20,698 research outputs found
Random graphs containing arbitrary distributions of subgraphs
Traditional random graph models of networks generate networks that are
locally tree-like, meaning that all local neighborhoods take the form of trees.
In this respect such models are highly unrealistic, most real networks having
strongly non-tree-like neighborhoods that contain short loops, cliques, or
other biconnected subgraphs. In this paper we propose and analyze a new class
of random graph models that incorporates general subgraphs, allowing for
non-tree-like neighborhoods while still remaining solvable for many fundamental
network properties. Among other things we give solutions for the size of the
giant component, the position of the phase transition at which the giant
component appears, and percolation properties for both site and bond
percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl
Where Graph Topology Matters: The Robust Subgraph Problem
Robustness is a critical measure of the resilience of large networked
systems, such as transportation and communication networks. Most prior works
focus on the global robustness of a given graph at large, e.g., by measuring
its overall vulnerability to external attacks or random failures. In this
paper, we turn attention to local robustness and pose a novel problem in the
lines of subgraph mining: given a large graph, how can we find its most robust
local subgraph (RLS)?
We define a robust subgraph as a subset of nodes with high communicability
among them, and formulate the RLS-PROBLEM of finding a subgraph of given size
with maximum robustness in the host graph. Our formulation is related to the
recently proposed general framework for the densest subgraph problem, however
differs from it substantially in that besides the number of edges in the
subgraph, robustness also concerns with the placement of edges, i.e., the
subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two
heuristic algorithms based on top-down and bottom-up search strategies.
Further, we present modifications of our algorithms to handle three practical
variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs
demonstrate that we find subgraphs with larger robustness than the densest
subgraphs even at lower densities, suggesting that the existing approaches are
not suitable for the new problem setting.Comment: 13 pages, 10 Figures, 3 Tables, to appear at SDM 2015 (9 pages only
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
Subgraphs in random networks
Understanding the subgraph distribution in random networks is important for
modelling complex systems. In classic Erdos networks, which exhibit a
Poissonian degree distribution, the number of appearances of a subgraph G with
n nodes and g edges scales with network size as \mean{G} ~ N^{n-g}. However,
many natural networks have a non-Poissonian degree distribution. Here we
present approximate equations for the average number of subgraphs in an
ensemble of random sparse directed networks, characterized by an arbitrary
degree sequence. We find new scaling rules for the commonly occurring case of
directed scale-free networks, in which the outgoing degree distribution scales
as P(k) ~ k^{-\gamma}. Considering the power exponent of the degree
distribution, \gamma, as a control parameter, we show that random networks
exhibit transitions between three regimes. In each regime the subgraph number
of appearances follows a different scaling law, \mean{G} ~ N^{\alpha}, where
\alpha=n-g+s-1 for \gamma<2, \alpha=n-g+s+1-\gamma for 2<\gamma<\gamma_c, and
\alpha=n-g for \gamma>\gamma_c, s is the maximal outdegree in the subgraph, and
\gamma_c=s+1. We find that certain subgraphs appear much more frequently than
in Erdos networks. These results are in very good agreement with numerical
simulations. This has implications for detecting network motifs, subgraphs that
occur in natural networks significantly more than in their randomized
counterparts.Comment: 8 pages, 5 figure
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