Real data show that interdependent networks usually involve intersimilarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent [Parshani et al. Europhys. Lett. 92, 68002 (2010)]. For example, the coupled worldwide port network and the global airport network are intersimilar since many pairs of linked nodes (neighboring cities), by direct flights and direct shipping lines, exist in both networks. Nodes in both networks in the same city are regarded as interdependent. If two neighboring nodes in one network depend on neighboring nodes in the other network, we call these links common links. The fraction of common links in the system is a measure of intersimilarity. Previous simulation results of Parshani et al. suggest that intersimilarity has considerable effects on reducing the cascading failures; however, a theoretical understanding of this effect on the cascading process is currently missing. Here we map the cascading process with intersimilarity to a percolation of networks composed of components of common links and noncommon links. This transforms the percolation of intersimilar system to a regular percolation on a series of subnetworks, which can be solved analytically. We apply our analysis to the case where the network of common links is an Erdős-Rényi (ER) network with the average degree K, and the two networks of noncommon links are also ER networks. We show for a fully coupled pair of ER networks, that for any K≥0, although the cascade is reduced with increasing K, the phase transition is still discontinuous. Our analysis can be generalized to any kind of interdependent random network systems.

Schools of fish and flocks of birds can move together in synchrony and decide on new directions of movement in a seamless way. This is possible because group members constantly share directional information with their neighbors. Although detecting the directionality of other group members is known to be important to maintain cohesion, it is not clear how many neighbors each individual can simultaneously track and pay attention to, and what the spatial distribution of these influential neighbors is. Here, we address these questions on shoals of Hemigrammus rhodostomus, a species of fish exhibiting strong schooling behavior. We adopt a data-driven analysis technique based on the study of short-term directional correlations to identify which neighbors have the strongest influence over the participation of an individual in a collective U-turn event. We find that fish mainly react to one or two neighbors at a time. Moreover, we find no correlation between the distance rank of a neighbor and its likelihood to be influential. We interpret our results in terms of fish allocating sequential and selective attention to their neighbors.

Through tuning the length of flexible bis(triazole) ligands and different metal ion coordination geometries, four Wells-Dawson polyoxoanion-based hybrid compounds, [Cu 6(btp) 3(P 2W 18O 62)].3H 2O ( 1) (btp = 1,3-bis(1,2,4-triazol-1-y1)propane), [Cu 6(btb) 3((P 2W 18O 62)].2H 2O ( 2), [Cu 3(btb) 6(P 2W 18O 62)].6H 2O (btb = 1,4-bis(1,2,4-triazol-1-y1)butane) ( 3), and [Cu 3(btx) 5.5((P 2W 18O 62)].4H 2O (btx = 1,6-bis(1,2,4-triazol-1-y1)hexane) ( 4), were synthesized and structurally characterized. In compound 1, the metal-organic motif exhibits a ladder-like chain, which is further fused by the ennead-dentate [P 2W 18O 62] (6-) anions to construct a 3D structure. In compound 2, the metal-organic motif exhibits an interesting Cu-btb grid layer, and the ennead-dentate polyoxoanions are sandwiched by two Cu-btb layers to construct a 3D structure. Compound 3 exhibits a (4 (2).6 (2).8 (2)) 3D Cu-btb framework with square and hexagonal channels arranged alternately. The hexa-dentate polyoxoanions incorporate only into the hexagonal channels. In compound 4, there exist two sets of (6 (1).10 (2)) 2(6 (1).8 (2).10 (3)) 3D Cu-btx frameworks to generate a 2-fold interpenetrated structure into which the penta-dentate polyoxoanions are inserted to construct a 3D structure. The structural analyses reveal that the length of flexible bis(triazole) ligands and metal ion coordination geometries have a synergic influence on the structures of this series. To our knowledge, they have the highest connectivity for the Wells-Dawson polyoxometalate coordination polymers to date.

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