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became freely available and formed unof cial reference datasets Part of one of these public series, pAug , appears in Figure 121 From the data a time series X(k) was extracted, corresponding to the number of bytes crossing the network during intervals of width = 12 ms We denote by X (m) the series X averaged over blocks of length m, a procedure called aggregation of level m, for example, X (3) (1) = (X(1) + X(2) + X(3))/3 Using this smoothing operator, an illustration as simple as it is fundamental to scale invariance is presented in Figure 121 From top to bottom, we trace the rst 512 points of four series X(k) = X (1) (k), X (8) (k), X (64) (k) and X (512) , with which varies from = 12 ms to = 12 8 8 8 ms, or 6, 144 s Apart from a reduction of variance, fortuitously compensated for by the automated setting of the graph scale, these three series present a quasi-uniform statistical face A smoothing of this kind, across nine orders of magnitude, if carried out on short memory data, would have revealed a strong evolution towards a constant traf c rate

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10000 5000 0 8000 6000 4000 2000 0 6000 4000 2000 0 4000 2000 0 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500

Figure 121 Visual demonstration of scale invariance Each plot shows the rst 512 points, in bytes per bin, of pAug Ethernet data From top to bottom, the resolution passes from = 12 ms to = 6144 s The visual appearance of variability remains the same; no smoothing effects are seen

This kind of scale invariance takes its pure form in the canonical process called fractional Gaussian noise (FGN) Z(t) This discrete time stationary process, which satis es E[Z(t)] = 0, has a parametric correlation function with a single parameter,

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the exponent , 0 < < 1: c ( ) = Z 1 ( + 1)2 2 2 + ( 1)2 2 1 (1 )(2 ) c 2

network One solution to this problem is the election of back-up leaders that monitor the main leader and interfere if they detect malicious behavior from the leader This requires some kind of a voting scheme to ensure that a single malicious leader cannot cause a signi cant amount of harm This of course increases the overhead of the approach because IDS data and alerts need to be sent both to the primary and back-up leaders, limiting the bene ts of the hierarchical approach To resist the compromise of multiple leaders, a large number of back-up leaders is required If every node ends up playing the role of a leader or a back-up leader then the hierarchical scheme becomes very resistant but ends up behaving as the peer-to-peer approach exhibiting the same advantages and disadvantages The other challenge of the hierarchical scheme is that, due to mobility, connectivity changes, failures, and so on, it may be challenging to maintain the hierarchy and ensure correct operation of the IDS hierarchy For example, it may be necessary in several instances for a new leader to be elected for a neighborhood either to increase ef ciency or just because a leader has moved out of range The result is that, for a brief period of time, a neighborhood may operate without a leader, making the IDS unavailable in that neighborhood This may introduce a window of vulnerability that a malicious user may attempt to exploit to launch an attack Another related problem with the hierarchical approach is that, as the intrusion detection data and alerts ow through a potentially changing hierarchy, it may take a signi cant amount of time for the hierarchy to correlate the data and detect an attack This results in increasing the detection latency of the architecture (especially compared with the peer-to-peer approach) 543 Key Novel Concepts for Cooperative Intrusion Detection in MANET.

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(121)

where = 1, 2, 3, and the asymptotic relation is valid for large The function c ( ) is a xed point of the aggregation operator, that is, each of the Z (m) have Z this same correlation function This perfect second order invariance is closely related to the power law decrease of c ( ), a decrease so slow that the correlation sum Z =1 cZ ( ) is in nite Moreover, if a process is not a fractional Gaussian noise, but has an asymptotic decrease of the same form: cX (k) c k , 0< <1 (122)

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