Abstract
The $k$ -core, a kind of structure of graphs, is a maximal connected subgraph with the minimum degree greater than or equal to $k$ , and has been used in many fields. The maximum $k$ such that a $k$ -core contains $u$ is the $K$ value of $u$ . Especially, for an edge-weighted graph, the degree of a vertex is the sum of weights of all its incident edges. The core decomposition problem on static graphs and the core maintenance problem on dynamic graphs have been studied in unweighted graphs. We improve the core decomposition algorithm to suit edge-weighted graphs, but it costs too much to update $K$ values of all vertices after the change of large graphs by using it directly. Then we find a small subgraph $H$ which contains all vertices whose $K$ values will change after the change of graphs. By operating on $H$ , the cost will be greatly reduced. Next, we design core maintenance algorithms for edge-weighted graphs in both insertion and deletion cases, which is the major work in this paper. In those core maintenance algorithms, a hierarchical process is added, which help us determine the new $K$ values of vertices in $H$ from the small ones to high. Finally, we conduct extensive experiments on real-world graphs to show the effectiveness and the efficiency that our algorithms have. The results show that our algorithms have the best performance.
Highlights
To describe the cohesive structures of a graph, a k-core of a graph which is a maximal connected vertex-induced subgraph with the minimum degree greater than or equal to k has been introduced in [1], where k is a positive integer
Appearing in a variety of fields, both research and industry, the k-core has a large number of applications, including analysing large graphs, finding the cohesive connected subgraphs which can help facilitate advertising in social networks [28]–[30], analysing the structures of the internet [36] and using in the biology area [31], [34]
Compared to the subgraphs that contain all vertices whose K values will change in unweighted graphs, the subgraphs we find is more complex in weighted graphs, since the K values of vertices whose K values will change are various
Summary
To describe the cohesive structures of a graph, a k-core of a graph which is a maximal connected vertex-induced subgraph with the minimum degree greater than or equal to k has been introduced in [1], where k is a positive integer. We need the new K values after the change to analysis the structures of the new network, and the linear time algorithm for K values calculation in a static graph can be used. This isn’t an efficient solution, since only a small part of vertices have their K values changed and the cost will be high in large graphs. If H is a maximal seed k-core of G, i.e., there is no other seed k-core H , H ⊃ H , H is a (weighted) k-core of G.
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