Abstract
Bootstrap percolation is a general representation of some networked activation process, which has found applications in explaining many important social phenomena, such as the propagation of information. Inspired by some recent findings on spatial structure of online social networks, here we study bootstrap percolation on undirected spatial networks, with the probability density function of longrange links’ lengths being a power law with tunable exponent. Setting the size of the giant active component as the order parameter, we find a parameterdependent critical value for the powerlaw exponent, above which there is a double phase transition, mixed of a secondorder phase transition and a hybrid phase transition with two varying critical points, otherwise there is only a secondorder phase transition. We further find a parameterindependent critical value around −1, about which the two critical points for the double phase transition are almost constant. To our surprise, this critical value −1 is just equal or very close to the values of many real online social networks, including LiveJournal, HP Labs email network, Belgian mobile phone network, etc. This work helps us in better understanding the selforganization of spatial structure of online social networks, in terms of the effective function for information spreading.
Introduction
Bootstrap percolation was originally introduced by Chalupa, Leath and Reich^{1} in the context of magnetic disordered systems in 1979. Since then, it has been studied extensively by physicists and sociologists, mainly due to its connections with various physical models and a variety of applications such as neuronal activity^{2} and jamming transitions^{3}. Bootstrap percolation can be essentially considered as an activation process on networks: (i) Nodes are either active or inactive; (ii) Once activated, a node remains active forever; (iii) Initially, each node is in an active state with a given probability p; (iv) Subsequently, inactive nodes become active if they have at least k active neighbors; (v) Nodes are activated in an iterative manner according to the condition in (iv), until no more nodes can be activated. This process has been investigated on different kinds of networks including lattices^{4,5}, trees^{6,7}, random networks^{8,9,10,11} and so on.
Bootstrap percolation has found applications in explaining many important social phenomena, such as the spreading of information^{12}, the propagation of infection^{13,14}, the adoption of new products and social behaviors^{15,16,17,18} including trends, fads, political opinions, belief, rumors, innovations and financial decisions. For instance, in emergence of cultural fads and adoption of new technology or objects, an individual can be positively influenced when there is a sufficient number of its close friends who have also done so^{19}. In other words, one may decide to buy a product when recommended by at least k users and trust a message when told by at least k neighbors; cf. the wellknown rule, “What I tell you three times is true”^{20}. In this way, the process leads initially localized effects propagating throughout the whole network. Moreover, a broad range of generalized formulations of bootstrap percolation on social networks are investigated, such as Watts’ model of opinions^{21}, in which k is replaced by a certain fraction of the neighbors and disease transmission models with different degrees of severity of infection^{22}.
Real networks are often embedded in space^{23} and social networks are no exception. Previous empirical studies of online social networks^{24}, email networks^{25} and mobile phone communication networks^{26} have confirmed a spatial scaling law, namely, the probability density function (PDF) of an individual to have a friend at distance r scales as , ^{27}. In fact, prior to these empirical observations, Kleinberg^{28} has proposed a spatial network model by adding longrange links to a 2dimensional lattice and he has proved that when , the structure is optimal for navigation. Recently, Hu et al.^{27} suggested the optimization of information collection as a possible explanation for the origin of this spatial scaling law.
Extensive studies have shown that the spatial organization can change the dimension^{29,30,31,32,33}, which dominates many important physical properties of networks^{34,35,36,37,38,39,40,41}. Moukarzel et al.^{42} studied kcore percolation on longrange spatial networks, which is built by taking a 2dimensional lattice and adding to each node one or more longrange links. In the networks, the probability density function (PDF) of longrange links with length r scales as . By numerical simulation, they found that the 3core transition is of firstorder for α > −1.75 (it is equivalent to the scaling of 2.75 found by Moukarzel et al.^{42}) and of secondorder for smaller α. In fact, kcore percolation has close relation to bootstrap percolation^{43,44}, nevertheless the two processes have different features from each other, being strongly dependent on the network structure^{8,45}. Although there is a deeper understanding of percolation processes and spatial networks now, how spatial organization influences the spreading process on social networks under the framework of bootstrap percolation remains further investigation.
In this paper, we numerically study bootstrap percolation on undirected Kleinberg’s spatial networks, which is a typical artificial social network. Setting the relative size of the giant active component to the network size as the order parameter, we find that the distribution of longrange links’ lengths can change the order of phase transition. In particular, our main findings are as follows: (i) We find a parameterdependent critical value α_{c}, above which a double phase transition^{46} is observed. Here, the socalled double phase transition means a mixture of two transitions at different critical points, consisting of a hybrid phase transition and a secondorder one. In this paper, we use the hybrid phase transition to indicate a firstorder phase transition in which the order parameter has a discontinuous jump between two nonzero values. (ii) Surprisingly, we find a parameterindependent critical value , about which the two critical points for the double phase transition are almost constant. (iii) When , the firstorder critical point decreases and the secondorder critical point increases as α decreases. When α < α_{c}, there is only a secondorder phase transition with an increasing critical point as the decreasing of α. Furthermore, we test the universality of our findings by drawing the phase diagram and give a possible explanation of the rich phase transition phenomena by simulating on related networks. Our findings indicate that the spatial scaling , observed in real social networks, may be resulted from some deepgoing principles in addition to the optimization of navigation and information collection, which is not yet fully understood now.
Results
Kleinberg model^{28} is a typical spatial network model, which has been well justified by empirical data^{24,25,26}. Here, the undirected Kleinberg’s spatial network is constrained on a 2dimensional periodic square lattice consisting of N = L × L nodes. In addition to its initially connected four nearest neighbors, each node i has a random longrange link to a node j with probability , where α is a tunable exponent and r_{ij} denotes the Manhattan distance, which quantifies the length of the shortest path between node i and node j, following strictly the horizontal and/or vertical links in lattices. Since the number of nodes at distance r to a given node is proportional to r^{d−1} in a ddimensional lattice, the probability Q(r_{ij}) can be mapped to a probability density function (PDF), . In the present 2dimensional case, where d = 2, the probability density function(PDF) scales as . An illustration of a 2dimensional undirected Kleinberg’s spatial network can be found in Fig. 1.
In the following, we focus on three indicators: (i) The relative size of the giant active component (S_{gc}) at the equilibrium, i.e., the probability that an randomly selected node belongs to the giant active component; (ii) The number of iterations (NOI) to reach the equilibrium, which is usually used to determine the critical point for the firstorder phase transition^{47,48}; (iii) The relative size of the second giant active component (S_{gc2}), which is usually used to determine the critical point for the secondorder phase transition^{32,48}.
Figure 2 shows rich phase transition phenomena when taking S_{gc} as the order parameter. When α ≥ −1, the curves of S_{gc}(p) are well overlapped and the system undergoes a double phase transition, mixed of a hybrid phase transition and a secondorder one as shown in Fig. 2a. Notice that S_{gc} has a continuous increase at (the secondorder critical point), where the transition is of secondorder. In contrast, S_{gc} has an abrupt jump directly from around 0.58 to almost 1 at (the firstorder critical point), where there is a hybrid phase transition. Surprisingly, the two critical points seem to be constant when , as indicated by the four overlapped S_{gc}(p) curves in Fig. 2a. When α < −1, there is only a secondorder phase transition with an increasing p_{c2} as the decreasing of α (see Fig. 2b). Specifically, when α = −2 and when α = −5. Although S_{gc} goes up sharper after p exceeds p_{c2} as α getting smaller, simulations justify that the curve of S_{gc}(p) is still continuous, meaning that the transition is indeed of secondorder when α < −1.
Finding critical points via simulations is always a difficult task that requires high precision. When α ≥ −1, where a part of the double phase transition is a hybrid phase transition, we can determine the critical point p_{c1} by calculating the number of iterations (NOI) in the cascading process, since NOI sharply increases when p approaches p_{c1} for the firstorder phase transitions^{47,48}. Accordingly, p_{c1} is calculated by plotting NOI as a function of p. As shown in Fig. 2c, NOI reaches its maximum at the same p when α ≥ −1, which is the evidence that is almost a constant value. Analogously, by plotting S_{gc2} as a function of p, we can precisely identify p_{c2}^{32,48}, at which S_{gc2} reaches its maximum (see Fig. 2d). We can see that p_{c2} increases as α decreases, as (α ≥ −1), 0.176 (α = −2) and 0.256 (α = −5).
Although to justify the hybrid phase transition and to determine the critical value α_{c} by simulations in a finite discrete system are not easy, we solve this problem by a crossvalidation on the critical point p_{c1} and the critical value α_{c}. Firstly, we fix α = −1 to determine p_{c1}. On the one hand, there is an intersection for curves of S_{gc}(p) at under different network sizes as shown in Fig. 3a, which can be considered as the critical point according to the finitesize analysis^{48}. On the other hand, the corresponding NOI reaches its maximum at when L = 800 as shown in Fig. 3b. Combining these two observations, a more appropriate critical point is identified as the average value . Conversely, we fix p = 0.263 to determine α_{c}. From Fig. 3c, we can see that S_{gc} has two phases: about 0.58 or close to 1 when , which is a strong evidence that S_{gc} undergoes a hybrid phase transition. If the increasing of S_{gc} is continuous, there is no such gap between the two phases. From Fig. 3d, we note that the corresponding averaging NOI reaches its maximum at . Combining these two evidences, we appropriately identify the critical value as the average value .
In addition, p_{c1} for the hybrid phase transition should be almost constant when α ≥ α_{c}, otherwise we cannot observe the separation of two phases in Fig. 3c for a fixed value p = 0.263. To verify that, we estimate errors of p_{c1} by varying α under fixed network size L. As shown in Fig. 4a, p_{c1} slightly decreases when α approaches the critical value . Even though, when α is in the range [−1,4], the difference between maximum and minimum value of p_{c1} is only 0.0008 after over 1000 realizations, which is very small compared to the whole range of p (i.e. [0,1]), indicating that the value of p_{c1} is not sensitive to the parameter α when α ≥ −1. Based on these evidences, p_{c1} can be roughly considered as a constant and its value is about 0.2634, which is the mean value of p_{c1} when α in the range [−1,4]. Furthermore, taking α = −1 as an example, we consider the effects of finitesize of networks on these results. As shown in Fig. 4b, the mean value of p_{c1} gradually approaches an extreme value around 0.263 and the standard deviation of p_{c1} decreases as L goes to infinity. Similar results hold for the analysis of the critical point p_{c2} and its value is also almost constant as 0.134.
A representative phase diagram for S_{gc} in the p − α plane is shown in Fig. 5. We find that the varying of α, which dominates the distribution of longrange links’ lengths, can change the order of phase transition. Overall, is confirmed to be a critical value, above which a double phase transition (region II) is present. When α ≥ −1, the curves of S_{gc}(p) are overlapped, suggesting that the properties of bootstrap percolation on these spatial networks are alike. When α < −1, the hybrid phase transition vanishes and S_{gc} only undergoes a secondorder phase transition (region I) with an increasing critical point as the decreasing of α. The maximum of p_{c2} is about 0.259, which is obtained when , i.e., all longrange links’ lengths are 2.
To test the universality of the findings, we simulated on undirected Kleinberg’s spatial networks in parameter spaces (k, α, k_{l}) and determined the critical points. Results are shown in Fig. 6. According to the relationship between the threshold k and half of the average degree of the network , where , there are three regions in the phase diagram:

When k is remarkably smaller than , e.g, k = 1 compared to , there is only a trivial firstorder phase transition at .

When k is around , e.g., k = 3 compared to , there is a critical value α_{c}, above which a double phase transition is observed. The value of α_{c} depends on the choice of both k and k_{l}. In particular, is found to be a parameterindependent critical value, about which the two critical points for the double phase transition are almost constant. Specifically, as shown in the phase diagram of Fig. 6, the color of data points for the same parameter k is nearly unchanged when α ≥ −1, which is a strong evidence that the values of p_{c1} and p_{c2} are almost constant. When , p_{c1} decreases and p_{c2} increases as α decreases. Note that α_{c} can be equal to in some parameter spaces, such as and .

When k is remarkably larger than , e.g, k = 5 compared to , the hybrid phase transition is absent and S_{gc} only undergoes a secondorder phase transition with an increasing p_{c2} as the decreasing of α (see Supplementary Figs S1S3 for the detailed shapes of S_{gc}(p) curves).
Moreover, simulations confirm that our main results also hold for Kleinberg’s spatial networks with directed longrange links since is still a critical value. However, there is only a firstorder phase transition with p_{c1} being almost constant instead of the formal double transition when α ≥ −1 (see Supplementary Fig. S4). In addition, simulations on undirected Kleinberg’s spatial networks without periodic boundary conditions suggest that whether the square lattice has periodic boundary conditions does not essentially affect our main results (see Supplementary Fig. S5).
To provide the insights on the mechanism of the transition, we simulate on different networks and compare with other related transitions. These networks include a simple 2dimensional lattice (Lattice), networks with all 5 links being longrange (LR) and networks without spatial structure, i.e. random 5regular networks (RR). In the LR network, which is a special case of longrange percolation model in the 2dimensional space^{49,50,51}, each node is associated with only k_{l} = 5 undirected longrange links instead of initially connected shortrange links based on a 2dimensional periodic lattice. As shown in Fig. 7, the curves of S_{gc}(p) on the spatial networks are between the ones on Lattice network and RR network. When α = −4, the S_{gc}(p) curve on the spatial network has similar trend with the one on Lattice network since the very longrange links are rare and the transition is of secondorder. When α ≥ −1, there is a double phase transition and the curves of S_{gc}(p) are almost overlapped with the one on RR network. These observations indicate that, to turn the value of α, we can change the bootstrap percolation properties of spatial networks from Lattice network to RR network, or vice versa. More specifically, when α = −4, all longrange links are highly localized and the structure of spatial networks is similar to Lattice network, whereas when α ≥ −1, mainly due to the existence of very longrange links, the spatial networks behave like RR network.
Together, it should be noted that the S_{gc}(p) curve on LR network when α = −1 acts like the ones on RR network and spatial networks when α ≥ −1. To better understand how does α affect the transition on LR network, taking k_{l} = 5 as an example, we show the phase diagram after k = 3 bootstrap percolation in Fig. 8. The diagram is divided into three regions by critical values and . As α decreases, the transition is of secondorder with an increasing p_{c2} when (region I). There is a double phase transition when (region II), where p_{c1} decreases and p_{c2} increases as α decreases. Once again, a double phase transition with two almost constant critical points, and , is observed when (region III). Further simulations suggest that similar main results also hold under other combinations of k and k_{l} (see Supplementary Fig. S6 for the phase diagram and Supplementary Figs S7–S11 for the shapes of S_{gc}(p) curves).
In fact, for the original Kleinberg’s spatial networks with directed longrange links, Sen et al.^{52} found that the varying of α can change the network structure, namely, the network is regularlatticelike when α < −2, smallworldlike when −2 < α < −1 and randomlike when α >−1. More recent studies^{30,31,32} also proposed three regimes: (i) When α > −1, the dimension of the spatial network is and the percolation transition belongs to the university class of percolation in ErdösRényi networks. (ii) When , d decreases continuously from to d = 2 and the percolation shows new intermediate behavior. (iii) When α < −3, the dimension is d = 2 and the percolation transition belongs to the university class of percolation in regular lattices. These previous findings suggest that the properties of spatial networks have qualitative changes when α is around −1, which is corresponding to the observation of these phase transition phenomena here.
Discussion
In summary, we have studied bootstrap percolation on spatial networks, where the distribution patterns of longrange links’ lengths can change the order of phase transition. In particular, we find a parameterdependent critical value α_{c}, above which a double phase transition, mixed of a hybrid phase transition at a higher p and a secondorder phase transition at a lower p, is present. It is particularly interesting that we find a almost parameterindependent critical value , about which the curves of S_{gc}(p) are well overlapped, indicating that the two critical points for the double phase transition are almost constant when α ≥ −1. As bootstrap percolation has not been studied yet on undirected Kleinberg’s spatial networks, our novel findings indicate that the topological properties of undirected Kleinberg’s spatial networks are alike when α ≥ −1 in the 2dimensional space. In fact, the scaling law has been empirically observed in many real networks^{24,25,26}, which may be resulted from complex selforganizing processes toward optimal structures for information collection^{27} and/or navigation^{28}. Since the cascading processes on spatial networks are almost the same when α ≥ −1, is indeed corresponding to the structure with the smallest average geographical length of links, which can exhibit as effective spreading of information as networks with even longer shortcut links. This is to some extent relevant to the principle of least effort in human behavior^{53}.
We find the varying of α can change the bootstrap percolation critical behavior from random regular networks to lattices, or vice versa. In particular, when , the spatial networks behave like random regular networks and there is a double phase transition. When , there is a richer phase transition phenomena. More specifically, the double phase transition is still present when , however, instead of being constant, the firstorder critical point decreases and the secondorder critical point increases as α decreases. The observation of such results may be mainly due to the smallworldlike network structure^{52,54}, which leads the transition showing a intermediate behavior. When α < α_{c}, the hybrid phase transition vanishes and there is only a secondorder phase transition with an increasing critical point as the decreasing of α. When α goes to negative infinity, where all longrange links are highly localized, the spatial networks degenerates into regular lattices and the transition is of secondorder. In this way, we give a possible explanation of the emergence of these phase transition phenomena.
Moreover, our results are, to some extent, relevant to the control of information spreading. For example, when α ≥ α_{c}, if we would like to make as many people as possible to know the information, the optimal choice of the fraction of initially informed people should be p_{c1}, since larger initially informed population provides no more benefit but requires higher cost as shown in Fig. 2a. Therefore, the results help us in better understanding the selforganization of spatial structure of online social networks, in terms of the effective function for information spreading. Besides, our work can possibly find applications in studying the diffusion of virus spread in other spatial networks such as ad hoc, wireless sensor networks and epidemiological graphs. However, in the context of a numerical study, it is really hard to tell whether the critical value is exactly −1 and whether the critical points are completely independent of α when . In addition, how does the transition depend on the spreading processes and what kind of transitions does it belong to for real social networks are still open questions. Hence, we expect to verify our findings in an analytical way and based on other generalized network models, including spatially constrained ErdösRényi networks^{55}, network of networks (NON)^{56}, multiplex networks^{57} and real social networks. Besides, to assign each node one longrange link is a highcost strategy when generating artificial spatial social networks, we leave individualized number of longrange links associated with each node and partial spatial embedding as future works.
Methods
To numerically implement the spatial scaling α when generating Kleinberg’s spatial networks, we add undirected longrange links to a 2dimensional periodic square lattice in a smart way as follows. First, a random length r between 2 and L/2 is generated with probability , which ensures the scaling in advance. Second, random segmentations of length r to Δx and Δy with the only constraint that are done to determine candidate nodes, where Δx and Δy are both integers. Namely, for an uncoupled node i with coordinates (x, y), named target node, all candidate nodes are these with coordinates (x + Δx, y + Δy) such that . The above procedure ensures all candidate nodes at distance r from the target node i are uniformly distributed. Hence, we can randomly choose an uncoupled candidate node (i.e., a node without any longrange link) to link with target node i. Be noted that, on a large system, a finite fraction of the nodes will have all candidate nodes already connected when α is very small. To deal with this problem, we additionally adopt an alternate procedure referring to the distance coarse graining procedure^{58}, in which we randomly choose an uncoupled nearest neighbor node of these candidate nodes until the linking is accomplished. We repeat such procedure for the rest uncoupled nodes until each node of the network has one undirected longrange link such that the degree of each node is exactly 5.
Additional Information
How to cite this article: Gao, J. et al. Bootstrap percolation on spatial networks. Sci. Rep. 5, 14662; doi: 10.1038/srep14662 (2015).
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Acknowledgements
The authors acknowledge Jun Wang and Panhua Huang for useful discussions. This work is partially supported by National Natural Science Foundation of China under Grants Nos. 61203156 and 11222543. J.G. acknowledges support from Tang Lixin Education Development Foundation by UESTC. T.Z. acknowledges the Program for New Century Excellent Talents in University under Grant No. NCET110070 and Special Project of Sichuan Youth Science and Technology Innovation Research Team under Grant No. 2013TD0006.
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J.G. and Y.H. designed the research. J.G. executed the experiments and prepared the figures. J.G., T.Z. and Y.H. analyzed the results. J.G. and T.Z. wrote the manuscript. All authors reviewed the manuscript.
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Gao, J., Zhou, T. & Hu, Y. Bootstrap percolation on spatial networks. Sci Rep 5, 14662 (2015). https://doi.org/10.1038/srep14662
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DOI: https://doi.org/10.1038/srep14662
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