Deleting edges to restrict the size of an epidemic in temporal networks

Abstract

Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information or behaviour spread over social networks, biological diseases spreading over contact or trade networks, and the potential flow of goods over logistical infrastructure. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with graphs whose structure is subject to discrete changes over time, i.e. with temporal graphs. Here, we consider temporal graphs in which edges are available at specified timesteps, and study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path (i.e. a path, along which the times of the edge availabilities increase). We introduce a natural deletion problem for temporal graphs and we provide positive and negative results on its computational complexity, both in the traditional and the parameterised sense (subject to various natural parameters), as well as addressing the approximability of this problem.

Finding Temporal Paths Under Waiting Time Constraints

“…In the context of temporal flows, Akrida et al [4] considered a concept of "vertex buffers", which however pertains to the quantity of information that a vertex can store, rather than a duration. Enright et al [26] considered deletion problems for reducing temporal connectivity. More closely related to our work, Himmel et al [11] studied a variant of restless temporal paths where several visits to the same vertex are allowed, i.e., restless temporal walks.…”

Temporal Connectivity: Coping with Foreseen and Unforeseen Delays

“…There has been extensive research on many other connectivity-related problems on temporal graphs [4,9,10,11,12,17,18,20,25]. Delays in temporal graphs have been considered in terms of manipulating reachability sets [7,21].…”

The Complexity of Growing a Graph

“…A temporal graph is a graph that changes with time [35]. Starting with the works of Berman [6] and Kempe et al [26] in the single-labeled case, in which every edge of an underlying graph can be available at most once, and continuing with subsequent work on the multi-labeled case [18,33,53], a temporal extension of the algorithmic principles of graph theory is currently under development. The vast majority of temporal graphs considered in the literature have a static set of nodes and the dynamics concern only the edges.…”

“…At the same time, there is an ongoing effort to set the algorithmic foundations of dynamic networks. This has started with passively dynamic networks, such as population protocols [3], other dynamic distributed computing models [28,40], and the algorithmic and structural properties of temporal graphs [2,6,8,18,26,33,35,53] and has been recently expanding to the investigation of actively dynamic networks, ranging from geometric reconfiguration models [14,15,20,36] to abstract reconfigurable networks [4,37] and even hybrid models [21,24,38].…”