Lili Rong

A deterministic small-world network created by edge iterations

Small-world networks are ubiquitous in real-life systems. Most previous models of small-world networks are stochastic. The randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections. Here we present a model that generates a small-world network in a simple deterministic way. Our model has a discrete exponential degree distribution. We solve the main characteristics of the model.

High-dimensional random Apollonian networks

We propose a simple algorithm which produces a new category of networks, high-dimensional random Apollonian networks, with small-world and scale-free characteristics. We derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network. The values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks. We estimate also analytically that the average path length of the networks increases at most logarithmically with the number of vertices.

High-dimensional Apollonian networks

We propose a simple algorithm which produces high-dimensional Apollonian networks with both small-world and scale-free characteristics. We derive analytical expressions for the degree distribution, the clustering coefficient and the diameter of the networks, which are determined by their dimension.

Evolving small-world networks with geographical attachment preference

We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model, the network growth is determined by the attachment of new nodes to already existing nodes that are geographically close. We analyse several topological properties for our model both analytically and by numerical simulations. The resulting network shows some important characteristics of real-life networks such as small-world effect and high clustering.

Evolving Apollonian networks with small-world scale-free topologies

We propose two types of evolving networks: evolutionary Apollonian networks (EANs) and general deterministic Apollonian networks (GDANs), established by simple iteration algorithms. We investigate the two networks by both simulation and theoretical prediction. Analytical results show that both networks follow power-law degree distributions, with distribution exponents continuously tuned from 2 to 3. The accurate expression of clustering coefficient is also given for both networks. Moreover, the investigation of the average path length of EAN and the diameter of GDAN reveals that these two types of networks possess small-world feature. In addition, we study the collective synchronization behavior on some limitations of the EAN.

A general geometric growth model for pseudofractal scale-free web

We propose a general geometric growth model for pseudofractal scale-free web (PSW), which is controlled by two tunable parameters. We derive exactly the main characteristics of the networks: degree distribution, second moment of degree distribution, degree correlations, distribution of clustering coefficient, as well as the diameter, which are partially determined by the parameters. Analytical results show that the resulting networks are disassortative and follow power-law degree distributions with a more general degree exponent tuned from 2 to 1+ln3ln2; the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order; the diameter grows logarithmically with the number of network nodes. All these reveal that the networks described by our model have small-world effect and scale-free topology.

Local-world evolving networks with tunable clustering

We propose an extended local-world evolving network model including a triad formation (TF) step. In the process of network evolution, random fluctuation in the number of new edges is involved. We derive analytical expressions for degree distribution, clustering coefficient and average path length. Our model can unify the generic properties of real-life networks: scale-free degree distribution, high clustering and small inter-node separation. Moreover, in our model, the clustering coefficient is tunable simply by changing the expected number of TF steps after a single local preferential attachment step.

A MODEL FOR CASCADING FAILURES IN COMPLEX NETWORKS WITH A TUNABLE PARAMETER

In this paper, based on the local preferential redistribution rule of the load after removing a node, we propose a cascading model and explore cascading failures on four typical networks, i.e. the BA with scale-free property, the WS small-world network, the NW network and the ER random network. Assume that a failed node leads only to a redistribution of the load passing through it to its neighboring nodes. We find that all networks reach the strongest robustness level against cascading failures in the case of α=1, which is a tunable parameter in our model, where the robustness is quantified by the critical threshold Tc, at which a phase transition occurs from a normal state to collapse. To a constant network size, we further discuss the correlations between the average degree 〈k〉 and Tc, and draw the conclusion that Tc has a negative correlative with 〈k〉, i.e. the bigger the value of 〈k〉, the smaller the critical threshold Tc. These results may be very helpful for real-life networks to avoid cascading-failure-induced disasters.

Vertex labeling and routing in expanded Apollonian networks

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale free, small world and highly clustered. We introduce a labeling of their nodes that allows one to determine the shortest path routing between any two nodes of the network based only on the labels.

EFFECT OF ATTACK ON SCALE-FREE NETWORKS DUE TO CASCADING FAILURE

In this paper, based on the local preferential redistribution rule of the load after removing a node, we propose a cascading model and explore cascading failures on scale-free networks. Assuming that a failed node leads only to a redistribution of the load passing through it to its neighboring nodes, we study the response of scale-free networks subject to attacks on nodes. The network robustness against cascading failures is quantitatively measured by the critical threshold Tc, at which a phase transition occurs from normal state to collapse. For each case of attacks on nodes, four different attack strategies are used: removal by the descending order of the degree, attack by the ascending order of the degree, random removal of breakdown, and removal by the ascending order of the average degree of neighboring nodes of a broken node. Compared with the previous result, i.e. the robust-yet-fragile property of scale-free networks on random failures of nodes and intentional attacks, our cascading model has totally different and interesting results. On the one hand, as unexpected, choosing the node with the lowest degree is more efficient than the one with the highest degree when α < 1, which is a tunable parameter in our model. On the other hand, the robustness against cascading failures and the harm order of four attack strategies strongly depends on the parameter α. These results may be very helpful for real-life networks to protect the key nodes and avoid cascading-failure-induced disasters.