Introduction

In the last decade, spreading dynamics in complex networks has attracted much attention from disparate disciplines, including mathematics, physics, social sciences, etc.1,2,3,4. Spreads of rumors5,6,7, innovations8,9,10, credits11, behaviors12 and epidemics13,14 were studied both in theoretical and empirical aspects. Spreading models, such as susceptible-infected (SI)15,16,17, susceptible-infected-susceptible (SIS)18,19 and susceptible-infected-recovered (SIR)20,21,22 have been studied to investigate the essential aspects of spreading processes in complex networks23. Theoretical studies revealed that underlying network structure have significant impacts on the outbreak threshold as well as outbreak size3. Specially, for scale-free networks with degree exponent γ ≤ 3, the outbreak threshold vanishes in the thermodynamic limit18,19,24,25. Further studies revealed that the degree heterogeneity promotes spreading outbreaks, however limits the outbreak size at large transmission probability13.

Utilizing network information to effectively enhance the spreading speed and outbreak size is an important topic in spreading dynamics studies26,27,28,29,30. The studies on effective information spreading can provide inspiration for epidemic controlling31,32,33, as well as marketing strategies optimization34,35,36. Methods for effective spreading roughly fall into two categories: one is to choose influential nodes as the spreading sources37,38, while the other is to employ proper contact strategies to optimize spreading paths39. Noticeable methods have been proposed for both the two classes. For the identification methods of influential nodes, Kitsak et al. revealed that selecting nodes with high k-shells as spreading sources can effectively enhance the spreading size26. Recently, Morone et al. proposed an optimal percolation method to identify the influential nodes40. As for the contact process (CP) without bias in heterogenous networks, scholars found that the spreading process follows a precise hierarchical dynamics, i.e., the hubs are firstly informed, and the information pervades the network in a progressive cascade across smaller degree classes41. Yang et al. proposed a biased contact process by using the local structure information in uncorrelated networks, and their results indicate that the spreading can be greatly enhanced if the small-degree nodes are preferentially selected39,42. Rumor spreading and random walk models with biased contact strategy were also studied in refs 43 and 44.

Previous results have manifested that in uncorrelated networks, designing a proper contact strategy can effectively promote the information spreading. However, degree-degree correlations (i.e., assortative mixing by degree) are ubiquitous in real world networks45,46,47. A positive degree-degree correlation coefficient indicates that nodes tend to connect to other nodes with similar degrees. While for negative correlation coefficients, large-degree nodes are more likely to connect to small-degree nodes. The degree-degree correlations have significant impacts on spreading dynamics. For instance, assortative (dissortative) networks have a smaller (larger) outbreak threshold, however outbreak size is on the contrary inhibited (promoted) at large transmission rates48,49.

Although correlations are prevalent in real-world systems, there still lack studies of effective spreading strategy focusing on correlated networks. To promote the information spreading in correlated networks is the motivation of this paper. We propose a preferential contact strategy (PCS), based on the local information of network structure and informed nodes densities. Our findings demonstrate that, when the strategy only consider local structure (LS) information, small-degree nodes should be preferentially contacted to promote the spreading speed, irrespective to the values of degree correlation coefficients. For highly assortative or disassortative networks, small-degree nodes should be more strongly favored to achieve the fastest spreading. Actually, the minimum convergence time of information spreading depends non-monotonically on the correlation coefficient. In addition, we find that the spreading can be further promoted when the information of informed density is incorporated into the PCS. The local informed density (LID) based strategy can better accelerate the spreading, as compared with the global informed density (GID) case.

Results

Model of correlated network

To study the interplay of degree correlations and contact strategies, we build correlated networks with adjustable correlation coefficients by employing a degree-preserving edge rewiring procedure. First we generate uncorrelated configuration networks (UCN)50 with power-law degree distributions and a targeted mean degree. Then, we adjust the degree correlation coefficient by using the biased degree-preserving edge rewiring procedure51. Details about the network generation can be found in the Methods Section.

Model of information spreading

We consider a contact process (CP) of susceptible-informed (SI)52 as the information spreading model. For the SI model, each node can either be in susceptible (S) state or informed (I) state. Initially, a small fraction of nodes are chosen uniformly as informed nodes, while the remainings are in the S state. At each time step, each informed node i select one of its neighbors j to contact with a pre-defined contact probability Wij. If the node j is in S state, then it will become I state with the transmission probability λ. During the spreading process, the synchronous updating rule is applied, i.e., all nodes will update their states synchronously in each time step53. Repeat this process till all nodes are informed, and the network converges to an unique all-informed state. Thus for the model we consider, spreading efficiency can be evaluated by the convergence time T, which is defined as the number of time steps that all nodes become informed.

Preferential contact strategy based on local information

In real spreading processes, it is hard for nodes to known explicitly the states of neighbors. The lack of information may arise many redundant contacts between the two informed nodes in the CP, which will greatly reduce the spreading efficiency. Thus, we propose a preferential contact strategy (PCS), which combines the local structure (LS) and local informed density (LID) information in a comprehensive way. The probability Wij that an informed node i selecting a neighbor j for contact is given by

Here Γi is the set of neighbors of i and ki its degree. In addition, α and β are two tunable parameters. The preferential structure exponent α determines the tendency to contact small-degree or large-degree nodes. Large-degree neighbors are preferentially contacted when α > 0, while small-degree neighbors are favored when α < 0. When α = 0 all neighbors are randomly chosen, and it reduces to the classical CP process52. The preferential dynamic exponent β reflects whether the neighbors with small or large LIDs are favored. For a specific node j, the LID is defined as

where Ij(t) is the number of informed neighbors of node j at time t.

Taking into the contact strategy is based on several reasons. Firstly, suppose there are two neighbors with the same degree, clearly the neighbor with a higher has a larger probability to be already informed. It is reasonable to preferentially choose the neighbor of the smaller as contact target by setting a suitable negative β. Secondly, contacting neighbors with low LIDs can further provide more latent chances to inform the next-nearest neighbors. Third, the LID is relatively easier to obtain, as compared with the global informed density (GID) of network ρG(t) = I(t)/N, where I(t) is the total number of informed nodes in the network at time t. For comparison, we also investigate the performance of GID information based strategy, where the contact probability is given by replacing with ρG(t) in Eq. (1).

Case of α = β = 0 in uncorrelated networks

We investigate the time evolutions of information spreading with unbiased contact strategy in heterogeneous random networks. When α = β = 0, hubs have a higher probability to be contacted since they have more neighbors. As a result, the hubs will become informed quickly. In contrast, small-degree nodes with fewer neighbors are less likely to be contacted and informed. To be concrete, the above scenario is illustrated in Fig. 1. We show the time evolutions of the informed density ρG(t), mean degree of newly informed nodes 〈kI(t)〉, and the degree diversity of the newly informed nodes D(t)54. Here D(t) is defined as

Figure 1
figure 1

For unbiased contacts, the time evolutions of information spreading in random scale-free networks.

The mean degree 〈kI(t)〉 of newly informed nodes (red circles), the density of informed nodes ρG(t) (black hollow squares), and the informed diversity of degrees D(t) (blue diamonds) versus time steps t. Other parameters are set as N = 104, γ = 3.0, 〈k〉 = 8, λ = 0.1, and respectively.

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where Ik(t) is the number of informed nodes with degree k at time t. The larger values of degree diversity D(t) indicate that the newly informed nodes are from diverse degree classes.

At initial time steps, the informed density ρG(t) is small, and a large value of 〈kI(t)〉 indicates that hubs are quickly informed. The value of D(t) is also very large during this stage since most nodes are in S state and nodes from all degree classes can be get informed. With the rapid increase of ρG(t), both 〈kI(t)〉 and D(t) decreases, which indicates intermediate degree nodes are gradually informed. In the late stage of spreading, small values of 〈kI(t)〉 and D(t) reveal that the time-consuming part of the spreading is to reach some small-degree nodes. Previous studies showed that the optimal biased contact strategy basing on the neighbor degrees is which when α ≈ −152. In this case, the small-degree nodes will be informed more easily, while the central role of the hubs for transmitting information is not excessively weakened. Therefore, balancing the contacts to small-degree and large-degree nodes is essential to the problem of facilitating the spreading.

Assortative and disassortative networks display distinct structure characteristics, with small-degree nodes play different roles23. For assortative networks, many small-degree nodes locate in the periphery of the network. While for disassortative networks, some small-degree nodes act as bridges of connecting two large-degree nodes, and with more small-degree nodes act as leaf nodes in the star-like structures (see illustration in Fig. S1 of Supporting Information). While the locations of small degree nodes have been altered by the degree correlations, transmitting information effectively to small degree nodes is essential for facilitating the spreading as discussed above. This suggests that we should treat small degree nodes more carefully in correlated networks. In the LS based PCS, nodes are identified by their degrees, and all neighbors with the same degree are treated as equivalent. This motivates that we could further distinguish the small degree nodes to better enhance the spreading. To this end, we incorporate the LIDs of neighbors into the PCS and favor transmitting information to low informed regions.

Cases of β = 0 in correlated networks

We first study the effect of PCS by only considering the LS information, i.e., β = 0. We verify the performance of the contact strategies in scale-free networks with given mean degrees and degree correlation coefficients. The networks are generated according to the method described in the Model Section. The size of networks is set to N = 104 and average degree 〈k〉 = 8. In addition, we apply the method to two empirical networks which are the Router55 and CA-Hep56. Initially, nodes are randomly chosen as the seeds for spreading. Without lose of generality, the transmission probability is set as λ = 0.1. All the results are obtained with averaging over 100 different network realizations, with 100 independent runs on each realization.

For a specific network, there always exists an optimal value of preferential structure exponent αo, which will lead to the minimum convergence time To [see the inset of Fig. 2(a)]. Figure 2 shows αo, To versus r for networks with different degree exponent γ. From Fig. 2(a), we find that the values of αo are negative irrespective of r. In other words, preferentially contacting small-degree neighbors will promote the spreading efficiency, which is consist with the previous studies for uncorrelated networks52. More importantly, αo depends non-monotonically on r. In particular, when r is either very large or small, αo tends to be smaller than that for the intermediate values of r. Thus, for highly assortative and disassortative networks, it requires stronger tendency to contact small-degree nodes to achieve optimal spreading. This is due to the distinct local structure characteristics of small degree nodes (see details in Sec. S1 of Supporting Information). We test the method for networks with different values of degree exponent γ in Fig. 2(a). It can be seen that for different γ the behaviors are similar. However, one noticeable difference is that for γ = 2.1, αo is significantly larger than the other three cases when r is small. We argue that this anomaly is caused by the structural constrains imposed by the strong heterogeneity of degrees for γ = 2.1. Some structural properties for γ = 2.1 and γ = 3.0 are summarized in Table S1 (see details in Sec. S2 of Supporting Information). To further clarify the effects of correlation coefficient r on the convergence time T, we plot the minimum convergence time To as a function of r in Fig. 2(b). One can see that To also depends non-monotonically on r. Specifically, To first decreases with r and then increases. For those highly disassortative networks, many small clusters are interconnected via some small-degree nodes. The inter-cluster transmissions of information delay the spreading and lead to a large value of To. When r is very large for assortative networks, though the core composed of large-degree nodes is easily informed, small nodes in the periphery are harder to be contacted. The core-periphery structure also gives rise to a slightly large value of To.

Figure 2
figure 2

The optimal performance of LS information based PCS in correlated configuration networks.

(a) The optimal value of preferential structure exponent αo and (b) the convergence time To versus correlation coefficient r for degree exponents γ = 2.1 (orange up triangles), γ = 2.5 (blue diamonds), γ = 3.0 (red circles), and γ = 4.0 (green down triangles), respectively. The inset of (a) shows the relationship between T and α when γ = 3.0 and r = 0.4. We set other parameters as N = 104, 〈k〉 = 8, and λ = 0.1, respectively.

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To complete the above discussions, we study the time evolution properties of the spreading process. Figure 3(a) depicts the informed density ρG(t) versus time t for the case of r = 0.55 and γ = 3.0. Note that r = 0.55 minimize To when γ = 3, as shown in Fig. 2(b). The three different lines correspond to different values of α, which are α = −1.5, −0.8, 0.0, respectively. It’s clear from Fig. 3(a) that the case for αo = −0.8 spreads faster than the two other cases. The number of newly informed nodes nI(t) as a function of t is given in Fig. 3(b). One can observe that, compared with α = 0 and α = −1.5, the nI(t) for αo = −0.8 is larger (smaller) than the other two cases at the early (late) stages, indicating the fastest spread of information. When the network is almost fully informed at late stages, the inset in Fig. 3(b) demonstrates that nI(t) decays faster with time for αo = −0.8. Figure 3(c) and (d) respectively show the time evolutions of mean degree of new informed nodes 〈kI(t)〉 and the corresponding degree diversity D(t). For αo = −0.8, the 〈kI(t)〉 and D(t) remain relatively stable with time. In other words, nodes with different degrees almost have uniform probabilities of being informed, which is close to the ideal situation for effective spreading52. However, for α = 0 large-degree nodes are first informed and then the small-degree ones, while for α = −1.5 the order is reversed. For the two cases, the degree diversity becomes small at the late stages of information spreading. Together with the 〈kI(t)〉 we can conclude that the spreading is delayed by small-degree (large-degree) nodes for α = 0 (α = −1.5). Correspondingly, the results of informed degree diversity D(t) in Fig. 3(d) validate the advantage of αo = −0.8 again, which is more stable than that for α = 0 and α = −1.5.

Figure 3
figure 3

The effect of LS information based PCSs on the time evolution of information spreading in correlated networks.

(a) The informed density ρG(t), and (b) the number nI(t), (c) mean degree 〈kI(t)〉, and (d) degree diversity D(t) of newly informed nodes versus t for different values of α. Different colors indicate different values of α. The inset of (a) shows the time evolution of ρG(t) in the time interval [80, 100]. The inset of (b) shows nI(t) in the time interval [150, 500]. We set other parameters as N = 104, γ = 3.0, 〈k〉 = 8, λ = 0.1, and r = 0.55, respectively.

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We also apply the LS information based PCS to two empirical networks. (1) Router. The router level topology of the Internet, collected by the Rocketfuel Project55. (2) CA-Hep. Giant connected component of collaboration network of arxiv in high-energy physics theory56. More details about the two networks can be found in Sec. S3 of Supporting Information. We find the similar phenomena that observed in Fig. 2, i.e., the non-monotonic dependence of αo and To on r for the case of the empirical networks (see Fig. 4). Nevertheless, some abnormal bulges of αo and To emerge at certain values of r. By analyzing the structures of the networks, we find that the networks are very similar to the original networks as there are few rewiring edges in the networks at these certain values of r with abnormal bulges. Owing to the structural complexity of the real networks, which are significantly different from the synthetic networks, leading to abnormal bulges at certain values of r. To prove that the above abnormal phenomenon comes from the structural complexity of the empirical networks, we randomize the empirical networks by sufficient rewiring process but do not change the original degree distribution and the degree of each node. After sufficient times of randomization, we then check the contact strategy in the randomized networks, and one can see the abnormal bulges disappear. Moreover, the curves become more smooth and the non-monotonic phenomenon becomes more evident.

Figure 4
figure 4

The optimal performance of LS information based PCS in correlated networks by rewiring real-world networks (red circles) and randomized networks (blue diamonds).

The optimal value of preferential structure exponent αo (a) and convergence time To (b) versus correlation coefficient r for the Router network. The αo (c) and To (d) versus r for the CA-Hep network.

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Cases of α = αo and β < 0 in correlated networks

When α = αo, the LS information based PCS can effectively enhance the spreading efficiency. We further incorporate the LID information, i.e., with β < 0 and α = αo in Eq. (1). The spreading efficiency ΔTβ is measured by , where Tβ represents the convergence time when β < 0, and To denotes the convergence time when β = 0. Thus ΔTβ > 0 (ΔTβ < 0) indicates that introducing LID information can enhance (inhibit) the spreading efficiency. For comparison we also study the effects of GID information, by replacing with ρG(t) in Eq. (1). The results in Fig. 5(a), (b) and (c) manifest that, the effective use of LID information can further reduce the convergence time when β is set as a small negative value (e.g., β = −0.1 and −0.2). Yet, β with larger magnitude (e.g., β = −0.5) will increase the convergence time. For different values of degree correlation coefficient r, there is obviously an optimal value βo at which the information spreading can be effectively enhanced. Moreover, compared with the GID information, utilizing the LID information not only speeds up the spreading more significantly but also has a wider range of β with ΔTβ > 0. For disassortative networks, as shown in Fig. 5(d) with r = −0.3, the LID based PCS can speed up the spreading of information for a wide range of β. Such an improvement is more evident as compared to uncorrelated [Fig. 5(a)] and assortative networks [Fig. 5(b) and (c)]. We conclude that LID based PCS performs better than the GID case in reducing the convergence time.

Figure 5
figure 5

In correlated networks, the effects of LID and GID based PCSs on the convergence time.

The relative ratio of the convergence time ΔTβ versus β for different r values: (a) r = 0, αo = −1.2, (b) r = 0.55, αo = −0.8, (c) r = 0.75, αo = −0.95, and (d) r = −0.3, αo = −2.5, respectively. We set other parameters as N = 104, γ = 3.0, 〈k〉 = 8, and λ = 0.1. It is noted that the results for the GID case are not shown when in subfigure (c), because their values exceed the minimum value of vertical coordinate. Obviously, ΔTβ monotonically increases with β.

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Figure 6 presents time evolutions of some statistics of the spreading process in disassortative networks with r = −0.3. Figure 6(a) and the inset suggest that the optimal value βo = −1.7 can better improve the speed of spreading. Figure 6(b) emphasizes that, for the case of β < 0, the number of newly informed nodes nI(t) increases faster than the case of β = 0 at the initial stage. However, the inset of Fig. 6(b) illustrates that, for the case of βo =−1.7, nI(t) goes to zero faster than the case of β = −3.5. Similar to Fig. 3(c) and (d), the results in Fig. 6(c) and (d) also manifest that too large (small) values of β make the small-degree (large-degree) nodes uneasy to be informed, which will inhibit the spreading. In strongly disassortative networks, complex local structures and dynamical correlations cause nodes with the same degree to be in different local dynamical statuses. The LS information based PCS can not effectively reflect and overcome the difference of local dynamical status. The optimal value βo guarantee the probability of being informed more homogeneous and steady for different degree classes, leading to the fastest spreading of information. Similarly, we also explain why the PCS based on the LID information yields better performance than the GID case (see details in Sec. S4 of Supporting Information).

Figure 6
figure 6

The effect of LID based PCS on the time evolution of information spreading in disassortative networks.

(a) The informed density ρG(t), and the number nI(t) (b), mean degree 〈kI(t)〉 (c) and degree diversity D(t) (d) of newly informed nodes versus t for different values of β. Different colors indicate different values of β. The inset of (a) shows the time evolution of ρG(t) in the time interval [80, 140]. The inset of (b) shows nI(t) in the time interval [550, 3000]. We set other parameters as γ = 3.0, N = 104, 〈k〉 = 8, λ = 0.1, r = −0.3, and α = −2.5, respectively.

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Finally, we verify the effectiveness of the informed density information based strategies in Router network and CA-Hep network. Figures S3–S7 show that the LID case performs better in improving the speed of information diffusion, and there exists an optimal value of β (see details in Sec. S5 of Supporting Information).

Conclusions

To effectively promote the information spreading in correlated networks, we proposed a PCS by considering both the LS and LID information. Based on extensive simulations in artificial and real-world networks, we verified the effectiveness of the proposed strategy, and generally found that preferentially selecting nodes with smaller degrees and lower LIDs is more likely to promote information spreading in a given network. First, we studied the strategy which only considers the LS information. For a given network, there generally exists an optimal preferential exponent αo, at which the small-degree nodes are favored and the convergence time To reaches its minimum value. Especially, the small-degree nodes should be favored more strongly to achieve optimal spreading when networks are highly assortative or disassortative. Also, the optimal convergence time To depends non-monotonically on the correlation coefficient r. Then, we induced the informed density into the LS information based PCS with optimal exponent αo. Compared to the strategy with GID, the LID information based PCS reduces the convergence time more significantly. For the LID case, an optimal value βo can be observed and minimize the convergence time. We further discussed the effects of different parameter combinations (α, β) and source nodes on the effectiveness of our proposed PCS based on LID information (see details in Sec. S6 of Supporting Information). We found that these factors do not qualitatively affect the above stated results, i.e., the LID based PCS can effectively accelerate the spreading.

Utilizing network information to improve the spreading is an important topic in spreading dynamics studies. In this work, we study the effect of correlated networks on the effective contact strategy basing on the local structure and informed density. Our results would stimulate further works about contact strategy in the more realistic situation of networks such as community networks57,58, weighted networks13,59, temporal networks60,61, and multiplex networks33,62. And this work maybe provide reference for the promotion of social contagions such as technical innovations, healthy behaviors, and new products10,11,12. The results presented in our work are based on extensive numerical simulations, how to get accurate theoretical results needs further study. We may get some meaningful insights of the theoretical approaches from other dynamics on complex networks, especially the similar dynamics zero range process63,64,65,66,67,68,69, in which a jumping-rate with biasing low-degree nodes would possibly avoid particle condensation.

Methods

Uncorrelated configuration model

We generate uncorrelated configuration networks (UCN)50 with power-law degree distributions and targeted mean degrees as follows: (1) A degree sequence of N nodes is drawn from the power-law distribution P(k) ~ k−γ, with all the degrees confined to the region , where γ is the degree exponent. Note that the average of the degrees is un-controlled but depends on γ. (2) Adjust the average of the degree sequence to a targeted value to eliminate the difference of mean degree between synthetic networks with different degree exponents70. In detail, to transform mean degree from original mean degree 〈k〉now to targeted mean degree 〈k〉tar, the degree of each node i is re-scaled as . Now the new degrees may be not integers, therefore we need to convert then to integers while preserving the degree distribution and the mean degree. Since can be written as with b ∈ [0, 1), we take with probability 1 − b, while with probability b. (3) The nodes with updated degrees are randomly connected via standard procedure of the UCN model.

Adjusting degree correlation coefficient

We use the biased degree-preserving edge rewiring procedure to adjust the degree correlation coefficient51. Note that this procedure is also applicable to empirical networks. The procedure is as follows: (1) At each step, two edges of the network are randomly chosen and disconnected. (2) Then we place another two edges among the four attached nodes, according to their degrees. To generate assortative (dissortaive) networks, the highest degree node is connected to the second highest (lowest) degree node, and also connect the rest pair of nodes. If one or both of these new edges are already exist in the network, the step will be discarded and a new pair of edges will be randomly chosen. (3) Repeat this procedure till the degree correlation coefficient reaches the target value. Here the degree correlation coefficient23 is defined as:

where m is the total number of edges in the network, A is the adjacency matrix (If there is an edge between nodes i and j, Aij = 1; otherwise, Aij = 0.) and δij is the Kronecker delta (which is 1 if i = j and 0 otherwise.). When r = 0 there is no degree-degree correlation in the network, while r > 0 and r < 0 indicate positive and negative degree-degree correlations respectively.

Additional Information

How to cite this article: Gao, L. et al. Effective information spreading based on local information in correlated networks. Sci. Rep. 6, 38220; doi: 10.1038/srep38220 (2016).

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