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Many complex systems can be represented as networks and separating a network into communities could simplify the functional analysis considerably. Recently, many approaches have been proposed for finding communities, but none of them can evaluate the communities found are significant or trivial definitely. In this paper, we propose an index to evaluate the significance of communities in networks. The index is based on comparing the similarity between the original community structure in network and the community structure of the network after perturbed, and is defined by integrating all the similarities. Many artificial networks and real-world networks are tested. The results show that the index is independent from the size of network and the number of communities. Moreover, we find the clear communities always exist in social networks, but don't find significative communities in proteins interaction networks and metabolic networks. The study of the community structure of networks has become a very important part of researches of complex networks. Nodes belonging to a tight-knit community are more likely to have particular properties in common. In social relationship network, communities usually represent different friend subgroups. In the world wide web, community analysis has uncovered thematic clusters. In biochemical or neural networks, different communities may represent different functional groups, and separating the network into such groups could simplify the functional analysis considerably. As a result, the problem of identification of communities has been the focus of many recent efforts. So two questions are proposed, the first is, how to detected communities in the networks? In recent studies, plenty of algorithms are proposed [1,2,3,4, 5,6,7,8,9,10,11,12,13,14,15] (see [5] as a review). The second question is coming hand in hand with the first question: how to evaluate the communities detected? We believe that there exist clear communities in some networks while no clear communities in the other networks. But almost all algorithms could find the "community structure" in networks in their ways, without thinking about whether the community structure actually exists or not. Even many algorithms can also find the community in random networks, in which are considered having no community. For the existence of such a situation, the discussion on the "significative communities" is needed. As a network is given, it is meaningless to detect the community when the community structure is not significative at all.Scientists try to propose a universal index to evaluate the partitions. And the modularity Q [16] was presented as an index of community structure and by now it has been widely accepted [5,10,11,14] as a measure for the community structure. Modularity Q was presented as * yanqing.hu.sc@gmail.com † zdi@bnu.edu.com a index of community structure by Newman and Grive, which was introduced as Q = r (e rr − a 2 r ), where e rr are the fraction of links that connect two nodes inside the community r, a r the f...

Empirical studies on the spatial structures in several real transport networks reveal that the distance distribution in these networks obeys power law. To discuss the influence of the power-law exponent on the network's structure and function, a spatial network model is proposed. Based on a regular network and subject to a limited cost C, long range connections are added with power law distance distribution P(r) = ar −δ . Some basic topological properties of the network with different δ are studied. It is found that the network has the smallest average shortest path when δ = 2. Then a traffic model on this network is investigated. It is found that the network with δ = 1.5 is best for the traffic process. All of these results give us some deep understandings about the relationship between spatial structure and network function.

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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.

Deep neural networks have been demonstrated to be vulnerable to adversarial examples, which fool networks by adding human-imperceptible perturbations to benign examples. At present, the practical transfer-based black-box attacks are attracting significant attention. However, most existing transfer-based attacks achieve only relatively limited success rates. We propose to improve the transferability of adversarial examples through the use of a virtual step and auxiliary gradients. Here, the “virtual step” refers to using an unusual step size and clipping adversarial perturbations only in the last iteration, while the “auxiliary gradients” refer to using not only gradients corresponding to the ground-truth label (for untargeted attacks), but also gradients corresponding to some other labels to generate adversarial perturbations. Our proposed virtual step and auxiliary gradients can be easily integrated into existing gradient-based attacks. Extensive experiments on ImageNet show that the adversarial examples crafted by our method can effectively transfer to different networks. For single-model attacks, our method outperforms the state-of-the-art baselines, improving the success rates by a large margin of 12%~28%. Our code is publicly available at https://github.com/mingcheung/Virtual-Step-and-Auxiliary-Gradients.

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