Emergence of Blind Areas in Information Spreading
Zi-Ke Zhang1,2., Chu-Xu Zhang1,3., Xiao-Pu Han1,2, Chuang Liu1,2*
1 Institute of Information Economy, Hangzhou Normal University, Hangzhou, People’s Republic of China, 2 Alibaba Research Center for Complexity Sciences, Hangzhou
Normal University, Hangzhou, People’s Republic of China, 3 Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, People’s Republic of
China

Abstract
Recently, contagion-based (disease, information, etc.) spreading on social networks has been extensively studied. In thispaper, other than traditional full interaction, we propose a partial interaction based spreading model, considering that the
informed individuals would transmit information to only a certain fraction of their neighbors due to the transmission ability
in real-world social networks. Simulation results on three representative networks (BA, ER, WS) indicate that the spreading
efficiency is highly correlated with the network heterogeneity. In addition, a special phenomenon, namely Information Blind
Areas where the network is separated by several information-unreachable clusters, will emerge from the spreading process.
Furthermore, we also find that the size distribution of such information blind areas obeys power-law-like distribution, which
has very similar exponent with that of site percolation. Detailed analyses show that the critical value is decreasing alongwith the network heterogeneity for the spreading process, which is complete the contrary to that of random selection.
Moreover, the critical value in the latter process is also larger than that of the former for the same network. Those findings
might shed some lights in in-depth understanding the effect of network properties on information spreading.
Citation: Zhang Z-K, Zhang C-X, Han X-P, Liu C (2014) Emergence of Blind Areas in Information Spreading. PLoS ONE 9(4): e95785. doi:10.1371/journal.pone.
0095785
Editor: Jesus Gomez-Gardenes, Universidad de Zarazoga, Spain
Received January 3, 2014; Accepted March 31, 2014; Published April 24, 2014
Copyright: ß 2014 Zhang et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was partially supported by the National Natural Science Foundation of China (grant numbers 11105024, 11205040, 1147015, 11301490 and
11305043), the EU FP7 Grant 611272 (project GROWTHCOM), the start-up foundation and Pandeng project of Hangzhou Normal University. The funders had norole in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: liuchuang@hznu.edu.cn. These authors contributed equally to this work.

proved to be approximate to 80% for the SIR model on random
networks when the population size is infinite [16]. Previous works
have also revealed that there indeed exists a propagation thresholdof information spreading on the small-world network [17,18], andthe spreading is much faster and broader than that on regular
network for the existence of the long-range edges [19]. Moreno etal. [5,20] studied the rumor spreading on scale-free networks, andfound that the existence of hub nodes can enhance the spreading
speed rather than the influenced scope. Recently, there is a vastclass of studies focusing on the spreading dynamics on interconnected networks [21], and there is a mixed phase below the critical
infection strength in weakly coupled networks [22]. The study ofcooperative behaviour spreading across the interdependent
networks [23] indicated that interdependent links are more likely
to connect two cooperators than the regular links. Furthermore,some interesting phenomena are discovered based on theinformation diffusion on real social systems, such as the telephoneinteractions [24,25], tweets [26,27] and emails [24]. Empiricalresults indicate that the human active patterns [28,29] and the roleof weak ties [30] would strongly affect the information spreading.
However, there are also plenty of researches arguing that theunderlying mechanism of information diffusion should be fundamentally different from that of epidemic spreading. Lü et al. [31]summarized the significant differences between them, andconcluded that the information, by considering the social
enhancement, would spread more effectively in regular networks
than that in random networks. This would to some extent support
the real human experiment reported by Centola [7]. Most

Introduction
With the advent of various online Social Networking Services(SNS), the dissemination of information through Internet, also
known as word-of-mouth or peer-to-peer spreading, has attracted muchattention for researchers in recent years [1,2]. Following the rapid
development of database technology and computational power,
various spreading phenomena on large-scale social networks, suchas the news [3], rumors [4,5], innovation [6], behavior [7,8],
culture [9], viral marketing [10], cooperative behaviors [11] etc,
can now be deeply studied in terms of theoretical models as well as
empirical analyses.
Generally, the information spreading dynamics is usuallystudied under the framework of epidemic spreading [12].
Therefore, in many studies, the process of information diffusion
is regarded equally as the disease propagation, which informs
agents to transmit information to their neighbors via social
connections [13]. Among them, the Susceptible-Infected-Recovered (SIR) model is the most commonly used method to describe
the information spreading process, where individuals would lose
interest of further contribution due to a variety of less predictablefactors, which is very similar to the R state of the SIR model in
epidemic spreading. The interplay between the spreading dynamics and network structure is a key insight in the study on network
spreading dynamics [14,15]. The network structure affects boththe spreading speed and prevalence through features such as theshortest path length, degree distribution, degree correlations, andso on. In addition, the upper bound of informed proportion is
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Emergence of Blind Areas in Information Spreading

spreading process [35], that is to say, individuals would not be able
to interact with all their own neighbours. Therefore, we propose atunable parameter a[½0,1, representing the interaction strength.
Thus, the number of neighbors will be informed, ak, is
proportional to the spreader’s degree k, neglecting the neighbors’
states. And the probability of each neighbor of being selected is k1 ateach round, and the repeating selection is prohibited. Figure 1
shows the proposed spreading rule. By setting a = 0.6, theinformed node I1 can only transmit information to threeneighbors, two uninformed nodes S1 , S3 and one exhausted node
R1 . Therefore, the present rule implies a different feature of
information spreading, Partial Interaction, which is usually neglectedin the standard SIR model and its variants for information
spreading. In addition, the immediate influence in the present
model corresponds to the two parameters in traditional SIR
model, the infected probability b and the recovered probability m,
are both equal to one in the proposed model.
To better investigate the effect of the present model, we perform
analyses on three kinds of network: (i) ER network [43]: a randomgraph where N nodes connected by E edges which are chosenrandomly from all the N(N{1)=2 possible edges; (ii) BA network[44]: a growing network where each newly added node connects tom old nodes by preferential attachment mechanism; (iii) WS
network [45]: Randomly reshuffle links of a regular network withprobability c and result in the small-world network. As aconsequence, we generate corresponding BA, ER and WS
networks with the same network size N = 10000, m = 3, c = 0.5
and the average degree SkT = 6. In addition, to alleviate the effectof randomly selecting the spreading seed, all simulation results are
obtained by averaging over 1000 independent realizations.

theoretical models assumed that the informed agents would
transmit the information to all their neighbors [32,33] or justone randomly chosen neighbor [34] in one single time step.
However, since passing messages along would take a perceived
transmission cost [35] in real societies, the diffusion targets would
be selected among individuals with potential interests [36]. Thus,
the information flow would travel through social connections
thereby depending on the properties of observed networks [37].
In this paper, we propose a theoretical model with considering
the effect of partial interaction, where the number of interacted
neighbors is in proportion to the spreader’s degree. Simulation
results show that the spreading process percolates on all threeclassical networks (ER, BA and WS networks), which is quite
similar with the site percolation in statistical physics [38]. In
addition, it is observed that the percolation peaks later on WS
network than other two networks, which could be regarded as agood explanation of why information can spread more widely on
WS network. Furthermore, it is also astonishing to find that thesize distribution of unreachable clusters, namely the Information
Blind Area, exhibits the power-law property, which is the universal
phenomenon in the community distribution [39,40].

ModelsIn this paper, we consider a synthetic network G(N,E), where
N is the number of nodes and E is the number of links,
representing the individuals and their interactions, respectively.
Analogous with the SIR model, every individual would be andonly be at one of the three following states during the process of
information spreading,

N Uninformed (S). The individual has not yet received theN
N

Results and Analysis

information, and is analogous to the susceptible state of the
SIR model;
Informed (I). The individual is aware of the information but has
not transmitted it, and is analogous to the infected state in the
SIR model;
Exhausted (R). After transmitting the information, the individual
will probably lose interest and no longer transmit it [41], thus
is analogous to the recovered state of the SIR model.

Exhausted Rate
Denote P(R) as the fraction of exhausted nodes to the network
size, obviously, larger P(R) at the stable state indicates broader
information spreading, and vice verse. To determine how the
introduced parameter a affects the spreading results, we start ouranalysis from observing the relationship between P(R) and a.
Figure 2 shows that, for all the three networks, P(R) is
monotonically increasing with a, suggesting a positive correlation

Subsequently, we are mainly interested in the effect of partial
interaction. That is to say, each infected node, so-called spreader[42], will only influence a certain fraction of its neighbors. Thus,
the model can be described as follows.

N Initially, one arbitrary node is randomly picked as theN
N
N
N

Information Seed (I-state) and the rest remain uninformed (Sstate).
Then, the seed will transmit information to a fraction of its
neighbors and then becomes exhausted (R-state), wherea[½0,1;For simplicity, we assume that all individuals fully trust theirsocial connections. Consequently, each individual will approve
the information once s/he receives it;
After approval, s/he becomes an informed individual or a
spreader (I-state), and will transmit the information to all afraction of her/his neighbors and becomes exhausted (R-state);
The above process will repeat until there is no individualtransmits the information any more.

Figure 1. (Color online) Illustration of the spreading rule in the
proposed model with three uninformed nodes, S1 , S2 and S3 ,two informed nodes I1 and I2 , and one exhausted node R1 . Eachof them has the same probability 15 to receive information from theinformed node I1 at each round. When a = 0.6, only ak = 3 nodes, e.g.
S1 , S3 and R1 , will receive information.
doi:10.1371/journal.pone.0095785.g001

In this model, the popular individuals (nodes with large degree)
tend to interact with more neighbors when they approve theinformation. However, since there is transmission cost in the
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Emergence of Blind Areas in Information Spreading

Information Blind Areas
As can be seen from Figure 3a, the exhausted rate is always lessthan 1, suggesting that some nodes will never be informed during
the whole spreading process. Specifically, in this paper, we namethe region consists of such unreachable nodes as the Information
Blind Area. Actually, such blind areas emerge regularly due to the
spreading process. Figure 4 shows two typical kinds of information
blind areas. One is composed by a single uninformed nodesurrounded by exhausted neighbors and thus it could not receive
news. As shown in Figure 4a, the uninformed node S1 is envelopedby three exhausted nodes R1 , R2 and R3 , who would neither
directly transmit information to S1 nor allow the informed agentsto contact S1 . The other one is composed of multiple uninformed
nodes connected with each other and surrounded together by
exhausted nodes. Thus, the information will never be transmitted
to such a cluster as no path is available. As shown in Figure 4b,
three uninformed nodes, S1 , S2 and S3 , compose a connectedcluster which is surrounded by five exhausted nodes, R1 , R2 , R3 ,R4 and R5 . Since no single uniformed node in this cluster canaccess the outside information world, they have to passively keep
blind for the spreading information.
In order to describe those two kinds of information blind areas
more clearly, we conduct experiments with N = 100 and SkT = 6.
Figure 5 shows the visualization results on the three representative
networks. It can be seen that most uninformed individuals at stablestate (Figure 5a, 5d, 5g) are generally small-degree ones, which isconsistent with the foregoing analysis. In addition, we also show
the two information blind areas. The single-node case is displayedin Figure 5b, Figure 5e, Figure 5h. In Figure 5b, the uninformed
individual #66 is surrounded by exhausted neighbors #7, #10,
#36, #71, #75 and #83. As a consequence, node #66 isprevented from hearing news by its neighbors. Figure 5e andFigure 5h are similar to that of Figure 5b. On the other hand,
Figure 5c, 5f, and 5i report the information blind areas of multinodes cases on BA, ER and WS networks, respectively. As shownin Figure 5c, three uninformed individuals #49, #86 and #97
compose a connected cluster, which is surrounded by exhausted
individuals #1, #4, #7, #17, #18, #35, and #61. Therefore, all
the nodes in this relatively large area will not be informed andkeep blind to spread information. Similar phenomena can also be
discovered in Figure 5f and Figure 5i on ER and WS networks,
respectively.

Figure 2. (Color online) The ratio of exhausted nodes P(R) as afunction of a for three observed networks.
doi:10.1371/journal.pone.0095785.g002

P(R) and a. As a consequence, in the following, we will use P(R)instead of a to give comprehensive discussions.
Figure 3 reports the model results. In Figure 3a, it shows the
dynamics of P(R) on three different networks with the fixedparameter a~0:35. It can be seen that WS network yields themost efficient spreading as the largest P(R) at the stable state,while BA network exhibits the narrowest spreading, and ER
network stays moderately. This result agrees with previous studies[31]. In addition, the inset shows the dynamic difference of P(R)betweentwo
continuous
time
steps,
denotedas
DP(R)~P(Rtz1 ){P(Rt ), which can be considered as the
spreading speed. It indicates that information spreads faster at
its initial stage while slows down at later time steps due to the
influence of hub nodes. From the degree distribution of Figure 3c,
it can be seen that BA network is occupied by more large-degree
nodes than other two. Therefore, information spreading on BA
network is the fastest while slowest on WS network. According to
the partial interaction in the present model, informed agents
transmit information to only a certain fraction of their neighbors.
Consequently, there would be also some uninformed individuals
remaining without receiving any news because they are surrounded by exhausted neighbors (see Figure 4). Considering that the
spreading process may influence differently to different type ofnodes, we observe the relationship between exhausted rate P(Rk )and node degree k of three networks at the stable state (seeFigure 3b), where P(Rk ) is the ratio of the total number of
exhausted nodes with degree k to the overall number of nodes with
degree k. It clearly shows that P(Rk ) and k are apparently
positively correlated, and nodes with smaller degree have lessprobability to receive information. Furthermore, Figure 3c
displays the degree distributions of three networks. In general,
BA network shows the power-law degree distribution where mostnodes are of low degree but still a few hub nodes exist due to therich-get-richer mechanism. Poisson degree distribution emerges in
ER network because of purely random attachment. Comparatively, most nodes in WS have moderate degree and only a few
small and large degree nodes, resulting from the rewiring process.
In a word, the generally different fraction of hub nodes (Figure 3c)and different spreading effects on different type of nodes (Figure 3b)
support the result of Figure 3a from the perspectives of structureand function, respectively.

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Scale-free Effect
In this section, we shall explain how such blind areas emerge
from the perspective of site percolation. Generally, in a typical site
percolation system, each node in the given network would be at
one of two given states: empty with probability p or occupied withprobability 1{p, and all the edges connecting with empty nodes
are cut off. Previous studies have proved that the site percolation
system would display a phase transition phenomenon fromconnected phase to disconnected phase when p grows to a criticalpoint pc [46]. At this point, the behavior of the ratio of giant
cluster size (the largest connected cluster) rg (rg ~Ng =N, where Ng
is the size of the giant cluster) indicates the network’s sudden
disintegration, and the scaling distribution of connected clusters
would emerge [47–51]. Figure 6a–6c respectively show how rgchanges according to different p, as well as the critical point (seeinsets) in three different networks (BA, ER and SW networks) withthe same network size N = 10000.
Note that, the role of the empty nodes in site percolation is quite
similar with that of the R-state individuals in the present model,
where the information blind areas are divided by those individuals.
For the R-state individuals are all connected in the proposed
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Emergence of Blind Areas in Information Spreading

Figure 3. (Color online) The dynamics of spreading on three networks. (a) Exhausted rate P(R) as a function of simulation time. Inset is DP(R)as the function of simulation time; (b) Exhausted rate as a function of the node degree; (c) Degree distributions of three networks in log-log scale. The
parameter a is set to 0.35 for (a) and (b).
doi:10.1371/journal.pone.0095785.g003

SNc2 T{SNc T2, where Nc is the size of each
SNc T
connected cluster except the giant one [52], for it would result inlarge fluctuation when p or R approaches the critical point, andthe insets of Figure 6a–6f show the corresponding results.
Furthermore, in different networks (BA, ER and WS), we findgood agreements between the size distributions of connectedclusters and information blind areas of the two respective models.
Figure 6g–6i show that the distributions of the connected clusters’size of the two models are coincident with almost the same
exponent 22.9 at their respective critical points, indicating thatthey have very similar critical phenomena.
Despite those similar properties, we should not regard thepresent model as an equivalent process as the site percolation. One
significant difference between them is that, the node states arechanged according to the diffusion process based on the given
network structure in the spreading model, while the site
percolation process only randomly label the node states regardlessof the network structure. Therefore, the critical points exhibit very
differently for the two models. In the insets of Figure 6a–6c, theERWS
critical values follow pBA
for the site percolation,c wpc wpc
while a completely opposite sequence for the proposed model asERWSshown in the insets of Figure 5d–5f (that is RBA
c vRc vRc ). For
the random labeling process, the more heterogeneous the network
is, the larger the critical point will be, as there are more smalldegree nodes when each one is treated equally. On the contrary, if
there exist some large-degree nodes dominating the network
center, messages can be easily spread out via those core nodes,
hence the critical point will be smaller for such large heterogeneous network in the diffusion process. Similar statements can also
be applied in illustrating that, for the same network, the criticalvalue of the interaction model is always smaller than that of site
BA
ER
ER
WS
percolation, e.g. pBA
and pWSc wRc , pc wRc
c wRc . In aword, although the site percolation principle can not be fully
projected to spreading dynamics, it still provides a promising andversatile tool to explain the critical phenomenon of the emergenceof blind areas in information spreading.

model, we set nodes in the largest connected subgraph in site
percolation system as the empty state. Analogous with observing
the giant cluster in site percolation, we consequently calculate theratio of the largest blind area size rs (rs ~Ns =N, where Ns is thesize of the largest information blind area) for various exhausted
rate R[½0,1. The results are shown in Figure 6d–6f, where thecurve exhibits a very similar trend with that of rg . In addition, we
numerically calculate the thresholds of both pc and Rc , according
to the peak of the relative variance of the size distribution. We

denote it by x~

Conclusions & Discussion
In this paper, we have applied the partial interaction effect in
the classical SIR model where three types of node states are
considered: (1) Uninformed (S); (2) Informed (I); (3) Exhausted (R).
Subsequently, we adopt it in an information spreading scenario,
where the interaction strength is proportional to each spreader’s
degree. Numerical experiments show that there is a clearly positive
relationship between the interaction strength and final coverage of

Figure 4. (Color online) Illustration of two typical kinds ofinformation blind areas: (a) one single-node case; and (b)example of multiple nodes.
doi:10.1371/journal.pone.0095785.g004

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Emergence of Blind Areas in Information Spreading

Figure 5. (Color online) Visualization of information blind areas resulted from designed experiments. Yellow and blue circles represent
the uninformed and exhausted nodes, respectively. The size of each node is proportional to its degree. In addition, (a–c) are the results on BA
network; (d–f) are the results on ER network; (g–i) are the results on WS network.
doi:10.1371/journal.pone.0095785.g005

that the diffusion process is significantly different from random
selection. Further numerical analyses on three representative
networks (BA, ER and WS) demonstrate that, for the spreading
process, the more heterogeneous the network is, the smaller thecritical point will be. By contrast, the random labeling process ofsite percolation is completely opposite. Moreover, the critical valueof network-based diffusion is smaller than that of purely random
selection. Those findings can be regarded as additional explana-

spreading. In addition, we find that the spreading effect is highly
influenced by the network structure. One interesting property will
result from the spreading process, where the network is divided
into several information-unreachable areas, namely Information
Blind Areas. Analysis reveals that a phase transition of such blind
areas’ size distribution will emerge when the fraction of R-state
individuals grows to a certain critical point. In spite of quite similar
results from site percolation analysis, detailed experiments show

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Emergence of Blind Areas in Information Spreading

Figure 6. (Color online) Comparisons between the present model and the site percolation system. (a–c) The relationship between the sizeof giant cluster and empty probability p. Inset shows x versus p, and the peak is the critical point pc ; (d–f) The relationship between the size of thelargest information blind areas and the exhausted rate R. Inset shows x versus R, and the peak is the critical point Rc ; (g–i) The size distributions ofconnected clusters of the two models.
doi:10.1371/journal.pone.0095785.g006

tion why small-world network yields the most efficient information/epdemic spreading in previous studies [19,31].
Recently, the research of both contagion-based spreading
models and applications has attracted more and more attention[24,37]. Numerical results in this paper demonstrate that, due tothe various network structure, there are always unreachable
individuals. Human communication pattern analysis [25,53]
would be a promising method to help in understanding how tolighten the Dark Corners. In addition, activation of Long-tailindividuals and products would also enhance the efficiency ofInformation Filtering [54] in the era of big data to solve the cold-start
dilemma [55,56].

Author Contributions
Conceived and designed the experiments: ZKZ CXZ XPH CL. Performed
the experiments: ZKZ CXZ. Analyzed the data: ZKZ CXZ XPH CL.
Contributed reagents/materials/analysis tools: CXZ CL. Wrote the paper:ZKZ CXZ XPH CL.

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April 2014 | Volume 9 | Issue 4 | e95785