75 research outputs found
Fast and simple decycling and dismantling of networks
Decycling and dismantling of complex networks are underlying many important
applications in network science. Recently these two closely related problems
were tackled by several heuristic algorithms, simple and considerably
sub-optimal, on the one hand, and time-consuming message-passing ones that
evaluate single-node marginal probabilities, on the other hand. In this paper
we propose a simple and extremely fast algorithm, CoreHD, which recursively
removes nodes of the highest degree from the -core of the network. CoreHD
performs much better than all existing simple algorithms. When applied on
real-world networks, it achieves equally good solutions as those obtained by
the state-of-art iterative message-passing algorithms at greatly reduced
computational cost, suggesting that CoreHD should be the algorithm of choice
for many practical purposes
Dismantling sparse random graphs
We consider the number of vertices that must be removed from a graph G in
order that the remaining subgraph has no component with more than k vertices.
Our principal observation is that, if G is a sparse random graph or a random
regular graph on n vertices with n tending to infinity, then the number in
question is essentially the same for all values of k such that k tends to
infinity but k=o(n).Comment: 7 page
Network dismantling
We study the network dismantling problem, which consists in determining a
minimal set of vertices whose removal leaves the network broken into connected
components of sub-extensive size. For a large class of random graphs, this
problem is tightly connected to the decycling problem (the removal of vertices
leaving the graph acyclic). Exploiting this connection and recent works on
epidemic spreading we present precise predictions for the minimal size of a
dismantling set in a large random graph with a prescribed (light-tailed) degree
distribution. Building on the statistical mechanics perspective we propose a
three-stage Min-Sum algorithm for efficiently dismantling networks, including
heavy-tailed ones for which the dismantling and decycling problems are not
equivalent. We also provide further insights into the dismantling problem
concluding that it is an intrinsically collective problem and that optimal
dismantling sets cannot be viewed as a collection of individually well
performing nodes.Comment: Source code and data can be found at
https://github.com/abraunst/decycle
On the -hull number and infecting times of generalized Petersen graphs
The -hull number of a graph is the minimum cardinality of an infecting
set of vertices that will eventually infect the entire graph under the rule
that uninfected nodes become infected if two or more neighbors are infected. In
this paper, we study the -hull number for generalized Petersen graphs and
a number of closely related graphs that arise from surgery or more generalized
permutations. In addition, the number of components of the complement of an
infecting set of minimum cardinality is calculated for the generalized Petersen
graph and shown to always be or . Moreover, infecting times for
infecting sets of minimum cardinality are studied. Bounds are provided and
complete information is given in special cases.Comment: 8 page
Complexity and algorithms related to two classes of graph problems
This thesis addresses the problems associated with conversions on graphs and editing by removing a matching. We study the f-reversible processes, which are those associated with a threshold value for each vertex, and whose dynamics depends on the number of neighbors with different state for each vertex. We set a tight upper bound for the period and transient lengths, characterize all trees that reach the maximum transient length for 2-reversible processes, and we show that determining the size of a minimum conversion set is NP-hard. We show that the AND-OR model defines a convexity on graphs. We show results of NP-completeness and efficient algorithms for certain convexity parameters for this new one, as well as approximate algorithms. We introduce the concept of generalized threshold processes, where the results are NP-completeness and efficient algorithms for both non relaxed and relaxed versions. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all cycles. We show that this problem is NP-hard even for subcubic graphs, but admits efficient solution for several graph classes. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all odd cycles. We show that this problem is NP-hard even for planar graphs with bounded degree, but admits efficient solution for some graph classes. We also show parameterized results.Esta tese aborda problemas associados a conversões em grafos e de edição pela remoção de um emparelhamento. Estudamos processos f-reversĂveis, que sĂŁo aqueles associados a um valor de limiar para cada vĂ©rtice e cuja dinâmica depende da quantidade de vizinhos com estado contrário para cada vĂ©rtice. Estabelecemos um limite superior justo para o tamanho do perĂodo e transiente, caracterizamos todas as árvores que alcançam o transiente máximo em processos 2-reversĂveis e mostramos que determinar o tamanho de um conjunto conversor mĂnimo Ă© NP-difĂcil. Mostramos que o modelo AND-OR define uma convexidade sobre grafos. Mostramos resultados de NP-completude e algoritmos eficientes para certos parâmetros de convexidade para esta nova, assim como algoritmos aproximativos. Introduzimos o conceito de processos de limiar generalizados, onde mostramos resultados de NP-completude e algoritmos eficientes para ambas as versões nĂŁo relaxada e relaxada. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos. Mostramos que este problema Ă© NP-difĂcil mesmo para grafos subcĂşbicos, mas admite solução eficiente para várias classes de grafos. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos Ămpares. Mostramos que este problema Ă© NP-difĂcil mesmo para grafos planares com grau limitado, mas admite solução eficiente para algumas classes de grafos. Mostramos tambĂ©m resultados parametrizados
Solving the undirected feedback vertex set problem by local search
An undirected graph consists of a set of vertices and a set of undirected
edges between vertices. Such a graph may contain an abundant number of cycles,
then a feedback vertex set (FVS) is a set of vertices intersecting with each of
these cycles. Constructing a FVS of cardinality approaching the global minimum
value is a optimization problem in the nondeterministic polynomial-complete
complexity class, therefore it might be extremely difficult for some large
graph instances. In this paper we develop a simulated annealing local search
algorithm for the undirected FVS problem. By defining an order for the vertices
outside the FVS, we replace the global cycle constraints by a set of local
vertex constraints on this order. Under these local constraints the cardinality
of the focal FVS is then gradually reduced by the simulated annealing dynamical
process. We test this heuristic algorithm on large instances of Er\"odos-Renyi
random graph and regular random graph, and find that this algorithm is
comparable in performance to the belief propagation-guided decimation
algorithm.Comment: 6 page
Theories for influencer identification in complex networks
In social and biological systems, the structural heterogeneity of interaction
networks gives rise to the emergence of a small set of influential nodes, or
influencers, in a series of dynamical processes. Although much smaller than the
entire network, these influencers were observed to be able to shape the
collective dynamics of large populations in different contexts. As such, the
successful identification of influencers should have profound implications in
various real-world spreading dynamics such as viral marketing, epidemic
outbreaks and cascading failure. In this chapter, we first summarize the
centrality-based approach in finding single influencers in complex networks,
and then discuss the more complicated problem of locating multiple influencers
from a collective point of view. Progress rooted in collective influence
theory, belief-propagation and computer science will be presented. Finally, we
present some applications of influencer identification in diverse real-world
systems, including online social platforms, scientific publication, brain
networks and socioeconomic systems.Comment: 24 pages, 6 figure
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