The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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Community detection in graphs

Intersection over Union (IoU) is the most popular evaluation metric used in the object detection benchmarks. However, there is a gap between optimizing the commonly used distance losses for regressing the parameters of a bounding box and maximizing this metric value. The optimal objective for a metric is the metric itself. In the case of axis-aligned 2D bounding boxes, it can be shown that IoU can be directly used as a regression loss. However, IoU has a plateau making it infeasible to optimize in the case of non-overlapping bounding boxes. In this paper, we address the this weakness by introducing a generalized version of IoU as both a new loss and a new metric. By incorporating this generalized IoU ( GIoU) as a loss into the state-of-the art object detection frameworks, we show a consistent improvement on their performance using both the standard, IoU based, and new, GIoU based, performance measures on popular object detection benchmarks such as PASCAL VOC and MS COCO.

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Generalized Intersection Over Union: A Metric and a Loss for Bounding Box Regression

Computational geometry

Survey: Graph clustering

This research uses football online betting odds of a broad variety of matches and bookmakers to identify known biases in odds pricing, namely the favorite-longshot bias and the away-favorite bias. Furthermore, it tries to answer the question whether a naive strategy of betting against these biases can be profitable. Our findings are consistent with previous research, confirming the existence of the favorite-longshot bias, but turn out to be in contrast with regard to the away-favorite bias, for which there is little evidence in the data. Also, we find that a strategy of betting against the favorite-longshot bias can be marginally profitable, whereas bets in favor of it result in large losses on average.

Biases in the Football Betting Market

We introduce the boolean hierarchy of k-partitions over NP for k ≥ 3 as a generalization of the boolean hierarchy of sets (i.e., 2- partitions) over NP. Whereas the structure of the latter hierarchy is rather simple the structure of the boolean hierarchy of k-partitions over NP for k ≥ 3 turns out to be much more complicated. We establish the Embedding Conjecture which enables us to get a complete idea of this structure. This conjecture is supported by several partial results.

The Boolean Hierarchy of NP-Partitions

This work draws attention to combinatorial network abstraction problems which are specified by a class $\mathcal{P}$ of pattern graphs and a real-valued similarity measure $\varrho$ based on certain graph properties. For fixed $\mathcal{P}$ and $\varrho$, the optimization task on any graph G is to find a subgraph G′ which belongs to $\mathcal{P}$ and minimizes $\varrho(G,G^{\prime})$. We consider this problem for the natural case of trees and distance-based similarity measures. In particular, we systematically study spanning trees of graphs that minimize distances, approximate distances, and approximate closeness-centrality with respect to some standard vector and matrix norms. The complexity analysis shows that all considered variants of the problem are NP-complete, except for the case of distance-minimization with respect to the L∞ norm. We further show that unless P = NP, there exist no polynomial-time constant-factor approximation algorithms for the distance-approximation problems if a subset of edges can be forced into the spanning tree.

Combinatorial network abstraction by trees and distances

Though complexity theory already extensively studies path-cardinality-based restrictions on the power of nondeterminism, this paper is motivated by a more recent goal: To gain insight into how much of a restriction, it is of nondeterminism to limit machines to have just one contiguous (with respect to some simple order) interval of accepting paths. In particular, we study the robustness-the invariance under definition changes-of the cluster class CL#P. This class contains each #P function that is computed by a balanced Turing machine whose accepting paths always form a cluster with respect to some length-respecting total order with efficient adjacency checks. The definition of CL#P is heavily influenced by the defining paper's focus on (global) orders. In contrast, we define a cluster class, CLU#P, to capture what seems to us a more natural model of cluster computing. We prove that the naturalness is costless: CL#P=CLU#P. Then we exploit the more natural, flexible features of CLU#P to prove new robustness results for CL#P and to expand what is known about the closure properties of CL#P. The complexity of recognizing edges-of an ordered collection of computation paths or of a cluster of accepting computation paths-is central to this study. Most particularly, our proofs exploit the power of unique discovery of edges-the ability of nondeterministic functions to, in certain settings, discover on exactly one (in some cases, on at most one) computation path a critical piece of information regarding edges of orderings or clusters.

Cluster computing and the power of edge recognition

We contribute to the study of inferring commercial relationships between autonomous systems (AS relationships) from observable BGP routes We deduce several forbidden patterns of AS relationships that impose a certain type of acyclicity on the AS graph We investigate algorithms for solving the acyclic all-paths type-of-relationship problem, ie, given a set of AS paths, find an orientation of the edges according to some types of AS relationships such that the oriented AS graph is acyclic (with respect to the forbidden patterns) and all AS paths are valley-free As possible AS relationships we include customer-to-provider, peer-to-peer, and sibling-to-sibling Moreover, we examine a number of problem versions parameterized by sets K and U where K is the set of edge types available for describing explicit pre-knowledge and U is the set of edge types available for completion of partial orientations A complete complexity classification of all 56 cases (8 type sets for pre-knowledge and 7 type sets for completion) is given The most relevant practical result is a linear-time algorithm for finding an acyclic and valley-free completion using customer-to-provider relations given any kind of pre-knowledge Interestingly, if we allow sibling-to-sibling relations for completions then most of the non-trivial inference problems become NP-hard

Sven Kosub

Papers

Biases in the Football Betting Market

The Boolean Hierarchy of NP-Partitions

Combinatorial network abstraction by trees and distances

Cluster computing and the power of edge recognition

Acyclic Type-of-Relationship Problems on the Internet