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.

/pdf/generalized-intersection-over-union-a-metric-and-a-loss-for-1m6sjmes8c.pdf

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

Computational geometry

Survey: Graph clustering

Given a p-order A over a universe of strings (i.e., a transitive, reflexive, antisymmetric relation such that if (x, y) is an element of A then |x| is polynomially bounded by |y|), an interval size function of A returns, for each string x in the universe, the number of strings in the interval between strings b(x) and t(x) (with respect to A), where b(x) and t(x) are functions that are polynomial-time computable in the length of x. 
By choosing sets of interval size functions based on feasibility requirements for their underlying p-orders, we obtain new characterizations of complexity classes. We prove that the set of all interval size functions whose underlying p-orders are polynomial-time decidable is exactly #P. We show that the interval size functions for orders with polynomial-time adjacency checks are closely related to the class FPSPACE(poly). Indeed, FPSPACE(poly) is exactly the class of all nonnegative functions that are an interval size function minus a polynomial-time computable function. 
We study two important functions in relation to interval size functions. The function #DIV maps each natural number n to the number of nontrivial divisors of n. We show that #DIV is an interval size function of a polynomial-time decidable partial p-order with polynomial-time adjacency checks. The function #MONSAT maps each monotone boolean formula F to the number of satisfying assignments of F. We show that #MONSAT is an interval size function of a polynomial-time decidable total p-order with polynomial-time adjacency checks. 
Finally, we explore the related notion of cluster computation.

The Complexity of Computing the Size of an Interval

We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed walks of length k starting at a given vertex v. The first is a direct implication of a matrix inequality by Marcus and Newman and generalizes two inequalities by Lagarias et al. and Dress & Gutman. We then use an inequality of Blakley and Dixon to show the inequality wk2e+p ≤ w2e+pk · wk−12e which also generalizes the inequality by Dress and Gutman and also an inequality by Erdos and Simonovits. Both results can be translated directly into the corresponding forms using the higher order densities, which extends former results.

In the second part, we provide a new family of lower bounds for the largest eigenvalue λ1 of the adjacency matrix based on closed walks and apply the before mentioned inequalities to show monotonicity in this and a related family of lower bounds of Nikiforov. This leads to generalized upper bounds for the energy of graphs.

In the third part, we demonstrate that a further natural generalization of the inequality w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) is not valid for general graphs. We show that wa+b · wa+b+c ≤ wa · wa+2b+c does not hold even in very restricted cases like w1 · w2 ≤ w0 · w3 (i.e., d · w2 ≤ w3) in the context of bipartite or cycle free graphs. In contrast, we show that surprisingly this inequality is always satisfied for trees and show how to construct worst-case instances (regarding the difference of both sides of the inequality) for a given degree sequence. We also provide a proof for the inequality w1 · w4 ≤ w0 · w5 (i.e., d · w4 ≤ w5) for trees and conclude with a corresponding conjecture for longer walks.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973020.4

Inequalities for the number of walks in graphs

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, i.e., 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.

/pdf/acyclic-type-of-relationship-problems-on-the-internet-y73ue2peoj.pdf

Acyclic type-of-relationship problems on the internet

This work studies (lowest) common ancestor problems in (weighted) directed acyclic graphs. We improve previous algorithms for the all-pairs representative LCA problem to O(n2.575) by using fast rectangular matrix multiplication. We prove a first non-trivial upper bound of O(min{n2m, n3.575}) for the all-pairs all lowest common ancestors problem. Furthermore, classes of dags are identified for which the problem can be solved considerably faster. Our algorithms scale with the maximal number of LCAs for one pair and--based on the famous Dilworth's theorem--with the size of a maximum antichain (i.e., width) of the dag. We extend and generalize previous results on computing shortest ancestral distances. It is shown that finding shortest distance common ancestors in weighted dags is not harder than computing all-pairs shortest distances, up to a polylogarithmic factor. Finally, we present a solution for the general all-pairs shortest distance LCA problem based on computing all-pairs all LCAs.

All-pairs ancestor problems inweighted dags

We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let γ : N → Q+ be any density function, i.e., γ is computable in polynomial time and satisfies γ(k) ≤ k - 1 for all k ∈ N. Then γ-Cluster is the problem of deciding, given an undirected graph G and a natural number k, whether there is a subgraph of G on k vertices which has average degree at least γ(k). For γ(k) = k-1, this problem is the same as the well-known clique problem, and thus NP-complete. In contrast to this, the problem is known to be solvable in polynomial time for γ(k) = 2. We ask for the possible functions γ such that γ-CLUSTER remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: γ-Cluster is NP-complete if γ = 2+Ω(1/k1-ɛ) for some ɛ > 0 and has a polynomial-time algorithm for γ = 2+O(1/k).

Sven Kosub

Papers

The Complexity of Computing the Size of an Interval

Inequalities for the number of walks in graphs

Acyclic type-of-relationship problems on the internet

All-pairs ancestor problems inweighted dags

The complexity of detecting fixed-density clusters