Promoting information spreading by using contact memory

TL;DR

This paper introduces a non-Markovian model for information spreading that incorporates human memory, revealing an effective contact strategy that enhances spreading efficiency on various networks.

Contribution

It proposes a novel contact strategy based on contact memory, improving understanding of spreading dynamics and providing a mean-field theory that aligns with simulations.

Findings

Preferential contact with less-contacted neighbors promotes spreading.

High-degree nodes are targeted early, small-degree nodes later.

The mean-field theory accurately predicts spreading behavior.

Abstract

Promoting information spreading is a booming research topic in network science community. However, the exiting studies about promoting information spreading seldom took into account the human memory, which plays an important role in the spreading dynamics. In this paper we propose a non-Markovian information spreading model on complex networks, in which every informed node contacts a neighbor by using the memory of neighbor's accumulated contact numbers in the past. We systematically study the information spreading dynamics on uncorrelated configuration networks and a group of $22$ real-world networks, and find an effective contact strategy of promoting information spreading, i.e., the informed nodes preferentially contact neighbors with small number of accumulated contacts. According to the effective contact strategy, the high degree nodes are more likely to be chosen as the contacted neighbors in the early stage of the spreading, while in the late stage of the dynamics, the nodes with small degrees are preferentially contacted. We also propose a mean-field theory to describe our model, which qualitatively agrees well with the stochastic simulations on both artificial and real-world networks.

Tables

Table 1. Table 1: Statistical characteristics of the 22 22 22 real-world networks. The statistical characteristics including the network size ( N 𝑁 N ), number of edges ( E 𝐸 E ), maximum degree ( k m ​ a ​ x subscript 𝑘 𝑚 𝑎 𝑥 k_{max} ), first and second moments of the degree distribution ( ⟨ k ⟩ delimited-⟨⟩ 𝑘 \langle k\rangle ) and second moments ( ⟨ k 2 ⟩ delimited-⟨⟩ superscript 𝑘 2 \langle k^{2}\rangle ), degree-degree correlations ( r 𝑟 r ), clustering ( c 𝑐 c ), and modularity ( Q 𝑄 Q ).

Category	Networks	Statistical Characteristics of Networks
		$N$	$E$	$k_{max}$	$⟨k⟩$	$⟨k^2⟩$	$r$	$c$	$Q$
Social	Advogato Massa et al. (2009)	5042	39227	803	15.56	1284	$-0.096$	0.092	0.337
	Hamsterster friendships Kunegis	1788	12476	272	13.955	635.61	$-0.089$	0.09	0.396
	Hamsterster full Kunegis	2000	16098	273	16.098	704.71	0.023	0.23	0.45
Metabolic	Caenorhabditis elegans Duch and Arenas (2005)	453	2025	237	8.9404	358.49	$-0.226$	0.124	0.401
	Reactome Joshi-Tope et al. (2005)	5973	145778	855	48.812	6995.1	0.241	0.606	0.719
	Human protein (Figeys) Ewing et al. (2007)	2217	6418	314	5.79	324.93	$-0.332$	0.008	0.472
	Human protein (Vidal) Rual et al. (2005)	2783	6007	129	4.317	68.103	$-0.137$	0.035	0.615
Infrastructure	Air traffic control Kunegis	1226	2408	34	3.928	28.899	$-0.015$	0.064	0.686
	OpenFlights Opsahl et al. (2010)	2905	15645	242	10.771	601.45	0.049	0.255	0.581
	Power Watts and Strogatz (1998)	4941	6594	19	2.669	10.333	0.003	0.08	0.932
Collaboration	CA-Hep Leskovec et al. (2007a)	8638	24806	65	5.744	74.601	0.239	0.482	0.752
	Netsci Newman (2006)	379	914	34	4.823	38.686	$-0.082$	0.741	0.846
Citation	DBLP Ley (2002)	12495	49563	709	7.933	347.28	$-0.046$	0.062	0.538
HumanSocial	Jazz musicians Gleiser and Danon (2003)	198	2742	100	27.697	1070.2	0.02	0.52	0.439
	Adolescent health Moody (2001)	2539	10455	27	8.236	86.414	0.251	0.142	0.597
Computer	Route views Leskovec et al. (2007b)	6474	12572	1458	3.884	640.08	$-0.182$	0.01	0.612
	Route Spring et al. (2004)	5022	6258	106	2.492	34.181	$-0.138$	0.012	0.898
Lexical	King James Kunegis	1707	9059	364	10.614	441.85	$-0.052$	0.162	0.461
OnlineContact	Pretty Good Privacy Boguñá et al. (2004)	10680	24340	206	4.558	86.287	0.239	0.266	0.877
	Email Guimera et al. (2003)	1133	5451	71	9.622	179.18	0.078	0.22	0.565
	Emailcontact Kitsak et al. (2010)	12625	20362	576	3.226	356.36	$-0.387$	0.109	0.674
	Blog Xie (2006)	3982	6803	189	3.417	47.145	$-0.133$	0.284	0.853

Equations

$$w_{ij}(t)=\frac{[n_j(t)+1{]}^{\alpha }}{\sum _{v\in \Gamma (i)}[n_v(t)+1{]}^{\alpha }},$$

$${\Phi }_j(t)=1-\prod _{i=1}^N[1-\lambda A_{ij}{w}_{ij}(t){\rho }_i(t)],$$

$${\rho }_j(t+1)={\rho }_j(t)+s_j(t){\Phi }_j(t).$$

$$n_j(t+1)=n_j(t)+{\Theta }_j(t),$$

$${\Theta }_j(t)=\sum _{i=1}^N{A}_{ij}{w}_{ij}(t){\rho }_i(t)$$

$$\rho (t)=\frac{1}{N}\sum _{j=1}^N{\rho }_j(t),$$

$$D_I(t)=\frac{1}{\sum _k{\left[\frac{I_k(t)-I_k(t-1)}{I(t)-I(t-1)}\right]}^2},$$

Full text

I Introduction

A wide range of propagation phenomenon in the real world, such as the spreading of information, rumor, disease and behavior, can be described by spreading dynamics on complex networks Pastor-Satorras et al. (2015); Wang et al. (2017); Pastor-Satorras and Vespignani (2001); May and Lloyd (2001); Wang et al. (2016); Wang and Li (2014). The problem of how to enhance or promote the spreading has attracted much attention in the last few years in many disciplines Pei and Makse (2013). Promoting spreading speed and understanding its effects in the outbreak size of the reached nodes are two important features to study in both theoretical and empirical aspects. Theoretically existing studies found that promoting the spreading dynamics could induce distinct critical phenomena with different outbreak thresholds and critical exponents Ghoshal and Barabási (2011). Practically speaking, promoting the spreading dynamics could shed some lights on the propagation of information Iribarren and Moro (2009); Miritello et al. (2011); Hambrick et al. (2012), marketing management Watts and Dodds (2007); Goel et al. (2010), disease spreading Granell et al. (2013); Wang et al. (2014); Buono et al. (2013); Buono and Braunstein (2015); Alvarez-Zuzek et al. (2016), etc. Many strategies for promoting spreading dynamics have been proposed, such as choosing influential seeds Morone and Makse (2015); Kitsak et al. (2010) and designing effective contact strategies Yang et al. (2008); Gao et al. (2016). Kitsak et al. found that selecting nodes with high k-shells as spreading sources can effectively enhance the size of the spreading in most cases Kitsak et al. (2010), however the k-shell index cannot reflect the importance of nodes in the core-like group in which the nodes are connected very locally Liu et al. (2015). Yang et al. proposed a contact strategy based on the degree of neighbors nodes to promote the information spreading Yang et al. (2008). They found that the information spreading can be greatly promoted in uncorrelated networks if the small-degree neighbors are preferentially contacted. In addition, if the reached nodes, denoted as informed nodes, preferentially contact nodes with few informed neighbors, the information spreading could be further promoted Gao et al. (2016).

In real-world, the activities of humans exhibit the characteristic of having memory Barabási (2005); Zhou et al. (2012). This characteristic has significant impacts on the spreading dynamics of information, epidemic, and behavior Vazquez (2007); Wang et al. (2015); Yang et al. (2011). For instance, human’s memory produces a larger prevalence in the exponential decay time of new informed nodes than processes without memory Vazquez et al. (2007). Wang et al. Wang et al. (2015) found that memory affects the growth pattern of the final outbreak size in the dynamics of social contagion. However, previous works about promoting information spreading always neglected the memory of individuals. In this paper, we propose a non-Markovian information spreading model, in which each individual (node) keeps memory of the number of accumulated contact (NAC) with informed neighbors in the past. We assume that every informed node contacts a neighbor based on the values of neighbors’ NACs. To describe our model, we develop a novel mean-field theory. Our theoretical predictions are in good qualitative agreement with the stochastic simulations on both artificial networks and $22$ real-world networks. Through theoretical analysis and extensive numerical simulations, we find an efficient strategy to promote the spreading which consists on preferentially contacting neighbors with small NACs. This strategy markedly promotes the information spreading, since it increases the number of effective contacts with susceptible nodes. With our effective strategy, we find that the informed nodes are more likely to contact high degree nodes in the early stage, while small degree nodes are preferentially contacted in the late stage. As a result, our strategy unifies the probability that nodes of different degrees will be contacted enhancing the information spreading remarkably.

II Model

We propose a generalized susceptible-informed model Pastor-Satorras et al. (2015) to describe the information spreading. In this model, each node can either be in the susceptible or informed state. Initially, we randomly choose a small fraction of nodes in the informed state, while the remaining nodes are in the susceptible state. At each time step $t$, each informed node $i$ contacts one of its neighbors $j$ with a probability $w_{ij}(t)$ (to be defined later). If node $j$ is susceptible, it will become informed with a transmission probability $\lambda$. During the spreading process, we adopt the synchronous updating rule, i.e., all nodes will update their states synchronously at each time step Shu et al. (2016). The dynamical process evolves until time $T$, at which we compute the density of informed nodes $\rho(T)$. The change of this magnitude with the parameters will allow us to evaluate the efficiency of our strategy Liu et al. (2016).

An effective information spreading strategy should increase the effective contacts between informed and susceptible nodes Toyoizumi et al. (2012). To achieve this goal our strategy assumes that every node remembers the number of accumulated contact (NAC) with informed neighbors in the past. An informed node $i$ contacts a neighbor $j$ at time $t$ with probability

[TABLE]

where $n_{j}(t)$ is the value of NAC of node $j$ at time $t$, and $\Gamma_{i}$ is the neighbor set of node $i$ not . The parameter $\alpha$ determines the tendency of node $i$ to contact a neighbor $j$ with small or large value of $n_{j}(t)$. For the case of $\alpha>0$, neighbors with larger $n_{j}(t)$ are preferentially contacted, and for the case of $\alpha<0$, the opposite situation happens. When $\alpha=0$, all neighbors are randomly chosen and we recover the classical susceptible-informed model. As every node remembers its NAC before time $t$, the information spreading process is a non-Markovian process. In Figs. 1 we show a schematic of the information spreading dynamics.

III Results

III.1 Theoretical analysis

To describe our model of information spreading dynamics, we develop a mean-field theory. We denote as $s_{i}(t)$ and $\rho_{i}(t)$ to the probabilities that node $i$ is susceptible and informed at time $t$ respectively. Since each node can only be in susceptible or informed state, we have $s_{i}(t)=1-\rho_{i}(t)$. A susceptible node $j$ will become informed at time $t$ if it fulfills two conditions simultaneously : (1) being contacted by an informed neighbor $i$ with probability $w_{ij}(t)$, and (2) being informed successfully with probability $\lambda$. The probability of node $j$ to make a transition to the informed state by neighbors at time $t$ is Gómez et al. (2010)

[TABLE]

where $N$ is the system size and $A_{ij}$ is the element of the adjacency matrix of a given network. If there is an edge between nodes $i$ and $j$, $A_{ij}=1$; otherwise, $A_{ij}=0$. Thus the evolution of the probability of informed node $j$ is

[TABLE]

In order to obtain the value of $w_{ij}(t)$, we compute $n_{j}(t)$ which is given by

[TABLE]

where

[TABLE]

is the increase of node $j$’s NAC at time $t$. From Eqs. (2)-(5), we obtain the evolution equations for the information spreading dynamics. At time $t$, the density of informed nodes is given by

[TABLE]

and the density of susceptible nodes is $s(t)=1-\rho(t)$.

III.2 Stochastic Simulations

We perform extensive numerical simulations on both artificial and real-world networks. All the obtained results are averaged over $10^{3}$ independent realizations of seeds on a fixed network. Initially, we randomly select 0.005 nodes as informed seeds. The information transmission probability is set as $\lambda=0.1$. Notice that other values of $\lambda>0$ do not qualitatively affect our results.

We first study the information spreading dynamics on artificial networks. We built the artificial networks using the uncorrelated configuration model Molloy and Reed (1998); Catanzaro et al. (2005) with power-law degree distributions $P(k)\sim k^{-\gamma}$ with $k_{min}\leq k\leq\sqrt{N}$, where $k_{min}$ is the smaller value of the degree and $\gamma$ is the degree exponent. In all our simulations the network size is $N=10^{4}$ and $k_{min}=4$ in order to have a high average degree and therefore a relative high number of contacts.

In Fig. 2 we show the results on artificial networks with different degree exponents. For a given value of $T$, we find that $\rho(T)$ decreases with $\alpha$, as shown in Figs. 2(a) and (c). Specifically, if every informed node $i$ preferentially contacts a neighbor $j$ with small value of $n_{j}(t)$, i.e., $\alpha<0$, more nodes will be informed. On the other hand, if neighbors with large NACs are preferentially contacted, i.e., $\alpha>0$, there will be few nodes to be informed. From the figures we can see that for $T=100$ almost all nodes are in the informed state when $\alpha\leq-1$, however, there are few informed nodes when $\alpha\geq 1$. We can explain the above phenomena as follows. The larger the value of $n_{j}(t)$, the larger the probability that node $j$ has been informed. As a result, an informed node $i$ should contact a neighbor with small value of NAC to increase the number of effective contacts (i.e., contact with a susceptible neighbor), and further promotes the information spreading. In Figs. 2 (b) and (d) we show the time evolution of $\rho(t)$ for $\gamma=2.1$ and $3.0$, respectively. We find that for small values of $\alpha$, $\rho(t)$ is large, nevertheless when $\alpha=1.0$ the information spreads slowly and it is hard to expand the information to the whole network. Our theoretical predictions agree with the stochastic simulations in most cases. The deviations between the theoretical and the numerical results arise from disregarding the strong dynamical correlations among the states of neighbors in the theory Wang et al. (2017); Pastor-Satorras et al. (2015). From the figures we can also see that the phenomena for $\gamma=2.1$ and $\gamma=3.0$ are similar, which indicates that the heterogeneity of the degree distribution does not qualitatively affect the results.

In order to explain our above results we compute the average values of NAC of nodes with degree $k$ $\langle n_{k}(T)\rangle$ at time $T$. From Figs. 3(a) and (c), we can observe that at the early stage of the information spreading dynamics (i.e., $T=10$), $\langle n_{k}(T)\rangle$ increases linearly with $k$ for all the values of $\alpha$, i.e., large degree nodes are more likely to be contacted by informed neighbors than low degree nodes. Note that $\langle n_{k}(T)\rangle$ is slightly smaller for $\alpha=-2.0$ than for $\alpha=1.0$. At later stage of the information spreading dynamics (i.e., $T=100$), we find some interesting phenomena as shown in Figs. 3(b) and (d). For the case of $\alpha=0.0$, $\langle n_{k}(T)\rangle$ still increases linearly with $k$. When $\alpha=1.0$, the values of $\langle n_{k}(T)\rangle$ of the largest degree nodes are more than $200$ times larger than the ones for nodes with minimum degree, and the number of the effective contacts are decreased in this case. However, for the case of $\alpha=-2.0$, almost all the nodes have the same values of $\langle n_{k}(T)\rangle$. This indicates that, compared to the case of $\alpha=1.0$, in the latter case informed nodes are more likely to contact neighbors with small degrees. Since nodes with small degrees have small probabilities to be informed, preferentially contacting them increases the number of effective contacts. As a result, nodes with different degrees almost have uniform probabilities of being contacted and the information spreading can be accelerated significantly.

In Fig. 4, we study the time evolution properties of the information spreading. In Figs. 4(a) and (e), we show the time evolution of the average number of accumulated contacts $\langle n_{S}(t)\rangle$ for the effective contacts (i.e., the newly contacted susceptible nodes) for $\gamma=2.1$ and $3.0$. For the case of $\alpha=-2.0$, most nodes with small number of contacts are informed at the early stage, while the remaining few nodes with large number of contacts are informed at the late stage. However, for the case of $\alpha=1.0$, nodes with small number of contacts are hard be informed. We also find that the ratio of the number of effective contacts to all contacts at time $t$, i.e., $p_{S}(t)$, is very high in the early stage when $\alpha=-2.0$ [see Figs. 4(b) and (f)]. To clarify the types of informed nodes at different stages of the process, we study the average degree $\langle k_{I}(t)\rangle$ and the degree diversity $D_{I}(t)$ Gao et al. (2016) of the newly informed nodes at time $t$. Here $D_{I}(t)$ is defined as

[TABLE]

where $I(t)$ is the number of newly informed nodes at time $t$, and $I_{k}(t)$ is the number of those with degree $k$. The large values of $D_{I}(t)$ indicate that the newly informed nodes have heterogeneous degrees. From Figs. 4(c) and (g), we can see that for the case of $\alpha=-2.0$, $\langle k_{I}(t)\rangle$ is very large at the early stage, and then decreases at the later stage. $D_{I}(t)$ shows the same tendency as seen in Figs. 4(d) and (h). Thus, when $\alpha=-2.0$ high degree nodes are more likely to be informed in the early stage, while in the late stage small degree nodes are often informed. When $\alpha=1.0$ both $\langle k_{I}(t)\rangle$ and $D_{I}(t)$ are large in the whole spreading process, which means that nodes with small degrees are hard to be informed. These results corroborate our finding that the heterogeneity of degree distribution does not qualitatively affect the above stated results.

Finally, we study our suggest information spreading dynamics on $22$ real-world networks, which includes metabolic networks, infrastructure networks, collaboration networks and citation networks, etc. For simplicity, the directed networks are treated as undirected ones and the weighted networks are treated as unweighted ones. Table 1 displays their statistical characteristics in details.

Fig. 5 shows the information spreading on two representative networks. Strikingly, we find that the results on real-world networks are qualitatively similar to the one found in artificial networks as shown in Fig. 2. Specifically, if the informed nodes preferentially contact neighbors $j$ with small $n_{j}(t)$, the information spreading will be markedly facilitated. On the contrary, the information spreads is slower if nodes with large NACs are favored, and it is hard to spread the information to the network.

We further verify the effectiveness of our suggested mean-field theory on the real-world networks at $T=100$ for different values of $\alpha$, as shown in Fig. 6. We can see that our theoretical predictions qualitatively agree with the numerical simulations, although the theoretical predictions are slightly larger than the simulation results. The deviations between theoretical and numerical predictions derive from the strong dynamical correlations among the states of neighbors Wang et al. (2017); Pastor-Satorras et al. (2015).

IV Discussions

In this paper, we proposed a novel non-Markovian information spreading model by introducing the individuals’ memory, which assumes that every node remembers the number of accumulated contacts (NAC) with informed neighbors in the past, and an informed node contacts a neighbor with different biases values of NAC. To describe the non-Markovian spreading dynamics, we developed a novel mean-field theory. Through extensive numerical simulations on artificial networks and real-world networks, we found that the memory characteristic markedly affects the spreading dynamics. If the informed nodes preferentially contact neighbors with low NACs, the information spreading is markedly promoted due to the increasing of the number of effective contacts between susceptible and informed nodes. In the effective contact strategy, the high degree nodes are more likely to be contacted in the early stage, while the small degree nodes are preferentially contacted in the late stage. As a result, nodes with different degrees almost have uniform probabilities of being contacted and the information spreading is accelerated significantly. Importantly, the effectiveness of the promoting strategy weakly depends on the structures of networks. The agreements between our theoretical predictions and numerical results are verified on artificial networks as well as the real-world networks. Our effective strategy to promote the information spreading on complex networks, could be used for other spreading dynamics, such as technical innovations, healthy behaviors, and new products adoption. Our theoretical method of non-Markovian spreading model can be extended to the study of other social dynamics.

Acknowledgements.

This work was supported by the National Natural Science Foundation of China (Grant No. 61673086). LAB thanks UNMDP and FONCyT, Pict 0429/2013 for financial support.