A limitation of the Graph Convolutional Networks (GCNs) is that it assumes at a particular @math layer of the neural network model only the @math order neighbourhood nodes of a social network are influential. Furthermore, the GCN has been evaluated on citation and knowledge graphs, but not extensively on friendship-based social graphs. The drawback associated with the dependencies between layers and the order of node neighbourhood for the GCN can be more prevalent for friendship-based graphs. The evaluation of the full potential of the GCN on friendship-based social network requires openly available datasets in larger quantities. However, most available social network datasets are not complete. Also, the majority of the available social network datasets do not contain both the features and ground truth labels. In this work, firstly, we provide a guideline on simulating dynamic social networks, with ground truth labels and features, both coupled with the topology. Secondly, we introduce an open-source Python-based simulation library. We argue that the topology of the network is driven by a set of latent variables, termed as the social DNA (sDNA). We consider the sDNA as labels for the nodes. Finally, by evaluating on our simulated datasets, we propose four new variants of the GCN, mainly to overcome the limitation of dependency between the order of node-neighbourhood and a particular layer of the model. We then evaluate the performance of all the models and our results show that on 27 out of the 30 simulated datasets our proposed GCN variants outperform the original model.
The scale-free model shows power law node degree distribution @math (where @math is the node degree and typically @math ) for a social network. This kind of distribution was first discussed by . Price, in turn, was inspired by Herbert Simon, who discusses power law in a variety of non-network economic data @cite_16 .
@cite_16
2137465421 2016986674 2150041461 2080478510
A power law degree distribution is established for a graph evolution model based on the graph class of k-trees. This k-tree-based graph process can be viewed as an idealized model that captures some characteristics of the preferential attachment and copying mechanisms that existing evolving graph processes fail to model due to technical obstacles. The result also serves as a further cautionary note reinforcing the point of view that a power law degree distribution should not be regarded as the only important characteristic of a complex network, as has been previously argued [D. Achlioptas, A. Clauset, D. Kempe, C. Moore, On the bias of traceroute sampling, or power-law degree distribution in regular graphs, in: Proceedings of the 37th ACM Symposium on Theory of Computing, STOC'05, 2005, pp. 694-703; L. Li, D. Alderson, J. Doyle, W. Willinger, Towards a theory of scale-free graphs: Definition, properties, and implications, Internet Mathematics 2 (4) (2005) 431-523; M. Mitzenmacher, The future of power law research, Internet Mathematics, 2 (4) (2005) 525-534]. We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability of getting connected to a site at distance d is proportional to @math where is a tunable parameter of the model. We show that the properties of the networks grown with @math are close to those of the genuine scale-free construction, while for @math the structure of the network is quite different. Thus, in this regime, the node degree distribution is no longer a power law, and it is well represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of . We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of @math nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus @math scales as @math , where @math is the @math th moment of the degree distribution. For a power-law degree distribution @math , @math thus scales as @math for @math , as @math for @math , as @math for @math , as @math for @math , and as @math for @math . These results agree with simulation data for networks with both uncorrelated and correlated node degrees. We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent γ=1 + In 3 ln 2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with y in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network ( In N>>1) the distribution tends to a Gaussian of width ∼ N centered at ∼ ln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2 + y.