19 research outputs found
KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation
We present KADABRA, a new algorithm to approximate betweenness centrality in
directed and undirected graphs, which significantly outperforms all previous
approaches on real-world complex networks. The efficiency of the new algorithm
relies on two new theoretical contributions, of independent interest. The first
contribution focuses on sampling shortest paths, a subroutine used by most
algorithms that approximate betweenness centrality. We show that, on realistic
random graph models, we can perform this task in time
with high probability, obtaining a significant speedup with respect to the
worst-case performance. We experimentally show that this new
technique achieves similar speedups on real-world complex networks, as well.
The second contribution is a new rigorous application of the adaptive sampling
technique. This approach decreases the total number of shortest paths that need
to be sampled to compute all betweenness centralities with a given absolute
error, and it also handles more general problems, such as computing the
most central nodes. Furthermore, our analysis is general, and it might be
extended to other settings.Comment: Some typos correcte
KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation
International audienceWe present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks. The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest. The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time |E| 1 2 +o (1) with high probability, obtaining a significant speedup with respect to the Θ(|E|) worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well. The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the k most central nodes. Furthermore, our analysis is general, and it might be extended to other settings
Hyperbolicity Measures "Democracy" in Real-World Networks
We analyze the hyperbolicity of real-world networks, a geometric quantity
that measures if a space is negatively curved. In our interpretation, a network
with small hyperbolicity is "aristocratic", because it contains a small set of
vertices involved in many shortest paths, so that few elements "connect" the
systems, while a network with large hyperbolicity has a more "democratic"
structure with a larger number of crucial elements.
We prove mathematically the soundness of this interpretation, and we derive
its consequences by analyzing a large dataset of real-world networks. We
confirm and improve previous results on hyperbolicity, and we analyze them in
the light of our interpretation.
Moreover, we study (for the first time in our knowledge) the hyperbolicity of
the neighborhood of a given vertex. This allows to define an "influence area"
for the vertices in the graph. We show that the influence area of the highest
degree vertex is small in what we define "local" networks, like most social or
peer-to-peer networks. On the other hand, if the network is built in order to
reach a "global" goal, as in metabolic networks or autonomous system networks,
the influence area is much larger, and it can contain up to half the vertices
in the graph. In conclusion, our newly introduced approach allows to
distinguish the topology and the structure of various complex networks
On Computing the Hyperbolicity of Real-World Graphs
International audienceThe (Gromov) hyperbolicity is a topological property of a graph, which has been recently applied in several different contexts, such as the design of routing schemes, network security, computational biology, the analysis of graph algorithms, and the classification of complex networks. Computing the hyperbolicity of a graph can be very time consuming: indeed, the best available algorithm has running-time O(n^{3.69}), which is clearly prohibitive for big graphs. In this paper, we provide a new and more efficient algorithm: although its worst-case complexity is O(n^4), in practice it is much faster, allowing, for the first time, the computation of the hyperbolicity of graphs with up to 200,000 nodes. We experimentally show that our new algorithm drastically outperforms the best previously available algorithms, by analyzing a big dataset of real-world networks. Finally, we apply the new algorithm to compute the hyperbolicity of random graphs generated with the Erdös-Renyi model, the Chung-Lu model, and the Configuration Model
Telling Stories: Enumerating Maximal Directed Acyclic Subgraphs with a Specified Set of Sources and Targets
Questa tesi presenta un algoritmo che enumera con ritardo lineare tutti i sottografi aciclici di un dato digrafo G=(V,E), le cui sorgenti e i cui pozzi sono contenuti rispettivamente in due sottoinsiemi fissati S e T di V.
Da questi sottografi, chiamati pitches, quelli massimali, chiamati storie, possono essere estratti in modo nettamente più efficiente rispetto all’unico algoritmo già esistente. Il miglioramento può diventare ancora più significativo se viene applicata una tecnica di pruning, che evita che siano generati molti pitches da cui non si può ottenere una storia.
Queste affermazioni verranno provate sperimentalmente nell’ultimo capitolo, utilizzanzo un database di reti metaboliche abbastanza esteso.
Tutti i risultati originali presenti in questa tesi saranno presentati dall’autore al 12th International Simposium on Exponential Algorithms (SEA), 5-7 giugno 2013, Roma. Saranno inoltre pubblicati da Lecture Notes in Computer Science
Into the Square - On the Complexity of Quadratic-Time Solvable Problems
This paper will analyze several quadratic-time solvable problems, and will classify them into two classes: problems that are solvable in truly subquadratic time (that is, in time for some ) and problems that are not, unless the well known Strong Exponential Time Hypothesis (SETH) is false. In particular, we will prove that some quadratic-time solvable problems are indeed easier than expected. We will provide an algorithm that computes the transitive closure of a directed graph in time , where denotes the number of edges in the transitive closure and is the exponent for matrix multiplication. As a side effect, we will prove that our algorithm runs in time if the transitive closure is sparse. The same time bounds hold if we want to check whether a graph is transitive, by replacing m with the number of edges in the graph itself. As far as we know, this is the fastest algorithm for sparse transitive digraph recognition. Finally, we will apply our algorithm to the comparability graph recognition problem (dating back to 1941), obtaining the first truly subquadratic algorithm. The second part of the paper deals with hardness results. Starting from an artificial quadratic-time solvable variation of the k-SAT problem, we will construct a graph of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the graph falsifies SETH. The analyzed problems are the following: computing the subset graph, finding dominating sets, computing the betweenness centrality of a vertex, computing the minimum closeness centrality, and computing the hyperbolicity of a pair of vertices. We will also be able to include in our framework three proofs already appeared in the literature, concerning the graph diameter computation, local alignment of strings and orthogonality of vectors
Into the Square - On the Complexity of Quadratic-Time Solvable Problems
This paper will analyze several quadratic-time solvable problems, and will classify them into two classes: problems that are solvable in truly subquadratic time (that is, in time for some ) and problems that are not, unless the well known Strong Exponential Time Hypothesis (SETH) is false. In particular, we will prove that some quadratic-time solvable problems are indeed easier than expected. We will provide an algorithm that computes the transitive closure of a directed graph in time , where denotes the number of edges in the transitive closure and is the exponent for matrix multiplication. As a side effect, we will prove that our algorithm runs in time if the transitive closure is sparse. The same time bounds hold if we want to check whether a graph is transitive, by replacing m with the number of edges in the graph itself. As far as we know, this is the fastest algorithm for sparse transitive digraph recognition. Finally, we will apply our algorithm to the comparability graph recognition problem (dating back to 1941), obtaining the first truly subquadratic algorithm. The second part of the paper deals with hardness results. Starting from an artificial quadratic-time solvable variation of the k-SAT problem, we will construct a graph of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the graph falsifies SETH. The analyzed problems are the following: computing the subset graph, finding dominating sets, computing the betweenness centrality of a vertex, computing the minimum closeness centrality, and computing the hyperbolicity of a pair of vertices. We will also be able to include in our framework three proofs already appeared in the literature, concerning the graph diameter computation, local alignment of strings and orthogonality of vectors