Efficient limited-time reachability estimation in temporal networks

Open Access

Efficient limited-time reachability estimation in temporal networks

Arash Badie-Modiri, Márton Karsai, and Mikko Kivelä

Phys. Rev. E 101, 052303 – Published 7 May 2020

Abstract

Time-limited states characterize many dynamical processes on networks: disease-infected individuals recover after some time, people forget news spreading on social networks, or passengers may not wait forever for a connection. These dynamics can be described as limited-waiting-time processes, and they are particularly important for systems modeled as temporal networks. These processes have been studied via simulations, which is equivalent to repeatedly finding all limited-waiting-time temporal paths from a source node and time. We propose a method yielding an orders-of-magnitude more efficient way of tracking the reachability of such temporal paths. Our method gives simultaneous estimates of the in- or out-reachability (with any chosen waiting-time limit) from every possible starting point and time. It works on very large temporal networks with hundreds of millions of events on current commodity computing hardware. This opens up the possibility to analyze reachability and dynamics of spreading processes on large temporal networks in completely new ways. For example, one can now compute centralities based on global reachability for all events or can find with high probability the infected node and time, which would lead to the largest epidemic outbreak.

Received 3 September 2019
Accepted 23 March 2020

DOI:https://doi.org/10.1103/PhysRevE.101.052303

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Evolving networks Network structure Spreading

NetworksStatistical Physics & Thermodynamics

Authors & Affiliations

Arash Badie-Modiri ¹, Márton Karsai^2,3, and Mikko Kivelä ¹

¹Department of Computer Science, School of Science, Aalto University, FI-0007, Finland
²Department of Network and Data Science, Central European University, H-1051 Budapest, Hungary
³Université de Lyon, ENS de Lyon, Inria, CNRS, Université Claude Bernard Lyon 1, LIP, F-69342, LYON Cedex 07, France

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 101, Iss. 5 — May 2020

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
An undirected temporal network and its corresponding static event graph. Two events $e_{i}$ and $e_{j}$ on a temporal network are adjacent if they share at least one node and $Δ t (e_{i}, e_{j}) \leq δ t$ , where in this instance we have chosen a value of $δ t = 3$ for the purpose of illustration. One valid temporal path is highlighted red on the temporal network and on the event graph.
Reuse & Permissions
Figure 2
Static event graph representation of a temporal network. Weakly connected components are ${e_{1}, e_{3}}$ and ${e_{2}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}}$ . Out-component set of the event $e_{2}$ is ${e_{2}, e_{4}, e_{6}, e_{7}, e_{8}}$ . Note that event $e_{5}$ is in the same weakly connected component as $e_{2}$ , but it is not a member of its out-component set. The algorithm finds the out-component by going backwards through a topological ordering of events (i.e., reverse time order); at each step, the out-component of each event is calculated by getting the union of the out-component sets of all events from the set of events in its out-edges plus the event itself. Since the ordering is reverse topological order, all the events in the out-edge set will already have their out-components calculated.
Reuse & Permissions
Figure 3
(a) Estimates of mean relative bias values, (b) relative standard error of our implementation of HyperLogLog cardinality estimation algorithm with $m = 2^{10}$ registers and random multisets. Relative standard error is calculated by dividing standard deviation of estimated cardinality over a set of 1000 estimators with different seeds divided by true values of cardinalities. Relative bias is measured as the difference of the estimated cardinalities from the true cardinality divided by true cardinality averaged over 1000 estimators. Panel (c) displays mean (blue dashed line) and the standard error of the mean (orange band) of 1000 independent estimations of out-component size estimations on a realization of a 1024-node random temporal network. The markers $*, +$ , and $\times$ denote the estimation errors for a subset of individual out-component sizes for three estimators out of the total 1000. Panel (d) shows relative standard error of estimation, similar to (b), but based on 1000 independent out-component size estimation on the same network as (c). Solid lines in panels (b), (c), and (d) indicate the theoretical accuracy of the HyperLogLog algorithm with $2^{10}$ registers, which is $1.04 / \sqrt{m} = 1.04 / \sqrt{2^{10}} = 0.0325$ relative to true cardinality [24]. For a more in-depth study of different variations of HyperLogLog and the role and reasoning about bias estimation, see Ref. [18].
Reuse & Permissions
Figure 4
Component sizes for a random temporal network model with an Erdös-Rényi static base and events created on each link with a Poisson process. Maximum (a) weakly connected and (b) out-component sizes measured in events as a function of average degree and $δ t$ . Solid line indicates the analytically estimated critical point for percolation of out-components [15]. (c) The growth of the maximum out-component for a fixed average degree $k = 9$ for component sizes measured in events ( $ρ_{o, e}$ ), temporal network nodes ( $ρ_{o, g}$ ), and the lifetime of the component ( $ρ_{o, l t}$ ). (d) Distributions of out-component sizes measured in events for networks with average degree $k = 9$ at three different values of $δ t$ . These $δ t$ values are marked with vertical lines in panel (c). Networks have $| V | = 1024$ nodes evolving for $T = 512$ for heat maps (a and b) and $T = 128$ time units for panel (c). Events were generated through a Poisson process with an expected interevent time of $α = 1$ time units. Each point is averaged over 10 realizations for heat maps (a and b) and 50 realizations for panel (c) with $2^{14}$ registers for each HyperLogLog counter, implying a relative standard error of $8.1 \times 10^{- 3}$ .
Reuse & Permissions
Figure 5
Top row displays maximum out-component sizes based on number of events ( $ρ_{o, e}$ ), number of unique nodes ( $ρ_{o, g}$ ), and lifetime of the out-component ( $ρ_{o, l t}$ ), and bottom row shows the median runtime of the algorithm for the estimation of out-component sizes (number of unique events, nodes and lifetime) for different $δ t$ values for (a) air transportation, (b) public transportation, (c) Twitter, and (d) mobile datasets. The vertical lines corresponds to the $δ t^{*}$ values in Table 1.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics