Starting an artificial 'epidemic' to hinder the other: an inspection on correlated network layers Abstract As is introduced in class, by selectively vaccinating a group of nodes that are more central can mean prevention or smaller infection scale to the possbile contagion. However, when it comes to the real outburst of an unpredictable or temporarily untreatable epidemic, for instance, a variation of a deadly flu or computer virus, the immunisation actually could only happen after the epidemic begins, at a time when people start to be aware of it or get prepared for it. So, instead of discussing the static group vaccination(pre-immunisation), I try to take inspection into the so-called post-immur which means that by nature, there exsits a time delay between the outbreak and the immunisation Then we may start to ask how to hinder the ongoing epidemic when we have already been left behind by it. Here, we try to set up another artificial'epidemic' on networks to interact with the former one. In this passage, I used SIR model for two interacting spreading processes on two coupled network layers and simulated the spreading processes on both homogenous, heterogenous and empirical network. Many different parameters and factors are covered, such as time delay, interaction coefficient and layer-to-layer correlation coefficient, and efforts are made to find out the correlation between these factors Introduction Review Newman [2] studied the coexistence threshold of two pathogens spreading on the same network. He used the SIS model and mapped the giant component scale of the network as a function of disease transmissibility. He found that in scale-free networks, if the first pathogen ' s transmissibility is larger than the coexistence threshold, then the second one cannot spread because thethe first process has immunised considerable number of nodes in the network and left a rather sparse and homogenous network for the second. Funk et al B3] widened the research by analyzing the case of partial immunity on ',networks and gave more sophisticated phase diagrams. Their works have provided me with much inspiration on this topic. However, although they did not point out, it is the first spreading process that represents the artificially started immunisation process, not the second one. This means that they both focus on vaccination(pre-immunisation)instead of post-immunisation. Also, the time dimension parameter is not taken into consideration. There is no other specific connections on time dimension between the two spreading processes except that one goes first and the other follows two processes are separated on timeline. On the contrary, in this passage, the time dimension of two spreading processes is made to overlap Spreading model I used SIR model in this passage Two spreading processes respectively have a infection strength A1=B/r and A2= B2/r2, where p is the rate of susceptible nodes s)turning into the state of infection(D and y is the rate of recovery/removal (R)after infection. The scale of the spreading process is measured through the ratior= R/N at the end of simulation, where R means the number of recovered/removed nodes and n means the total number of nodes in the network Interaction coefficient
Starting an artificial 'epidemic' to hinder the other: an inspection on correlated network layers Abstract As is introduced in class, by selectively vaccinating a group of nodes that are more central can mean prevention or smaller infection scale to the possbile contagion. However, when it comes to the real outburst of an unpredictable or temporarily untreatable epidemic, for instance, a variation of a deadly flu or computer virus, the immunisation actually could only happen after the epidemic begins, at a time when people start to be aware of it or get prepared for it. So, instead of discussing the static group vaccination(pre-immunisation), I try to take inspection into the so-called post-immunisation, which means that by nature, there exsits a time delay between the outbreak and the immunisation process. Then we may start to ask how to hinder the ongoing epidemic when we have already been left behind by it. Here, we try to set up another artificial 'epidemic' on networks to interact with the former one. In this passage, I used SIR model for two interacting spreading processes on two coupled network layers and simulated the spreading processes on both homogenous, heterogenous and empirical network. Many different parameters and factors are covered, such as time delay, interaction coefficient and layer-to-layer correlation coefficient, and efforts are made to find out the correlation between these factors. Introduction Review Newman [2] studied the coexistence threshold of two pathogens spreading on the same network. He used the SIS model and mapped the giant component scale of the network as a function of disease transmissibility. He found that in scale-free networks, if the first pathogen's transmissibility is larger than the coexistence threshold, then the second one cannot spread because the the first process has immunised considerable number of nodes in the network and left a rather sparse and homogenous network for the second. Funk et al [3] widened the research by analyzing the case of partial immunity on 'overlay' networks and gave more sophisticated phase diagrams. Their works have provided me with much inspiration on this topic. However, although they did not point out, it is the first spreading process that represents the artificially started immunisation process, not the second one. This means that they both focus on vaccination (pre-immunisation) instead of post-immunisation. Also, the time dimension parameter is not taken into consideration. There is no other specific connections on time dimension between the two spreading processes except that one goes first and the other follows— two processes are separated on timeline. On the contrary, in this passage, the time dimension of two spreading processes is made to overlap. Spreading model I used SIR model in this passage. Two spreading processes respectively have a infection strength and , where is the rate of susceptible nodes turning into the state of infection and is the rate of recovery/removal after infection. The scale of the spreading process is measured through the ratio at the end of simulation, where means the number of recovered/removed nodes and means the total number of nodes in the network. Interaction coefficient
The artificial 'epidemic' can represent many empirical settings, e.g. the spread of a anti-virus programme or patch on the Internet to fix a bug. Moreover, the social or personal behavior restrictions recommended by the government medical department can also be viewed as spreading processes that have hindering effect on the spread of disease. Comparing these two cases, we can see that the artificial epidemic's immunising effect on the first one may vary under different circumstances. Computers that have installed the patch and fixed the bug are impossible to get infected afterwards, however people who obey the restrictions are still partially exposed to the danger of infection. That' s why the interaction coefficient n is defined to measure the level of interaction Once a node is infected with B, next time, Bl is made to become n Bl From another equal perspective instead of letting the n affect B, n can also be made to affect y. Since they are essentially the same, here I take the first definition Type of interaction Immunity As is introduced above, it means that if a node is infected with B then it gets immune to A.(Here A is the first process and B is the second, b has a time delay after A )In this case, when n lies between 0 and 1, it means partial immunity, and when it is O, it means complete immunity Infection facilitation It means that if a node is infected with a then it is much easier to get infected by B.(e.g. AIDS can greatly damage carrier's immune system and makes him exposed to infection from othe pathogens. )In this case, n is bigger than 1 Infection facilitation or immunity between process a and B can be either one-sided or mutual Moreover, the levels of influence from a to B or from b to A are not necessarily the same, meaning that nB->A can be different from n4->B. Considering that we would like to hinder a disease or virus from further propagation, in this passage we only consider the case of one-sided immunity(from B to A), which means 0 A b=l(and denote nB->4=: n Network of networks(Correlated network layers) Different spreading processes usually take place in different channels. As a typical example, AIDS virus spreads through sexual relationship, blood transfusion and mother-and-child relationship rather than spreading through air and water. This means by hugging or kissing an AIDS carrier, one is not likely to be infected. On the contrary, some flu can spread easily via close contact with the carriers. The characteristics of these two viruses promise different spreading networks within the same group of hosts So I took the idea called overlay networks'in []or network of networks. For brevity concerns, it is assumed that two network have the same nodeset but different connection patterns. Therefore, they are more like two correlated network layers Here I assume that two network layers A& B have exactly the same topological properties. Admittedly, there could be some significant topological difference between two layers, however, the major concern here is the influence of correlation between layer A and B on the spreading processes, so it is easonable to neglect the topological difference here. To measure the similarity between two network layers, the correlation coefficient between two discrete random variables is employed. It is denoted as P= VD(AD(B) 3). The value of plies between-1 and +1. p=l means two variables are fully, linearly correlated to each other and p=0 shows that two variables are linearly independent. As is pointed out in class, on heterogenous networks, those who are in the central part of the network are more likely to get infected at a early time. Therefore, the centrality model is used to represent the correlation strength of two network layers. Applying different centrality models can help us get different correlation coefficient. So when A and B are two degree centrality sequences of LayerA and LayerB, we can get the Pd as degree correlation coefficient. Time Delay
The artificial 'epidemic' can represent many empirical settings, e.g. the spread of a anti-virus programme or patch on the Internet to fix a bug. Moreover, the social or personal behavior restrictions recommended by the government medical department can also be viewed as spreading processes that have hindering effect on the spread of disease. Comparing these two cases, we can see that the artificial 'epidemic's immunising effect on the first one may vary under different circumstances. Computers that have installed the patch and fixed the bug are impossible to get infected afterwards, however people who obey the restrictions are still partially exposed to the danger of infection. That's why the interaction coefficient is defined to measure the level of interaction. Once a node is infected with B, next time, is made to become . From another equal perspective, instead of letting the affect , can also be made to affect . Since they are essentially the same, here I take the first definition. Type of interaction Immunity As is introduced above, it means that if a node is infected with B then it gets immune to A. (Here A is the first process and B is the second, B has a time delay after A.) In this case, when lies between 0 and 1, it means partial immunity, and when it is 0, it means complete immunity. Infection facilitation It means that if a node is infected with A then it is much easier to get infected by B.(e.g. AIDS can greatly damage carrier's immune system and makes him exposed to infection from other pathogens.) In this case, is bigger than 1. Infection facilitation or immunity between process A and B can be either one-sided or mutual. Moreover, the levels of influence from A to B or from B to A are not necessarily the same, meaning that can be different from . Considering that we would like to hinder a disease or virus from further propagation, in this passage we only consider the case of one-sided immunity (from B to A), which means and (and denote ). Network of networks (Correlated network layers) Different spreading processes usually take place in different channels. As a typical example, AIDS virus spreads through sexual relationship, blood transfusion and mother-and-child relationship rather than spreading through air and water. This means by hugging or kissing an AIDS carrier, one is not likely to be infected. On the contrary, some flu can spread easily via close contact with the carriers. The characteristics of these two viruses promise different spreading networks within the same group of hosts. So I took the idea called 'overlay networks' in [3] or network of networks. For brevity concerns, it is assumed that two network have the same nodeset but different connection patterns. Therefore, they are more like two correlated network layers. Here I assume that two network layers A & B have exactly the same topological properties. Admittedly, there could be some significant topological difference between two layers, however, the major concern here is the influence of correlation between layer A and B on the spreading processes, so it is reasonable to neglect the topological difference here. To measure the similarity between two network layers, the correlation coefficient between two discrete random variables is employed. It is denoted as [3]. The value of lies between -1 and +1. means two variables are fully, linearly correlated to each other and shows that two variables are linearly independent. As is pointed out in class, on heterogenous networks, those who are in the central part of the network are more likely to get infected at a early time. Therefore, the centrality model is used to represent the correlation strength of two network layers. Applying different centrality models can help us get different correlation coefficient. So when and are two degree centrality sequences of LayerA and LayerB, we can get the as degree correlation coefficient. Time Delay
As is mentioned above, usually the artificial 'epidemic needs a period of reaction time to intervene Therefore, it is assumed that there is a at delay period between two spreading processes During th delay period, the first process A spreads independently. After that, Bs influence on A is taken into account Methods The simulation could only be based on discrete rounds instead of being continuous. So in order to mimic the continuous circumstance, having balanced between the reduction of noise and errors and computation capability, I set each discrete round as a short time period AT= 0.01. As a result, in each round, the possibility for a susceptible node to get infected is BAT. Similarly, the possibility for a infected node to recover is yAT. As is pointed out above, changing respective B and y (as possibility. not rate)for each round only affect the time dimension of the spreading process. It is the ratio X that is really responsible for the final infection scale Since two network layers are supposed to be topologically the same the generation of two correlated network layers becomes simply mapping one network to another. This means that one layer is just a projection of another via a certain kind of transformation. This has provided the design of simulation programme with great convenience. Layer B is generated by firstly making a copy of layer A and rewiring the copy s node reference sequence. This is just equivalent to generating a network via a shuffled degree sequence of layer A. After the rewiring process, a one-to-one 'look-up-table'is returned The starters of the two spreading processes are randomly chosen and takes up as small as 0.5%6-1% of the total number of nodes Results Simulation on computer generated network The simulation is conducted on both homogenous network and heterogenous network. A connected WS small-world network with rewiring possibility p= 0.05 is used as a typical homogenous network and a BA scale-free network with m =5 is used as a typical heterogenous network. They share the same N=3000 and k>= 10. Their heterogenity is characterized by the variance of degree sequence. Here ra A 140 and owg=0 It is assumed that the artificial immunising 'epidemic is powerfully pushed and is much easier to pread than the disease. Considering this, I set A2= l and 1=0.5 as fundamental and fixed parameters for simulation, since if not so, the analysis will become very complicated and not straightforward enough Note: each simulation is repeated for at least 10 times and the average is used as the final result in order to minimize possible errors Infection dynamics at time dimension: on BA scale-free network, n=0,Pd=1
As is mentioned above, usually the artificial 'epidemic' needs a period of reaction time to intervene. Therefore, it is assumed that there is a delay period between two spreading processes. During the delay period, the first process A spreads independently. After that, B's influence on A is taken into account. Methods The simulation could only be based on discrete rounds instead of being continuous. So in order to mimic the continuous circumstance, having balanced between the reduction of noise and errors and computation capability, I set each discrete round as a short time period . As a result, in each round, the possibility for a susceptible node to get infected is . Similarly, the possibility for a infected node to recover is . As is pointed out above, changing respective and (as possibility, not rate) for each round only affect the time dimension of the spreading process. It is the ratio that is really responsible for the final infection scale. Since two network layers are supposed to be topologically the same, the generation of two correlated network layers becomes simply mapping one network to another. This means that one layer is just a projection of another via a certain kind of transformation. This has provided the design of simulation programme with great convenience. Layer B is generated by firstly making a copy of layer A and rewiring the copy's node reference sequence. This is just equivalent to generating a network via a shuffled degree sequence of layer A. After the rewiring process, a one-to-one 'look-up-table' is returned. The starters of the two spreading processes are randomly chosen and takes up as small as 0.5% - 1% of the total number of nodes. Results Simulation on computer generated network The simulation is conducted on both homogenous network and heterogenous network. A connected WS small-world network with rewiring possibility is used as a typical homogenous network and a BA scale-free network with is used as a typical heterogenous network. They share the same and . Their heterogenity is characterized by the variance of degree sequence. Here and . It is assumed that the artificial immunising 'epidemic' is powerfully pushed and is much easier to spread than the disease. Considering this, I set and as fundamental and fixed parameters for simulation, since if not so, the analysis will become very complicated and not straightforward enough. Note: each simulation is repeated for at least 10 times and the average is used as the final result in order to minimize possible errors. Infection dynamics at time dimension: on BA scale-free network
time delay At/AT=0 Infection dynamics at time dimension 0.6 0.5 0.4 +十+ 0.3 0.1 0.0 1000 t/AT Figure 1-1 · time delay△t/△T=30 Infection dynamics at time dimension 0.6 A 0.5 0.4 +++ 3 0.2 0.1 0.0 400 800 1000 t/AT Figure 1-2
time delay Figure 1-1 time delay Figure 1-2
We can see the infection rate grows rapidly at the beginning of spreading process and gradually decreases after it reaches its peak since at that time, the network has already been partially saturated. Also, the infection rate of A is quite sensitive to time delay. Short time delay between a& B can greatly suppress the infection rate growth of epidemic A Scale of epidemic as a function of time delay The final scale of spreading process a, represented by the ratio r, is used to measure the immunising effect of spreading process B as a function of time delay at 1. Cases under different interaction coefficient and network heterogenity Here the simulation is conducted under full layer correlation Pd= 1, which is equivalent to A and B spreading together on the same network
We can see the infection rate grows rapidly at the beginning of spreading process and gradually decreases after it reaches its peak since at that time, the network has already been partially saturated. Also, the infection rate of A is quite sensitive to time delay. Short time delay between A & B can greatly suppress the infection rate growth of epidemic A. Scale of epidemic as a function of time delay The final scale of spreading process A, represented by the ratio , is used to measure the immunising effect of spreading process B as a function of time delay . 1. Cases under different interaction coefficient and network heterogenity Here the simulation is conducted under full layer correlation , which is equivalent to A and B spreading together on the same network
Scale of epidemic as a function of time delay on Ba scale- free network n=0 +n=0.5 0.8 0.6 0.4 0.2 0.0 100 △t△T Figure 2-1 Scale of epidemic as a function of time delay on ws small-world network n=0 n=0.5 0.8 0.6 0.4 0.2 0.0 100125150175200 △t/△T As is showed in the figures, cases are quite different under distinct network heterogenity. Firstly, under the same n, an epidemic in homogenous network spreads much faster and a relatively larger scale of nodes will get infected than that in the case of homogenous network. Secondly, heterogenous networks are more sensitive to both the intervention of the artificial 'epidemic, both the time delay
Figure 2-1 Figure 2-2 As is showed in the figures, cases are quite different under distinct network heterogenity. Firstly, under the same , an epidemic in homogenous network spreads much faster and a relatively larger scale of nodes will get infected than that in the case of homogenous network. Secondly, heterogenous networks are more sensitive to both the intervention of the artificial 'epidemic', both the time delay
and the interaction coefficient. This can be inferred from the fact below: in the heterogenous case, the final scale of epidemic a grows dramatically as the time delay increases, especially at the beginning of its spreading process. However, in the homogenous case, the curve is close to linear with a much gentler gradient. There is not much difference between a timely intervention and a untimely one Moreover, when raising n under the same time delay, the r of the heterogenous network increases significantly, especially in the case of early intervention On the contrary, in homogenous case, the shape of the curve bears no significant change. Even the change of n can only cause the curve to slightly shift upwards. Therefore, in the context of heterogenous network, which is commonly seen in empirical settings, earlier intervention and stronger interaction capability is very important for the artificial immunising process to hinder the spread of an epidemic. 2 Cases under different correlation coefficient complete immunity n=0 Scale of epidemic as a function of time delay on Ba scale- free network 1.0 +pd=0 0.8 0.15 0.6 0.4 0.2 0.0 100 △t/△T Figure 3-1-1 · partial immunity n=0.5
and the interaction coefficient. This can be inferred from the fact below: in the heterogenous case, the final scale of epidemic A grows dramatically as the time delay increases, especially at the beginning of its spreading process. However, in the homogenous case, the curve is close to linear with a much gentler gradient. There is not much difference between a timely intervention and a untimely one. Moreover, when raising under the same time delay, the of the heterogenous network increases significantly, especially in the case of early intervention. On the contrary, in homogenous case, the shape of the curve bears no significant change. Even the change of can only cause the curve to slightly shift upwards. Therefore, in the context of heterogenous network, which is commonly seen in empirical settings, earlier intervention and stronger interaction capability is very important for the artificial immunising process to hinder the spread of an epidemic. 2. Cases under different correlation coefficient complete immunity Figure 3-1-1 partial immunity
Scale of epidemic as a function of time delay on ba scale-free network 1.0 0 0.8 0.6 0.4 0.2 0.0 t△T Figure 3-2 As is showed in the figures above, the suppressing effect against a provided by b greatly decreases as he correlation coefficient becomes zero or below. A possible explanation could be that in heterogenous networks, when two coupled network layers are poorly correlated, a central node in layer A is more likely to be less central in layer B. Therefore, since the early infection incidences tend to happen on central nodes, those who are firstly influenced by B are more likely to be located at the edge of layer A. When the correlation coefficient is close to -1, chances are that B may not be able to spread to those central nodes of a if B is not powerful enough, since these nodes are located at the edge of B and are hard to reach by nature. Therefore, even if B takes instant action, the scale of infection is still relatively high. This is somehow equivalent to the extension of time delay before B's action or the reduction of B's infection strength Similarly, we can also infer that in homogenous networks, the correlation coefficient is trivial to this problem. Since every node in a homogenous network is of the same priority and centrality, changing correlation does not lead to the situation we have mentioned above 3. Remark There may be doubt that the spreading process b may start from a larger group of people, not necessarily the same size as spreading process A does. However, this is to some extent equivalent to reducing the time delay between two spreading processes, because adding members to the starter group is just equivalent to giving the artificial'epidemic' more time to spread, thus narrowing the time delay It seems that these factors here: interaction coefficient, correlation coefficient and time delay although discussed separately, are to some extent the same towards this problem. They can be unified as one factor that affects process B's ability to occupy the central nodes of layer A Further work should be done to understand how these factors are correlated to each other Simulation on empirical dataset Background
Figure 3-2 As is showed in the figures above, the suppressing effect against A provided by B greatly decreases as the correlation coefficient becomes zero or below. A possible explanation could be that in heterogenous networks, when two coupled network layers are poorly correlated, a central node in layer A is more likely to be less central in layer B. Therefore, since the early infection incidences tend to happen on central nodes, those who are firstly influenced by B are more likely to be located at the edge of layer A. When the correlation coefficient is close to -1, chances are that B may not be able to spread to those central nodes of A if B is not powerful enough, since these nodes are located at the edge of B and are hard to reach by nature. Therefore, even if B takes instant action, the scale of infection is still relatively high. This is somehow equivalent to the extension of time delay before B's action or the reduction of B's infection strength. Similarly, we can also infer that in homogenous networks, the correlation coefficient is trivial to this problem. Since every node in a homogenous network is of the same priority and centrality, changing correlation does not lead to the situation we have mentioned above. 3. Remark There may be doubt that the spreading process B may start from a larger group of people, not necessarily the same size as spreading process A does. However, this is to some extent equivalent to reducing the time delay between two spreading processes, because adding members to the starter group is just equivalent to giving the artificial 'epidemic' more time to spread, thus narrowing the time delay. It seems that these factors here: interaction coefficient, correlation coefficient and time delay, although discussed separately, are to some extent the same towards this problem. They can be unified as one factor that affects process B's ability to occupy the central nodes of layer A. Further work should be done to understand how these factors are correlated to each other. Simulation on empirical dataset Background
Here we assume that a is a new type of blackmail computer virus spreading on the Internet(e. g. the recent virus calledWannacry,). The 'R' state of A could mean that the user information on computer is completely wiped out by the virus since its user is not willing to pay the money, or it could mean that the user finally pay the money within required period of time and the computer is fully recovered The spreading process B is a kind of anti-virus approach, for instance, a software patch(n=0)or a warning delivered through e-mail or other media channels to uninfected users(0 <n< 1). The R state of B could mean that the virus' self-update has drived the approach of b useless or that the user begins to disobey and ignore the restrictions warned A and B spread on two layers of network, one is based on the router-level connection between computers, another can be based on the user-to-user connection(since security warnings and software patches can be delivered peer-to-peer or informed by college information security department). So, I assume here the Pd is partially positive, although not good, the correlated layer is till generated by mapping For brevity, the simulation is done on an Internet network based on the AS(autonomous system)level under different time delay and interaction coefficient Network description The dataset is downloaded from the 'SNAP website. Total Nodes number N= 10670: Total Edges number E=22002: The degree distribution satisfies the powerlaw with exponent y A 1.6 Simulation parameters Let AT=0.01, \2=1 and 1=0.5; correlation coefficient Pd A 0.75. Randomly chosen starter group scale: 0. 2%of N. Note: Each simulation is repeated for at least 10 times and the average is used as the final result to minimize possible errors Results
Here we assume that A is a new type of blackmail computer virus spreading on the Internet(e.g. the recent virus called 'Wannacry'). The 'R' state of A could mean that the user information on computer is completely wiped out by the virus since its user is not willing to pay the money, or it could mean that the user finally pay the money within required period of time and the computer is fully recovered. The spreading process B is a kind of anti-virus approach, for instance, a software patch( ) or a warning delivered through e-mail or other media channels to uninfected users( ). The 'R' state of B could mean that the virus' self-update has drived the approach of B useless or that the user begins to disobey and ignore the restrictions warned. A and B spread on two layers of network, one is based on the router-level connection between computers, another can be based on the user-to-user connection (since security warnings and software patches can be delivered peer-to-peer or informed by college information security department). So, I assume here the is partially positive , although not good, the correlated layer is still generated by mapping. For brevity, the simulation is done on an Internet network based on the AS(autonomous system) level under different time delay and interaction coefficient. Network description The dataset is downloaded from the 'SNAP' website. Total Nodes number ; Total Edges number ; The degree distribution satisfies the powerlaw with exponent . Simulation Parameters Let , and ; correlation coefficient . Randomly chosen starter group scale: 0.2% of . Note: Each simulation is repeated for at least 10 times and the average is used as the final result to minimize possible errors. Results
1.0 +n=0 +n=0.3 0.8 +n=0.6 n=1 0.6 丰丰 0.4 0.2 0.0 △t/T Figure 4 When n= l, which means that epidemic A spreads independently, the final infection scale is approximately 0. 54. It is showed that increasing n can only make significant difference to the final scale of epidemic only when the intervention is timely. The curves marked by scattered points tend to gradually coincide with each other, showing that the intervention of B and its countermeasure has little use to suppress A Discussion and further works 1. Other interaction models and multiple layers. In this passage, only the case of one-sided immunity is considered More inspection needs to be made to understand the mixed interaction between multiple processes, both the case of immunity and infection facilitation. Also, what will the situations in three or much more networks layers be? This universial problem is left to be een 2. Better ways to generate correlated network layers. In this passage I assume that two orrelated layers have exactly the same set of nodes and topological properties Different orrelation coefficient is reached simply by mapping one layer's nodeset to another. Although it makes the analysis easier and straightforward, many other factors are neglected. For example, the nodeset of A is not necessarily fully covered by B. To describe partial cover, some other parameters can be defined, such as the ratio of covered nodes against uncovered ones or the average shortest path length for an uncovered node to reach a covered one. The study of cascading behaviour may provide more inspiration to this topic Also, topological properties like density, heterogenity or community patterns may be distinct between layers. Therefore, we would need a model that can take all these factors into account when generating correlated network layers. For some studies in specific areas, efforts should also be made to collect empirical data of correlated networks
Figure 4 When , which means that epidemic A spreads independently, the final infection scale is approximately 0.54. It is showed that increasing can only make significant difference to the final scale of epidemic only when the intervention is timely. The curves marked by scattered points tend to gradually coincide with each other, showing that the intervention of B and its countermeasure has little use to suppress A. Discussion and further works 1. Other interaction models and multiple layers. In this passage, only the case of one-sided immunity is considered. More inspection needs to be made to understand the mixed interaction between multiple processes, both the case of immunity and infection facilitation. Also, what will the situations in three or much more networks layers be? This universial problem is left to be seen. 2. Better ways to generate correlated network layers. In this passage I assume that two correlated layers have exactly the same set of nodes and topological properties. Different correlation coefficient is reached simply by mapping one layer's nodeset to another. Although it makes the analysis easier and straightforward, many other factors are neglected. For example, the nodeset of A is not necessarily fully covered by B. To describe partial cover, some other parameters can be defined, such as the ratio of covered nodes against uncovered ones or the average shortest path length for an uncovered node to reach a covered one. The study of cascading behaviour may provide more inspiration to this topic. Also, topological properties like density, heterogenity or community patterns may be distinct between layers. Therefore, we would need a model that can take all these factors into account when generating correlated network layers. For some studies in specific areas, efforts should also be made to collect empirical data of correlated networks
