735 research outputs found
On the Number of Maximal Cliques in Two-Dimensional Random Geometric Graphs: Euclidean and Hyperbolic
Maximal clique enumeration appears in various real-world networks, such as
social networks and protein-protein interaction networks for different
applications. For general graph inputs, the number of maximal cliques can be up
to . However, many previous works suggest that the number is much
smaller than that on real-world networks, and polynomial-delay algorithms
enable us to enumerate them in a realistic-time span. To bridge the gap between
the worst case and practice, we consider the number of maximal cliques in two
popular models of real-world networks: Euclidean random geometric graphs and
hyperbolic random graphs. We show that the number of maximal cliques on
Euclidean random geometric graphs is lower and upper bounded by
and with high
probability for any . For a hyperbolic random graph, we give the
bounds of and
where is the power-law degree
exponent between 2 and 3.Comment: 22 pages, 6 figure
Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs
In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph G in O(m + n^{4.5(1-?)}) expected time if a geometric representation is given or in O(m + n^{6(1-?)}) expected time if a geometric representation is not given, where n and m denote the numbers of vertices and edges of G, respectively, and ? denotes a parameter controlling the power-law exponent of the degree distribution of G. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently
Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs
In this paper, we study the maximum clique problem on hyperbolic random
graphs. A hyperbolic random graph is a mathematical model for analyzing
scale-free networks since it effectively explains the power-law degree
distribution of scale-free networks. We propose a simple algorithm for finding
a maximum clique in hyperbolic random graph. We first analyze the running time
of our algorithm theoretically. We can compute a maximum clique on a hyperbolic
random graph in expected time if a geometric
representation is given or in expected time if a
geometric representation is not given, where and denote the numbers of
vertices and edges of , respectively, and denotes a parameter
controlling the power-law exponent of the degree distribution of . Also, we
implemented and evaluated our algorithm empirically. Our algorithm outperforms
the previous algorithm [BFK18] practically and theoretically. Beyond the
hyperbolic random graphs, we have experiment on real-world networks. For most
of instances, we get large cliques close to the optimum solutions efficiently.Comment: Accepted in ESA 202
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
Degree correlations in scale-free null models
We study the average nearest neighbor degree of vertices with degree
. In many real-world networks with power-law degree distribution
falls off in , a property ascribed to the constraint that any two vertices
are connected by at most one edge. We show that indeed decays in in
three simple random graph null models with power-law degrees: the erased
configuration model, the rank-1 inhomogeneous random graph and the hyperbolic
random graph. We consider the large-network limit when the number of nodes
tends to infinity. We find for all three null models that starts to
decay beyond and then settles on a power law , with the degree exponent.Comment: 21 pages, 4 figure
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