Core Percolation in Scalefree Cross Layer Networks Nan

Core Percolation in Scalefree Cross Layer Networks Nan Wang

Overview Introduction Theoretical Basis Simulations & Experiments Conclusion

Introduction • Core percolation, as a fundamental structural transition resulted from preserving core nodes in the network, is crucial in network controllability and robustness. • We can consider core nodes as stable nodes in a network.

Single-Network Interdependent- More stringent demand on network stability.

Theoretical Basis •

Theoretical Basis Greedy Leaf Removal (GLR) Procedure. • Single, non-interacting network where core nodes are obtained by a classic Greedy Leaf Removal (GLR) procedure that takes off leaf nodes along with their neighbors iteratively. • α-removable: can become isolated without directly removing themselves; β-removable: which can become a neighbor of a leaf; γ-removable: which can be removable but neither α-removable nor β-removable.

Alternating GLR Procedure • Consider Two networks A and B. Assume that each node in A depends on a node in B with a probability q(0<q≤ 1) and vice versa. We use Alternating GLR Procedure to get the core of the interdependent networks.

Fully-Interdependent Networks • Firstly consider the condition of q=1, which means fully interdependency. GLR procedure in two interdependent networks

Partially-Interdependent Networks •

One-to-Multiple Interdependent Networks

Experimental Setup Simulations & Experiments Single-Layer Network Real-World Networks Partial-Interdependent Networks Synthetic Networks One-to-multiple Interdependent Networks

Real-World Data Sets

Fitting to our model We use degree distribution P(k) to fit the real-world networks in static-free model and obtain the fitting parameters.

Fitting results

Single-Layer Network ER model Synthetic Networks Real-world Networks

Revised AGLR Procedure The original theory is difficult for simulation and experiments on real-world data set. The revised AGLR : Iterate rounds until no nodes can be removed in both networks, and the rest nodes are the core.

Partial-Interdependent Networks Non-corresponding part The results of the first layer in double-layer networks where figure (b) are sections of figure (a)

Partial-Interdependent Networks Real-world data sets. P denotes the corresponding possibility of two networks, and when p=1, it is equa fully – interdependent networks.

One-to-multiple Interdependent Networks The results of theoretical derivation and experiments on synthetic networks, where figure (b) is the sections of (a).

One-to-multiple Interdependent Networks layer. A The results of in the case of one-to-multiple interdependent networks lay

Conclusion • The topology of the network under different models is analyzed. • Further expand from the single layer of the complex network to the double-layer network and analyze the stability of two layers of different correspondence. • In the double-layer network, the core proportion of the network will change at a jumping point with the change of the network mean degree. • The final stability of the network will depend on the node's average (k 1, k 2) in the network.

Q & A Thanks!