13,432 research outputs found
Maximum Cliques in Graphs with Small Intersection Number and Random Intersection Graphs
In this paper, we relate the problem of finding a maximum clique to the
intersection number of the input graph (i.e. the minimum number of cliques
needed to edge cover the graph). In particular, we consider the maximum clique
problem for graphs with small intersection number and random intersection
graphs (a model in which each one of labels is chosen independently with
probability by each one of vertices, and there are edges between any
vertices with overlaps in the labels chosen).
We first present a simple algorithm which, on input finds a maximum
clique in time steps, where is an
upper bound on the intersection number and is the number of vertices.
Consequently, when the running time of this algorithm is
polynomial.
We then consider random instances of the random intersection graphs model as
input graphs. As our main contribution, we prove that, when the number of
labels is not too large (), we can use the label
choices of the vertices to find a maximum clique in polynomial time whp. The
proof of correctness for this algorithm relies on our Single Label Clique
Theorem, which roughly states that whp a "large enough" clique cannot be formed
by more than one label. This theorem generalizes and strengthens other related
results in the state of the art, but also broadens the range of values
considered.
As an important consequence of our Single Label Clique Theorem, we prove that
the problem of inferring the complete information of label choices for each
vertex from the resulting random intersection graph (i.e. the \emph{label
representation of the graph}) is \emph{solvable} whp. Finding efficient
algorithms for constructing such a label representation is left as an
interesting open problem for future research
Small world networks and clustered small world networks with random connectivity
International audienceThe discovery of small world properties in real-world networks has revolutionized the way we analyze and study real-world systems. Mathematicians and physicists in particular have closely studied and developed several models to artificially generate networks with small world properties. The classical algorithms to produce these graphs artificially make use of the fact that with the introduction of some randomness in ordered graphs, small world graphs can be produced. In this paper, we present a novel algorithm to generate graphs with small world properties based on the idea that with the introduction of some order in a random graph, small world graphs can be generated. Our model starts with a randomly generated graph. We then replace each node of the random graph with cliques of different sizes. This ensures that the connectivity between the cliques is random but the clustering coefficient increases to a desired level. We further extend this model to incorporate the property of community structures (clusters) found readily in real-world networks such as social, biological and technological networks. These community structures are densely connected regions of nodes in a network that are loosely connected to each other. The model generates these clustered small world graphs by replacing nodes in the random graph with densely connected set of nodes. Experimentation shows that these two models generate small world and clustered small world graphs, respectively, as we were able to produce the desired properties of a small world network with high clustering coefficient and low average path lengths in both cases. Furthermore, we also calculated relative density and modularity to show that the clustered networks indeed had community structures
A mathematical foundation for the use of cliques in the exploration of data with navigation graphs
Navigation graphs were introduced by Hurley and Oldford (2011) as a graph-theoretic framework for exploring data sets, particularly those with many variables. They allow the user to visualize one small subset of the variables and then proceed to another subset, which shares a few of the original variables, via a smooth transition. These graphs serve as both a high level overview of the dataset as well as a tool for a first-hand exploration of regions deemed interesting.
This work examines the nature of cliques in navigation graphs, both in terms of type and magnitude, and speculates as to what their significance to the underlying dataset might be. The questions answered by this body of work were motivated by the belief that the presence of cliques in navigation graphs is a potential indicator for the existence of an interesting, possibly unanticipated, relationship among some of the variables.
In this thesis we provide a detailed examination of cliques in navigation graphs, both in terms of type, size and number. The study of types of cliques informs us of the potential significance of highly connected structures to the underlying data and guides our approach for examining the possible clique sizes and counts. On the other hand, the prevalence of large clique sizes and counts is suggestive of an interesting, possibly unexpected, relationship between the variates in the data.
To address the challenges surrounding the nature of cliques in navigation graphs, we develop a framework for the derivation of closed-form expressions for the moments of count random variables in terms of their underlying indecomposable summands is established. We use this framework in conjunction with a connection between intersecting set families to obtain edge counts within a clique cover and thus, obtain closed-form expressions for the moments of clique counts in random graphs
Large cliques and independent sets all over the place
We study the following question raised by Erd\H{o}s and Hajnal in the early
90's. Over all -vertex graphs what is the smallest possible value of
for which any vertices of contain both a clique and an independent set
of size ? We construct examples showing that is at most
obtaining a twofold sub-polynomial
improvement over the upper bound of about coming from the natural
guess, the random graph. Our (probabilistic) construction gives rise to new
examples of Ramsey graphs, which while having no very large homogenous subsets
contain both cliques and independent sets of size in any small subset
of vertices. This is very far from being true in random graphs. Our proofs are
based on an interplay between taking lexicographic products and using
randomness.Comment: 12 page
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