Bo Jiao

Accurately and quickly calculating the weighted spectral distribution

The weighted spectral distribution (WSD) is a measure defined on the normalized Laplacian spectrum. It can be used for comparing complex networks with different sizes (number of nodes) and provides a sensitive discrimination of the structural robustness of complex networks. In this paper, we design an algorithm for the accurate calculation of the WSD in large-scale complex networks by utilizing characteristics of the graph structure. As an extension to Sylvester’s Law of Inertia for the calculation of the WSD based on spectral characteristics, our algorithm exhibits a higher time efficiency when applied to large-scale complex networks.

Correlation between weighted spectral distribution and average path length in evolving networks

The weighted spectral distribution (WSD) is a metric defined on the normalized Laplacian spectrum. In this study, synchronic random graphs are first used to rigorously analyze the metrics scaling feature, which indicates that the metric grows sublinearly as the network size increases, and the metrics scaling feature is demonstrated to be common in networks with Gaussian, exponential, and power-law degree distributions. Furthermore, a deterministic model of diachronic graphs is developed to illustrate the correlation between the slope coefficient of the metrics asymptotic line and the average path length, and the similarities and differences between synchronic and diachronic random graphs are investigated to better understand the correlation. Finally, numerical analysis is presented based on simulated and real-world data of evolving networks, which shows that the ratio of the WSD to the network size is a good indicator of the average path length.

The 3-cycle weighted spectral distribution in evolving community-based networks

One of the main organizing principles in real-world networks is that of network communities, where sets of nodes organize into densely linked clusters. Many of these community-based networks evolve over time, that is, we need some size-independent metrics to capture the connection relationships embedded in these clusters. One of these metrics is the average clustering coefficient, which represents the triangle relationships between all nodes of networks. However, the vast majority of network communities is composed of low-degree nodes. Thus, we should further investigate other size-independent metrics to subtly measure the triangle relationships between low-degree nodes. In this paper, we study the 3-cycle weighted spectral distribution (WSD) defined as the weighted sum of the normalized Laplacian spectral distribution with a scaling factor n, where n is the network size (i.e., the node number). Using some diachronic community-based network models and real-world networks, we demonstrate that the ratio of the 3-cycle WSD to the network size is asymptotically independent of the network size and strictly represents the triangle relationships between low-degree nodes. Additionally, we find that the ratio is a good indicator of the average clustering coefficient in evolving community-based systems.

A heuristic nonlinear operator for the aggregation of incomplete judgment matrices in group decision making

The weighted-average operator and ordered-weighted-average operators are typically used in group decision making (GDM) problems to aggregate individual expert opinions to a collective opinion. However, the existing aggregation operators pay more attentions on the determination of the weights, and neglect the information about the relationship between the values being fused. In this paper, we develop a heuristic-nonlinear-aggregation (HNA) operator based on two metrics of similarity and consistency for the GDM based on incomplete judgment matrices. The similarity and consistency respectively measure the differences between a collective matrix and two optimum matrices, i.e. the optimum similarity matrix and the optimum consistency matrix, which can be calculated by quadratic programming models and the relationship between the values being fused. The validity and practicability of the HNA operator are illustrated by numerical examples.

Stability of weighted spectral distribution in a pseudo tree-like network model*

The comparison of networks with different orders strongly depends on the stability analysis of graph features in evolving systems. In this paper, we rigorously investigate the stability of the weighted spectral distribution (i.e., a spectral graph feature) as the network order increases. First, we use deterministic scale-free networks generated by a pseudo tree-like model to derive the precise formula of the spectral feature, and then analyze the stability of the spectral feature based on the precise formula. Except for the scale-free feature, the pseudo tree-like model exhibits the hierarchical and small-world structures of complex networks. The stability analysis is useful for the classification of networks with different orders and the similarity analysis of networks that may belong to the same evolving system.

Performance Evaluations of Graph Spectra on Evolving Systems

Evolving complex networks are abundant in the real world, i.e., Networks with different sizes (number of nodes) may originate from the same evolving system. In this paper, we use some typical evolving models of complex networks to evaluate the performances of graph spectra. The experimental results verify that the normalized Laplacian spectrum is a good indicator to distinguish between different evolving systems.

Comparison of Biased and Unbiased Sampling Algorithms Using Graph Metrics

Degree distribution, hierarchy, clustering and small-world are typical graph features to measure the structure of complex networks. This paper uses these features to compare and evaluate the performances of biased and unbiased sampling algorithms on two types of scale-free networks. The numerical analysis verifies that the biased sampling performs better than the unbiased sampling on networks in which high-degree nodes are loosely interconnected. Our work is useful for the in-depth understanding of the biased and unbiased samplings.

Comparison of graph sampling algorithms using normalized Laplacian spectra

The normalized Laplacian spectrum is a good indicator of connectivity for comparing graphs with different sizes (i.e., the number of nodes). This paper shows the performances of several random-walk sampling algorithms using the spectral indicator. Based on two types of complex network models with exponential and power-law degree distributions, we determine that the sampling algorithms do not perform well on the spectral indicator, and find that the spectral indicator of biased sampling graphs is much closer to that of original graphs when the assortative mixing coefficient of original graphs decreases. Especially, for disassortative networks, biased sampling graphs perform better than unbiased sampling graphs on the spectral indicator.

Network simulation and evaluation for the size reduction of a given topology

Running on a size-reduction network simulated by a given large-scale topology can obviously enhance the time efficiency. In this paper, we realize the size-reduction using the input extracting of a network simulator and evaluate the similarity between the size-reduction and given topologies using multiple criterions. The evaluation result verifies that the network simulation with input extracting is effective for the size reduction.

Application of normalized Laplacian spectra for multicast routing evaluations

The multiplicity of the eigenvalue 1 (ME1) and the weighted spectral distribution (WSD), i.e., two weakly-related metrics defined on the normalized Laplacian spectrum, capture the size-independent features of the Internet structure. In this paper, we evaluate the multicast routing protocol in the size-independent Internet structure, and numerically find that the protocol indexes tend to stable as the network size increases if the normalized Laplacian spectrum of the network topology is asymptotically independent of the network size. The real-world Internet topology evolves (i.e., the network size increases) over time, which indicates that the protocol should be evaluated in the size-independent network structure. Our studies provide a specific application of the normalized Laplacian spectrum, which verifies that the size-independent structure captured by the spectrum is useful for the evaluation of network protocols.