network robustness. More precisely, it has F′/F′′ = 6.08 and 3.39 for the Email-Enron network and the loc-Gowalla network. Though the improvements are not significant as the one that WayEnhS has, WayEnhSr is much faster than WayEnhS, e.g., it could acquire an enhanced network within 1 minutes in our simulation environment regarding the loc-Gowalla network.
4.2.3 Summary
As we have discussed, the true robustness of a network is detected by the most advanced attack strategy which, however, is usually too time-consuming to be a criterion for the cut-add strategy. To overcome this problem, we firstly study which naive method could be an alternative of the advanced one and reach ARRSe (our basic method ARRS with early stop), which could effectively check the critical point during the enhancement process based on HubS. We then further investigate that process and acquire our first method, WayEnhS, which has a better performance and similar time consumption against the one based on HubS. But WayEnhS is still hardly employed to tackle large networks, since it has to calculate F(S)pert. Hence, we further investigate the effects of assortativity on WayEnhS, an attribute that highly correlates with the network robustness and could be controlled locally. Such trick not only helps us improve the performance of WayEnhS again but also gives us the second method, WayEnhSr, which is totally local. To demonstrate the effectiveness of both WayEnhS and WayEnhSr, four real-world networks are considered. The results regarding attacks from over ten state-of-the-art strategies illustrate that both of them have the ability to greatly improve the robustness of a network. Nevertheless, both WayEnhS and WayEnhSr are indirect methods to increase the bound of the robustness against the most advanced attack strategy, e.g., ARRSe withT =103 is more effective than the one withT=10, since the first has larger correlation with EvolF. Thus, this problem is still open and more studies are needed, including itself of the most advanced attack strategy and its more reliable representative.
4.3 Influences of Acquaintances on the Containment of Epidemics
In this part, rather than consider the case in Section 3.1.2, i.e., the influence of the degree distribution on the epidemic spreading, we mainly focus on existing networks and study effects of the change of the corresponding structure on the spreading patterns under immunization. In particular, we will see how we tune the network structure to make the immunization strategy work more effectively.
4.3.1 Ways to weaken the robustness of a network
From Section 3, we know that the immunization problem and the robustness problem in a degree are equivalent to each other. Therefore7, for the case of keeping the degree sequence fixed, we could actually directly use the strategies that we introduced in Section
7One potential application is that, e.g., one can temporarily reconstruct the contact network of a company or institute, so that the immunization of such network could be achieved by a small amount of staffs instead of all working from home during a crisis such as a pandemic.
4.2 but in a reverse process, i.e., ξ−(g)instead of ξ+(g). In other words, we replace step 4) in Section 4.2 with: letG← G′ andS ← S′ if F′(S′)< F(S), where F′(S′)and F(S)are associated withG′ and G, respectively. To distinguish from WayEnhS and WayEnhSr, we mark their corresponding inverse processes with ‘i’, namely, WayEnhSi and WayEnhSri.
Table 4.1 shows the associated results. As we can see from there, ARRSe only works in the power grid network (S2, S3, and S5), which indicates that WayEnhSi is not suitable for this problem. By contrast, ξ−(g) based on HubS (S1) works well. But again, it would suffer a problem of the time consumption. As an alternative,ξ−(r)(S8) has comparable results against S1 in the yeast network and the global airline network, but it is much worse than S1 in the power grid network and the as-733 network. Indeed, one could study more regarding this problem following a routine similar to Section 4.2. But here we are more interested in another problem.
Conf. r Random HubS ARRSe Power Yeast as-733 Airline
S1 ✓ 0.0123 0.0315 0.0020 0.0206
S2 ✓ 0.0084 0.0754 0.0088 0.0363
S3 ✓ ✓ 0.0084 0.0772 0.0089 0.0366
S4 ✓ ✓ 0.0259 0.0325 0.0072 0.0216
S5 ✓ ✓ 0.0082 0.0762 0.0089 0.0359
S6 0.0600 0.1617 0.0107 0.1174
S7 ✓ 0.0751 0.0940 0.0148 0.0618
S8 ✓ 0.0638 0.0328 0.0098 0.0218
Table 4.1: Performances of different configurations (Conf.) regarding WayEnhSi, WayEnhSri, and other strategies on the power grid network (Power), the yeast network (Yeast), the as-733 network (as-733), and the global airline network (Airline). S1 isξ−(g)withg=F(HubS). S2 is WayEnhSi(1). S3 is WayEnhSi(Random). S4 isξ−(g)withg=F(HubS)and only pairs leading to decreases ofrare considered. S5 follows the same constrain fromras S4 but with ARRSe criteria. S6 is randomly rewiring the network a number of times. S7 is WayEnhSri (Random).
And S8 isξ−(g)withg=r. The numbers are the related results, i.e.,Fof ARRS (withT=103) on networks drawn based on each configuration.
Supposing that a number of edges, saymr =|Mu|, are allowed to be removed from a network, we ask how this removal would influence the effectiveness of an immunization strategy. Specifically, considering a networkG(N,M), we are interested in the following problems:
i) ifmris given, then tackle
arg min
Mu
F′(·), (4.6)
whereF′(·)corresponds to the remaining networkG′ =G(N,M \ Mu); ii) if a certain F′(·)needs to be achieved, then tackle
minmr; (4.7)
iii) a bi-objective problem regarding both Eqs. (4.6) and (4.7).
4.3 Influences of Acquaintances on the Containment of Epidemics
In what follows, we particularly focus on HubS as a case study, and others could be investigated in the similar way8.
From the results in Section 4.2 and Table 4.1, we know that the assortativityr truly has effects on F(HubS). In tandem with the fact that a more practicable strategy should usually be local, an edgeeij in our framework is removed if it has
Prodi1: maximumki×kj, Prodi2: minimumki×kj,
Subti1: minimum|ki−kj|, or Subti2: maximum|ki−kj|
amongarcandidates. Indeed, it again looks similar to the explosive percolation problem. Fig.
A.25 (Appendix A.3.1) illustrates the associated results. As we can see from there, Prodi1 only works on the power grid network, where it is also better than Prodi2. But on other networks, Prodi2 surpasses Prodi1 and holds the best on two of them. Subti2 is slightly worse than Prodi2 on the yeast network and the global airline network, but it works on all four networks. Note that Prodi2 considers both the degree itself and the difference of corresponding nodes while Subti2 only captures the difference.
4.3.2 The role of less connected acquaintance
Based on those results, now we could further study the following scenario9: assuming that you are asked to temporarily stop contacting one or a few of your acquaintances (including all people whom you have connections with), who of them would contribute more to the protection of the whole network? Since aiming to design a local strategy, the below part would consider the degree of each node from the original network rather than keep updating as the removal of edges (e.g., the case in Section 4.3.1). Specifically, for a given network G(N,M), the following processes are conducted: i) randomly pick up a node i fromG; ii) arrandom selections (might account for the irrational decision) are conducted on its corresponding nearest neighbors; iii) choose the node jfrom the arcandidates and then removeeij based on Prodi1, Prodi2, Subti1, or Subti2; iv) repeat i)-iii) a number of times.
As illustrated in Fig. A.26 (Appendix A.3.1), still only Prodi2 and Subti2 work. It is worth mentioning that Prodi2 is simply equivalent to choosing the node with the smallest degree10 among the ar candidates, which greatly facilitates the possibility of its implementation.
Hence, in what follows, we would mainly consider and verify Prodi2.
8All three problems are actually NP-hard.
9In general, an epidemic would die out if we effectively control the basic reproductive numberη0(see also Eq.
(3.10)), such asηiacould be curbed by social distancing, face masks, or / and hands hygiene etc. Indeed, when the spread is mild, the Test-Track-Treat (test communities for diagnosis, track contacts of infection, and treat by the quarantine of those cases) strategy might be the most efficient approach. But when the situation becomes severe, that strategy would only have limited effect and a national restriction might be needed. Hence, here we discuss a local approach could play potential role in such as the suppression of a severe spread.
10This could be achieved by the aid of machine learning methods, such as train a model correlated with node degrees based on information like age, career, etc. Here we consider the case that the degrees of neighbors are inferred by the focused node.
4.3.3 Applications
Similar to Sections 3.4.5.5 and 3.5.7.4, we still employ the SIR model and follow the same settings (i.e.,ηr =0.05 andb=105) to demonstrate the effectiveness of Prodi2. Specifically, we conduct the verification through the below processes: i) based on Prodi2, acquire a new networkG′ by removingmr edges from the given networkG; ii) immunizeG′ using RanS, AcqI, or HubS, and then independently run the SIR model on the associatedG′. The sequence with respect to AcqI is obtained through: i) for a nodeiinG′, randomly target one of its nearest neighbors; ii) check every node ofG′ and then score each node based on the number of targeted times; iii) independently repeat ii) 20 times and sum up their corresponding scores; iv) acquire the sequence based on the related score.
ar 0.5n 1.0n 1.5n 2.0n 2.5n 3.0n
1 0.3546 0.2382 0.1563 0.0903 0.0461 0.0231 2 0.3570 0.2445 0.1555 0.0888 0.0453 0.0208 8 0.3533 0.2357 0.1484 0.0795 0.0380 0.0161
Table 4.2:The mean of the average infected frequency⟨αinf⟩,⟨⟨αinf⟩⟩, over the infected probability ηi ∈ [0.01, 0.20](with interval 0.01) and the immunized fraction q∈ [0.01, 0.16] (with interval 0.0075) of RanS, in regard to networks drawn at mr = 0.5n, 1.0n, ..., 3.0n through Prodi2 with ar =1, 2 and 8 on the Email-Enron network.
ar 0.5n 1.0n 1.5n 2.0n 2.5n 3.0n
1 0.0459 0.0342 0.0225 0.0124 0.0057 0.0025 2 0.0428 0.0304 0.0178 0.0087 0.0038 0.0014 8 0.0394 0.0241 0.0121 0.0051 0.0018 0.0005
Table 4.3:⟨⟨αinf⟩⟩of Prodi2 regarding AcqI on the Email-Enron network.
ar 0.5n 1.0n 1.5n 2.0n 2.5n 3.0n
1 0.0457 0.0341 0.0225 0.0125 0.0060 0.0026 2 0.0434 0.0304 0.0185 0.0094 0.0041 0.0015 8 0.0398 0.0248 0.0131 0.0057 0.0021 0.0005
Table 4.4:⟨⟨αinf⟩⟩of Prodi2 regarding HubS on the Email-Enron network.
Tables 4.2, 4.3, and 4.4 exhibit the associated results on the Email-Enron network, where the mean of the average infected frequency⟨αinf⟩(see also Section 3.4.5.5 for its definition) are given, say⟨⟨αinf⟩⟩. As we can see from there, for all three cases (i.e., networks accordingly under the immunization of RanS, AcqI, and HubS), Prodi2 withar=8 is much better than the one that randomly removes edges (namely, Prodi2 with ar = 1), especially when mr becomes large, e.g., the fraction of ⟨⟨αinf⟩⟩ of ar = 8 and the one of ar = 1 is less 0.5 for mr=2.0nregarding AcqI and HubS (Tables 4.3 and 4.4). Besides, Prodi2 withar=2 is also much better than the random one for most cases. Meanwhile, for a specificar,⟨⟨αinf⟩⟩would hugely decrease asmrincreases, even though each node is only asked to collapse one edge with its neighbors on average.
We then consider the contours of ⟨αinf⟩ as a function of qand ηi atmr = 2.0non the Email-Enron network and the loc-Gowalla network. As depicted in Figs. A.27, A.28, and A.29 (Appendix A.3.1), on both networks, Prodi2 with largerarcould achieve better results,