Non-consensus opinion model on directed networks

Non-consensus opinion model on directed networks

Bo Qu Delft University of Technology, Delft, Netherlands Qian Li Department of Physics and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA Shlomo Havlin Department of Physics, Bar Ilan University, Ramat Gan, Israel H. Eugene Stanley Department of Physics and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA Huijuan Wang
4 April 2014
4 April 2014
Abstract

Dynamic social opinion models have been widely studied on undirected networks, and most of them are based on spin interaction models that produce a consensus. In reality, however, many networks such as Twitter and the World Wide Web are directed and are composed of both unidirectional and bidirectional links. Moreover, from choosing a coffee brand to deciding who to vote for in an election, two or more competing opinions often coexist. In response to this ubiquity of directed networks and the coexistence of two or more opinions in decision-making situations, we study a non-consensus opinion model introduced by Shao et al. [1] on directed networks. We define directionality as the percentage of unidirectional links in a network, and we use the linear correlation coefficient between the indegree and outdegree of a node to quantify the relation between the indegree and outdegree. We introduce two degree-preserving rewiring approaches which allow us to construct directed networks that can have a broad range of possible combinations of directionality and linear correlation coefficient and to study how and impact opinion competitions. We find that, as the directionality or the indegree and outdegree correlation increases, the majority opinion becomes more dominant and the minority opinion’s ability to survive is lowered.

1 Introduction

Network theory based on graph theory uses a graph to represent symmetric or asymmetric relations between objects shown by undirected and directed links, respectively. The study of social networks is one of the most important applications of graph theory. Social scientists began refining the empirical study of networks in the 1970s, and many of the mathematical and physical tools currently used in network science were originally developed by them [2]. Social network science has been used to understand the diffusion of innovations, news, and rumors as well as the spread of disease and health-related human behavior [3, 4, 5, 6, 7, 8, 9]. The decades-old hot topic of opinion dynamics continues to be a central focus among researchers attempting to understand the opinion formation process. Although it may seem that treating opinion as a variable or a set of variables is too reductive and the complexity of human behavior makes such an approach inappropriate, often human decisions are in response to limited options: to buy or not to buy, to choose Windows or Linux, to buy Procter & Gamble or Unilever, to vote for the Republican or the Democrat.

Treating opinion as a variable allows us to model patterns of opinion formation as a dynamic process on a complex network with nodes as agents and links as interactions between agents. Although the behavior dynamics of human opinion are complex, statistical physics can be used to describe the “opinion states” within a population and also the underlying processes that control any transitions between them [10, 11, 12, 13, 14, 15, 16]. Over the past decade numerous opinion models have combined complex network theory and statistical physics. Example include the Sznajd model [17], the voter model [18, 19, 20], the majority rule model [21, 22], the social impact model [23, 24], and the bounded confidence model [25, 26]. All of these models ultimately produce a consensus state in which all agents share the same opinion. In most real-world scenarios, however, the final result is not consensus but the coexistence of at least two differing opinions.

Shao et al. [1] proposed a non-consensus opinion (NCO) model that achieves a steady state in which two opinions can coexist. Their model reveals that when the initial population of a minority opinion is above a certain critical threshold, a large steady-state spanning cluster with a size proportional to the total population is formed [1]. This NCO complex network model belongs to the same universality class as percolation [1, 27, 28], and it and its variants, have received much attention. Among the variants are a NCO model with inflexible contrarians [29] and a NCO model on coupled networks [30, 31].

To date the model has not been applied on directed networks. Directed networks are important because many real-world networks, e.g., Twitter, Facebook, and email networks, are directed [32]. In contrast to undirected networks, directed networks contain unidirectional links. In opinion models, a unidirectional link between two nodes indicates that the influence passing between the two nodes is one-way. A real-world example might be a popular singer who influences the opinions the fans hold, but the fans do not influence the singer’s opinion. In contrast, bidirectional links occur when the influence between two agents is both ways. Real-world unidirectional links are ubiquitous and strongly influence opinion formation, i.e., widespread one-way influence has a powerful effect on opinion dynamics within a society.

Our goal here is to examine how the NCO model behaves on directed networks. We compare the results of different networks in which we vary the proportion of unidirectional links. We also measure the influence of asymmetry between indegree and outdegree. We find that when the indegree and outdegree of each node are the same, an increase in the number of unidirectional links helps the majority opinion spread and when the fraction of unidirectional links is at a certain level, increasing the asymmetry between indegree and outdegree increases the minority opinion’s ability to survive. We also observe that changing the proportion of the unidirectional links or the relationship between the indegree and outdegree of the nodes causes phase transitions.

2 Basic definitions and notations

2.1 The NCO model

In a NCO model [1] on a single network with nodes, each with binary opinions, a fraction of nodes has opinion and a fraction has opinion . The opinions are initially randomly assigned to each node. At each time step, each node adopts the majority opinion, when considering both its own opinion and the opinions of its nearest neighbors (the agent’s friends). A node’s opinion does not change if there is a tie. Following this opinion formation rule, at each time step the opinion of each node is updated. The updates occur simultaneously and in parallel until a steady state is reached. Note that when the initial fraction is above a critical threshold, (even minority), both opinions continue to exist in the final steady state.

Figure 1 shows an example of the dynamic process of the NCO model on a small directed network with nine nodes. Here we consider the in-neighbors of a node as the friends influencing the node, and the out-neighbors as the friends influenced by the node. At time four nodes are randomly assigned the opinion (empty circle), and the other five nodes the opinion . At time node A has opinion but is in a local minority and thus updating it means changing its opinion to . At time node B belongs to a local minority and thus needs updating. At time all nodes hold the same opinion as their local majority, and the system has reached a final non-consensus steady state.

Figure 1: Schematic plot of the dynamics of the NCO model on a directed graph with 9 nodes.

2.2 The directionality and indegree outdegree correlation

To quantitatively measure the one-way influence in a network, we define the directionality as the ratio between unidirectional links and all links. The directionality is , where the normalization , because a bidirectional link can be considered as two unidirectional links. Because we want to determine how much one-way influence affects the NCO model, we consider as a variable the fraction of one-way links , where represents undirected networks. Although the sum of indegree and the sum of outdegree are equal in a directed network, the indegree and outdegree of a single node are usually not the same. To quantify the possible difference between the node’s indegree and outdegree, we use the linear correlation coefficient between them,

(2.1)

where and are the indegree and outdegree of node respectively. The average degree is the same for both indegree and outdegree. Note that when the indegree is linearly dependent on the outdegree for all nodes, and when the indegree and outdegree are independent of each other. In this paper we confine ourselves to the case in which the indegree and outdegree follow the same distribution. In this case, implies that holds for every node .

3 Algorithm Description

Inspired by earlier research on directed networks [33, 32, 34, 35, 36, 30], we propose two algorithms to construct directed networks. One is a rewiring algorithm that can be applied to any existing undirected network to obtain a directed network with any given directionality but each node has the same indegree and outdegree as the original undirected network. The other constructs directed networks with a given directionality and indegree-outdegree correlation, and with the same given indegree and outdegree distribution. Note that all networks considered in this paper contain neither self-loops nor multiple links in one direction between two nodes.

3.1 Directionality-increasing rewiring (DIR)

Here we introduce a rewiring approach that changes the directionality but does not change the indegree and outdegree of any node. It was first proposed in Ref. [37], and also employed by Ref. [32]. Here we improve it to gradually increase the directionality, via a technique we call directionality-increasing rewiring (DIR).

Many undirected network models with various properties have been designed. Examples include the Erdös-Rényi model [38], the Bárabasi-Albert scale-free model [39], and the small-world model [40]. If the links of an undirected graph are considered bidirectional, for an arbitrary undirected graph the indegree and outdegree correlation will be . Figure 2 demonstrates an approach that changes the directionality but does not change the indegree and outdegree of any node nor . We randomly choose two bidirectional links connecting four nodes and treat them as four unidirectional links. Note that this differs from the technique presented in Ref. [32] in that we choose two bidirectional links instead of two random links that may also contain unidirectional links so that the directionality increases after each step. Then we choose two unidirectional links, one from each bidirectional link, and rewire them as follows. For both unidirectional links the head of one link is replaced with the head of the other. If this rewiring introduces multiple links in any direction between any two nodes, we discard it and randomly choose two other bidirectional links. We can increase the number of unidirectional links by repeating the rewiring step and increasing the directionality in each step. The directionality can be varied from 0 to 1. In general, DIR can be applied to any directed network to further increase its directionality.

Figure 2: (Color online) Directionality-increasing rewiring (DIR)

3.2 Constructing an asymmetric indegree and outdegree network and rewiring it to decrease its directionality (ANC-DDR)

We have shown how to obtain a desired directionality when the indegree and outdegree correlation is . We further propose an algorithm to construct a network with a given combination of and , where . Inspired by the work presented in Ref. [35], which focuses on generating directed scale free (SF) networks with correlated indegree and outdegree sequences, we extend it to a scenario in which the indegree and outdegree sequences follow a distribution that is arbitrary but the same, and we control not just the correlation between the indegree and outdegree but also the directionality, which was ignored in Ref. [35]. We generate an indegree sequence (following a Poisson distribution or power law) and a null outdegree sequence. We then copy a fraction of the indegree sequence to the outdegree sequence, and shuffle the fraction of the indegree sequence as the rest of the outdegree sequence. We thus create an outdegree sequence, a fraction of which is identical to the corresponding part of the indegree sequence and a fraction of which is independent of the indegree sequence. After randomly connecting all nodes (given their indegree and outdegree), as in the configuration model [33], we obtain a network with a directionality111. and an indegree and outdegree correlation close to . Note that we can further control the indegree and outdegree correlation in a small range close to by discarding networks with indegree and outdegree correlations outside the expected range. This enables us to construct a network with the indegree and outdegree correlation (), a technique we call asymmetric indegree-outdegree network constructing (ANC).

We use the following rewiring steps to further tune the directionality without changing the indegree and outdegree of each node or the indegree and outdegree correlation . The goal is to decrease the directionality by repeatedly rewiring two unidirectional links into one bidirectional link. In each step, we choose four nodes linked with at least three directed links as shown on the top half of Fig. 3. We rewire these three links to the positions shown at the bottom of Fig. 3. If this rewiring introduces multiple links between any two nodes in any direction we discard the rewiring, select four new nodes, and repeat the step222An efficient rewiring program is available upon request. This rewiring produces at least one more bidirectional link and thus decreases the directionality. We call this procedure directionality-decreasing rewiring (DDR). We combine DDR with ANC and call the entire algorithm ANC-DDR.

Using ANC we can construct a network with a specified indegree and outdegree correlation , where the indegree and the outdegree follow the same given distribution and, using DDR, we can change the directionality in a range dependent on the given without changing the indegree and outdegree. The range within which we can tune depends on the given . For example, for binomial networks333Binomial networks are directed networks with the same Poissonian indegree and outdegree distributions., can be changed from to when , but the minimum value of must be approximately and any smaller value is disallowed when . We explore the relation between the minimal possible directionality and a given indegree and outdegree correlation first via numerical simulations444In each realization of the simulations, we apply DDR repeatedly on the network constructed by ANC until the four-node structure in Fig. 3 cannot be found after a number of consecutive attempts, then the directionality is considered the minimal directionality corresponding to the given . For each given , we perform realizations and calculate the average of the minimal directionality . in both binomial and SF networks555SF networks are directed networks whose indegree and outdegree distributions follow the same power law.. Figure 3 shows the linear relationship in both types of network. Binomial networks are characterized by a Poisson degree distribution with , where is the node degree and is the average degree. The degree distribution of SF networks is given by , where is the smallest degree, is the degree cutoff, and is the exponent characterizing the broadness of the distribution [39]. In this paper we use the natural cutoff at approximately [41] and .

Figure 3: (Color online) (a) The degree-preserving rewiring for decreasing the directionality. (b) Plot of the minimal directionality obtained by simulating ANC-DDR, for binomial networks (, , nodes) and SF networks (, , nodes) with 100 realizations, and the theoretical minimum possible directionality , Equ. (3.1), for binomial networks (the solid line) and for SF networks (the dash line) as a function of the indegree and outdegree correlation .

For any network constructed using ANC-DDR with an arbitrary given degree distribution (where the distribution is same in both indegree and outdegree), we can analytically prove (see Appendix A) the relationship between the minimal possible directionality and the indegree outdegree correlation ,

(3.1)

The simulation and theoretical results concerning the relationship between and are consistent, both indicating a linear relation. Because of the finite number of attempts and the random selection process determining the four-node structure, the obtained using simulations is slightly larger than the theoretical .

Although using ANC-DDR we can construct a network with a given and a given within a corresponding range to , it is computationally expensive to generate a large network with the minimal possible directionality. We thus apply DIR to undirected network models in order to generate directed networks with a directionality ranging over , but with the given indegree and outdegree correlation , to understand the effect of directionality on opinion competitions. We then use ANC-DDR to generate directed networks with a given indegree and outdegree distribution and correlation, and a given directionality, to explore the effect of both and on the opinion model.

4 The influence of the directionality

In order to examine how the directionality influences the NCO model, we apply DIR to undirected network models to generate directed binomial networks, SF networks, and random regular (RR) networks666In this paper, random regular (RR) networks are directed networks in which the indegrees of all nodes and outdegrees of all nodes are the same and the nodes are randomly connected. [42] with directionality ranging over . The NCO model is further simulated on each directed network instance. All simulation results are the average of networks with nodes and .

We use to denote the size of the largest cluster in the steady state (where is the initial opinion randomly assigned to a fraction of nodes) and to denote the size of the second largest cluster. For all three types of networks, we plot and as a function of for different values of the directionality in Fig.4(a), 4(b), and 4(c). Note that, depending on the value of , there is a critical threshold above which there is a giant steady-state component of opinion . The peak of indicates the existence of a second-order phase transition, where is the order parameter and is the control parameter. Note that as the value of increases, in all networks shifts to the right, a shift observable from the shift of the peak of . In RR networks we lose the peak of when the directionality is close to , which suggests the disappearance of the second order phase transition. The sharp jump of around also indicates the appearance of an abrupt phase transition. When these networks contain an increasing one-way influence (increasing directionality), in all cases the minority opinion will need a greater number of initial supporters if they are to survive when the steady state is reached.

(a) RR
(b) Binomial
(c) SF
(d) Binomial
Figure 4: (Color online) Plot of the normalized largest cluster of opinion as a function of the initial fraction for different values of the directionality : , and for different networks with nodes and : (a) RR, (b) binomial, (c) SF (). In the insets we plot as a function of with the same symbols and for the same networks as in the main figure. (d) Plot of the degree distribution of the nodes which keep the majority () and the minority opinion () in binomial networks (also with nodes and ), when the directionality (the main figure) and (the inset). All results are based on averaging realizations.

To further understand this change we consider two extreme cases, and . In the former, an agent influences only those who can influence the agent in return. In the latter, an agent influences only those who cannot influence the agent in return. This latter case allows a much more rapid spread of opinions, each agent interacts with a larger number of agents, each has in-neighbors as well as out-neighbors, and the opinion is diffused over a wider area. Note that both the majority and minority opinions can benefit from this wider diffusion, but there is a higher risk that the minority opinion will be devoured at some point. This is the case because the bidirectional link connecting two minority opinion agents benefits the minority opinion—the two agents can encourage each other to keep the minority opinion. When rewiring this kind of link there is a higher probability that the two agents will interact with the majority opinion and thus a higher probability that their opinion will be changed to the majority opinion. Thus rewiring makes it more difficult for the minority opinion to form a stable structure.

As directionality increases, it is easier for minority opinion agents to keep their minority opinion if they have fewer neighbors. Figure 4(d) plots the degree distributions (in which the indegree and outdegree follow the same distribution) of the minority-opinion nodes and majority-opinion nodes respectively in the steady state at the critical threshold when the directionality is and . Note that the degrees of most of the minority-opinion nodes that keep their minority opinion are equal to , , or . Minority-opinion nodes with a degree larger than 3 can keep their minority opinion when , but seldom when –i.e., as the value of increases, the number of nodes following the majority opinion increases, and only low-degree nodes are able to keep the minority opinion.

Figure 5: (Color online) (a) Plot of the critical threshold as a function of the directionality for different networks with nodes and : RR(), binomial() and SF() (). All results are based on averaging realizations. (b) Plot of the critical threshold as a function of the variance of the degree sequence of the networks ( nodes and ) with different values of the directionality: () and (). All results are based on averaging realizations.

It has been shown that network topology may significantly influence such dynamic processes in networks as epidemics or cascading failures [12, 43, 44, 45, 46]. We thus compare the critical threshold on directed binomial, RR, and SF networks in which the indegree and outdegree (i) follow the same binomial distribution, (ii) are a constant, and (iii) follow a power-law distribution. Figure 5 shows that, as the directionality increases, the critical threshold of the RR networks increases more rapidly than the others. As stated above, as increases, only nodes with degrees less than 4, the average degree, are likely to keep the minority opinion, and in RR networks all nodal degrees are . Figure 5 also shows that the existence of hubs (extremely high-degree nodes) in SF networks causes them, at , to have a much higher critical threshold than the others, and that the critical threshold in binomial networks is slightly larger than the critical threshold in RR networks. The existence of hubs benefits the majority-opinion nodes because the probability that an agent with many friends (i.e., a hub) will follow the majority opinion and influence many others is high. They thus strongly contribute to the diffusion of the majority opinion.

Reference [47, 48] describes how a second-order phase transition becomes first-order and the critical threshold is higher when the average degree increases. In fact, we find that in the networks with the same average degree, the larger the variance of the degree sequence, the larger will be its critical threshold. This is the case because networks with a wider degree variance are more likely to have majority-opinion hubs that can influence many other agents. Figure 5 shows simulation results that support this behavior. Note that as the variance of the degree sequence increases, the critical threshold increases. To change the variance in these simulations we select a SF network with an average degree , randomly remove an existing link, and randomly add a link between two nodes previously unconnected. As we remove and add links repeatedly, the variance of the degree sequence decreases and we stop at an excepted variance. To obtain the specified directionality, we apply DIR on the networks. This gives us a wide range of degree variance, which allows us to study the relationship between the variance and critical threshold .

5 The influence of indegree and outdegree asymmetry

We have discussed how the critical threshold increases as the directionality increases in networks in which the indegree and outdegree are the same for each node. The number of in-neighbors and out-neighbors of nodes in real-world networks often differ, however. We mentioned above how a popular singer can influence many people and not be influenced in return. The social network of the singer has many more out-neighbors than in-neighbors. Because this real-world phenomenon is so ubiquitous, we now examine how different correlations between the indegree and outdegree affect opinion competition.

In Section 3.2 we use ANC-DDR to construct a network with an arbitrary but identical indegree and outdegree distribution, together with a given combination of the directionality and the linear correlation coefficient between the indegree and outdegree. We perform simulations to study the influence of both the directionality and the correlation coefficient on the critical threshold . Figures 6(a) and 6(b) show that, given the directionality, the critical threshold increases for binomial and SF networks, respectively, as the indegree and outdegree correlation increases. Figure 7 shows that when the directionality and the correlation coefficient are increased in binomial networks, the critical threshold increases. The same behavior is observed in SF networks.

(a) Binomial
(b) SF
Figure 6: (Color online) Plot of the normalized largest cluster of opinion as a function of the initial fraction , when the directionality , for different values of the indegree and outdegree correlation : , and for different networks with nodes and : (a) Binomial, (b) SF. In the insets we plot as a function of with the same symbols and for the same networks as in the main figure. All results are based on averaging realizations.
Figure 7: (Color online) Plot of the critical threshold as a function of the linear correlation coefficient and the directionality for binomial networks. All results are based on averaging realizations.

The influence of the indegree and outdegree correlation on the critical threshold can be understood as follows. A smaller means a clearer inequality or asymmetry between the indegree and outdegree links. When the indegree and outdegree links are asymmetrical, a node with more in-neighbors than out-neighbors is more likely to follow the majority opinion and, because it has few out-neighbors, its own opinion will have little influence. Compared with the nodes which have the same number of in-neighbors and out-neighbors and tends to follow as well as spread the majority opinion, such nodes (with fewer out-neighbors) cannot help. Nodes with more out-neighbors than in-neighbors have greater influence and can thus hold the minority opinion and contribute to its spread. Thus the minority opinion benefits more from an inequality between the indegree and outdegree, or equivalently from a smaller , so the lower correlation coefficient leads to a smaller critical value .

Figure 8: (Color online) Plot of the average indegree and outdegree of the nodes in the largest and cluster for binomial networks, when the initial faction of the opinion equals as a function of . The representation of the four lines are as follows: the average indegree () and outdegree () of the nodes in the largest cluster; the average indegree() and outdegree() of the nodes in the largest cluster. All results are based on averaging realizations.

We now further explore the properties of nodes in the final steady state. We focus on binomial networks in the steady state and calculate as a function of , the average indegree and outdegree in the largest and clusters with a directionality (generated by ANC) when the initial fraction of opinion equals (minority). As discussed above, and seen in Fig. 8, the outdegree links of a node with the minority opinion in the steady state tends to be larger for all than the indegree links, because nodes with few in-neighbors are less influenced by other nodes and thus can more easily keep their minority opinion. On the contrary, the indegree of a node with the majority opinion tends to be larger than its outdegree. Note that, when the initial fraction of the opinion is , the average number of indegree links is smaller for the nodes in the largest cluster compared with the nodes in the largest cluster. Note also that in the majority clusters () both the indegree and the outdegree are close to , which is the average degree of the whole network. This is in marked contrast with the average indegree of the nodes in the largest minority cluster with degree approximately 2.5. The average outdegree of minority is larger than 4 when the linear correlation coefficient is . As increases there is a higher correlation between the indegree and outdegree and the average outdegree of minority decreases rapidly.

6 Conclusions

Because of the ubiquity of the non-consensus steady state in real-world opinion competitions and the dominance of unidirectional relationships in real-world social networks, we study a non-consensus opinion model on directed networks. To quantify the extent to which a network is directed, we use a directionality parameter , defined as the ratio between the number of unidirectional links and the total number of links. We also employ a linear correlation coefficient between the indegree and outdegree to quantify any asymmetry.

We propose two approaches to construct directed networks. The first is directionality-increasing rewiring (DIR) and is used to rewire the links of an undirected network to obtain a directed network with any directionality value without changing the indegree and outdegree, i.e., the indegree-outdegree correlation value is fixed at . The second is ANC-DDR, a combination of asymmetric indegree-outdegree network construction (ANC) and directionality-decreasing rewiring (DDR). Using ANC we construct a directed network () with an arbitrary but identical indegree and outdegree distribution and a given indegree-outdegree correlation . We then use DDR to further decrease the directionality of the network.

We use DIR and ANC-DDR to generate directed networks with a given combination of and and investigate how the directionality and the linear correlation coefficient between the indegree and outdegree links affect the critical threshold of the NCO model. We find that in both binomial and SF networks increasing or increases the critical threshold . We also find that as and increase, the phase transition becomes abrupt and is no longer second-order. We find that as a network becomes more directed it becomes more difficult for a minority opinion to form a cluster, while increasing the indegree-outdegree asymmetry makes the minority opinion more stable. Our work indicates that directionality and the asymmetry between indegree and outdegree play a critical role in real-world opinion competitions.

Acknowledgements

We wish to thank ONR (Grants No. N00014-09-1-0380 and No. N00014-12-1-0548), DTRA (Grants No. HDTRA-1-10-1-0014 and No. HDTRA-1-09-1-0035), NSF (Grant No. CMMI 1125290), the European EPIWORK, the Multiplex (No. 317532) EU projects, CONGAS (Grant No. FP7-ICT-2011-8-317672), BSF, and LINC (No. 289447), the Deutsche Forschungsgemeinschaft (DFG), and the Israel Science Foundation for support.

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Appendix A Proof of Eq. (3.1)

Given the indegree and outdegree of each node, the minimum directionality can be reached if all the unidirectional links of each node are either indegree links or outdegree links but not both, because unidirectional indegree links and outdegree links of a node may form bidirectional links by rewiring so that the directionality is further reduced. This means that the minimum directionality can be reached if an arbitrary node has only unidirectional indegree links or outdegree links but not both, where and represent the indegree and outdegree links of node , respectively. Hence the minimum possible directionality, given the number of indegree and outdegree links of each node, is

(A.1)

We denote the indegree and outdegree sequences by and with the same length . The indegree of each node is independent and follows the distribution with the mean . In order to introduce the indegree and outdegree correlation, is constructed from as follows: a fraction of the elements in equals that in (, for , without loss of generality, we assume is an integer), while a fraction of is obtained by copying and shuffling the rest of , such that for and large , and are independent but follow the same distribution . Hence,

(A.2)

where and are independent random variables following the same probability distribution , and indicates the value of when .

We then consider the case when , i.e.,

(A.3)

where Max and Min are the maximum and minimum functions, respectively.

The minimum of random variables has the distribution,

(A.4)

when the two random variables are independent. In the same way, we have

(A.5)

Hence,

(A.6)

Combining A.3 and A.2, we have

(A.7)
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