2,100 research outputs found
Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function
Synchronization is central to many complex systems in engineering physics
(e.g., the power-grid, Josephson junction circuits, and electro-chemical
oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms).
Despite these widespread applications---for which proper functionality depends
sensitively on the extent of synchronization---there remains a lack of
understanding for how systems evolve and adapt to enhance or inhibit
synchronization. We study how network modifications affect the synchronization
properties of network-coupled dynamical systems that have heterogeneous node
dynamics (e.g., phase oscillators with non-identical frequencies), which is
often the case for real-world systems. Our approach relies on a synchrony
alignment function (SAF) that quantifies the interplay between heterogeneity of
the network and of the oscillators and provides an objective measure for a
system's ability to synchronize. We conduct a spectral perturbation analysis of
the SAF for structural network modifications including the addition and removal
of edges, which subsequently ranks the edges according to their importance to
synchronization. Based on this analysis, we develop gradient-descent algorithms
to efficiently solve optimization problems that aim to maximize phase
synchronization via network modifications. We support these and other results
with numerical experiments.Comment: 25 pages, 6 figure
Efficient Centrality Maximization with Rademacher Averages
The identification of the set of k most central nodes of a graph, or
centrality maximization, is a key task in network analysis, with various
applications ranging from finding communities in social and biological networks
to understanding which seed nodes are important to diffuse information in a
graph. As the exact computation of centrality measures does not scale to
modern-sized networks, the most practical solution is to resort to rigorous,
but efficiently computable, randomized approximations. In this work we present
CentRA, the first algorithm based on progressive sampling to compute
high-quality approximations of the set of k most central nodes. CentRA is based
on a novel approach to efficiently estimate Monte Carlo Rademacher Averages, a
powerful tool from statistical learning theory to compute sharp data-dependent
approximation bounds. Then, we study the sample complexity of centrality
maximization using the VC-dimension, a key concept from statistical learning
theory. We show that the number of random samples required to compute
high-quality approximations scales with finer characteristics of the graph,
such as its vertex diameter, or of the centrality of interest, significantly
improving looser bounds derived from standard techniques. We apply CentRA to
analyze large real-world networks, showing that it significantly outperforms
the state-of-the-art approximation algorithm in terms of number of samples,
running times, and accuracy.Comment: Accepted to KDD '2
Scalable Algorithms for the Analysis of Massive Networks
Die Netzwerkanalyse zielt darauf ab, nicht-triviale Erkenntnisse aus vernetzten Daten zu gewinnen. Beispiele für diese Erkenntnisse sind die Wichtigkeit einer Entität im Verhältnis zu anderen nach bestimmten Kriterien oder das Finden des am besten geeigneten Partners für jeden Teilnehmer eines Netzwerks - bekannt als Maximum Weighted Matching (MWM).
Da der Begriff der Wichtigkeit an die zu betrachtende Anwendung gebunden ist, wurden zahlreiche Zentralitätsmaße eingeführt. Diese Maße stammen hierbei aus Jahrzehnten, in denen die Rechenleistung sehr begrenzt war und die Netzwerke im Vergleich zu heute viel kleiner waren. Heute sind massive Netzwerke mit Millionen von Kanten allgegenwärtig und eine triviale Berechnung von Zentralitätsmaßen ist oft zu zeitaufwändig. Darüber hinaus ist die Suche nach der Gruppe von k Knoten mit hoher Zentralität eine noch kostspieligere Aufgabe. Skalierbare Algorithmen zur Identifizierung hochzentraler (Gruppen von) Knoten in großen Graphen sind von großer Bedeutung für eine umfassende Netzwerkanalyse.
Heutigen Netzwerke verändern sich zusätzlich im zeitlichen Verlauf und die effiziente Aktualisierung der Ergebnisse nach einer Änderung ist eine Herausforderung. Effiziente dynamische Algorithmen sind daher ein weiterer wesentlicher Bestandteil moderner Analyse-Pipelines.
Hauptziel dieser Arbeit ist es, skalierbare algorithmische Lösungen für die zwei oben genannten Probleme zu finden. Die meisten unserer Algorithmen benötigen Sekunden bis einige Minuten, um diese Aufgaben in realen Netzwerken mit bis zu Hunderten Millionen von Kanten zu lösen, was eine deutliche Verbesserung gegenüber dem Stand der Technik darstellt. Außerdem erweitern wir einen modernen Algorithmus für MWM auf dynamische Graphen. Experimente zeigen, dass unser dynamischer MWM-Algorithmus Aktualisierungen in Graphen mit Milliarden von Kanten in Millisekunden bewältigt.Network analysis aims to unveil non-trivial insights from networked data by studying relationship patterns between the entities of a network. Among these insights, a popular one is to quantify the importance of an entity with respect to the others according to some criteria. Another one is to find the most suitable matching partner for each participant of a network knowing the pairwise preferences of the participants to be matched with each other - known as Maximum Weighted Matching (MWM).
Since the notion of importance is tied to the application under consideration, numerous centrality measures have been introduced. Many of these measures, however, were conceived in a time when computing power was very limited and networks were much smaller compared to today's, and thus scalability to large datasets was not considered. Today, massive networks with millions of edges are ubiquitous, and a complete exact computation for traditional centrality measures are often too time-consuming. This issue is amplified if our objective is to find the group of k vertices that is the most central as a group. Scalable algorithms to identify highly central (groups of) vertices on massive graphs are thus of pivotal importance for large-scale network analysis.
In addition to their size, today's networks often evolve over time, which poses the challenge of efficiently updating results after a change occurs. Hence, efficient dynamic algorithms are essential for modern network analysis pipelines.
In this work, we propose scalable algorithms for identifying important vertices in a network, and for efficiently updating them in evolving networks. In real-world graphs with hundreds of millions of edges, most of our algorithms require seconds to a few minutes to perform these tasks. Further, we extend a state-of-the-art algorithm for MWM to dynamic graphs. Experiments show that our dynamic MWM algorithm handles updates in graphs with billion edges in milliseconds
Optimizing spread dynamics on graphs by message passing
Cascade processes are responsible for many important phenomena in natural and
social sciences. Simple models of irreversible dynamics on graphs, in which
nodes activate depending on the state of their neighbors, have been
successfully applied to describe cascades in a large variety of contexts. Over
the last decades, many efforts have been devoted to understand the typical
behaviour of the cascades arising from initial conditions extracted at random
from some given ensemble. However, the problem of optimizing the trajectory of
the system, i.e. of identifying appropriate initial conditions to maximize (or
minimize) the final number of active nodes, is still considered to be
practically intractable, with the only exception of models that satisfy a sort
of diminishing returns property called submodularity. Submodular models can be
approximately solved by means of greedy strategies, but by definition they lack
cooperative characteristics which are fundamental in many real systems. Here we
introduce an efficient algorithm based on statistical physics for the
optimization of trajectories in cascade processes on graphs. We show that for a
wide class of irreversible dynamics, even in the absence of submodularity, the
spread optimization problem can be solved efficiently on large networks.
Analytic and algorithmic results on random graphs are complemented by the
solution of the spread maximization problem on a real-world network (the
Epinions consumer reviews network).Comment: Replacement for "The Spread Optimization Problem
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