The maximum number of induced open triangles in graphs of a given order

Pyatkin, A. V.; Lykhovyd, Eugene; Butenko, Sergiy

doi:10.1007/s11590-018-1330-2

Cited by 9 publications

(5 citation statements)

References 7 publications

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“…Therefore, (35) implies that for every n ≥ N 1 + N 2 we have D n (r) ≤ 𝜖. This proves that D n (r) → 0 when n → ∞ for any 0 ≤ r < 1, whereby Q n (p) → 1, n → ∞ for any 1∕2 < p ≤ 1 by virtue of (29). ▪…”

Section: Independent Union Of Cliquesmentioning

confidence: 72%

“…However, from Figure 3 one might see that for

p>0.5

, the probability of the set of vertices of

G\left(n,p\right)

to form an IUC quickly approaches the probability of forming a clique. Interestingly, the breaking point is not

2/3

, when the expected number of open triangles is maximized [29]. This observation will be established formally in the following theorem.…”

Section: Special Cases: Asymptotic Almost Sure Bounds For Independent...mentioning

confidence: 99%

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Asymptotic bounds for clustering problems in random graphs

Lykhovyd,

Butenko,

Krokhmal

2023

Networks

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Graph clustering is an important problem in network analysis. This problem can be approached by first finding a large cluster subgraph (i.e., a subgraph in which every connected component is a complete graph), perhaps in a relaxed form (connected components may have missing edges), and then assigning each of the remaining vertices to one of the connected components of the cluster subgraph according to some optimization criteria. The more vertices can be included in the initial cluster subgraph (also referred to as independent union of clusters), the more “clusterable” the graph is. This paper proposes a framework for establishing asymptotic bounds on the cardinality of independent unions of clusters in Erdős‐Rényi random graphs with constant , referred to as uniform random graphs. In particular, sufficient conditions ensuring (where is the number of nodes) upper bounds with probability 1 are developed and shown to be applicable for the maximum independent union of cliques as well as some clique relaxations. In addition, it is shown that every graph must have an independent union of cliques of cardinality at least . Since this bound is asymptotically tight on uniform random graphs, this suggests that these graphs can be viewed as a “least clusterable” class of graphs.

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Section: Independent Union Of Cliquesmentioning

confidence: 72%

“…However, from Figure 3 one might see that for

p>0.5

, the probability of the set of vertices of

G\left(n,p\right)

to form an IUC quickly approaches the probability of forming a clique. Interestingly, the breaking point is not

2/3

, when the expected number of open triangles is maximized [29]. This observation will be established formally in the following theorem.…”

Section: Special Cases: Asymptotic Almost Sure Bounds For Independent...mentioning

confidence: 99%

Asymptotic bounds for clustering problems in random graphs

Lykhovyd,

Butenko,

Krokhmal

2023

Networks

Self Cite

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“…Graphs with the maximal number of cherries were determined in [23] (cherries are called open triangles in that paper). We will use the same line of reasoning to find the maximal value of ns(A(G)).…”

Section: Declarationsmentioning

confidence: 99%

Measuring associativity: graph algebras of undirected graphs

Kátai-Urbán,

Waldhauser

2024

Algebra Univers.

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We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent.

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“…| and the number of edges equals the number of wedges in A, which is bounded by O(|V | 3 ), see [39], leading to a total running time and space complexity of O(|V | 3 ).…”

Section: Compute the Vertex-weighted Wedge Graphmentioning

confidence: 99%

Inferring Tie Strength in Temporal Networks

Oettershagen¹,

Konstantinidis²,

Italiano³

2022

Preprint

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Inferring tie strengths in social networks is an essential task in social network analysis. Common approaches classify the ties as weak and strong ties based on the strong triadic closure (STC). The STC states that if for three nodes, A, B, and C, there are strong ties between A and B, as well as A and C, there has to be a (weak or strong) tie between B and C. So far, most works discuss the STC in static networks. However, modern large-scale social networks are usually highly dynamic, providing user contacts and communications as streams of edge updates. Temporal networks capture these dynamics. To apply the STC to temporal networks, we first generalize the STC and introduce a weighted version such that empirical a priori knowledge given in the form of edge weights is respected by the STC. The weighted STC is hard to compute, and our main contribution is an efficient 2-approximative streaming algorithm for the weighted STC in temporal networks. As a technical contribution, we introduce a fully dynamic 2-approximation for the minimum weight vertex cover problem, which is a crucial component of our streaming algorithm. Our evaluation shows that the weighted STC leads to solutions that capture the a priori knowledge given by the edge weights better than the non-weighted STC. Moreover, we show that our streaming algorithm efficiently approximates the weighted STC in large-scale social networks.

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The maximum number of induced open triangles in graphs of a given order

Cited by 9 publications

References 7 publications

Asymptotic bounds for clustering problems in random graphs

Asymptotic bounds for clustering problems in random graphs

Measuring associativity: graph algebras of undirected graphs

Inferring Tie Strength in Temporal Networks

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