Efficiently identifying critical nodes in large complex networks

Mario Ventresca,Dionne Aleman

doi:10.1186/s40649-015-0010-y

Abstract

The critical node detection problem (CNDP) aims to fragment a graph G=(V,E) by removing a set of vertices R with cardinality |R|≤k, such that the residual graph has minimum pairwise connectivity for user-defined value k. Existing optimization algorithms are incapable of finding a good set R in graphs with many thousands or millions of vertices due to the associated computational cost. Hence, there exists a need for a time- and space-efficient approach for evaluating the impact of removing any v∈V in the context of the CNDP. In this paper, we propose an algorithm based on a modified depth-first search that requires O(k(|V|+|E|)) time complexity. We employ the method within in a greedy algorithm for quickly identifying R. Our experimental results consider small- (≤250 nodes) and medium-sized (≤25,000 nodes) networks, where it is possible to compare to known optimal solutions or results obtained by other heuristics. Additionally, we show results using six real-world networks. The proposed algorithm can be easily extended to vertex- and edge-weighted variants of the CNDP.

Highlights

Detecting important or critical vertices in a graph/network has many important applications. These critical vertices/nodes may be used to promote or mitigate a diffusive process that is acting upon the network
If promoting a spreading process, such as to spread market advertisements or public health warnings, the notion of ‘critical’ refers to the identification of individuals who are most likely to be influential spreaders and maximally permit information spread through the network
We focus on the Ventresca and Aleman Computational Social Networks (2015) 2:6 context of mitigation; the results are typically applicable for problems where the goal is to maximally aid the diffusive process [8,9,10,11]

Summary

Introduction

Detecting important or critical vertices in a graph/network has many important applications. The residual network will contain a relatively large set of connected components, each containing a similar number of vertices This problem has been shown to be N P hard [12,13]. The above four observations imply that v∗ of Algorithm 1 can be computed in linear time by augmenting a DFS for identifying cut vertices to calculate the impact of removing any vertex v ∈ V. O(n + m) and Equation 4 can be executed in constant time per node during the search, the proposed greedy algorithm requires O(k(n+m)) complexity to remove k vertices from G. A priority queue can be utilized to store the set of connected components C, which are represented and ordered by their respective root vertices and their impact on the objective value if removed, respectively. All networks are unweighted and simplified before use (no self-loops or multi-edges)

Collaboration Social

SEQ MIS Graph k LB SEQ MIS

ForestFire ForestFire ForestFire ForestFire

Conclusion

Slow greedy