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alternative to Kleinberg’s model in which the individuals are hierarchically grouped into categories, according to their social identities [867]. These categories are grouped into super-categories, and so forth, thus creating a tree-like hierarchy of organization that deďŹ nes a social distance between any two people, given by the height of the lowest level in the tree at which the two are connected. A network is then constructed by connecting two individuals with a probability which is larger for shorter social distances, and it is assumed to decay off exponentially with longer social distances. The greedy algorithm for communicating a message to a target person speciďŹ es that, at each time step, the message passes to that neighbor of the current holder who has the shortest social distance to the target. Like in the case of Kleinberg, the ability to search and ďŹ nd speciďŹ c targets was found to depend not only on the short network distances, but also on the existence of links constructed with probabilities that decay exponentially with social distances. Rosvall et al. have proposed to characterize the ease or difďŹ culty of navigation in different networks, in terms of the information needed to locate speciďŹ c addresses [272]. Suppose that an agent on node p wants to locate through the shortest path (or if there are several, through the degenerate shortest paths) a node q somewhere else in a connected network. When starting at node p, the agent has to ďŹ nd the right exit link, in the direction towards target q. The information value in knowing one particular exit channel is log2 kp , where kp is the degree of node p. At the subsequent node j along the shortest path to the target, the number of questions to ask to ďŹ nd the right exit link is reduced to log2 (kj − 1), since the incoming link is known. This means that the total information value of knowing any one of the degenerate paths between p and q is given by ⎛ ⎞ 1 1 ⎠S(p → q) = −log2 âŽ? (7.8) kp kj − 1 {path(p,q)}
j ∈path(p,q)
where the sum runs over the set {path(p, q)} of degenerate shortest paths between p and q. The authors have investigated the search information S for a number of networks, ďŹ nding that one needs more information to orient in real than in randomized networks (with the same degree distribution of the real networks) [272]. Moreover, the difference S(l) = S(l) − Srand (l) between the average search information for nodes separated by l links in the real networks, S(l), and the same quantity in the randomized counterparts, Srand (l), is positive and increasing for large distances l. This indicates that essentially all the contribution to the excess of S with respect to Srand comes from large distances. For some real networks, as, for example the Internet at the autonomous system level, S(l) is even negative at distances shorter than a certain horizon lhorizon , which implies that these real networks are organized to optimize the search at these short distances. Thus, local communication is favored, whereas communication for l > lhorizon is disfavored. A similar behavior is also found in some modelling topologies, as for instance in structured modular networks. Models of hierarchical networks, on the other hand, do worse than random networks on all scales [868]. GuimerĂ et al. have studied the optimal network structure related with the problem of avoiding congestion effects that arise when parallel searches are performed [333]. For a single search problem the optimal network is clearly a highly polarized starlike structure. This structure is indeed very simple and efďŹ cient in terms of searchability, since the average number of steps to ďŹ nd a given node is always bounded, independently of the size of the system. However, the polarized starlike structure becomes inefďŹ cient when many search processes coexist in the network, due to the limited capacity of the central node. In Ref. [333] the case was made of a node that can receive an inďŹ nite number of incoming packets (and store the packets in a queue), but it is limited on the number of packets that can be sent within a given time interval. Studies on congestion models of network trafďŹ c [331] have highlighted that, for low values of packets creation rate the system reaches a steady state in which the total number of oating packets in the network N (t) uctuates around a ďŹ nite value. As increases, the system undergoes a continuous phase transition to a congested phase in which N(t) âˆ? t, that is, packets accumulate in the network (for more details see Section 3.2.2). In Ref. ÂŻ [333] it is analytically shown that, for low values of , the average load of the network N (t) is proportional to N d, ÂŻ where d is the average search cost(number of steps) of a packet, while when approaches c , N (t) is inversely proportional to 1 − B ∗ /(N − 1), where B ∗ is the maximum of the effective betweenness Bj , which is given by the total number of packets passing through the node j when all possible pathways are considered for all possible pairs of vertices. Therefore, in order to minimize the cost of the whole network in the case of small , the network structure should possess small path lengths (as e.g. a starlike structure). Conversely, in the case of higher , the betweenness should be minimized. This means that only two classes of networks can be considered as optimal: starlike conďŹ gurations when the number of parallel searches is small, or homogeneous conďŹ gurations when the number of parallel searches is large.