Agent Based Simulation of Spreading in Social

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find an answer to the question: ,,How inactivity of actors of a social system affect spreading ... Simulation of Spreading in Social-Systems of Temporarily Active Actors. 331. Though .... At first we sit our agents on a square lattice topology with periodic boundary condition. .... see an almost linear trend [12] (see inset of Fig. 5).
Agent Based Simulation of Spreading in Social-Systems of Temporarily Active Actors Gergely Kocsis and Imre Varga University of Debrecen, Faculty of Informatics, Department of Informatics Systems and Networks, Kassai Str. 26. H-4028 Debrecen, Hungary {kocsis.gergely,imre.varga}@inf.unideb.hu http://www.inf.unideb.hu

Abstract. In this work a novel model of information spreading processes in systems of dynamic active-inactive actors is presented. In our model information can flow only through those actors of the system that are currently active. Based on this model we carried out computer simulations showing how the activity of agents affect the process. We also carried out some basic investigation of the effect of inhomogeneous activity. The results of the work can be used to qualitatively predict what would be the effect if the activity of agents would change in a social system. Keywords: ABM, spreading, active nodes.

1

Introduction

The investigation of social spreading phenomena is in the focus of research for a couple of decades and the importance of it is still increasing. The main reason of this is that however the fundamentals of the field have been invented at the beginning of the XXth century [1, 2], the importance of it again increased with the appearance of novel social systems of the information society such as online social networks. Information diffusion – and more generally social diffusion phenomena – have been investigated in a huge number of works [3–5]. However in these works the actors of the social system are modeled as always active entities or the activity of them is only marginally mentioned. This assumption however does not represent the real properties of these actors. Contrary to the above works here we investigate the effect of inactivity of agents on spreading phenomena e.g. on the spreading of information on a social network. To be able to focus on the effect of the temporal change of activity of agents itself, we extend a previously introduced model of information spreading and study the resulting differences. We examine homogeneous activity on different types of networks from regular lattices to scale-free networks. At the end we also analyze the effect of inhomogeneous activity based on realistic data. One final goal of ours is to find an answer to the question: ,,How inactivity of actors of a social system affect spreading phenomena in it and what can we do to make this spreading faster?”. J. Was, G.C. Sirakoulis, and S. Bandini (Eds.): ACRI 2014, LNCS 8751, pp. 330–338, 2014.  c Springer International Publishing Switzerland 2014 

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Though the exact validation of the answer of such a question is far from being trivial, based on our results one can get an idea e.g. how to make an advertising campaign more effective, by making some actors more active than others. Then later the results of these actions can be measured by advertising effectiveness measuring indicators such as number of visitors of a homepage, or number of downloads of a given paper.

2

Model

In our model the actors of the social system are represented by interacting agents put on a given network topology. All agents have the following properties to describe them: Si ∈ {0, 1} tells if agent i is informed or not. Ri ∈ {0, 1} describes whether an agent inactive or active. There are 3 system wide properties as well. These properties describe the agents, but we set the values of them to be equal for all of them so that analytical investigation of the model becomes more easy. α shows how sensitive an agent is for information coming from its neighbors. β shows the sensitivity of agents for outer information. These two parameters were presented in a previous model in [6]. Finally the probabilistic value γ tells how often agents go inactive. For all three values α, β, γ ∈ [0, 1] ∈ R. Larger values are for more sensitivity and more probability of being inactive respectively. Since α, β and γ are equal for all agents, and do not change during the evolution of the model agents can be at four different states that are the following: σ0 σ1 σ2 σ3

active-uninformed inactive-uninformed active-informed inactive-informed

R = 1, R = 0, R = 1, R = 0,

S S S S

=1 =1 =0 =0

Led by the states of their neighbors and their environment, agents can stochastically change their own states at discrete points of time following the below described state change rules. Active-uninformed (σ0 ) agents can go to inactive-uninformed state σ1 with possibility γ. They can get informed – meaning that they switch to activeinformed σ2 state with probability (1 − γ)A, where A tells the probability of getting informed based on the current state of the agent, its neighbors, and the environment. The meaning of A is described in details below. From the previous 2 rules it comes clearly, that agents at state σ0 can also stay in the same state with probability (1−γ)(1−A). Inactive-uninformed (σ1 ) agents can change their state to active-uninformed (σ0 ) with probability (1 − γ) while with probability γ they can also stay at the same state (σ1 ). In our model informed agents can not go uninformed, thus active-informed (σ2 ) agents can choose only between two possibilities at each timestep: With probability γ they can turn inactive-informed (σ3 ) and with probability (1 − γ) they can stay active. Finally inactive-informed (σ3 ) agents also stay inactive-informed with probability γ and change to activeinformed with probability (1 − γ). Summarizing the above rules we can say that agents go or stay inactive with probability γ and go or stay active with probability (1 − γ). Only active agents can affect and be affected by other agents. And

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only active-uninformed can turn informed with probability (1 − γ)A, where A is presented later in eq. 1. On Fig. 1 one can follow these rules more easily.

σ

γ

1

1-γ

σ

0

(1-γ)(1-A)

1-γ γ

(1-γ)A

σ

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1-γ γ

σ

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Fig. 1. State-change rules of agents of our model. Red circles are for uninformed agents while green ones are for informed agents. Grey filled circles are for inactive states. With probability γ agents go/stay inactive and with probability 1 − γ they go/stay active. The information of active-uninformed agents follow the rule presented in [6].

The most crucial state change in our model is the change from uninformed (σ0 ) state to informed (σ2 ). This rule is motivated by the model presented in a previous work [6]. This model describes information spreading as the consequence of the competition of two separate information channels. Namely agents get information from an outer channel, and from their neighbors. As it was written above in our model α and β tells how sensitive agents are to the information sources. Based on these parameters and the new ones catching the activity of agents, the probability that an active agent goes informed is again (1 − γ)A, where A is ni  ((1 − Sj )Rj ) + βSi Ri ) , (1) A = 1 − exp(αSi Ri j=1

where ni is the number of neighbors of agent i. It is important to emphasize again that in our case however all agents have the same values for α and β only active agents can spread and receive information. This phenomenon is also caught in eq. 1, where in the exp() function we describe the incoming information to an agent as the following: If an agent is not informed and active it can receive information from their active informed neighbors. The amount of information coming from them is multiplied by the respective sensitivity α. After this if again the agent is active-uninformed it receives information from the outside. This amount is again multiplied by the sensitivity to it β.

3

Results of Computer Simulations

To investigate the time evolution of our model for different parameters on different network topologies, we carried out computer simulations. In all our

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simulations the system size was set to N = 106 . the sensitivity values, based on previous studies of ours [6] were α = 10−2 and β = 10−4 . 3.1

Results on Square Lattice

At first we sit our agents on a square lattice topology with periodic boundary condition. This makes visualization of the system very easy so we can get a look into the evolution of the system through snapshots of it. Fig. 2 shows snapshots of the system at three different points of time. It can be seen clearly that the evolution of the system shows much similarity to the referred model. At first some random agents get informed due to the outer information channel. After a time these agents serve as nucleation points of growing clusters while as time passes the whole system tends to arrive to a homogeneous informed state. The effect of the activity property of agents comes visible if we take a look at the time needed for the system to arrive to its final state. It can be seen at the first sight that the dynamic activity of agents dramatically slows down spreading however the evolution of the system does not show other qualitative changes.

a)

b)

c)

d)

e)

f)

Fig. 2. Snapshots of the system on a square lattice taken at different points of time. a, b and c) shows the case of 0 inactivity. d, e and f ) are for non zero inactivity (γ = 0.5). t = 100, 250, 375 respectively for a, b and c while t = 350, 800, 1250 respectively for d, e and f . Note that while the qualitative behavior of the system does not change with the raised level of inactivity, the evolution of it dramatically slows down.

In order to get a better insight of this slowing we analyzed the amount of informed agents in the system during its evolution. The results of plotting the percentage of informed agents in the system (σ2 + σ3 ) as a function of time are presented on Fig. 3. The first important information of this figure is that the

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shape of the curves on Fig. 3 a) follows the well known logistic form that usually appears in the case of diffusion processes. It can also be seen clearly that however the spreading process shows the same fashion regardless of the value of γ the time needed to reach the state where 95% the agents are informed tS=0.95 is not a linear function of γ. Even, from the semi-logarithmic scale of Fig 3 b) it can be seen that this dependence is faster than exponential. The reason why we chose to examine tS=0.95 instead of tS=1 is that the spreading slows down so much close to the homogeneous state that tS=1 → ∞, however tS=0.95 still fits our needs while the respective value can be reached in a reasonable time.

a)

b) 2

4

tS=0.95

10

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10

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0.3

0.5

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0.7

0.9

Fig. 3. Simulation results on a square lattice. a) The ratio of informed agents in the system as a function of time for different levels of inactivity γ. With the increase of γ it takes non-linearly more and more time to reach the homogeneous informed state. b) The time needed to reach 95% domination of informed agents. Note that the dependence on γ is faster than exponential.

3.2

Simulation Results on Complex Networks

Since square lattice is far from being a good model of real world social networks, now that we got an insight to the behavior of the model on it, we also ran simulations on more complex networks. Namely we applied Watts-Strogatz rewired lattices [7], and scale-free networks [8]. The rewired lattices generated by the method invented by Watts and Strogatz are still pretty far from real social network topologies however they have two very important properties. i.) These are one of the most simple networks known to have small-world property that is a fundamental property of real social network topologies as well. ii.) By tuning the rewiring probability between two extreme cases of p (p = 0 and p = 1) we can map the behavior of our model continuously from regular lattices to networks much closer to the real form of social network topologies. Since the most crucial parameter of rewired lattices is the rewiring probability p we carried out simulations for different values of γ while we ran p continuously between 0 and 1. Fig. 4 a) clearly shows that the domination time tS=0.95

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decreases with the growing of p and this is independent of the actual value of γ. One can even possibly say that this dependence follows a power law at least at the first half of the interval [0, 1]. However because of the short range of values and the obvious differences at the very beginning and at the end we more likely state that these curves are just close to follow a power law. Comparing Fig. 3 b) and Fig. 4 b) shows that the qualitative time evolution of the model does not change if we switch from a regular square lattice to a rewired lattice (on the figure rewiring probability p = 0.9, however we found similar pictures for other values of p as well). As it is well-known in the literature that realistic social networks have scalefee property, we also performed simulations on these types of networks. Namely we used the model invented by Varga et. al. to generate networks with the proper parameters [9, 10]. To get realistic results, these networks are similar to the topology of Facebook (based on the Facebook sample of Koblenz Network Collection [11]). Our results showed hardly any difference between the case of a scale-free network and rewired lattices. This means that while rewiring probability p has some effect on the spreading (see Fig. 4 a)), other topological properties do not affect its nature. The result of these simulations are presented on Fig. 4 b). 3.3

The Effect of Inhomogeneous Activity

In all the previous cases we assumed that however the probability of being inactive γ can be set between 0 and 1 continuously, the chosen value is system wide. This mean that we handle all the agents of the system as if they were similar. It is easy to see that this assumption is far from being realistic. However previous

a)

b)

Fig. 4. Simulation results on complex networks. a) The time needed by the system to evolve to a state where 95% of the agents are informed tS=0.95 as a function of the rewiring probability p. tS=0.95 is decreasing almost following a power law. b) The information time of the system increases faster than exponential with the increase of γ for rewired lattice and for scale-free networks as well.

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. .

Fig. 5. Time evolution of the system in cases of homogeneous and inhomogeneous activity. Note that there is even quantitatively hardly any difference between the two curves. This means that the increased activity of high degree nodes does not make the spreading much faster without increasing activity of law degree nodes. inset: The activity of nodes depends linearly on their degree [12].

studies showed that the activity of agents of a social system depends linearly on the number of contacts. More precisely on a social network if we draw the amount of posts of agents as a function of the number of their connections we see an almost linear trend [12] (see inset of Fig. 5). Based on this result we modified our model, so that the value of γ is not a system-wide parameter anymore, but it can change of its value γi for different i agents. To catch the above mentioned linear dependence we set the inactivity property of agents to: ni γi = γ(1 − ), (2) nmax where nmax is the highest degree in the system and ni is the number of connections of agent i. As a result of this the agent with the highest degree will be almost always active and the agents with low degree go/stay inactive in each timestep with a probability γi → γ. Or more simply, we make higher degree nodes linearly more active, so that the system gets closer to the real case, while it can be still parametrized through γ. The results of our simulations showed that making the agents of the system inhomogeneous in this sense does not affect the evolution as much as one would expect (see Fig. 5). Based on our previous knowledge that emphasize the importance of high degree nodes of the system it would not be surprising if we would find that making these nodes more active makes the spreading dramatically faster. In contrary however according to our simulations this speeding up does not happen. The reason of this is that however high degree nodes are connected

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to many low degree nodes, because the inactivity of these latter ones information can not be sent or received. The result of this is that however high degree nodes try to be much more active than before, they find no agents to interact with. From the above results it comes clearly that if one wants to increase the speed of spreading on a social network he has to take care to increase or at least do not decrease the activity of low degree nodes as well. This statement also has a practical meaning for example advertisers on social networks. It tells us that if one would like to improve the effectiveness of an advertising campaign it is not enough to focus on central nodes. The increasing of the activity of marginal parts of the network is also essential.

4

Discussion

In this paper we introduced a novel model of spreading phenomena on social systems where the actors are active only in a given part of their time. Based on computer simulations we showed that the increased inactivity of agents slows down the spreading however this change does not have a qualitative effect. We ran our simulations on systems with different network topologies (square lattice, rewired lattice, scale-free network). In all cases we found that the time needed for the system to reach an almost homogeneous informed state depends on the probability of an agent to be inactive as a faster than exponential function. In the most realistic case, where the network topology was a scale-free network, we also examined the effect of inhomogeneous inactivity. We found that the increased activity of central nodes can not make the spreading faster if in the same time the activity of peripheral actors is not increased. Based on these results, in the future we would like to investigate the system using different sensitivity of actors. In our later plans we would also like to test our model on more dynamic networks. We would also like to apply our model outside social networks e.g. in the case of wireless sensor networks. ´ Acknowledgments. The publication was supported by the TAMOP-4.2.2. C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund.

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