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The nodes and links connecting them are the two main geometric elements of a complex network. If the nodes and links change over time, all the nodes and all the links are regarded the dynamic systems, respectively, and thus, a complex dynamical network (CDN) can be regarded to be composed of the two interconnected subsystems: the nodes subsystem (NS) and the links subsystem (LS). The outer synchronization happens in two CDNs, which means that the state error between the two NSs converges to zero. However, the dynamics of LS is ignored in the existing literature about the outer synchronization. In addition, the existing literature rarely consider the unknown interaction between two CDNs into the outer synchronization. In view of this, this paper investigates the outer synchronization of two CDNs with the unknown interaction. By using the dynamics of LSs and synthesizing the adaptive control scheme, the outer synchronization is asymptotically achieved. The results in this paper show that the outer synchronization happens when the two LSs can track, respectively, the synthesized auxiliary dynamic tracking targets. Finally, the numerical simulation is given to show the effectiveness of the theoretical results in this paper.

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Spatial topological relation is an important and typical multilayer spatial relation, when Apriori is used to mining spatial constraint topology association rules, it will has some repeated computing. And so this paper proposes an algorithm of spatial constraint topology association rules mining based on complement set, which is used to mining spatial multilayer transverse association rules with constraint condition from spatial database. This algorithm generates candidate frequent topological itemsets with constraint condition not only by down-top search strategy as Apriori, but also by computing complement set of candidate from down-top search strategy, which is suitable for mining any spatial topological frequent itemsets with constraint condition. This algorithm compresses a kind of spatial topological relation to form an integer. By the way, firstly, the algorithm may efficiently reduce some storage space when creating mining database. Secondly, the algorithm is fast to obtain topological relation between two spatial objects, namely, it may easily compute support of candidate frequent itemsets. Finally, the algorithm may fast generate candidate via double search strategy, i.e. one is that it connects (k+1)-candidate frequent itemsets with constraint condition of k-frequent itemsets as down-top search strategy, the other is that it computes complement set of (k+1)-candidate frequent itemsets with constraint condition. The result of experiment indicates that the algorithm is able to extract spatial multilayer transverse association rules with constraint condition from spatial database via efficient data store, and it is very efficient to extract any frequent topology association rules with constraint condition.

Since the algorithms of constraint frequent neighboring class set mining based on Apriori has some redundancy candidate constraint frequent neighboring class set and some repeated computing, so its efficiency isn’t improved. Hence, this paper proposes an algorithm of constraint frequent neighboring class set mining based on interval mapping, which may efficiently extract short constraint frequent neighboring class set from large spatial database via up search. The algorithm uses binary weights to change neighboring class set into integer, which is looked on as a spatial transaction, and it uses interval mapping to generate constraint frequent neighboring class set via up search, i.e. the algorithm creates an interval to map a range of generating candidate, up search is mapping candidate from minimum to maximum of the interval. The method is different from traditional up search or down-up search. The experimental result indicates that the algorithm is more efficient than the constraint frequent neighboring class set mining algorithm based on Apriori when mining short constraint frequent neighboring class set.