Sensitivity of directed networks to the addition and pruning of edges and vertices

Abstract

We study the sensitivity of directed complex networks to the addition and pruning of edges and vertices and introduce the susceptibility, which quantifies this sensitivity. We show that topologically different parts of a directed network have different sensitivity to the addition and pruning of edges and vertices and, therefore, they are characterized by different susceptibilities. These susceptibilities diverge at the critical point of the directed percolation transition, signaling the appearance (or disappearance) of the giant strongly connected component in the infinite size limit. We demonstrate this behavior in randomly damaged real and synthetic directed complex networks, such as the World Wide Web, Twitter, the Caenorhabditis elegans neural network, directed Erdős-Rényi graphs, and others. We reveal a non-monotonous dependence of the sensitivity to random pruning of edges or vertices in the case of Caenorhabditis elegans and Twitter that manifests specific structural peculiarities of these networks. We propose the measurements of the susceptibilities during the addition or pruning of edges and vertices as a new method for studying structural peculiarities of directed networks.

The central role of peripheral nodes in directed network dynamics

“…2). Additionally, it has been recently shown that, in directed uncorrelated random complex networks, the number of nodes s in the finite in-component of any node scales as , where s * is a characteristic parameter that depends on 38 . This means that only a small number of IN nodes can be reached by traveling backwards from any other IN node, and at least one of these must be a SOURCE node.…”

“…For example, a synchronized brain network may be desynchronized through the random destruction of synapses, e.g. in Alzheimer’s disease, as the random destruction of links alters the global organization of the network 38 . Similarly, a targeted attack on low in-degree centrality nodes may allow an attacker to exert considerable influence on a network, by controlling the dynamics of newly-created SOURCE nodes.…”

“…The results of analytical calculations presented in Fig. 2 were obtained following standard methods (see 38 and the references therein for further details.) In particular, for any directed uncorrelated random network with N nodes, and an in-degree ( q in ) out-degree ( q out ) distribution , the fraction of SOURCE nodes N SOURCE / N can be calculated from the probability of randomly selecting a SOURCE node.…”

“…For any node with out-degree q out , this is equal to the probability of selecting a node with , given by , and that at least one of its out-links leads to the CORE, given by , where y c is the probability that an out-link leads to a finite component. Taking into account all possible degrees,where y c can be determined self-consistently 38 . The average number of IN-CORE links can also be calculated using this formalism, considering a node randomly selected with probability , and the probability that it receives m in-links from nodes in finite components, given its in-degree, the probability that an in-link comes from a finite component x c 38 , and the probability that at least one out-link leads to a CORE node .…”

“…Taking into account all possible degrees,where y c can be determined self-consistently 38 . The average number of IN-CORE links can also be calculated using this formalism, considering a node randomly selected with probability , and the probability that it receives m in-links from nodes in finite components, given its in-degree, the probability that an in-link comes from a finite component x c 38 , and the probability that at least one out-link leads to a CORE node . To ensure that such a node belongs to the CORE, it is sufficient to ensure its m in-links from the IN component account at most for q i − 1 of its total in-links, i.e.…”