On several partitioning problems of Bollobás and Scott

AbstractJudicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let G be a hypergraph with mi edges of size i for i=1,2. We show that for any integer k⩾1, V(G) admits a partition into k sets each containing at most m1/k+m2/k2+o(m2) edges, establishing a conjecture of Bollobás and Scott. We also prove that V(G) admits a partition into k⩾3 sets, each meeting at least m1/k+m2/(k−1)+o(m2) edges, which, for large graphs, implies a conjecture of Bollobás and Scott (the conjecture is for all graphs). For k=2, we prove that V(G) admits a partition into two sets each meeting at least m1/2+3m2/4+o(m2) edges, which solves a special case of a more general problem of Bollobás and Scott

Fast and Accurate Species Trees from Weighted Internode Distances

Species tree estimation is a basic step in many biological research projects, but is complicated by the fact that gene trees can differ from the species tree due to processes such as incomplete lineage sorting (ILS), gene duplication and loss (GDL), and horizontal gene transfer (HGT), which can cause different regions within the genome to have different evolutionary histories (i.e., "gene tree heterogeneity"). One approach to estimating species trees in the presence of gene tree heterogeneity resulting from ILS operates by computing trees on each genomic region (i.e., computing "gene trees") and then using these gene trees to define a matrix of average internode distances, where the internode distance in a tree T between two species x and y is the number of nodes in T between the leaves corresponding to x and y. Given such a matrix, a tree can then be computed using methods such as neighbor joining. Methods such as ASTRID and NJst (which use this basic approach) are provably statistically consistent, very fast (low degree polynomial time) and have had high accuracy under many conditions that makes them competitive with other popular species tree estimation methods. In this study, inspired by the very recent work of weighted ASTRAL, we present weighted ASTRID, a variant of ASTRID that takes the branch uncertainty on the gene trees into account in the internode distance. Our experimental study evaluating weighted ASTRID shows improvements in accuracy compared to the original (unweighted) ASTRID while remaining fast. Moreover, weighted ASTRID shows competitive accuracy against weighted ASTRAL, the state of the art. Thus, this study provides a new and very fast method for species tree estimation that improves upon ASTRID and has comparable accuracy with the state of the art while remaining much faster. Weighted ASTRID is available at https://github.com/RuneBlaze/internode

Find Your Place: Simple Distributed Algorithms for Community Detection

International audienceGiven an underlying graph, we consider the following dynamics: Initially, each node locally chooses a value in {−1, 1}, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of the values held by its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node. We prove that the process resulting from this dynamics produces a clustering that exactly or approximately (depending on the graph) reects the underlying cut in logarithmic time, under various graph models that exhibit a sparse balanced cut, including the stochastic block model. We also prove that a natural extension of this dynamics performs community detection on a regularized version of the stochastic block model with multiple communities. Rather surprisingly, our results provide rigorous evidence for the ability of an extremely simple and natural dynamics which is non-trivial even in a centralized setting

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Algorithms and Software for the Analysis of Large Complex Networks

The work presented intersects three main areas, namely graph algorithmics, network science and applied software engineering. Each computational method discussed relates to one of the main tasks of data analysis: to extract structural features from network data, such as methods for community detection; or to transform network data, such as methods to sparsify a network and reduce its size while keeping essential properties; or to realistically model networks through generative models