Characterizing the Dynamical Importance of Network Nodes and Links

Abstract

The largest eigenvalue of the adjacency matrix of the networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.In recent years, there has been much interest in the study of the structure of networks arising from real world systems, of dynamical processes taking place on networks, and of how network structure impacts such dynamics [1]. Remarkably, the largest eigenvalue of the network adjacency matrix (which we denote λ) has recently emerged as the key quantity determining many important properties for the study of a variety of different dynamical network processes. Some examples are the following: (i) for a heterogeneous collection of chaotic and/or periodic dynamical systems coupled by a network of connections, the critical coupling strength [2] for the emergence of coherence is proportional to 1/λ (ii) the critical disease contagion probability for the onset of an epidemic [3] scales as 1/λ; (iii) in percolation on a network, the condition for the emergence of a giant component also involves λ [4]. In addition to these, there are other notable examples where λ plays a similar role [5,6,7].In many situations it might be desirable to control dynamical processes that take place on networks. For example, in epidemic spreading, one would like to increase the threshold for epidemic transmission. In percolation, one might like to identify the key nodes holding the network together and protect them (e.g., in the transportation network or the internet) or disrupt them (e.g., in the case of a terrorist network or pathogen protein network). Such strategies would greatly benefit from a quantitative characterization of the effect of the removal of the different nodes or edges in the network. We will define the dynamical importance of nodes and edges as the relative change in the largest eigenvalue of the network adjacency matrix upon their removal. This provides an objective quantification of the relative importance of the different elements of the network that could potentially be used to formulate control strategies for those network processes * Electronic address: juanga@math.umd.edu that are governed by the largest eigenvalue of the network adjacency matrix. We also will describe an efficient way to approximate the dynamical importance.We consider a network as a directed graph with N nodes, and we associate to it a N × N adjacency matrix whose elements A ij are positive if there is a link going from node i to node j with i = j and zero otherwise (A ii ≡ 0). We denote the largest eigenvalue of A b...

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