282 research outputs found
Centrality measures for graphons: Accounting for uncertainty in networks
As relational datasets modeled as graphs keep increasing in size and their
data-acquisition is permeated by uncertainty, graph-based analysis techniques
can become computationally and conceptually challenging. In particular, node
centrality measures rely on the assumption that the graph is perfectly known --
a premise not necessarily fulfilled for large, uncertain networks. Accordingly,
centrality measures may fail to faithfully extract the importance of nodes in
the presence of uncertainty. To mitigate these problems, we suggest a
statistical approach based on graphon theory: we introduce formal definitions
of centrality measures for graphons and establish their connections to
classical graph centrality measures. A key advantage of this approach is that
centrality measures defined at the modeling level of graphons are inherently
robust to stochastic variations of specific graph realizations. Using the
theory of linear integral operators, we define degree, eigenvector, Katz and
PageRank centrality functions for graphons and establish concentration
inequalities demonstrating that graphon centrality functions arise naturally as
limits of their counterparts defined on sequences of graphs of increasing size.
The same concentration inequalities also provide high-probability bounds
between the graphon centrality functions and the centrality measures on any
sampled graph, thereby establishing a measure of uncertainty of the measured
centrality score. The same concentration inequalities also provide
high-probability bounds between the graphon centrality functions and the
centrality measures on any sampled graph, thereby establishing a measure of
uncertainty of the measured centrality score.Comment: Authors ordered alphabetically, all authors contributed equally. 21
pages, 7 figure
Ranking hubs and authorities using matrix functions
The notions of subgraph centrality and communicability, based on the
exponential of the adjacency matrix of the underlying graph, have been
effectively used in the analysis of undirected networks. In this paper we
propose an extension of these measures to directed networks, and we apply them
to the problem of ranking hubs and authorities. The extension is achieved by
bipartization, i.e., the directed network is mapped onto a bipartite undirected
network with twice as many nodes in order to obtain a network with a symmetric
adjacency matrix. We explicitly determine the exponential of this adjacency
matrix in terms of the adjacency matrix of the original, directed network, and
we give an interpretation of centrality and communicability in this new
context, leading to a technique for ranking hubs and authorities. The matrix
exponential method for computing hubs and authorities is compared to the well
known HITS algorithm, both on small artificial examples and on more realistic
real-world networks. A few other ranking algorithms are also discussed and
compared with our technique. The use of Gaussian quadrature rules for
calculating hub and authority scores is discussed.Comment: 28 pages, 6 figure
Scalable Katz ranking computation in large static and dynamic graphs
Network analysis defines a number of centrality measures to identify the most central nodes in a network. Fast computation of those measures is a major challenge in algorithmic network analysis. Aside from closeness and betweenness, Katz centrality is one of the established centrality measures. In this paper, we consider the problem of computing rankings for Katz centrality. In particular, we propose upper and lower bounds on the Katz score of a given node. While previous approaches relied on numerical approximation or heuristics to compute Katz centrality rankings, we construct an algorithm that iteratively improves those upper and lower bounds until a correct Katz ranking is obtained. We extend our algorithm to dynamic graphs while maintaining its correctness guarantees. Experiments demonstrate that our static graph algorithm outperforms both numerical approaches and heuristics with speedups between 1.5Ă and 3.5Ă, depending on the desired quality guarantees. Our dynamic graph algorithm improves upon the static algorithm for update batches of less than 10000 edges. We provide efficient parallel CPU and GPU implementations of our algorithms that enable near real-time Katz centrality computation for graphs with hundreds of millions of nodes in fractions of seconds
Distribution of centrality measures on undirected random networks via cavity method
The Katz centrality of a node in a complex network is a measure of the node's
importance as far as the flow of information across the network is concerned.
For ensembles of locally tree-like and undirected random graphs, this
observable is a random variable. Its full probability distribution is of
interest but difficult to handle analytically because of its "global" character
and its definition in terms of a matrix inverse. Leveraging a fast Gaussian
Belief Propagation-cavity algorithm to solve linear systems on a tree-like
structure, we show that (i) the Katz centrality of a single instance can be
computed recursively in a very fast way, and (ii) the probability that a
random node in the ensemble of undirected random graphs has centrality
satisfies a set of recursive distributional equations, which can be
analytically characterized and efficiently solved using a population dynamics
algorithm. We test our solution on ensembles of Erd\H{o}s-R\'enyi and
scale-free networks in the locally tree-like regime, with excellent agreement.
The distributions display a crossover between multimodality and unimodality as
the mean degree increases, where distinct peaks correspond to the contribution
to the centrality coming from nodes of different degrees. We also provide an
approximate formula based on a rank- projection that works well if the
network is not too sparse, and we argue that an extension of our method could
be efficiently extended to tackle analytical distributions of other centrality
measures such as PageRank for directed networks in a transparent and
user-friendly way.Comment: 14 pages, 11 fi
Eigenvector-Based Centrality Measures for Temporal Networks
Numerous centrality measures have been developed to quantify the importances
of nodes in time-independent networks, and many of them can be expressed as the
leading eigenvector of some matrix. With the increasing availability of network
data that changes in time, it is important to extend such eigenvector-based
centrality measures to time-dependent networks. In this paper, we introduce a
principled generalization of network centrality measures that is valid for any
eigenvector-based centrality. We consider a temporal network with N nodes as a
sequence of T layers that describe the network during different time windows,
and we couple centrality matrices for the layers into a supra-centrality matrix
of size NTxNT whose dominant eigenvector gives the centrality of each node i at
each time t. We refer to this eigenvector and its components as a joint
centrality, as it reflects the importances of both the node i and the time
layer t. We also introduce the concepts of marginal and conditional
centralities, which facilitate the study of centrality trajectories over time.
We find that the strength of coupling between layers is important for
determining multiscale properties of centrality, such as localization phenomena
and the time scale of centrality changes. In the strong-coupling regime, we
derive expressions for time-averaged centralities, which are given by the
zeroth-order terms of a singular perturbation expansion. We also study
first-order terms to obtain first-order-mover scores, which concisely describe
the magnitude of nodes' centrality changes over time. As examples, we apply our
method to three empirical temporal networks: the United States Ph.D. exchange
in mathematics, costarring relationships among top-billed actors during the
Golden Age of Hollywood, and citations of decisions from the United States
Supreme Court.Comment: 38 pages, 7 figures, and 5 table
Delinquent Networks
Delinquents are embedded in a network of relationships. Social ties among delinquents are modelled by means of a graph where delinquents compete for a booty and benefit from local interactions with their neighbors. Each delinquent decides in a non cooperative way how much delinquency effort he will exert. Using the network model developed by Ballester et al. (2006), we characterize the Nash equilibrium and derive an optimal enforcement policy, called the key-player policy, which targets the delinquent who, once removed, leads to the highest aggregate delinquency reduction. We then extend our characterization of optimal single player network removal for delinquency reduction, the key player, to optimal group removal, the key group. We also characterize and derive a policy that targets links rather than players. Finally, we endogenize the network connecting delinquents by allowing players to join the labor market instead of committing delinquent offenses. The key-player policy turns out to be much more complex since it depends on wages and on the structure of the network.Social networks, delinquency decision, key group, NP-hard problem, crime policies
- âŚ