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Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many realworld graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior andmore »

Counting homomorphisms of a constant sized pattern graph H in an input graph G is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input G and the pattern H. Given the significance of this problem and the large sizes of modern inputs, we investigate when nearlinear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of H, Hhomomorphisms can be counted exactly in nearlinear time in bounded degeneracy graphs. Can we precisely characterize the patterns H for which nearlinear time algorithms are possible? We completely resolve this problem, discovering a clean dichotomy using finegrained complexity. Let m denote the number of edges in G. We prove the following: if the largest induced cycle in H has length at most 5, then there is an O(m log m) algorithm for counting Hhomomorphisms in bounded degeneracy graphs. If the largest induced cycle in H has length at least 6, then (assuming standard finegrained complexity conjectures) there is a constant γ > 0,more »