Thermal properties of structurally balanced systems on classical random graphs

FPT algorithms for domination in sparse graphs and beyond

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $\lambda:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph ${\cal G}=(G,\lambda)$ is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity---a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of $\Theta(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. The natural question is then whether sparse temporal spanners can always be found in some classes of dense graphs. In this paper, we answer this question affirmatively, by showing that if the underlying graph $G$ is a complete graph, then one can always find temporal spanners of density $O(n \log n)$. The best known result for complete graphs so far was that spanners of density ${n \choose 2}- \lfloor n/4 \rfloor = O(n^2)$ always exist. Our result is the first positive answer as to the existence of $o(n^2)$ sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.

Minimal separators in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>P</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math>-tidy graphs

The dynamics of networks on Heider's balance theory moves toward reducing the tension by constantly reevaluating the interactions to achieve a state of balance. Conflict of interest, however, is inherent in most complex systems; frequently, there are multiple ideals or states of balance, and moving towards one could work against another. In this paper, by introducing the competitive balance theory, we study the evolution of balance in the presence of conflicts of interest. In our model, the assumption is that different states of balance compete in the evolution process to dominate the system. We ask whether, through these interactions, different states of balance compete to prevail their own ideals or a set of coexisting ideals in a balanced condition is a possible outcome. The results show that although there is a symmetry in the type of balance the system either evolves towards a symmetry breaking, where one of the states of balance dominates the system, or, less frequently, the competing states of balance coexist in a jammed state.

In recent years, a variety of randomized constructions of sketching matrices have been devised, that have been used in fast algorithms for numerical linear algebra problems, such as least squares regression, low-rank approximation, and the approximation of leverage scores. A key property of sketching matrices is that of subspace embedding. In this paper, we study sketching matrices that are obtained from bipartite graphs that are sparse, i.e., have left degree s that is small. In particular, we explore two popular classes of sparse graphs, namely, expander graphs and magical graphs. For a given subspace ${\mathcal{U}} \subseteq {{\mathbb{R}}^n}$ of dimension k, we show that the magical graph with left degree s = 2 yields a (1 ± ϵ) ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -subspace embedding for ${\mathcal{U}}$, if the number of right vertices (the sketch size) $m = {\mathcal{O}}\left( {{k^2}/{\varepsilon ^2}} \right)$. The expander graph with $s = {\mathcal{O}}(\log k/\varepsilon )$ yields a subspace embedding for $m = {\mathcal{O}}\left( {k\log k/{\varepsilon ^2}} \right)$. We also discuss the construction of sparse sketching matrices with reduced randomness using expanders based on error-correcting codes. Empirical results on various synthetic and real datasets show that these sparse graph sketching matrices work very well in practice.

Two graphs G and H with the same vertex set Vare P4 -isomorphic if every four vertices {a,b,c,d} \subseteq V$ induce a chordless path (denoted by P4 ) in G if and only if they induce a P4 in H. We call a graph split-perfect if it is P4 -isomorphic to a split graph (i.e., a graph being partitionable into a clique and a stable set). This paper characterizes the new class of split-perfect graphs using the concepts of homogeneous sets and p-connected graphs and leads to a linear time recognition algorithm for split-perfect graphs, as well as efficient algorithms for classical optimization problems on split-perfect graphs based on the primeval decomposition of graphs. The optimization results considerably extend previous ones on smaller classes such as P4 --sparse graphs, P4 -lite graphs, P4 --laden graphs, and (7,3)-graphs. Moreover, split-perfect graphs form a new subclass of brittle graphs containing the superbrittle graphs for which a new characterization is obtained leading to linear time recognition.

CGMAE: Self-supervised Masked Auto-Encoder with Cross-Graph node alignment for node classification

The intensity of traffic flows increases every year and intensifies its impact significantly on urban territory's ecologic situation and natural environment. In Ukrainian cities, the growth of noise pollution is quite a topical problem as it has a significant impact on the environment because of the increase in car numbers. Traffic flow intensity and densification of the built-up area cause the increase of the noise level. As analytical methods of calculation of determination of traffic noise level have several disadvantages, then for assessing the level of noise on arterial streets of Lviv city, research is carried out with consideration of traffic flow indicators. One of the critical criteria that have an impact on noise pollution in the urban environment is traffic flow composition. Conducted research indicates that the speed of traffic flow creates significant noise pollution, especially on those streets where cobblestone is a road pavement. Freight and passenger transport that moves along arterial streets causes the increase of noise level and increases its impact on the city territory. On sections of arterial streets in peak periods of the day with high values of volume-capacity ratio and general share of freight and public transport of more than 30% in a flow, the noise exceeds the acceptable norms and has a harmful impact on the environment. That is why it is necessary to assess noise pollution and the development of protection measures from noise during the planning and construction of some urban regions or residential areas. It is established that to lower the noise level on streets, it is necessary to consider existing intersections of arterial streets and the regime of movement on them. To lower transport noise, it is necessary to distribute freight and transit transport by city streets more rationally, use qualitative public transport, and limit the maximal speed regime on city streets. Further research on the road network will provide an opportunity to assess the noise level distribution of traffic flows in Lviv city. It will help to determine the problem places and propose methods of combating traffic noise.

We overview algorithmic results for classes of sparse graphs emphasizing new developments in this area. We focus on recently introduced classes of graphs with bounded expansion and nowhere-dense graphs and relate algorithmic meta-theorems for these classes of graphs to their analogues for proper minor-closed classes of graphs, classes of graphs with bounded tree-width, locally bounded tree-width and locally excluding a minor.

It is a well established fact, that—in the case of classical random graphs like (variants of) G n,p or random regular graphs-spectral methods yield efficient algorithms for clustering (e. g. coiouring or bisection) problems. The theory of large networks emerging recently provides convincing evidence that such networks, albeit looking random in some sense, cannot sensibly be described by classical random graphs. A variety of new types of random graphs have been introduced. One of these types is characterized by the fact that we have a fixed expected degree sequence, that is for each vertex its expected degree is given. Recent theoretical work confirms that spectral methods can be successfully applied to clustering problems for such random graphs, too—provided that the expected degrees are not too small, in fact≥log6 n. In this case however the degree of each vertex is concentrated about its expectation. We show how to remove this restriction and apply spectral methods when the expected degrees are bounded below just by a suitable constant. Our results rely on the observation that techniques developed for the classical sparse G n,p random graph (that is p=c/n) can be transferred to the present situation, when we consider a suitably normalized adjacency matrix: We divide each entry of the adjacency matrix by the product of the expected degrees of the incident vertices. Given the host of spectral techniques developed for G n,p this observation should be of independent interest.

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r+2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings.Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems.

Atualmente, não apenas os elementos do ambiente térmico influenciam a produtividade dos suínos, mas, também, a interação com o ambiente aéreo gerado na instalação, junto com o nível de ruídos. Estudou-se, neste trabalho, o perfil do ambiente acústico dentro de um galpão de creche de suínos usando-se mapas de distribuição da média dos níveis de ruído, em função da temperatura e umidade relativa do ar. O ambiente construído foi dividido em células/quadrantes de 2 m² e medidos, no centro geométrico no nível do leitão, o nível de ruído, a temperatura de bulbo seco e a umidade relativa do ar. Com os resultados mapeados, notou-se que o nível médio de ruído da creche foi de 12,49 dB mas não se encontraram correlações entre o ambiente térmico e o ambiente acústico. Os níveis de ruído obtidos estão abaixo dos valores preconizados internacionalmente para alojamento de suínos e dentro das normas nacionais vigentes para os trabalhadores da granja. Os resultados apontam que os leitões recém transferidos para a creche vocalizaram mais que aqueles que já estavam habituados ao ambiente (10,98 dB), resultando em nível médio de ruído superior (12,58 dB).

How spacetime emerges from featureless nothingness is one of the most intriguing questions in fundamental physics. In this thesis, we take on the random geometry approach to study discretized spacetime and follow the assumption that, in the sim- plest form, geometric structures may arise from random connections between dots under certain rules. We study a family of random graph models called Exponential Random Graph Models (ERGMs). Although this family was extensively investigated in the network science community as a proxy to study real-world social networks, its strength is in its formulation as a Gibbs-Boltzmann distribution in equilibrium sta- tistical mechanics. Thus, one can modify, analyze, and simulate the ensemble using familiar tools from statistical mechanics. We are interested in modifying the basic ERGMs to arrive at a random graph model that possesses emergent geometric properties. This would serve as a proof-of-principle that geometric spacetime may emerge from randomly connected dots. Our study leads to novel classes of random graphs whose edges can self-assemble themselves into both simple geometric primitives (e.g. triangles) and more complex structures (e.g. hexagons). The number of such structures is relatively large compared to the amount of dots and connections available in the graph. Lastly, but interestingly, our model is free from the graph collapse problem that is often observed in the traditional ERGMs.

We study the detection and the reconstruction of a large very dense subgraph in a social graph with n nodes and m edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$ . A subgraph S is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$ . Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$ , then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.

The ability to exchange secret information is critical to many commercial, governmental, and military networks. The intrinsically secure communications graph (iS-graph) is a random graph which describes the connections that can be securely established over a large-scale network, by exploiting the physical properties of the wireless medium. This paper provides an overview of the main properties of this new class of random graphs. We first analyze the local properties of the iS-graph, namely the degree distributions and their dependence on fading, target secrecy rate, and eavesdropper collusion. To mitigate the effect of the eavesdroppers, we propose two techniques that improve secure connectivity. Then, we analyze the global properties of the iS-graph, namely percolation on the infinite plane, and full connectivity on a finite region. These results help clarify how the presence of eavesdroppers can compromise secure communication in a large-scale network.