Approximating the Temporal Neighbourhood Function of Large Temporal Graphs

Pierluigi Crescenzi,Andrea Marino,Clémence Magnien

doi:10.3390/a12100211

Abstract

Temporal networks are graphs in which edges have temporal labels, specifying their starting times and their traversal times. Several notions of distances between two nodes in a temporal network can be analyzed, by referring, for example, to the earliest arrival time or to the latest starting time of a temporal path connecting the two nodes. In this paper, we mostly refer to the notion of temporal reachability by using the earliest arrival time. In particular, we first show how the sketch approach, which has already been used in the case of classical graphs, can be applied to the case of temporal networks in order to approximately compute the sizes of the temporal cones of a temporal network. By making use of this approach, we subsequently show how we can approximate the temporal neighborhood function (that is, the number of pairs of nodes reachable from one another in a given time interval) of large temporal networks in a few seconds. Finally, we apply our algorithm in order to analyze and compare the behavior of 25 public transportation temporal networks. Our results can be easily adapted to the case in which we want to refer to the notion of distance based on the latest starting time.

Highlights

Temporal networks are graphs in which nodes and edges can appear or disappear over time, due to failures or malfunctioning of the entities participating to the system represented by the temporal graph, but mostly to the “normal” behaviour of the system itself
We show how the sketch approach, which has already been widely used in the case of non-temporal graphs [10,11,12,13], can be applied to the case of temporal networks in order to approximately compute the cardinalities of the reverse temporal cones of a temporal network
We experimentally evaluate the quality of the approximation performed by ATNF by comparing the approximate value of the temporal neighborhood function with the exact one on a data-set containing several medium-size temporal networks

Summary

Introduction

Temporal networks are graphs in which nodes and edges can appear or disappear over time, due to failures or malfunctioning of the entities participating to the system represented by the temporal graph, but mostly to the “normal” behaviour of the system itself. A typical temporal network is a person-to-person communication network within a company. In such a network, for example, nodes can appear or disappear (depending on the recruitment policy of the company), and edges appear whenever an employee of the company sends an e-mail message to another employee of the company. We will focus on temporal networks in which the set of nodes does not change over time (at least over a specified interval of time). We consider only the case in which edges are available at discrete time instants, so that the dynamics of the network are specified only by the appearance times of the edges. Several different types of temporal networks have, been analyzed: person-to-person

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