A k-shell decomposition method for weighted networks (1205.3720v2)

Abstract: We present a generalized method for calculating the k-shell structure of weighted networks. The method takes into account both the weight and the degree of a network, in such a way that in the absence of weights we resume the shell structure obtained by the classic k-shell decomposition. In the presence of weights, we show that the method is able to partition the network in a more refined way, without the need of any arbitrary threshold on the weight values. Furthermore, by simulating spreading processes using the susceptible-infectious-recovered model in four different weighted real-world networks, we show that the weighted k-shell decomposition method ranks the nodes more accurately, by placing nodes with higher spreading potential into shells closer to the core. In addition, we demonstrate our new method on a real economic network and show that the core calculated using the weighted k-shell method is more meaningful from an economic perspective when compared with the unweighted one.

• The paper introduces a weighted k-shell decomposition method that integrates edge weights into network analysis.
• The method calculates a weighted degree using the geometric mean of node connectivity and total edge weight.
• Empirical validation on diverse networks shows its superior ability to identify influential nodes and predict spreading dynamics.

A $k$-shell Decomposition Method for Weighted Networks

The paper "A $k$-shell decomposition method for weighted networks" by Antonios Garas, Frank Schweitzer, and Shlomo Havlin introduces a novel approach to understanding the intricate structures of weighted networks. The method extends the traditional $k$-shell decomposition, which is widely used in network analysis to ascertain central nodes within unweighted networks. This new advancement integrates edge weights into the analysis, allowing for a richer and more nuanced interpretation of network topology and node centrality.

Methodological Advancements

The traditional $k$-shell decomposition partitions a network into hierarchical layers, assigning to each node a $k_s$ value indicative of its location, with higher values corresponding to more central nodes. This methodology is valuable for identifying influential nodes in terms of connectivity. However, the limitation of ignoring link weights can overlook critical aspects of many real-world networks where interactions carry different strengths, such as economic or communication networks.

The authors propose a generalized weighted $k$-shell ($W_{k-{\rm shell}}$) method. The crux of the approach lies in defining a "weighted degree" $k'_i$ for a node $i$ which considers both the node's degree and the sum of the weights of its edges. Specifically, $k'_i$ is calculated by the geometric mean of the degree and the total weight sum of the node's connections, allowing for a balance between node connectivity and the strength of interactions.

Empirical Validation

The paper applies the $W_{k-{\rm shell}}$ method to four diverse weighted networks: a Corporate Ownership Network, a scientific collaboration network, the neural network of the nematode C. Elegans, and the U.S. air transportation network. Each showcases different aspects of the proposed method's efficacy.

For the Corporate Ownership Network, the weighted method reveals a smaller, economically meaningful core that aligns better with influential global economies than the unweighted counterpart. This strengthened core identification is consistent across other network types, highlighting the robustness of the method.

Dynamic Implications

A significant aspect of the paper is its exploration of dynamic processes over networks. Using a Weighted Susceptible-Infectious-Recovered (W-SIR) model, the authors simulate spreading processes to assess node roles within different $k$-shell layers. Results consistently show that the $W_{k-{\rm shell}}$ method places nodes with higher spreading potential closer to the network core, indicating superior performance in stratifying nodes by dynamic influence.

Conclusion and Implications

The $W_{k-{\rm shell}}$ method offers a nuanced tool for the decomposition of networks where weights play a pivotal role. By overcoming the limitations of binary connectivity, this method allows researchers to dissect network intricacies with greater precision, particularly for applications involving spread dynamics, economic connectivity, or communication efficiency.

Future research could explore the parameter space of the weighting scheme, possibly adapting it for networks where degree or weight should predominate, based on specific contexts or applications. The flexibility and depth provided make this method a vital addition to the toolkit for analyzing complex weighted networks, enhancing our theoretical understanding and practical ability to exploit these systems' underlying structures.