Stochastic block models (SBMs) have been playing an important role in modeling clusters or community structures of network data. But, it is incapable of handling several complex features ubiquitously exhibited in real-world networks, one of which is the power-law degree characteristic. To this end, we propose a new variant of SBM, termed power-law degree SBM (PLD-SBM), by introducing degree decay variables to explicitly encode the varying degree distribution over all nodes. With an exponential prior, it is proved that PLD-SBM approximately preserves the scale-free feature in real networks. In addition, from the inference of variational E-Step, PLD-SBM is indeed to correct the bias inherited in SBM with the introduced degree decay factors. Furthermore, experiments conducted on both synthetic networks and two real-world datasets including Adolescent Health Data and the political blogs network verify the effectiveness of the proposed model in terms of cluster prediction accuracies.
There is also other literature on addressing degree heterogeneity in networks using spectral methods, for which we leave the references @cite_44 @cite_60 @cite_63 @cite_54 for interested readers.
@cite_44 @cite_54 @cite_63 @cite_60
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Heterogeneous networks consisting of small cells (e.g. femtocells) are capable of achieving high capacity and improving indoor cellular coverage area, while the fractional frequency reuse (FFR) scheme has been proposed for upcoming and future cellular systems to improve spectral efficiency in cellular OFDM networks. In two-tier networks of macrocells layered with femtocells, the resource allocation will most likely be sharing the same licensed spectrum. In this paper, we formulate a simple femtocells resource allocation strategies that allows a femtocells base station (FBS) to allocate more resources (co-channel) in high usage areas such as commercial FBS (cFBS) and orthogonal resource allocation for randomly deployed home user FBS (hFBS). We analyze our strategy in a multi-cell systems using area spectral efficiency (ASE) in composite fading consisting of Nakagami-m fading, path-loss and log-normal shadowing. Analytical and simulation results show that our simple resource allocation strategies are able to reduce inter-tier interferences and to offer an improvement in the overall spectral efficiency of the two-tier systems. We consider wireless sensor networks under a heterogeneous random key predistribution scheme and an on-off channel model. The heterogeneous key predistribution scheme has recently been introduced by Yagan - as an extension to the Eschenauer and Gligor scheme - for the cases when the network consists of sensor nodes with varying level of resources and or connectivity requirements, e.g., regular nodes vs. cluster heads. The network is modeled by the intersection of the inhomogeneous random key graph (induced by the heterogeneous scheme) with an Erdős-Renyi graph (induced by the on off channel model). We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model so that with high probability all of its nodes are connected to at least k other nodes; i.e., the minimum node degree of the graph is no less than k. We also present numerical results to support our results in the finite-node regime. The numerical results suggest that the conditions that ensure k-connectivity coincide with those ensuring the minimum node degree being no less than k. Heterogeneity arises in a wide range of scenarios in mobile opportunistic networks and is one of key factors that govern the performance of packet forwarding algorithms. While the heterogeneity has been empirically investigated and exploited in the design of new forwarding algorithms, it has been typically ignored or marginalized when it comes to rigorous performance analysis of such algorithms. In this paper, we develop an analytical framework to the performance gain achievable by exploiting the heterogeneity in mobile nodes' contact dynamics. In particular, we derive a delay upper bound of a heterogeneity-aware forwarding policy per a given number of message copies and obtain its closed-form expression, which enables our quantitative study on the benefit of leveraging underlying heterogeneity structure in the design of forwarding algorithms. We then analytically show that less than 20 of total (unlimited) message copies is only enough under various heterogeneous network settings to achieve the same delay as that obtained using the unlimited message copies when the networks become homogeneous. We also provide independent simulation results including real trace-driven evaluation to support our analytical results. Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks.