Maximum Clique Size Research Articles

Overlapping community detection in weighted networks via hierarchical clustering.

In real-world networks, community structures often appear as tightly connected clusters of nodes, with recent studies suggesting a hierarchical organization where larger groups subdivide into smaller ones across different levels. This hierarchical structure is particularly complex in trade networks, where actors typically belong to multiple communities due to diverse business relationships and contracts. To address this complexity, we present a novel algorithm for detecting hierarchical structures of overlapping communities in weighted networks, focusing on the interdependency between internal and external quality metrics for evaluating the detected communities. The proposed Graph Hierarchical Agglomerative Clustering (GHAC) approach utilizes maximal cliques as the basis units for hierarchical clustering. The algorithm measures dissimilarities between clusters using the minimal closed trail distance (CT-distance) and the size of maximal cliques within overlaps, capturing the density and connectivity of nodes. Through extensive experiments on synthetic networks with known ground truth, we demonstrate that the adjusted Silhouette index is the most reliable internal metric for determining the optimal cut in the dendrogram. Experimental results indicate that the GHAC method is competitive with widely used community detection techniques, particularly in networks with highly overlapping communities. The method effectively reveals the hierarchical structure of communities in weighted networks, as demonstrated by its application to the OECD weighted trade network, which describes the balanced trade value of bilateral trade relations.

Efficient non-isomorphic graph enumeration algorithms for several intersection graph classes

Intersection graphs are well-studied in the area of graph algorithms. Some intersection graph classes are known to have algorithms enumerating all unlabeled graphs by reverse search. Since these algorithms output graphs one by one and the numbers of graphs in these classes are vast, they work only for a small number of vertices. Binary decision diagrams (BDDs) are compact data structures for various types of data and useful for solving optimization and enumeration problems. This study proposes enumeration algorithms for five intersection graph classes, which admit O(n)-bit string representations for their member graphs. Our algorithm for each class enumerates all unlabeled graphs with n vertices over BDDs representing the binary strings in time polynomial in n. Moreover, our algorithms are extended so that it enumerates those with constraints on the maximum (bi)clique size and/or the number of edges.

Classification of schizophrenic traits in transcriptions of audio spectra from patient literature: artificial intelligence models enhanced by geometric properties

Schizophrenia is a severe mental illness that affects approximately 1% of the global population and presents significant challenges for patients, families, and healthcare professionals. Characterized by symptoms such as delusions, hallucinations, disorganized speech or behavior, and cognitive impairment, this condition has an early onset and chronic trajectory, making it a debilitating challenge. Schizophrenia also imposes a substantial burden on society, exacerbated by the stigma associated with mental disorders. Technological advancements, such as computerized semantic, linguistic, and acoustic analyses, are revolutionizing the understanding and assessment of communication alterations, a significant aspect in various severe mental illnesses. Early and accurate diagnosis is crucial for improving prognosis and implementing appropriate treatments. In this context, the advancement of Artificial Intelligence (AI) has provided new perspectives for the treatment of schizophrenia, with machine learning techniques and natural language processing allowing a more detailed analysis of clinical, neurological, and behavioral data sets. The present article aims to present a proposal for computational models for the identification of schizophrenic traits in texts. The database used in this article was created with 139 excerpts of patients' speeches reported in the book “Memories of My Nervous Disease” by German judge Daniel Paul Schreber, classifying them into three categories: 1 - schizophrenic, 2 - with schizophrenic traits and 3 - without any relation to the disorder. Of these speeches, 104 were used for training the models and the others 35 for validation.Three classification models were implemented using features based on geometric properties of graphs (number of vertices, number of cycles, girth, vertex of maximum degree, maximum clique size) and text entropy. Promising results were observed in the classification, with the Decision Tree-based model [1] achieving 100% accuracy, the KNN- k-Nearest Neighbor model observed with 62.8% accuracy, and the 'centrality-based' model with 59% precision. The high precision rates, observed when geometric properties are incorporated into Artificial Intelligence Models, suggest that the models can be improved to the point of capturing the language deviation traits that are indicative of schizophrenic disorders. In summary, this study paves the way for significant advances in the use of geometric properties in the field of psychiatry, offering a new data-based approach to the understanding and therapy of schizophrenia.

Convergence rates for sums-of-squares hierarchies with correlative sparsity

AbstractThis work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.

Computing L(p,1)-Labeling with Combined Parameters

Given a graph, an $L(p,1)$-labeling of the graph is an assignment $f$ from the vertex set to the set of nonnegative integers such that for any pair of vertices $u$ and $v$, $|f (u) - f (v)| \ge p$ if $u$ and $v$ are adjacent, and $f(u) \neq f(v)$ if $u$ and $v$ are at distance $2$. The \textsc{$L(p,1)$-labeling} problem is to minimize the span of $f$ (i.e.,$\max_{u\in V}(f(u)) - \min_{u\in V}(f(u))+1$). It is known to be NP-hard even for graphs of maximum degree $3$ or graphs with tree-width 2, whereas it is fixed-parameter tractable with respect to vertex cover number. Since the vertex cover number is a kind of the strongest parameter, there is a large gap between tractability and intractability from the viewpoint of parameterization. To fill up the gap, in this paper, we propose new fixed-parameter algorithms for ${L(p,1)-{\rm L{\small ABELING}}}$ by the twin cover number plus the maximum clique size and by the tree-width plus the maximum degree. These algorithms reduce the gap in terms of several combinations of parameters.

Cellular goore game with application to finding maximum clique in social networks

Abstract The goore game (GG) is a model for collective decision making under uncertainty, which can be used as a tool for stochastic optimization of a discrete variable function. The GG has a fascinating property that can be resolved in a distributed manner with no intercommunication between the players. The game has found applications in many network applications, including sensor networks, quality-of-service routing, and social networks. In this paper, we introduce an extension of GG called cellular goore game (CGG) for the first time. The CGG is a network of GGs. In this network, each node (or subset of nodes in the network) plays the rule of referees, each of which participates in a GG with its neighboring players (voters) at any time. Like in GG, each player independently selects its optimal action between two available actions based on their gains and losses received from its adjacent referee. Players in CGG know nothing about how other players are playing or even how/why they are rewarded/penalized. The potential of the CGG is shown by providing an algorithm for finding a maximum clique in social networks. Our research provides the first-time study of the CGG for finding a maximum clique in graphs. The performance of the CGG-based algorithm for finding maximum clique is studied on the standard clique benchmark called DIMACS by several experiments. The obtained result shows that the CGG-based algorithm is superior to the existing algorithms in terms of finding maximum clique size and time.

Exact Maximum Clique Algorithm for Different Graph Types Using Machine Learning

Finding a maximum clique is important in research areas such as computational chemistry, social network analysis, and bioinformatics. It is possible to compare the maximum clique size between protein graphs to determine their similarity and function. In this paper, improvements based on machine learning (ML) are added to a dynamic algorithm for finding the maximum clique in a protein graph, Maximum Clique Dynamic (MaxCliqueDyn; short: MCQD). This algorithm was published in 2007 and has been widely used in bioinformatics since then. It uses an empirically determined parameter, Tlimit, that determines the algorithm’s flow. We have extended the MCQD algorithm with an initial phase of a machine learning-based prediction of the Tlimit parameter that is best suited for each input graph. Such adaptability to graph types based on state-of-the-art machine learning is a novel approach that has not been used in most graph-theoretic algorithms. We show empirically that the resulting new algorithm MCQD-ML improves search speed on certain types of graphs, in particular molecular docking graphs used in drug design where they determine energetically favorable conformations of small molecules in a protein binding site. In such cases, the speed-up is twofold.

On the complexity of coloring ‐graphs

AbstractAn ‐graph is a graph that can be partitioned into r independent sets and ℓ cliques. In the Graph Coloring problem, we are given as input a graph G, and the objective is to determine the minimum integer k such that G admits a proper vertex k‐coloring. In this work, we describe a Poly versus NP‐hard dichotomy of this problem regarding the parameters r and ℓ of ‐graphs, which determine the boundaries of the ‐hardness of Graph Coloring for such classes. We also analyze the complexity of the problem on ()‐graphs under the parameterized complexity perspective. We show that given a (2, 1)‐partition of a graph G, finding an optimal coloring of G is ‐complete even when K1 is a maximal clique of size 3; XP but W[1]‐hard when parameterized by ; fixed‐parameter tractable (FPT) and admits a polynomial kernel when parameterized by . Besides, concerning the case where K1 is a maximal clique of size 3, a P versus NPh dichotomy regarding the neighborhood of K1 is provided; furthermore, an FPT algorithm parameterized by the number of vertices having no neighbor in K1 is presented.

Equitable Clique-Coloring in Claw-Free Graphs with Maximum Degree at Most 4

A clique of a graph G is a set of pairwise adjacent vertices of G. A clique-coloring of G is an assignment of colors to the vertices of G in such a way that no inclusion-wise maximal clique of size at least two of G is monochromatic. An equitable clique-coloring of G is a clique-coloring such that any two color classes differ in size by at most one. Bacso and Tuza proved that connected claw-free graphs with maximum degree at most four, other than chordless odd cycles of order greater than three, are 2-clique-colorable and a 2-clique-coloring can be found in $$O(n^{2})$$ Bacso and Tuza (Discrete Math Theor Comput Sci 11(2):15–24, 2009). In this paper we prove that every connected claw-free graph with maximum degree at most four, not a chordless odd cycle of order greater than three, has an equitable 2-clique-coloring. In addition we improve the algorithm described in the paper mentioned by giving an equitable 2-clique-coloring in linear time for this class of graphs.

Cliques and Chromatic Number in Multiregime Random Graphs

In this paper, we study cliques and chromatic number in the random subgraph Gn of the complete graph Kn on n vertices, where each edge is independently open with a probability pn. Associating Gn with the probability measure ℙn, we say that the sequence {ℙn} is multiregime if the edge probability sequence {pn} is not convergent. Using a recursive method we obtain uniform bounds on the maximum clique size and chromatic number for such multiregime random graphs.

Dynamic data structures for interval coloring

We consider the dynamic graph coloring problem restricted to the class of interval graphs in the incremental and fully dynamic setting. The input consists of a sequence of intervals that are to be either colored, or deleted, if previously colored. For the incremental setting, we consider the well studied optimal online algorithm (KT-algorithm) for interval coloring due to Kierstead and Trotter [1]. We present the following results on the dynamic interval coloring problem.■Any direct implementation of the KT-algorithm requires Ω(Δ2) time per interval in the worst case.■There exists an incremental algorithm which supports insertion of an interval in amortized O(log⁡n+Δ) time per update and maintains a proper coloring using at most 3ω−2 colors.■There exists a fully dynamic algorithm which supports insertion of an interval in O(log⁡n+Δlog⁡ω) update time and deletion of an interval in O(Δ2log⁡n) update time in the worst case and maintains a proper coloring using at most 3ω−2 colors. The KT-algorithm crucially uses the maximum clique size in an induced subgraph in the neighborhood of a given vertex. We show that the problem of computing the induced subgraph among the neighbors of a given vertex has the same hardness as the online boolean matrix vector multiplication problem [2]. We show that■Any algorithm that computes the induced subgraph among the neighbors of a given vertex requires at least quadratic time unless the OMv conjecture [2] is false. Finally, we obtain the following result on the OMv conjecture.■If the matrix and the vectors in the online sequence have the consecutive ones property, then the OMv conjecture [2] is false.

Analysis of Financial Network Topological Dynamics of the Russian Stock Market from 2012 to 2019

In this paper we study some properties of the Russian stock market with the application of network approaches and data-driven science. Complex networks theory allows us to construct and analyse topological network structures of the market. Among the important information which is possible to acquire from it is the relationships between stocks returns with the analysis of hidden information and market dynamics. This paper is focused on the analysis of the market network dynamics over time. We construct market networks for 75 consecutive overlapping 250-day periods to analyze the dynamics of the structural properties of the market rank-correlation-based network. Degree distribution and maximum clique size are considered as the important structural characteristics of the market network. In our opinion these parameters are the essential graph attributes and give insight into Russian financial market structure.

Clique-Coloring of $$K_{3,3}$$-Minor Free Graphs

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this paper, we generalize these results to $$K_{3,3}$$ -minor free ( $$K_{3,3}$$ -subdivision free) graphs.

Property testing in high-dimensional Ising models

This paper explores the information-theoretic limitations of graph property testing in zero-field Ising models. Instead of learning the entire graph structure, sometimes testing a basic graph property such as connectivity, cycle presence or maximum clique size is a more relevant and attainable objective. Since property testing is more fundamental than graph recovery, any necessary conditions for property testing imply corresponding conditions for graph recovery, while custom property tests can be statistically and/or computationally more efficient than graph recovery based algorithms. Understanding the statistical complexity of property testing requires the distinction of ferromagnetic (i.e., positive interactions only) and general Ising models. Using combinatorial constructs such as graph packing and strong monotonicity, we characterize how target properties affect the corresponding minimax upper and lower bounds within the realm of ferromagnets. On the other hand, by studying the detection of an antiferromagnetic (i.e., negative interactions only) Curie–Weiss model buried in Rademacher noise, we show that property testing is strictly more challenging over general Ising models. In terms of methodological development, we propose two types of correlation based tests: computationally efficient screening for ferromagnets, and score type tests for general models, including a fast cycle presence test. Our correlation screening tests match the information-theoretic bounds for property testing in ferromagnets in certain regimes.

The (theta, wheel)-free graphs Part III: Cliques, stable sets and coloring

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins, and consequently obtain a polynomial time recognition algorithm for the class. In this paper we further use this decomposition theorem to obtain polynomial time algorithms for maximum weight clique, maximum weight stable set and coloring problems. We also show that for a graph $G$ in the class, if its maximum clique size is $\omega$, then its chromatic number is bounded by max$\{\omega,3\}$, and that the class is 3-clique-colorable.

Efficient Algorithm for Maximal Clique Size Evaluation

A large dataset network is considered for computation of maximal clique size (MC). Additionally, its link with popular centrality metrics to decrease uncertainty and complexity and for finding influential points of any network has also been investigated. Previous studies focus on centrality metrics like degree centrality (DC), closeness centrality (CC), betweenness centrality (BC) and Eigenvector centrality (EVC) and compare them with maximal clique size however, in this study Katz centrality measure is also considered and shows a pretty robust relation with maximal clique size (MC). Secondly, maximal clique size (MC) algorithm is also revised for network analysis to avoid complexity in computation. Association between MC and five centrality metrics has been evaluated through recognized methods that are Pearson’s correlation coefficient (PCC), Spearman’s correlation coefficient (SCC) and Kendall’s correlation coefficient (KCC). The strong strength of association between them is seen through all three correlation coefficients measure.

A simple and space efficient segment tree implementation

The segment tree is an extremely versatile data structure. In this paper, a new array based implementation of segment trees is proposed. In such an implementation of segment tree, the structural information associated with the tree nodes can be removed completely. Some primary computational geometry problems such as stabbing counting queries, measure of union of intervals, and maximum clique size of Intervals are used to demonstrate the efficiency of the new array based segment tree implementation. Each interval in a set S = {I1, I2, ⋯ , In} of n intervals can be insert into or delete from the heap based segment tree in O(log n) time. All the primary computational geometry problems can be solved efficiently.

A new decomposition technique for maximal clique enumeration for sparse graphs

Given a graph, we are interested in the question of finding all its maximal cliques. This question models the community detection problem and has been extensively studied. Here, we approach it under the light of an important graph parameter. The degeneracy of a graph G is the smallest integer k such that every subgraph of G contains a vertex of degree at most k. Using a new decomposition technique, we present two output sensitive algorithms for the maximal clique enumeration problem. The first one has enumeration time depending only on the degeneracy of the graph. This is the first such result in the literature. This algorithm requires that the cliques are stored in memory. Thus, we propose a second one, which has enumeration time depending on the degeneracy and the maximum degree, and which only requires memory polynomial in the degeneracy of the graph (besides the space to store the graph itself). Then we show that this algorithm can be easily parallelized. As a by-product of our decomposition technique, we propose new algorithms for the maximum clique and p-clique problems as well as an approximation algorithm counting maximal cliques, with approximation ratio the maximum clique size.

Graphs vertex-partitionable into strong cliques

A clique in a graph is strong if it intersects all maximal independent sets. A graph is localizable if it has a partition of the vertex set into strong cliques. Localizable graphs were introduced by Yamashita and Kameda in 1999 and form a rich class of well-covered graphs that coincides with the class of well-covered graphs within the class of perfect graphs. In this paper, we give several equivalent formulations of the property of localizability and develop polynomially testable characterizations of localizable graphs within three non-perfect graph classes: triangle-free graphs, C4-free graphs, and line graphs. Furthermore, we use localizable graphs to construct an infinite family of counterexamples to a conjecture due to Zaare-Nahandi about k-partite well-covered graphs having all maximal cliques of size k.

Complex Network Analysis of the Contiguous United States Graph

We model the contiguous states (48 states and the District of Columbia) of the United States (US) as an undirected network graph with each state represented as a node and there is an edge between two nodes if the corresponding two states share a common border. We determine a ranking of the states in the US with respect to a suite of node-level metrics: the centrality metrics (degree, eigenvector, betweenness and closeness), eccentricity, maximal clique size, and local clustering coefficient. We propose a normalization-based approach to obtain a comprehensive centrality ranking of the vertices (that is most likely to be tie-free) encompassing the normalized values of the four centrality metrics. We have applied the proposed normalization-based approach on the US States graph to obtain a tie-free ranking of the vertices based on a comprehensive centrality score. We observe the state of Missouri to be the most central state with respect to all the four centrality metrics. We have also analyzed the US States graph with respect to a suite of network-level metrics: bipartivity index, assortativity index, modularity, size of the minimum connected dominating set, algebraic connectivity and degree metrics. The approach taken in this paper could be useful for several application domains: transportation networks (to identify central hubs), politics (to identify campaign venues with larger geographic coverage), cultural and electoral studies (to identify communities of states that are relatively proximal to each other) and etc.