Due to the heterogeneity on degree distribution, we consider two cases onthe setting of parameter n . The first one is n = 1. And another one is10 -14 -12 -10 -8 -6 -4 -2 (a) Digg20090-order Null model1-order Null model P () (b) Brightkite0-order Null model1-order Null model
Figure 6: (Color online) Simulation results on two real-world social networks: (a) Digg2009and (b) Brightkite. The three curves in each panel shows P ( τ ) of the original friendshipnetwork, the corresponding zero-order Null model and the one-order Null model, respec-tively. Simulations runs on parameter settings p = 1, q = 0, n = 1. n i = k i , namely, the sender broadcasts messages to all neighbors. Simulationresults on the random scale-free networks with different power law exponent α of degree distribution are shown in Fig. 5. Generally, the inter-event timedistributions P ( τ ) on random scale-free networks are in bimodal type, andthe second peak mainly depends on the value of n : broadcasting messagesto all neighbors will leads to more higher peak. However, the power lawexponent α of degree distribution almost has no effect on P ( τ ) (Fig. 5(a)),indicating that exponent of power-law degree distribution dose not play thedominant role in the dynamics of the system, which is sharply different tomany other network-based dynamics. Similar phenomena is also observed onthe waiting time distribution P ( τ w ) (Fig. 5(b)). We also run the model on real-world media. The datasets of real-worldsocial networks are the friendship network of Digg users and Brightkite users,respectively. The detailed information of the real-world social networks canbe found in
Appendix . Both of the two networks are scale-free like.Under the typical parameter setting n = 1, p = 1, q = 0, as shown11n Fig. 6, both of the two inter-event time distributions P ( τ ) on the real-world media are bimodal like. Nonetheless, the tail of P ( τ ) decays slowlyand is close to power functional form, in contrast to the scenario on randomscale-free networks shown in Fig. 5(a). Analogous bimodal phenomenon wasobserved in the inter-event time distribution of on-line book-marking [25].In order to dig out the potential source of this difference, we compare theresults on the original friendship networks with the cases on zero-order Nullmodel and one-order Null model of the networks.Null model is an usual method to investigate impact of the network struc-tures on study. A zero-order Null model is almost completely different withthe original network except for the same numbers of nodes and edges. Weconstruct Zero-order Null model by the following methodat each step, a ran-domly selected edge is removed and then we randomly select two nodes, addan edge if they are unconnected, repeat this procedure enough times. Andone-order Null model is defined as that, the degree of each node remainsunchanged, but links are randomly reassigned. The method of constructingone-order Null model is: at each step, we randomly select two edges (edges E ij which connect nodes i and j , and E lm , say) which are do not possess samenode, delete this two edges, add two new edges E il and E jm if neither E il nor E jm does not exist, and repeat this procedure until all edges are treated.As shown in Fig. 6, on both of two datesets, P ( τ ) of one-order Nullmodel is very close to the results on the original networks, indicating thatthe differences on degree distribution drives the difference on the modelingresults between the the scenario on random scale-free networks and the oneon real-world media.
4. Discussion
Taking account into the activation effect in social interactions, our modelbridges queueing models and criticality phenomena, obtains rich statisticalproperties consistent with empirical studies. The results of the model mainlyincludes the following points. One is rich non-Poisson properties in individ-ual’s activities. , which corresponds to human activities in communications.Noticed that, since in the model only one event occurs in each time step, thetime interval effectively is the interval of event. Therefore all the results intemporal patterns are in the mean of the “relative clock” proposed in Ref.[31] that eliminates the effect of season fluctuation. Another point is the scal-ing patterns of bursts, which highly relates to many information spreading12rocesses, such as the spreading range of meme and duration time of rumor.In view of similar bimodal property is widely observed in many naturalsystems, our model is likely to provide a new perspective on the bimodalproperty in earthquakes [32, 33], as well as other analogous natural systems,such as tsunami [34], rainfall [35], and forest fire [36].In summary, the results of our model imply that, the punctuated equi-librium dynamics in human social interactions would be an important mech-anism that drives the emergence of non-Poisson properties in human socialactivity patterns and many social dynamics. Although we arrange the schemebased on the considerations of human communications, with suitable mod-ification, our model could be readily applicable to other interacting socialsystems such as trading and other economic systems. This model wouldshed light on the studies on human communications and social dynamics,such as information spreading, rumor propagation and disease outbreak [37].
5. Acknowledgments
This work was funded by the National Important Research Project (GrantNo. 91024026), the National Natural Science Foundation of China (No.11205040, 11105024, 10975126, 11275186, 61403421), the Major ImportantProject Fund for Anhui University Nature Science Research (Grant No.KJ2011ZD07) and the Specialized Research Fund for the Doctoral Programof Higher Education of China (Grant No. 20093402110032).
Appendix A. Introduction of datasets and social networks ∼ lerman/downloads/digg2009.html. For each story,it collects the list of all Digg users who have voted for the story up to thetime of data collection. We also retrieved the voters’ friendship links. Wedenote k to be the number of friendship links of a user, and P ( k ) to be thedistribution of k. p ( k ) is a power-law regime P ( k ) = k − α with α = 1 .
64 forlow k and α = 2 .
34 for high k (see Fig. 7 (a)).The second real-world social network is the giant component which con-tains 212,950 friendship links of 56,712 users, extracted from Brightkite,13 -5 -4 -3 -2 -1 Slope = -2.34Slope = -1.64 (a)
Digg P ( k ) k (b) Brightkitek