Efficient network immunization under limited knowledge

Abstract Targeted immunization of centralized nodes in large-scale networks has attracted significant attention. However, in real-world scenarios, knowledge and observations of the network may be limited, thereby precluding a full assessment of the optimal nodes to immunize (or quarantine) in order to avoid epidemic spreading such as that of the current coronavirus disease (COVID-19) epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central among these n nodes is immunized. This process can globally immunize a network. We find that even for small n (≈10) there is significant improvement in the immunization (quarantine), which is very close to the levels of immunization with full knowledge. We develop an analytical framework for our method and determine the critical percolation threshold pc and the size of the giant component P∞ for networks with arbitrary degree distributions P(k). In the limit of n → ∞ we recover prior work on targeted immunization, whereas for n = 1 we recover the known case of random immunization. Between these two extremes, we observe that, as n increases, pc increases quickly towards its optimal value under targeted immunization with complete information. In particular, we find a new general scaling relationship between |pc(∞) − pc(n)| and n as |pc(∞) − pc(n)| ∼ n−1exp(−αn). For scale-free (SF) networks, where P(k) ∼ k−γ, 2 < γ < 3, we find that pc has a transition from zero to nonzero when n increases from n = 1 to O(log N) (where N is the size of the network). Thus, for SF networks, having knowledge of  ≈log N nodes and immunizing the most optimal among them can dramatically reduce epidemic spreading. We also demonstrate our limited knowledge immunization strategy on several real-world networks and confirm that in these real networks, pc increases significantly even for small n.

Early studies in networks found that immunizing real networks against an epidemic is highly challenging due to the existence of hubs which prevent eradication of the virus even if many nodes are immunized [29][30][31].At the same time, if these hub nodes, largest degree nodes, are targeted, the network can reach immunity [29,31,32].However, these previous models of targeted immunization have assumed full knowledge of the network structure which in many cases is not available.Recent research has shown that even those in control of a network often have knowledge only of a small part of the whole network structure [33][34][35].This was particularly demonstrated with the current COVID-19 epidemic where the detailed social network of individuals is unknown.
In this Letter, we study targeted immunization or removal in networks with limited knowledge.We assume that at each stage, a number n nodes are observed and the node with highest degree is immunized and thus unable to continue spreading infection.This procedure is repeated until 1 − p fraction of nodes are immunized.In particular, our model could apply to a situation where several cooperative teams are sent to immunize a network and each team has access to information on a small subset (n nodes) of the network.We develop a theoretical framework for this model of immunization with limited information using percolation theory for networks with arbitrary degree distribution.In the limit of n = N we recover prior work on targeted attack [32], whereas for n = 1 we recover the case of random failure [30,31].We observe excellent agreement between our theoretical and simulation results for the n dependence of the critical threshold p c and the size of the giant component P ∞ for p > p c .The giant component and p c characterize the efficiency of the immunization.The smaller is the giant component, the immunization strategy is better.The larger is p c the immunization is more efficient since less immunization doses are needed to stop the epidemics.We find an analytical relationship between n and p c for both Erdős-Rényi (ER) and Scale-Free (SF) networks.Surprisingly, we also find that p c quickly reaches a plateau even for relatively small n, after which increasing n has negligible effect on p c .This means that immunization with small n (not knowing the whole structure) can dramatically improve the immunization.
Let G(V, E) be a network where V and E are the set or removed yet will be observed.
of nodes and edges, respectively.N = |V | is the number of nodes in the network.We assume that a preventer or attacker has limited knowledge of the overall network structure and instead possesses only limited information on several nodes.Specifically, we randomly select n nodes for which the preventer or attacker is assumed to have information on the node degree.The preventer or attacker then targets the node with the highest degree among these n.This procedure is then repeated until a 1 − p fraction of nodes are immunized or removed from the network.
In Fig. 1, the limited information immunization (attack) is illustrated together with global targeted immunization on a network.Here a total of n = 3 nodes are observed.In panel (a), an individual with global information about the network structure chooses the highest degree node u to immunize (or remove).However, in panel (b), the individual only knows at a time the degree of 3 nodes in the network, i.e. v 1 , v 2 , v 3 .Consequently, node v 3 with the highest degree k = 4 (marked in red) is immunized (or removed).
Suppose the degree distribution of a network is given by P (k) and F (k) = k s=0 P (s) being the cumulative probability that the degree of a randomly chosen node is less than or equal to k.Furthermore, at an arbitrary time t during the iterative percolation process, assume the distribution of the original degree (including the immunized neighbors) of the remaining nodes is P (k, t).Then, the degree distribution of the node which is immunized at time t is given by where F (k, t) is the cumulative distribution of P (k, t).This formula can be recognized as being derived from order statistics giving the maximum of several independent random variables [36].For k = 0, Eq. ( 1) becomes P r (0, t) = F (0, t) n .Hence we define F (k − 1, t) = 0, and then Eq. ( 1) is valid for k ≥ 0.
In a limited knowledge immunization or attack, each node's immunization changes the degree distribution of the remaining nodes in the following way where N (k, t) is the number of nodes with degree k at time t, and P r (k, t) is the likelihood that a node immunized at time t has a degree k.
Then, plugging Eq. ( 1) into Eq.( 2) gives, which becomes in the continuous limit, and using P (k, t) = ∆F (k, t), we obtain, Note that F (k = −1, t) = 0, and thus the entire term inside the ∆ is 0 for k = −1.Similarly, this implies that for k = 0 and likewise for any k ≥ 0 this term is also 0. Thus, we get the following simple ordinary differential equation, with the initial condition F (k, t = 0) = F (k).It can be shown (see Sec I in SM) that the solution of this ODE, Eq. ( 3) is or equivalently, where F p (k) is the cumulative distribution of the degree after immunizing (removing) 1 − p fraction of nodes.For n = 1, the solution of Eq. ( 3) is F p (k) = F (k) as expected.Also Eq. ( 5) converges to F (k) in the limit n → 1.
We can now obtain the degree distribution of the occupied nodes after fraction 1 − p nodes are immunized, which is given by The equation for v, the probability of a randomly chosen link to lead to a node not in the giant component, is where P (Θ|k) is the probability of a node to be occupied given its degree is k.Using Bayes Theorem, we note that P (k)P (Θ|k) = P (Θ)P (k|Θ) = pP p (k). Hence Eq. ( 7) becomes The giant component S can be found by where v is found from Eq. ( 8).
At criticality, we take the derivative of both sides of Eq. ( 8) and substitute v = 1 representing the location where the first solution with v < 1 exists, as opposed to only the v = 1 solution.Thus, the critical condition is We now study our limited knowledge immunization (attack) strategy, i.e., the general result, Eqs. ( 8) and ( 9), on ER networks.First, we analyze the giant component P ∞ .For the case n = 1, limited knowledge immunization (or attack) reduces to the classical random attack, while for n → ∞ (meaning the global network is observed) corresponds to targeted attack [14,29,31,32].Using Eq. ( 9), the giant component P ∞ can be solved numerically for any given p.In Fig. 2(a), simulations and analytic results are shown for the giant component P ∞ as a function of 1 − p under limited information immunization with different n.As the knowledge index n increases from 1 to N , limited knowledge immunization moves from being like random immunization (attack) to being like targeted immunization (attack).The simulations are in excellent agreement with the theoretical results (lines).
Next, we focus on the critical threshold, p c , of limited knowledge immunization (attack).Overall, we find that one does not need a very large knowledge n ∼ 10 to achieve nearly the very close effect as targeted immunization (attack) with complete information.This can be seen by observing the critical threshold p c as a function of n in Fig. 2(b).In Fig. 2(c) we show the variation of p c with k for several fixed n.
Next, the behavior of p c in the limit of large n is derived analytically.By examining Eq. ( 5) we notice that when n → ∞ there are two distinct behaviors depending on whether k is small, F (k) < p; or k is large, F (k) > p.It can be shown (see Sec II in SM) that the leading term behaves as, where α k = |log [p/F (k)]|.In the limit n → ∞, we can get the expected result for targeted immunization (attack), F p (k) = min {F (k)/p, 1} [31,32].
Plugging Eq. ( 11) into Eq.( 10), and noting that from a sum of exponentials decaying with n only the lowest decay rate contributes to the leading term, we obtain (see Sec II in SM) where p ∞ c = p c (n → ∞), and the decay rate α is now and k slow is the degree which gives the lowest rate α.(See illustration in SM).
It is clear that k slow must be k < or k > because F (k) is monotonic.If F (k slow ) = F (k > ) = 1 then k < should be taken as k slow , and the corresponding α should be taken.It should also be noted that if k slow is not unique, it would simply change the pre-factor A in Eq. (12).Another special case is where As expected from the theory, one can see that ∆p c ∼ 1/n for small n and exponential decay for large n.When p c → 1 which occurs for ER network when k → 1, the power law regime becomes much broader as the Sec II of Next, study SF networks with is the normalization factor, and m and K are the minimum and maximum degree respectively [30].Similar to ER networks, the size of the giant component, P ∞ can be obtained from Eq. ( 9).Fig. 3(a), shows P ∞ as a function of 1 − p for different n values.The results demonstrate that SF networks become more immunized/vulnerable compared to ER networks under the immunization/attack as n increases.Compared with ER networks, one can observe that slightly higher values of n (more knowledge) are needed to reach the near-steady-state region of fully targeted strategy.  1respectively.This confirms Eq. ( 14) for γ = 2.1, and δ = (3 − γ)/2 = 0.45.
For SF networks with 2 < γ < 3, under random immunization/attack (n = 1), it has been shown that p c = 0 for an infinite system [30], while for high-degree immunization/attack (n → ∞), p c > 0 [31,32].Next we wish to find out for which n, p c becomes non-zero, and how it depends on the system size N .To this end, we analyze Eqs. ( 5) and (10) for large k (high degrees govern the behavior in SF), small n and p as follows (elaborated in SM).
It can be shown that for large degrees, Substituting this into Eq.( 5) and assuming (k/m) γ−1 n for large degrees, it can be concluded that In addition, we notice that P p (k) has a new natural cutoff, K p , which depends on p and N as follows (see Sec III in SM), 1) .
This helps us to evaluate the second moment of P p (k) where β = (3 − γ)/(γ − 1).Considering this, and plugging Eq. ( 13) into Eq.( 10), and keeping the leading terms in the limit of large N , we obtain (see Sec III in SM for more details) where From Eq. ( 14), it is easy to see that if n log N then p c → 0, while if n ∼ log N then p c is non-zero.The pre-factor C(n) depends on n but not in N .
Fig. 4(a) shows p c versus γ.It is known that for 2 < γ < 3 and n = 1, if N → ∞ then p c → 0 [30].Also for n = 5, we can see that system size matters and p c decreases as N increases.Fig. 4(b) shows that the scaling with n/ log N of Eq. ( 14) is valid.Furthermore, it is seen in Fig. 4(b) that when n is small or N is large, such that n/ log N 1 (in Fig. 4 it is 0.07), p c approaches 0. Fig. 4(c) supports the exponential scaling of p c versus n −1 log N obtained analytically in Eq. ( 14).
In summary, our results provide a framework for understanding and carrying out efficient immunization with limited knowledge.Especially in cases of global pandemics such as e.g., the current COVID-19, it is impossible to know the full interactions of all individuals.Thus an effective way to limit spreading is obtaining information on a few (n) individuals and targeting the most central of these.For example, testers could stand at a supermarket and select a group of people entering the store simultaneously.Information on the connections of these people e.g., the number of people they live with, where and how often they meet with other people etc. could be quickly obtained (such as through cell phone tracking) and then the individual with the most connections in the group could be quarantined or immunized.Our results demonstrate that even when this is done in small groups of people (low n), it is possible to obtain a significant improvement in global immunization compared to randomly selecting individuals.In our model, this was seen by the reduced size of the giant component and the large critical threshold p c .Overall, these findings could help to develop better ways for immunizing large networks and designing resilient infrastructure.

FIG. 2 .
FIG. 2. Results on ER networks.(a) The giant component P∞ of an ER network with k = 4 varies with the fraction of immunized (or removed) nodes 1 − p under limited knowledge.As the n is increased, limited knowledge immunization tends to have the same immunization effect as targeted immunization.(b) The critical threshold pc of limited knowledge immunization as a function of n on ER networks.Note that already for small n ∼ 10, pc is close to targeted immunization (global knowledge, n ∼ N ).(c) Critical threshold pc as a function the mean degree k of ER networks for limited knowledge immunization.(d) The scaling of |pc(∞) − pc| with n on ER networks.Symbols are average results of simulations over 100 independent realizations on ER networks with 10 6 nodes.All simulation results (symbols) agree well with theoretical results from Eq. (10) (dashed lines).

FIG. 3 .
FIG. 3. Results for SF networks.Comparison of theory (lines) and simulation (symbols) for limited knowledge immunization or attack, n, for SF networks.(a) The size of the giant comversus n for SF network with γ = 2.5.(b) Critical threshold pc versus n on SF networks.Simulations are obtained for a system with 10 6 nodes and averaged over 100 independent realizations.The minimum and maximum degree of network are m = 2 and K = 1000 respectively.

FIG. 4 .
FIG. 4. How pc for SF networks depends on γ and the system size N .(a) pc as a function of γ for different values of n and N .pc decreases with increasing N .For n = 1 and 2 < γ < 3, it is well-known that pc approaches zero for infinite system.(b) pc as a function of n/ log N for SF network with γ = 2.1.The enlarged figure for small n = 1, • • • , 5 are shown in the inset, and pc approaches zero for n/ log N 1.(c) The scaling of pc with N and n for large N and small n.Here C(n) is the pre-factor.The minimum and maximum degree of nodes are m = 2 and K = N 1/(γ−1) respectively.This confirms Eq. (14) for γ = 2.1, and δ = (3 − γ)/2 = 0.45.