3.5 Evolutionary Framework for the Identification of Influential Nodes
3.5.4 Optimization of the critical threshold
We further study (APe) and (APf). Similar to Fig. 3.23, the performance regarding those four pruning strategies are exhibited in Fig. 3.25. Compared to PruOrd, PruOrdq usesξg(·) =qc instead ofFon the slice where critical threshold is, otherwise, still considers ξg(·) = F. Others take the same change except for PruRangq. Since we are trying to optimize the thresholdqc, the condition a(Tp)>b1(Tp)of Eq. (3.37) would become useless. Hence, it is removed in PruRangq, that is, PruRangq chooses alla(Tp)following some probability. In addition, we conduct Eq. (3.38) in the below way
b1(Tp) =qcn+b1(0)−α3Tpandb2(Tp) = b1(Tp)
b2(0)αTp (3.39) to ensure that a(Tp)has more chances choosing a value larger thanqc, whereα3 > 1. As illustrated in Fig. 3.23 where b1(0) = 0.1n and α3 = (b1(0)−100)/Tˆp are conducted for PruRangq, all those four strategies share similar performance to ARRSq, which means that they do not really work, at least when they are initialized with those shallow methods.
But surprisingly, ABetS guides them to have much better results than the one of ARRSq.
Meanwhile, the optimal configurations regardingα1andα2are accordingly 0 and 1.
3.5.4.1 Effects of the average order parameter on the critical threshold
Now we further study the influence ofF onqc. Results from Table 3.3 tell us that the optimization ofF truly conduces to the acquirement of really goodqc. Meanwhile, we also learn that ARRSq has a smallerqc in most networks than ARRS from Fig. A.21. Moreover, ABetS would lock ARRSq (see Fig. 3.22d) and AHubS results in the bestqc. Besides, even RanS leads to a better result than ABetS. Therefore, we naturally ask what would happen regardingqcif we change the way to controlF.
A straightforward try is to ignore F, i.e., only the slice which qc belongs to is dealt with. Since having to ensure that every node has the possibility to be checked, we here mainly consider an variant of PruRangq, say PruRangqv1, that is, chooset1and t′1 following
3.5 Evolutionary Framework for the Identification of Influential Nodes
RanS HubS
AHubS APagS
ACIS ABe
tS qc
ARRSq PruOrdq(0,0) PruOrdq(1,0) PruOrdq(0,1) PruOrdq(1,1)
(a)
RanS HubS
AHubS APa
gS ACIS ABe
tS qc
ARRSq PruGriq(0,0) PruGriq(1,0) PruGriq(0,1) PruGriq(1,1)
(b)
RanS HubS
AHubS APa
gS ACIS ABe
tS qc
ARRSq PruRanq(0,0) PruRanq(1,0) PruRanq(0,1) PruRanq(1,1)
(c)
RanS HubS
AHubS APa
gS ACIS ABe
tS qc
ARRSq PruRangq(0,0) PruRangq(1,0) PruRangq(0,1) PruRangq(1,1)
(d)
Figure 3.25:Performance of PruOrdq, PruGriq, PruRanq, and PruRangq regardingqcof different initial sequences compared to ARRSq and ABetS (dashed lines). (a) PruOrdq. (b) PruGriq. (c) PruRanq with uniformly random selection of t1. (d) PruRangq considering Eq. (3.38). The difference ofqcbetween every two ticks is 0.002. Each result is the mean of 20 IIs.
t1< qc<t′1, andt1= a(Tp)(see Eqs. (3.37) and (3.38)) holding b1(Tp) =qcnandb2(Tp) = n
b2(0)Tp. (3.40)
Apparently, b2(Tp) decreases as Tp increases (see Fig. A.22 in Appendix A.2.14), which indicates that a(Tp)has higher probability to have a value closed to qc with the rise ofTp (based on the assumption that the sequence becomes more and more orderly). Besides, one can tuneb2(0)>0 to control the convergence rate of a(Tp). Since the whole sequence is likely to be considered in the beginning, PruRangqv1 with α1 = 0 always has better performance than the one withα1 =1. But still, this strategy with high probability has a local optimal solution because it could not effectively mix the whole sequence (i.e., the tail and head might not be visited often). Nevertheless, PruRangqv1 converges very fast and could obtain a better result than most state-of-the-art methods (those in Section 3.3) using much less time.
Another way of ignoringFis to disturb those groups whichqcdoes not belong to. Recall that the local goal functionξ(·)is designed to minimizeGp(q)(see Eq. (3.25)). Hence, the disorder of those groups is in a way equivalent to ignoringF. To achieve that, we only need
to slightly modify PruRangq, that is, randomly permute a slice whereqc does not locate, which we call PruRangqv2.
The third way is loosingFon those groups whichqcdoes not belong to. From the design of ARRS we know that a smallns cannot further optimize a sequence locally regardingF.
Therefore, ifnsis fixed at a small value on all groups excluding the one thatqc belongs to, then somehow this strategy could benefit from both order and disorder ofF, and the order ofF is weaker than PruRangq, which we name PruRangqv3.
RanS HubS
AHubS APa
gS ACIS ABe
tS qc
PruGriq PruRangqv1 PruRangqv2 PruRangqv3 PruRangqv4
(a)
RanS HubS
AHubS APa
gS ACIS ABe
tS qc
PruGriq PruRangqv4 PruRangqv5 PruGriqv4 PruGriqv5
(b)
Figure 3.26: Performance of PruRangqv1, PruRangqv2, PruRangqv3, PruRangqv4, PruRangqv5, PruGriqv4 and PruGriqv5 regardingqcof different initial sequences compared to PruGriq and ABetS (dashed lines). The difference ofqcbetween every two ticks is 0.002. Each result is the mean of 20 IIs.
The forth way still focuses on Fof those groups thatqcdoes not belong to but chooses a group with some probability relying onF. Considering ARRS on a sliceSp(t1,t′1)thatqc does not locate, one can accept a new slice with a probability following either
Sp(T−1)
Sp(T−1) +Sp(T) or
Sp(T)
Sp(T−1) +Sp(T), (3.41) and we accordingly call them PruRangqv4 and PruRangqv5. Obviously, PruRangqv4 would be more likely to choose a new slice if it has smallerFthan the old one, while PruRangqv5 would reject it with a higher probability.
The corresponding results with respect to these strategies are shown in Fig. 3.26, in which the early termination is conducted, and PruGriqv4 and PruGriqv5 are two strategies based on PruGriq and Eq. (3.41), respectively. Besides, to verify them, PruGriq is chosen as a comparison, which accounts for the best on average in Fig. 3.25. Obviously, all those strategies based on PruRangq heavily rely on the initial sequence (comparing RanS and others). Specifically, in Fig. 3.26a, only PruRangqv4 has better performance than PruGriq, which indicates thatF truly has influence onqc. Meanwhile, from Fig. 3.26b, we learn that both PruRangq and PruGriq based on Eq. (3.41) are more effective than PruGriq, which means that the disorder ofF also has an impact onqc. It is worth noting that PruGriqv5 is better than PruGriqv4 if the initial sequence is based on HubS, which is very important because HubS is much easier to compute compared to APagS, ACIS, ABetS, and many others.
This is also the reason that we choose PruGriqv5 in [L2].
3.5 Evolutionary Framework for the Identification of Influential Nodes
3.5.4.2 Mutation operators
The influence of disorder of F on qc motivates us to further introduce the mutation operators from the genetic algorithm. We conduct them on bothSpandS, i.e., local and global mutations. In detail, both of them, at each time, equally choose one with some probabilities from the following six mutation operators to produce the corresponding sequence:
i) the displacement mutation (DM) operator [103] usually randomly selects a fragment that would be moved from the sequence and eventually inserted in a random place;
ii) the exchange mutation (EM) operator [104] aims at choosing two nodes in the sequence at random and then exchanging them (a similar strategy could be found in ref. [5]);
iii) as for the insertion mutation (ISM) operator [103, 105], one random node is moved out the sequence and placed at a random position afterwards;
iv) the simple inversion mutation (SIM) operator [106] selects randomly two cut points in the sequence, and then reverses the fragment between these two cut points;
v) on the basis of SIM, we slightly change it by narrowing the cut points (S-SIM). Namely, the random selection happens in a narrow range;
vi) the inversion mutation (IVM) [107] operator works similarly to the DM. It also randomly selects a fragment, removes it from the sequence, and then inserts it in a randomly selected position, however, in the reversed order.
The corresponding performance can be found in Fig. A.23 in Appendix A.2.14.