Node connectivity plays a central role in temporal network analysis. We provide a broad study of various concepts of walks in temporal graphs, that is, graphs with fixed vertex sets but arc sets changing over time. Taking into account the temporal aspect leads to a rich set of optimization criteria for “shortest” walks. Extending and broadening state-of-the-art work of Wu et al. [IEEE TKDE 2016], we provide an algorithm for computing shortest walks that is capable to deal with various optimization criteria and any linear combination of these. It runs in O(|V|+|E|log|E|) time where |V| is the number of vertices and |E| is the number of time-arcs. A central distinguishing factor to Wu et al.’s work is that our model allows to, motivated by real-world applications, respect waiting-time constraints for vertices, that is, the minimum and maximum waiting time allowed in intermediate vertices of a walk. Moreover, other than Wu et al. our algorithm also allows to search for walks that pass multiple subsequent time-arcs in one time step, and it can deal with a richer set of optimization criteria. Our experimental studies indicate that our richer modeling can be achieved without significantly worsening the running time when compared to Wu et al.’s algorithms.

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Node connectivity plays a central role in temporal network analysis. We provide a comprehensive study of various concepts of walks in temporal graphs, that is, graphs with fixed vertex sets but edge sets changing over time. Importantly, the temporal aspect results in a rich set of optimization criteria for "shortest" walks. Extending and significantly broadening state-ofthe-art work of Wu et al. [IEEE TKDE 2016], we provide a quasi-linear-time algorithm for shortest walk computation that is capable to deal with various optimization criteria and any linear combination of these. A central distinguishing factor to Wu et al.'s work is that our model allows to, motivated by real-world applications, respect waiting-time constraints for vertices, that is, the minimum and maximum waiting time allowed in intermediate vertices of a walk. Moreover, other than Wu et al. our algorithm does not request a strictly increasing time evolvement of the walk and can optimize a richer set of optimization criteria. Our experimental studies indicate that our richer modeling can be achieved without significantly worsening the running time. * Supported by the DFG, projects DAMM (NI 369/13) and FPTinP (NI 369/16).

We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted n-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial time if one can find a set of obligatory edges that yield a spanning subgraph with O(log n) connected components. We also provide a linear-time algorithm that reduces any input graph that consists of a tree together with g additional edges to an equivalent graph with O(g) vertices. Based on this, we obtain a polynomial-time algorithm for g ∈ O(log n). On the negative side, we show that o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard and that there are presumably no exact algorithms running in 2 o(n) time or in f (d) · n O(1) time for any computable function f .

Many real-world networks evolve over time, that is, new contacts appear and old contacts may disappear. They can be modeled as temporal graphs where interactions between vertices (which represent people in the case of social networks) are represented by time-stamped edges. One of the most fundamental problems in (social) network analysis is community detection, and one of the most basic primitives to model a community is a clique. Addressing the problem of finding communities in temporal networks, Viard et al. [TCS 2016] introduced ∆-cliques as a natural temporal version of cliques. Himmel et al. [SNAM 2017] showed how to adapt the well-known Bron-Kerbosch algorithm to enumerate ∆-cliques. We continue this work and improve and extend the algorithm of Himmel et al. to enumerate temporal k-plexes (notably, cliques are the special case k = 1).We define a ∆-k-plex as a set of vertices with a lifetime, where during the lifetime each vertex has in each consecutive ∆ + 1 time steps edges to all but at most k − 1 vertices in the chosen set of vertices. We develop a recursive algorithm for enumerating all maximal ∆-k-plexes and perform experiments on real-world social networks that demonstrate the practical feasibility of our approach. In particular, for the special case of ∆-1-plexes (that is, ∆-cliques), we observe that our algorithm is on average significantly faster than the previous algorithms by Himmel et al. [SNAM 2017] and Viard et al. [IPL 2018] for enumerating ∆-cliques. * An extended abstract of this work appeared in the proceedings of the 2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM '18) [3]. This version contains all proof details, extended experimental findings, and the analysis of a new version of our algorithm that fixed a small bug in the code (which has no large impact on the results). † Supported by the DFG, projects DAMM (NI 369/13) and FPTinP (NI 369/16). ‡ Supported by the DFG, project MATE (NI 369/17).