Mapping the q -voter model: From a single chain to complex networks
Arkadiusz Jędrzejewski, Katarzyna Sznajd-Weron, Janusz Szwabiński
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Mapping the q -voter model: From a single chain to complexnetworks Arkadiusz J¸edrzejewski a , Katarzyna Sznajd-Weron a , Janusz Szwabi´nski b a Department of Theoretical Physics, Wroc law University of Technology, Poland b Institute of Theoretical Physics, University of Wroc law, Poland
Abstract
We propose and compare six different ways of mapping the modified q -voter model tocomplex networks. Considering square lattices, Barab´asi-Albert, Watts-Strogatz andreal Twitter networks, we ask the question if always a particular choice of the group ofinfluence of a fixed size q leads to different behavior at the macroscopic level. UsingMonte Carlo simulations we show that the answer depends on the relative average pathlength of the network and for real-life topologies the differences between the consideredmappings may be negligible. Keywords:
Opinion formation, Opinion dynamics, Q-voter model, Agent-basedmodelling, Social influence, Complex networks
1. Introduction
Ordering dynamics is not only a classical subject of non-equilibrium statistical physics[1, 2], but also one of the most studied issues in the field of sociophysics [3, 4, 5]. Itoften represents opinion dynamics under the most common type of the social influence,known as conformity. Among many others, models with binary opinions are of particularinterest.One of the most general models of binary opinion dynamics was introduced by Castel-lano et al. [6] under the name the q -voter model as a simple generalization of the originalvoter model [7]. In the proposed model, q randomly picked neighbors (with possiblerepetitions) influence a voter to change its opinion. If all q neighbors agree, the votertakes their opinion; if they do not have an unanimous opinion, the voter can still flipwith probability ǫ . It has been argued that for q = 2 and ǫ = 0 the q -voter model co-incides with the modified Sznajd model, in which unanimous pair (in one dimension) ofthe neighboring sites S i S i +1 influences one of two randomly chosen neighbors i.e. S i − or S i +2 [8].Following this reasoning in Ref. [9] we have introduced a modified one-dimensionalversion of the q -voter model, as a natural extension of the original voter and the Sznajd Email addresses: [email protected] (Arkadiusz J¸edrzejewski), [email protected] (Katarzyna Sznajd-Weron), [email protected] (Janusz Szwabi´nski)
Preprint submitted to Physica A August 13, 2018 odel: a panel of q neighboring spins S i , S i +1 , ..., S i + q − is picked at random. If all q neighbors are in the same state, they influence one of two neighboring spins S i − or S i + q . If not all spins in the q -panel are equal then nothing changes in the system. Thismodification has been later considered in Refs. [10, 11, 12].For ǫ = 0 both formulations of the q -voter model seem to be almost identical with theexception of the repetitions possible in the original version [6]. However, there is anotherdifference between formulations [6] and [8], namely the first belongs to the class of socalled inflow and the second to outflow dynamics [5, 13, 14]. There was a controversyrelated to the subject if the inflow and outflow dynamics are equivalent [8, 13, 15, 16, 17].Recently, it has been shown that they are equivalent for q = 2 [17], at least in respect tothe exit probability. However for larger values of q , even in one dimension the situationis not clear [9, 10, 11, 14, 18]. Moreover, differences between dynamics in respect to thephase transitions induced by the stochastic noise has not been investigated up till now.At first glance, it seems to be trivial that choosing a different set of interactionpartners will lead to different results on the macroscopic scale. However, as describedabove, the problem occurred to be not as simple as it seems and gained the attention inthe literature. Therefore, one of the aims of this paper is to contribute to the outflow-inflow discussion.The second, probably more significant, aim is related to applications of the q -votermodel in social sciences, because as noted by Macy and Willer there was a little effort toprovide analysis of how results differ depending on the model designs [19]. Moreover, inrespect to social applications one could ask the question – how to construct the groupof influence to create easier an order (consensus) in the system? Another question onecould ask – is the problem relevant for any type of a network or maybe for some networksdifferent types of the group of influence will lead to the same results at the macroscopicscale? Only in the case of a complete graph the definition of the q -voter model is straight-forward – since on this topology all spins are neighbors [20, 21], all proposed versions ofthe q -voter model are equivalent.In the context of opinion dynamics it would be however desirable to consider themodels on top of more complex networks, as they are better representations of contactpatterns observed in the social systems [22, 23]. There are already several attempts togeneralize the q -voter model to complex networks [12, 24, 25, 26]. However, as shown inRef. [27], even in the simple case of transferring the model from 1D chain to a 2D squarelattice there is no unique rule of choosing the group of influence. Thus, the main goalof this paper is to check, how different ways of picking up the group mayimpact the macroscopic behavior of the model.
Specifically, we will focus hereon the phase transitions induced by the stochastic noise that represents one type of thesocial response, known as independence [20, 21].
2. Model
Within the modified q -voter model we consider a set of N agents called spinsons.This name, being a combination of the words “spin” and “person” , is used to emphasizethat the Ising spins in our model represent persons characterized by only one binary trait(a detailed explanation of this notion may be found in Ref. [21]). Each i -th spinson hasan opinion on some issue that at any given time can take one of two values S i = ± , i =2 , , . . . , N (“up” and “down”). The opinion of a spinson may be changed under theinfluence of its neighbors according to two different types of the social response [28]: • Independence is a particular type of non-conformity. It should be understood asunwillingness to yield to the group pressure. Independence introduces indetermi-nation in the system through an autonomous behavior of the spinsons [29]. • Conformity is the act of matching spinson’s opinion to a group norm. The nature ofthis interaction is motivated by the psychological observations of the social impactdating back to Asch [30]: if a group of spinson’s neighbors unanimously shares anopinion, the spinson will also accept it.Other types of the social response are possible as well (see Ref. [21] for an overview), butthe above two are of particular interest for studying opinion dynamics.We study the model by means of Monte Carlo simulations with a random sequentialupdating scheme. Each Monte Carlo step consists of N elementary events, each ofwhich may be divided into the following steps: (1) pick a spinson at random, (2) decidewith probability p , if the spinson will act as independent, (3) if independent, change itsopinion with probability 1 /
2, (4) if not independent, let the spinson take the opinion ofits randomly chosen group of influence, provided the group is unanimous. More detailson the dynamic rules of the model may be found in Ref. [26]. It is worth to stress here thedifference between the modified q -voter model with independence p ≥ q -voter model with ǫ ≥ p ≥ ǫ ≥
0, in which each elementary time step is described by thefollowing algorithm:1. Choose at random one spinson S i located at site i .2. Decide with probability p , if the spinson will act independently.3. In case of independence, a spinson flips to the opposite state with probability 1/2.4. In other case (conformity), choose q neighbors of site i (a so called q -panel):(a) If all the q neighbors are in the same state, i.e. q -panel is unanimous, thespinson takes the state of the q neighbors.(b) Otherwise, i.e. if q -panel is not unanimous, spinson flips with probability ǫ .Clearly, the original q -voter model is a special case of the above algorithm with p = 0and the model considered here is a special case with ǫ = 0. Note, that in contrast to p ,parameter ǫ does not describe the independence. For p = 0, if only the unanimous q -panelexists, the spinson will take its state, which means that it never acts independently. Inconsequence, the state with all spinsons in the same state is the absorbing steady statefor the original q -voter model, whereas it is not for the model considered here unless p = 0. Therefore, the original q -voter model with p = 0 is not suitable to model e.g.diffusion of innovation, for which the initial state with all spinsons down (unadopted) isa typical one [26].Introduction of a group pressure as one of the rules governing the dynamics assumessome form of interactions between the spinsons. Those interactions are best illustratedas connections between nodes of a graph the spinsons are living on. In its original for-mulation, the q -voter model can be easily investigated on an arbitrary lattice and thedefinition is clear, since q individuals influencing the voter are chosen with repetitions3rom the nearest neighborhood of the voter. Therefore, even in one-dimension the param-eter q can have an arbitrary value [6]. Although such a definition of the model leads tointeresting results from the physical point of view, it seems to be sociologically unreliable.In the modified version, repetitions are forbidden, which is probably more sociologicallyjustified. Moreover, q influencing agents may form panels of different kinds, includingstructures proposed in [12, 26], which is also different from the original formulation ofthe model. This in turn allows to investigate the role of the group structure. However,such a modified definition causes ambiguity in mapping the model on an arbitrary graph.We use here both the Watts-Strogatz [31] and the Barab´asi-Albert networks [32] asthe underlying topology of spinson-spinson interactions, since they nicely recover thesmall world property of many real social systems [33]. We set q = 4 for two reasons:(1) to reflect the empirically observed fact that a group of four individuals sharing thesame opinion has a high chance to ’convince’ the fifth, even if no rational argumentsare available [30, 34] and (2) to compare our results with those obtained on the squarelattice.Chosen from the plethora of possibilities, in Fig. 1 six different groups of influenceon a complex network are schematically shown. We would like to stress here that thechoice of precisely such groups is not accidental and is dictated mainly by earlier papers[6, 12, 26, 20]: • Line - after picking up a random target spinson (marked with a double red circlein the figure), we randomly choose one of its neighbors, then one of the neighborsof the neighbor and finally a neighbor of the latter one. All members of the q -panelare indicated with a blue circle in the figure. This is the natural generalization ofthe 1D q -voter model and was used e.g. in Ref. [12]. • Block - the group consists of a random neighbor of the target spinson, and threeneighbors of the neighbor. This method resembles to some extent the 2 × • N N - four randomly chosen nearest neighbors of the target spinson are in thegroup. This method was used in the original q -voter model [6]. • N N
N N method leading to an extended rangeof the influence: the group is composed of three randomly chosen nearest neighborsof the target spinson, and a neighbor of one of those nearest neighbors. We haveintroduced this method just to investigate the impact of the range of an influencegroup on system’s ability to stay in an ordered state. • RandBlock - a spinson and its three neighbors build the the group of influenceas in the
Block method. However, the block may be located anywhere on thenetwork. This method has been chosen as a reference for the mean-field typeapproach represented by the next method and similarly as
N N • Rand - the group consists of four randomly chosen spinsons, not necessarily con-nected with the target spinson. This corresponds to the mean-field approach, forwhich analytical results on the phase transitions are already known [20].4 igure 1: (Color online) Different groups of influence on a complex network. A random spinson is markedwith a double red circle, the spinsons in its the group of influence - with a single blue one. Interactionswithin the group and between the target spinson and the group are represented by thick lines. See textfor explanations. .1 0.2 0.3 0.4 0.500.20.40.60.81 p m NNNN3BlockLineRandBlockRand
Figure 2: Magnetization m as a function of the independence factor p for different groups of influenceon the square lattice N = 100 × p increases with the interaction range, as expected. Note that in case of
RandBlock and
Rand methods we actually abstract away fromthe underlying network topology of the model. We expect both methods to be equivalentto the complete graph case if the minimum degree of a node in the network is biggerthan or equal to q = 4 (otherwise the RandBlock may differ slightly from the completegraph, because there will be not always enough spinsons in the neighborhood to build theinfluence group). Although we use
Rand mostly as a benchmark for our simulations, the
RandBlock is much more interesting because it corresponds to a situation often encoun-tered in many organizations, in which an informal and unknown network of interactionsis over imposed on the given formal communication structure.
3. Results
The main goal of this paper is to answer the questions if and how details at themicroscopic level manifest at the macroscopic scale. Among other macroscopic phenom-ena, phase transitions are certainly the most interesting ones. For the models of opiniondynamics, the most natural order parameter is an average opinion m , defined as magne-tization i.e. m = N P S i . It has been shown that in the case of the q -voter model thephase transition may be induced by the independence factor p [29, 20]. Below the crit-ical value of independence, p < p c , the order parameter m = 0. For high independence, p > p c , there is a status-quo, i.e. m = 0. Such results were obtained on the completegraph topology which corresponds to the mean field approach, as well as on the squarelattice but only for the one particular choice of the q-panel equivalent to the Sznajdmodel [29]. In [29] a 2 × N N , which corresponds to so called inflow dynamics [14]. As expected,6 .1 0.2 0.3 0.4 0.500.20.40.60.81 p m NNNN3BlockLineRandBlockRand
Figure 3: Magnetization m as a function of the independence factor p for six different groups of influenceon the Barab´asi-Albert network of size N = 10 and parameters M = 8 and M = 8. Four models (Block,Line, RandBlock and Rand) collapse into a single curve and only two (NN and NN3 models) can bedistinguished from others. RandBlock and
Block methods overlap and agree with MFA result found in [20], i.e. p c ( q ) = ( q − / ( q − q − ), which for q = 4 gives p c = 3 /
11. Methods for which therange of interaction is shorter tend to show lower critical value of p . This result is veryintuitive, since the infinite range of interactions usually corresponds to MFA and givesthe largest critical value.Results on the Barab´asi-Albert (BA) network (see Fig. 3) are less intuitive. Itoccurs that for this topology differences between methods are almost negligible. Thephase transition is observed for all six models and the critical value of p changes onlyslightly with method. Four models (Block, Line, RandBlock and Rand) collapse into asingle curve and only two (NN and NN3 models) can be distinguished from others. Thenatural question arises here - why differences between models are clearly visible on thesquare lattice and are almost negligible in the case of BA?It should be recalled that the average path length l for the square lattice increaseswith the system size N as l ∼ N / , whereas in the case of BA as l ∼ lnN/lnlnN [22]. It means that for the same system size the average path length is dramaticallyshorter on BA than on the square lattice. In result the range of interactions on BA iseffectively much larger. To check the role of the average path length we have simulatedall 6 methods on the Watts-Strogatz network. This topology is particularly convenientbecause for the fixed network size N it is possible to decrease the average path length l by increasing the rewiring probability β .Results for several values of β are presented in Fig. 4. As β increases the criticalpoint p c shifts towards higher values and simultaneously differences between 4 methodsvanish up to a threshold value β = 0 .
5. Results for larger values of β (i.e. β > .
5; notshown in Fig. 4) are identical with those obtained for β = 0 .
5. To check how resultsscale with the system size we simulated models on networks of sizes from N = 500 to N = 10 (see Fig. 5). Surprisingly, results are virtually independent on the system size.Analogous results has been obtained for the exit probability in one dimensional system7 .1 0.2 0.3 0.4 0.500.51 p m β =0.05 m β =0.10.1 0.2 0.3 0.4 0.500.51 p m β =0.2 0.1 0.2 0.3 0.4 0.500.51 p m β =0.5NNNN3BlockLineRand Figure 4: Magnetization m as a function of the independence factor p for different groups of influenceon the Watts-Strogatz network of size N = 10 with the average degree K = 8 and rewiring parameter β . The critical value of independence p c increases with the range of interactions, as expected. However,with increasing β differences between models vanish and for large β only two models (NN and NN3) canbe distinguished from others. with inflow [14] and outflow dynamics [12].The fact that results do not depend on the system size undermines our predictionsthat the average path length l itself determines if all mapping methods overlap or not,because the l increases with the system size [35]. However, one should probably not lookat the path length itself but at the relative path length, which is defined as the averagepath length of a given network divided by the average path length of a random networkof the same size and average degree. Normalizing networks’ characteristics by those ofthe corresponding random graphs is a procedure usually used to compare networks ofdifferent sizes [36, 37, 38]. Thus, we will use the relative path length to describe thenetworks under consideration and to compare them. For example for the Watts-Strogatznetwork with k = 8 and β = 0 .
05 the relative path length is equal l rel = 1 . N = 500 and l rel = 1 . N = 1000, i.e. almost size independent. ForBarab´asi-Albert of size N = 500 it is much shorter i.e. l rel = 0 .
974 and almost does notchange with the system size. Interestingly, if one consider relative path length l rel forWatts-Strogatz of size N = 500 and k = 8 with different β it occurs that relative pathlength approaches 1 for β = 0 . l rel ( β = 0 .
05) = 1 . , l rel ( β =0 .
1) = 1 . , l rel ( β = 0 .
2) = 1 . , l rel ( β = 0 .
5) = 1 . β = 0 . https://snap.stanford.edu/data/egonets-Twitter.html [39], becauseit includes about 1000 different networks with a broad spectrum of diverse characteristics.It was relatively easy to find in the dataset networks of the same size, but with different8 .1 0.2 0.3 0.4 0.500.51 p m NN N=500N=2500N=10000N=100000 0.1 0.2 0.3 0.4 0.500.51 p m Block0.1 0.2 0.3 0.4 0.500.51 p m Line 0.1 0.2 0.3 0.4 0.500.51 p m RandBlock
Figure 5: Magnetization m as a function of the independence factor p on the Watts-Strogatz networkwith the average degree K = 8 and rewiring parameter β = 0 .
05 for several system sizes. Results arenot influenced significantly by the system size and the critical value of independence p c increases withthe range of interactions, as expected. average path lengths and/or clustering coefficients. Thus the dataset was well suited fortesting our hypothesis about the path lengths. Magnetization m as a function of theindependence factor p on six different networks of size N = 233 is presented in Fig. 6.In the top row results on three real Twitter networks [39] are presented. Networks in theleft and middle top panels have almost identical path length l but different clusteringcoefficient C . On the other hand, middle and right networks have almost identicalclustering coefficient but slightly different path length. It seems that results for allmethods overlaps the best on the right network, which has the shortest path length.Simultaneously, it seems that results on left and middle networks are the most similarto each other, i.e. path length l is more significant than clustering coefficient C indetermining if all mapping methods will give the same result or not. However, becausethe differences between properties that we take into account (i.e. l and C ) do not varymuch from network to network, for all three Twitter networks almost all methods collapseinto a single curve and only N N and
N N l , each mapping gives a completely different result (bottom left). On the otherhand, if the path lengths are similar to those of real networks, for both WS and BAmodels we observe already known behavior - most methods collapse into one curve andonly N N and
N N .1 0.2 0.3 0.4 0.500.20.40.60.81 p m Twitter 14353392, L=2.116, C=0.458
NNNN3BlockLineRandBlockRand 0.1 0.2 0.3 0.4 0.500.20.40.60.81 p m Twitter 15507297, L=2.119, C=0.526
NNNN3BlockLineRandBlockRand 0.1 0.2 0.3 0.4 0.500.20.40.60.81 p m Twitter 351092905, L=2.005, C=0.524
NNNN3BlockLineRandBlockRand0.1 0.2 0.3 0.4 0.500.20.40.60.81 p m WS, L=8.894, C=0.450
NNNN3BlockLineRandBlockRand 0.1 0.2 0.3 0.4 0.500.20.40.60.81 p m WS, L=2.15, C=0.077
NNNN3BlockLineRandBlockRand 0.1 0.2 0.3 0.4 0.500.20.40.60.81 p m BA, L=2.1101, C=0.1715
NNNN3BlockLineRandBlockRand
Figure 6: Magnetization m as a function of the independence factor p on six different networksof size N = 233. In the top row results on three real Twitter ego networks [39] are presentedwhereas in the bottom row results on three artificial networks are shown. The Twitter network IDsin the titles of the plots correspond to file names (‘ego’ node IDs) in the Twitter dataset taken from https://snap.stanford.edu/data/egonets-Twitter.html [39]. In the bottom row results on threemodel networks are shown: a Watts-Strogatz network with the average degree K = 4 and the rewiringprobability β = 0 .
035 (bottom left), a WS network with K = 18 and β = 1 (bottom middle) and aBarab´asi-Albert one with the number of new edges M = M = 10 (bottom right). N N method differs the most from the
Rand and
Line is the most similar.
N N method has relatively the shortest range ofinteractions and
Line much larger.The differences between methods may be also explained, at least qualitatively, interms of probabilities of finding non-unanimous influence groups. For the sake of sim-plicity let us assume that our network has the topology of a Bethe lattice [40] with thecoordination number k . For convenience, we took a modified definition of the Bethelattice with the central node having only k − k − k − agents at the second level of itsego graph, and in general ( k − d nodes at distance d . Now, let us consider our model atan early stage of a simulation. Let us assume that there are only two spinsons includingthe central one in the “down” state due to independence and that the central spinson hasbeen chosen again in a basic Monte Carlo event (it will be referred as the target spinsonin the following). However this time it is not independent, i.e. it is exposed to the grouppressure. Since most of the spinsons are in the “up” state, the system has a naturaltendency to reduce disorder due to conformity. Nevertheless, we can ask the questionwhether there are significant differences between the methods in maintaining disorderin the system. In other words we can check if the methods differ in the probabilities offinding a non-unanimous group of influence in this situation.To this end, we can consider configurations with the other “down” spinson residingat different levels of the target’s ego graph. We start with the “down” spinson being inthe nearest neighborhood of the target node. In this case we expect the N N method togive the highest probability to build a group the “down” spinson belongs to. The reasonis simple: this is the only method which operates exclusively in the nearest neighborhoodof the target spinson. Thus we draw 4 agents out of k − N N k − Line and
Block methods require only one drawing fromthe first level (top right plot of Fig. 7). Hence it is less likely to hit the “down” spinson.Finally,
RandBlock and
Rand algorithms yield the smallest probabilities, because theyoperate on the whole network rather than in the close neighborhood of the target node.Since the problem at hand is nothing but a variation of an urn problem [41], we canactually calculate for each method the probability of finding a non-unanimous group ofinfluence. To focus our attention we set k = 9 and the system size N = 4680 meaningthat the ego graph of the target spinson consists of 4 levels. In the case of the N N igure 7: (Color online) A schematic representation of our model at an early stage of a simulation ona Bethe lattice with the coordination number k = 5 and the root node having k − NN method (left) gives much higherprobability of finding a non-unanimous group of influence than the Block one (right). Bottom: if thenot-adopted spinson is at the second level, the
Block method yields higher probability than NN (in NN case the probability is zero). q -panels is just the number of 4-combinations selectedfrom the nearest neighborhood of the target, | Ω | = (cid:18) k − (cid:19) = (cid:18) (cid:19) = 8!4!4! = 70 . (1)The not adopted agent at the first level has to belong to each non-unanimous group. Theother three members are selected from seven “up” spinsons residing at that level. Thus,the number of non-unanimous groups in the N N method is equal to | N U | = (cid:18) (cid:19)(cid:18) (cid:19) = 7!3!4! = 35 . (2)This yields the following probability of finding a non-unanimous group: P (1) NN = | N U || Ω | = 0 . . (3)The superscript (1) in the last expression indicates the level the other “down” spinsonbelongs to. In the N N | Ω | = (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) = 1344 (4)possible influence groups. Once the “down” spinson is chosen, we select two others fromthe first level, then pick one of the members of the group and add one of its neighborsto the group. The number of all possibilities is in this case given by | N U | = (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) = 504 . (5)Hence, P (1) NN = | N U || Ω | = 0 . . (6)Similar analysis leads to the following results for the other methods: P (1) Block = 0 . , P (1) Line = 0 . ,P (1) Rand = 0 . ∗ − , P (1) RandBlock = 0 . ∗ − . (7)We see that indeed the N N method gives the highest probability of leaving the targetspinson untouched in this case.If the other “down” spinson resides at the second level of target’s ego graph, the
Block method should give the highest probability of maintaining disorder in the system,because it consists of drawings mostly from that level (bottom row of Fig. 7). Again, wecan calculate the corresponding probabilities to get: P (2) Block = 0 . , P (2) Line = 0 . , P (2) NN = 0 . ,P (2) Rand = 0 . ∗ − , P (2) RandBlock = 0 . ∗ − , P (2) NN = 0 . (8)13he Block method gives indeed the highest probability, followed by
Line and
N N
N N is zero, because the method operates only at the first level. It isinteresting to note that the probabilities for the second level are much lower than thosefor the first one. With similar reasoning we can show that in general the farther thedistance between the two “down” spinsons, the smaller the chance to maintain disorder,i.e. to let the target spinson unchanged. As a consequence, only the state of the twoclosest levels is actually significant for the evolution of target’s opinion. For this reasonthe methods
N N and
N N
3, followed by
Block and
Line destroy the order in the systemsthe fastest, i.e. for relative small values of the independence parameter p .Note that the above conclusion is in accordance with our simulation results shownin Figs. 3, 4 and 6. Thus, although a Bethe lattice resembles ego graphs of model andreal networks on average only, the reasoning remains the same - if the “down” agents aresparse, the N N and
N N
N N and
N N
Block , Line , RandBlock and
Rand ) areessentially indistinguishable. The
N N and
N N p , i.e. destroying the order abit faster (see Fig. 3, bottom row of Fig. 4 and most of the plots in Fig. 6 for furtherreference).
4. Conclusions
From physical point of view it is always an interesting question how details at themicroscopic scale manifest at the macroscopic level. In the field of opinion dynamicssuch a macroscopic quantity is the opinion, defined in a case of binary models as themagnetization m . In this paper we examine six models that differ only in the way ofselecting a group of influence but the size of this group remains fixed. Therefore thereare no differences between models on the complete graph. For other topologies meth-ods for which range of interaction is shorter tend to show the lower critical value ofthe independence factor p below which m > m = 0. Only two methods, RandBlock and
Rand , give exactly the same results on all topologies and overlap MFAresult found in [20]. Remaining four methods give different results and differences be-tween methods increase with the relative path length, i.e. are the most visible on theregular lattices. With decreasing relative path length the differences between methodsvanish. One should notice that the average path length itself does not determine thedifferences between models, because results are virtually independent on the system size.What determines network properties is the relative path length defined as the ratio be-tween the average path length of considered network and the average path length of arandom graph of the same size L and average degree h k i . It should be noted that mostof the real-world networks are characterized by relatively short paths [22] and thereforedifferences between models should be negligible.14e believe that our results contribute also to the discussion on differences betweeninflow and ouflow dynamics. As noted by Dietrich Stauffer: The crucial difference of theSznajd model compared with voter or Ising models is that information flows outward: Asite does not follow what the neighbors tell the site, but instead the site tries to convincethe neighbors [42]. Of late, a debate on whether inflow dynamics is different from outflowdynamics has emerged [13, 17, 14]. Our findings indicate that not the direction of theinformation flow itself but the range of interactions is important, what coincides withresults obtained by Castellano and Pastor-Satorras [17]. It is worth to notice that someof rules, investigated here, may be viewed as inflow and other as outflow dynamics. Inparticular, the
N N method corresponds to the inflow dynamics. On the other hand,the
Block method was inspired by the two dimensional and the
Line model by theone dimensional outflow dynamics. Therefore both can be viewed as outflow dynamics.Both outflow rules (
Block and
Line ) give the same results on scale-free and real Twitternetworks, whereas the inflow rule
N N gives lower value of the critical value of p . However,it seems that the critical value of p increases with the relative range of interactions andtherefore it is understandable that N N (inflow) rule gives the lowest value of p . Soperhaps one should not think about the direction (in or out) of the information flow itselfbut the range of interactions, what would coincide with results obtained by Castellanoand Pastor-Satorras [17]. Summarizing, indeed inflow and outflow dynamics give differentresults but the reason is simply the difference in the range of interactions.As already mentioned in the introduction, the main motivation for this paper was theremark by Macy and Willer that there was a little effort to provide analysis of how resultsdiffer depending on the model designs [19]. In the context of the problem posed here,it would seem that the structure of the group of influence may be important from thesocial point of view. However, as we have shown in the case of many complex networks,including BA and real networks, the importance of the group structure is often negligible.In this paper we considered only static networks (not changing in time), which is acommon approach while studying dynamical processes like opinion spreading or diffusionof innovation. However, the characteristics of many real networks evolve in time and thereis more and more data available on temporal networks [43]. If the changes take placeat time scales comparable to those of studied processes, the temporal heterogeneities insuch networks may lead to big differences in the dynamics of the processes, even if thenetworks appear similar from the static perspective [44]. Thus, it could be worth tocheck what is the impact of the group structure in models put on top of real temporalnetworks. This issue will be addressed in one of the forthcoming papers. Acknowledgments
This work was supported by funds from the National Science Centre (NCN) throughgrant no. 2013/11/B/HS4/01061.
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