The heterogeneous structure implies that a very few nodes may play the critical role in maintaining structural and functional properties of a large-scale network. Identifying these vital nodes is one of the most important tasks in network science, which allow us to better conduct successful social advertisements, immunize a network against epidemics, discover drug target candidates and essential proteins, and prevent cascading breakdowns in power grids, financial markets and ecological systems. Inspired by the nested nature of real networks, we propose a decomposition method where at each step the nodes with the lowest degree are pruned. We have strictly proved that this so-called lowest degree decomposition (LDD) is a subdivision of the famous k-core decomposition. Extensive numerical analyses on epidemic spreading, synchronization and nonlinear mutualistic dynamics show that the LDD can more accurately find out the most influential spreaders, the most efficient controllers and the most vulnerable species than k-core decomposition and other well-known indices. The present method only makes use of local topological information, and thus has high potential to become a powerful tool for network analysis.
VV with a preset size | ' | Vn = , which has the maximal dynamical impact (e.g., the influence maximization problem [19]); (iii) to provide a ranking of nodes' functional significances based on their structural features, which is usually on the basis of a proper centrality measure. Accordingly, many centralities have been developed to characterize influences of individual nodes, ranging from simple measures like degree [1], closeness [20] and betweenness [16], to elaborately designed ones like PageRank [21], LeaderRank [22] and collective influence [23].
A particularly interesting group of methods is network decomposition, with an underlying hypothesis that nodes are organized in different levels and the ones at the nucleus are most influential, which is to some extent supported by recent empirical evidences about the nested organization [11] and core-periphery structure [24]. Aiming at identifying influential nodes, the most successful decomposition method till far is the k-core decomposition [14]. Considering a connected simple network G where multiple links and self-loops are not allowed, the k-core decomposition process starts by removing all nodes with degree 1 k = . This causes new nodes with degree 1 k to appear. These are also removed and the process stops when all remaining nodes are of degree 1 k . The removed nodes and their associated links form the 1-shell, and the nodes in the 1-shell are assigned a k-shell value ks=1. This pruning process is repeated to extract the 2-shell, that is, in each step the nodes with degree 2 k are removed. Nodes in the 2-shell are assigned a k-shell value ks=2. The process is continued until all higher-layer shells have been identified and all nodes have been removed. Recent empirical and theoretical studies [10,14] both suggest that the kshell index is a good measure of a node’s influence: a higher ks indicates a larger influence.
A severe drawback of the k-core decomposition is that ks is not sufficiently distinguishable as each shell may contain numerous nodes. Some modified methods are recently proposed, mainly via replacing degree in the decomposition process by other centralities or combining k-shell index with other centralities (see such variants of the k-core decomposition in Ref. [4]). These modifications bring some certain improvement in accuracy, together with complicated details that clouds our understanding about network organization. Here we propose a even simpler decomposition method named as lowest degree decomposition (LDD for short). Firstly, the nodes with the lowest degree are removed, which form the 1-shell under LDD and are assigned a value Ls=1. Then, the remaining nodes with the lowest degree are removed, which form the 2-shell with Ls=2. This pruning process stops when all nodes have been removed. A notable difference from k-core decomposition is that LDD peels off every shell at once, without any iterations. Taking the well-studied Zachary karate club network [25] as an example, figure 1 illustrates the results from k-core decomposition (Fig. 1a) and LDD (Fig. 1b
which takes into account the importance of i’s neighbors. The free parameter is set to balance the contributions from i itself and its neighbors. We can also define the k-shell+ index in a similar way. Notice that, though later we will show that LDD+ and k-shell+ indices perform better than LDD and k-shell indices, we do not think the former are better than the latter since we have to tune one parameter in LDD+ and k-shell+ indices, while LDD and k-shell indices are parameter-free.
To see whether LDD can be used to characterize individual nodes’ influences, we use 9 real networks from disparate fields for experimental analyses. They are all simple networks, where directionality and weight of any link are ignored and self-loops are not allowed. In brief, there are one word network (AdjNoun), two communication networks (Email-Enron and Email-URV), two biological networks (PPI and Enzyme), two social networks (Dublin and Hamsterster), and two power grids (Bcs and Ops). Detailed descriptions, corresponding references and topological statistics of these networks are presented in Supplementary Section II. We compare the proposed methods, LDD and LDD+, with six benchmark indices including k-shell [14], k-shell+, degree, betweenness [16], Hindex [15], and mixed degree decomposition (MDD) [17]. The precise definitions of betweenness, H-index and MDD are shown in Methods. In the later experimental analyses, if an index contains a tunable parameter, its value will be turned to the one corresponding to the best performance.
We first test whether LDD can well quantify a node’s influence in spreading dynamics by applying two standard spreading model, the susceptible-infected-recovered (SIR) model and the susceptibleinfected-susceptible (SIS) model [26] (see Methods for the model descriptions). For SIR model, the influence of a node i, say Ri, is defined as the number of eventual
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