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Spatial Social Networks Neighborhoods are usually formed on the basis of particle indices That is, assuming a ring social network, the immediate neighbors of a particle with index i are particles with indices (i 1 mod ns ) and (i 1 mod ns ), where ns is the total number of particles in the swarm Suganthan proposed that neighborhoods be formed on the basis of the Euclidean distance between particles [820] For neighborhoods of size nN , the neighborhood of particle i is de ned to consist of the nN particles closest to particle i Algorithm 164 summarizes the spatial neighborhood selection process Calculation of spatial neighborhoods require that the Euclidean distance between all particles be calculated at each iteration, which signi cantly increases the computational complexity of the search algorithm If nt iterations of the algorithm is executed, the spatial neighborhood calculation adds a O(nt n2 ) computational cost Determins ing neighborhoods based on distances has the advantage that neighborhoods change dynamically with each iteration

Figure 11 Economic development: Web 10 (distribute), Web 20 (participate), Web 30 (contribute), and Web 40 (transform) (used with permission from Marcus Cake, http:// wwwmarcuscakecom/key-concepts/internet-evolution [6])

Fitness-Based Spatial Neighborhoods Braendler and Hendtlass [81] proposed a variation on the spatial neighborhoods implemented by Suganthan, where particles move towards neighboring particles that have found a good solution Assuming a minimization problem, the neighborhood of

ball team through his company The market price for the tickets is $2,500, but he only pays his employer $750 for them The difference of $1,750 is considered income to Mr Harvest, and should be reported as such to the IRS

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Algorithm 164 Calculation of Spatial Neighborhoods; Ni is the neighborhood of particle i Calculate the Euclidean distance E(xi1 , xi2 ), i1 , i2 = 1, , ns ; S = {i : i = 1, , ns }; for i = 1, , ns do S = S; for i = 1, , nNi do Ni = Ni {xi : E(xi , xi ) = min{E(xi , xi ), xi S = S \ {xi }; end end

S };

.

Club dues are taxable income to the employee, except for that portion of the dues that are business related, which must be substantiated Clubs that fall into this category are airline and hotel clubs, as well as golf, athletic, and country clubs The portion of the club dues that are personal income to the employee can be treated as a wage expense to the employer for tax purposes

particle i is de ned as the nN particles with the smallest value of E(xi , xi ) f (xi ) (1648)

where E(xi , xi ) is the Euclidean distance between particles i and i , and f (xi ) is the tness of particle i Note that this neighborhood calculation mechanism also allows for overlapping neighborhoods Based on this scheme of determining neighborhoods, the standard lbest PSO veloc ity equation (refer to equation (166)) is used, but with yi the neighborhood-best determined using equation (1648)

Example: Brad Harvest is a member of an airline club, which allows him ac-

Growing Neighborhoods As discussed in Section 162, social networks with a low interconnection converge slower, which allows larger parts of the search space to be explored Convergence of the fully interconnected star topology is faster, but at the cost of neglecting parts of the search space To combine the advantages of better exploration by neighborhood structures and the faster convergence of highly connected networks, Suganthan combined the two approaches [820] The search is initialized with an lbest PSO with nN = 2 (ie with the smallest neighborhoods) The neighborhood sizes are then increased with increase in iteration until each neighborhood contains the entire swarm (ie nN = ns ) Growing neighborhoods are obtained by adding particle position xi2 (t) to the neighborhood of particle position xi1 (t) if ||xi1 (t) xi2 (t)||2 < dmax where dmax is the largest distance between any two particles, and = 3t + 06nt nt (1650) (1649)

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