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creased. Because of the small-world effect, it may result in the triangles far away compared to the regular net- work becoming of gauge variance. If the triangles far away compared to the regular network become of gauge variance, it will block the spread. So we expect the size of avalanche to decrease, and the exponent of the power-law distribution to decrease accordingly. Based on these anal- ysis, we set the edge adding probability q of the small- world from q = 0 . 05 to q = 0 . 3, and observe the exponent of the size distribution decrease from -1.91 to -2.79, as given in Fig. 9 (a) and Fig. 9 (b) respectively. 

As is well known, for the small-world network[15, 16], 

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when the edge adding probability q is big enough, the structure of the network becomes random. We note the diagrams are curved in Fig. 9 (a) and Fig. 9 (b) when s is small, but they take different forms. It means that the random networks are easier for the spread in the initial conditions. 

Then there is the time distribution of the avalanche on the small-world network, which is similar to the size distribution of the avalanche, as given in Fig. 10 (a) and Fig. 10 (b) for comparison. But we note, when q is big enough, with the structure of the network becom- ing random, the time distribution no longer satisfies the power-law form. Besides, there is condensation at the end of the diagram, which we want to study in the later work. 

At last, for the construction of the small-world net- 

