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Usually V is the set of all nodes of G or the set of nodes in a given set D of traf c demands Graph R is known as the logical topology and a connection request in R is called a logical link of the network The routing and wavelength assignment (RWA) problem: Given a set R of connection requests in a WDM network G, nd a routing path in G for every (u, v) R and assign each path a wavelength such that the paths with the same wavelength do not share any common link in G The routing path with an assigned wavelength is called a light path.

where c represents the activity coefficients of H+ and A and the activities of electrically neutral species are assumed equal to unity. In the first step, solvent molecules S act as a Lewis base (electron pair donor) to the H atom and as a Lewis acid (electron

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The traf c grooming problem: Given a set D of traf c demands and a set of light paths in a WDM network G, multiplex the traf c demands of D into the light paths subject to the bandwidth constraint of the light paths The multiplexing is realized by a device called add-drop multiplexer (ADM) at nodes of G A general goal in the study of the above problems is to determine the resources required to achieve a given connectivity as a function of network size and functionality of network nodes Much of the discussions for the goal is on minimizing the number of wavelength channels and the number of ADMs for realizing a given communication application These optimization problems have been extensively studied in both communication and graph algorithms communities for WDM networks [1,2,9,15,38,40,41].

M(Q 1)/N 1 = 06 This is called the probability that two nodes share a common key [28] Further, it can be checked that if two nodes do not share a common key, then they may communicate via another intermediate node Let nodes i , j do not share a common key, but i , k share a common key and k , j share a common key, i, j, k are all distinct Hence, the secret communication between i and k needs a key (encrypted by i , decrypted by k ) and that between k and j needs another secret key (encrypted by k , decrypted by j ) It has been shown that the communication between two nodes is possible in almost 099995 proportion of cases [28] However, the following problems are immediate: 1.

= Var(L (p n P )) Corr(L (p n P ),P )2 Var(L (p n P )) = 1 Corr(L (p n P ),P )2 SD(L (p n P )) = 1 Corr(R,P )2 SD(R )

Since the quality of the solutions for these optimization problems is critical for the performance of WDM networks, it is extremely important to have ef cient algorithms with good guaranteed performance for those problems However, it is challenging to develop such algorithms in most cases For example, the RWA problem is NP-hard for even very simple networks like rings and trees [9,19] Inapproximability results are known for the RWA problem in networks with more complex topologies when the routing paths are given [32] Similar hardness results are known for the other optimization problems as well For those problems on simple networks like rings and trees, many ef cient algorithms with good guaranteed performance have been known But for the problems on more complex networks, no effective approach has been developed and the existing algorithms are based on the integer linear programming or ad hoc heuristics.

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The performance of those algorithms are not guaranteed Readers may refer to other works on the subject [1,32,38,40,41] for more details on the minimization problems on complex networks In this chapter,.

Communication between any two nodes in 60 Percent of the cases will be in one step (no involvement of any other node), but the communication between any two of them needs two steps for the rest 40 Percent of the cases, making the average of 14 steps in each communication This is an overhead Thus, we need a design where we can guarantee that there is a common key between any two nodes 2 The direct communication between any two nodes can be decrypted by at most Q 2 other nodes However, if one takes the help of a third intermediate node, then the communication can be decrypted by at most 2(Q 2) nodes Thus, any communication can be decrypted by at most 14(Q 2) nodes on an average 3.

Table 3.3 Comparison of opinion leaders and innovators Acceptor Active Passive Opinion leader Innovator Rejector Opinion leader Innovator

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