Computing cliques and cavities in networks

Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to as simplices and cliques of different orders. Notably, cavities consisting of higher-order cliques play an important role in brain functions. Since searching for maximum cliques is an NP-complete problem, we use k-core decomposition to determine the computability of a given network. For a computable network, we design a search method with an implementable algorithm for finding cliques of different orders, obtaining also the Euler characteristic number. Then, we compute the Betti numbers by using the ranks of boundary matrices of adjacent cliques. Furthermore, we design an optimized algorithm for finding cavities of different orders. Finally, we apply the algorithm to the neuronal network of C. elegans with data from one typical dataset, and find all of its cliques and some cavities of different orders, providing a basis for further mathematical analysis and computation of its structure and function.

Clique Is Hard on Average for Regular Resolution

We prove that for k ≪ 4√ n regular resolution requires length n Ω( k ) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k -clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional n Ω( k ) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

The Szemerédi–Petruska conjecture for a few small values

Let H be a 3-uniform hypergraph of order n with clique number $$\omega (H)=k$$ ω ( H ) = k . Assume that the union of the k-cliques of H equals its vertex set, the intersection of all maximum cliques of H is empty, but the intersection of all but one k-clique is non-empty. For fixed $$m=n-k$$ m = n - k , Szemerédi and Petruska conjectured the sharp bound $$n\hbox {\,\,\char 054\,\,}{m+2\atopwithdelims ()2}$$ n 6 m + 2 2 . In this note the conjecture is verified for $$m=2,3$$ m = 2 , 3 and 4.

GCLIQUE: An Open Source Genetic Algorithm for the Maximum Clique Problem

<p>A clique in a graph is a set of vertices that are all connected to each</p><p>other. A maximum clique is a clique of maximum size. A graph may have</p><p>more than one maximum cliques. The problem of finding a maximum</p><p>clique is a strongly hard NP-hard problem. It is not possible to find an</p><p>approximation algorithm which finds a maximum clique that is a constant</p><p>factor of the optimum solution. In this work, we present a genetic algorithm</p><p>for the maximum clique problem that is able to find optimum or</p><p>close to optimum solutions to most DIMACS graphs. The genetic algorithm</p><p>uses new crossover mechanisms which are able to find reasonably</p><p>good cliques which can then be used in other applications downstream.</p><p>We also provide C++ code for our algorithm. Results show that our algorithm</p><p>is able to find maximum cliques for most DIMACS instances, and</p><p>if not, close to optimum solutions for the other instances.</p>

GCLIQUE: An Open Source Genetic Algorithm for the Maximum Clique Problem

<p>A clique in a graph is a set of vertices that are all connected to each</p><p>other. A maximum clique is a clique of maximum size. A graph may have</p><p>more than one maximum cliques. The problem of finding a maximum</p><p>clique is a strongly hard NP-hard problem. It is not possible to find an</p><p>approximation algorithm which finds a maximum clique that is a constant</p><p>factor of the optimum solution. In this work, we present a genetic algorithm</p><p>for the maximum clique problem that is able to find optimum or</p><p>close to optimum solutions to most DIMACS graphs. The genetic algorithm</p><p>uses new crossover mechanisms which are able to find reasonably</p><p>good cliques which can then be used in other applications downstream.</p><p>We also provide C++ code for our algorithm. Results show that our algorithm</p><p>is able to find maximum cliques for most DIMACS instances, and</p><p>if not, close to optimum solutions for the other instances.</p>

Clique Selection and its Effect on Paraclique Enrichment: An Experimental Study

The paraclique algorithm provides an effective means for biological data clustering. It satisfies the mathematical quest for density, while fulfilling the pragmatic need for noise abatement on real data. Given a finite, simple, edge-weighted and thresholded graph, the paraclique method first finds a maximum clique, then incorporates additional vertices in a controlled manner, and finally extracts the subgraph thereby defined. When more than one maximum clique is present, however, deciding which to employ is usually left unspecified. In practice, this frequently and quite naturally reduces to using the first maximum clique found. In this paper, maximum clique selection is studied in the context of well-annotated transcriptomic data, with ontological classification used as a proxy for cluster quality. Enrichment p-values are compared using maximum cliques chosen in a variety of ways. The most appealing and intuitive option is almost surely to start with the maximum clique having the highest average edge weight. Although there is of course no guarantee that such a strategy is any better than random choice, results derived from a large collection of experiments indicate that, in general, this approach produces a small but statistically significant improvement in overall cluster quality. Such an improvement, though modest, may be well worth pursuing in light of the time, expense and expertise often required to generate timely, high quality, high throughput biological data.