70,757 research outputs found
Average case analysis of DJ graphs
AbstractSreedhar et al. [V.C. Sreedhar, G.R. Gao, Y.-F. Lee, A new framework for elimination-based data flow analysis using DJ graphs, ACM Trans. Program. Lang. Syst. 20 (2) (1998) 388–435; V.C. Sreedhar, Efficient program analysis using DJ graphs, PhD thesis, School of Computer Science, McGill University, Montréal, Québec, Canada, 1995] have presented an elimination-based algorithm to solve data flow problems. A thorough analysis of the algorithm shows that the worst-case performance is at least quadratic in the number of nodes of the underlying graph. In contrast, Sreedhar reports a linear time behavior based on some practical applications.In this paper we prove that for goto-free programs, the average case behavior is indeed linear. As a byproduct our result also applies to the average size of the so-called dominance frontier.A thorough average case analysis based on a graph grammar is performed by studying properties of the j-edges in DJ graphs. It appears that this is the first time that a graph grammar is used in order to analyze an algorithm. The average linear time of the algorithm is obtained by classic techniques in the analysis of algorithms and data structures such as singularity analysis of generating functions and transfer lemmas
Average nearest neighbor degrees in scale-free networks
The average nearest neighbor degree (ANND) of a node of degree is widely
used to measure dependencies between degrees of neighbor nodes in a network. We
formally analyze ANND in undirected random graphs when the graph size tends to
infinity. The limiting behavior of ANND depends on the variance of the degree
distribution. When the variance is finite, the ANND has a deterministic limit.
When the variance is infinite, the ANND scales with the size of the graph, and
we prove a corresponding central limit theorem in the configuration model (CM,
a network with random connections). As ANND proved uninformative in the
infinite variance scenario, we propose an alternative measure, the average
nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic
function whenever the degree distribution has finite mean. We then consider the
erased configuration model (ECM), where self-loops and multiple edges are
removed, and investigate the well-known `structural negative correlations', or
`finite-size effects', that arise in simple graphs, such as ECM, because large
nodes can only have a limited number of large neighbors. Interestingly, we
prove that for any fixed , ANNR in ECM converges to the same limit as in CM.
However, numerical experiments show that finite-size effects occur when
scales with
Limit theorems for assortativity and clustering in null models for scale-free networks
An important problem in modeling networks is how to generate a randomly
sampled graph with given degrees. A popular model is the configuration model, a
network with assigned degrees and random connections. The erased configuration
model is obtained when self-loops and multiple edges in the configuration model
are removed. We prove an upper bound for the number of such erased edges for
regularly-varying degree distributions with infinite variance, and use this
result to prove central limit theorems for Pearson's correlation coefficient
and the clustering coefficient in the erased configuration model. Our results
explain the structural correlations in the erased configuration model and show
that removing edges leads to different scaling of the clustering coefficient.
We then prove that for the rank-1 inhomogeneous random graph, another null
model that creates scale-free simple networks, the results for Pearson's
correlation coefficient as well as for the clustering coefficient are similar
to the results for the erased configuration model
Sampling and Inference for Beta Neutral-to-the-Left Models of Sparse Networks
Empirical evidence suggests that heavy-tailed degree distributions occurring
in many real networks are well-approximated by power laws with exponents
that may take values either less than and greater than two. Models based on
various forms of exchangeability are able to capture power laws with , and admit tractable inference algorithms; we draw on previous results to
show that cannot be generated by the forms of exchangeability used
in existing random graph models. Preferential attachment models generate power
law exponents greater than two, but have been of limited use as statistical
models due to the inherent difficulty of performing inference in
non-exchangeable models. Motivated by this gap, we design and implement
inference algorithms for a recently proposed class of models that generates
of all possible values. We show that although they are not exchangeable,
these models have probabilistic structure amenable to inference. Our methods
make a large class of previously intractable models useful for statistical
inference.Comment: Accepted for publication in the proceedings of Conference on
Uncertainty in Artificial Intelligence (UAI) 201
Relation between Financial Market Structure and the Real Economy: Comparison between Clustering Methods
We quantify the amount of information filtered by different hierarchical
clustering methods on correlations between stock returns comparing it with the
underlying industrial activity structure. Specifically, we apply, for the first
time to financial data, a novel hierarchical clustering approach, the Directed
Bubble Hierarchical Tree and we compare it with other methods including the
Linkage and k-medoids. In particular, by taking the industrial sector
classification of stocks as a benchmark partition, we evaluate how the
different methods retrieve this classification. The results show that the
Directed Bubble Hierarchical Tree can outperform other methods, being able to
retrieve more information with fewer clusters. Moreover, we show that the
economic information is hidden at different levels of the hierarchical
structures depending on the clustering method. The dynamical analysis on a
rolling window also reveals that the different methods show different degrees
of sensitivity to events affecting financial markets, like crises. These
results can be of interest for all the applications of clustering methods to
portfolio optimization and risk hedging.Comment: 31 pages, 17 figure
Diversity of graphs with highly variable connectivity
A popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large-scale connectivity. While it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent, the relationship between competing descriptions is not entirely understood. This paper lends perspective on this problem by showing how the degree sequence and other constraints (e.g., connectedness, no self-loops or parallel edges) on a particular graph play a primary role in dictating many features, including its correlation structure. Building on recent work, we show how a simple structural metric characterizes key differences between graphs having the same degree sequence. More broadly, we show how the (often implicit) choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions
Spin models on random graphs with controlled topologies beyond degree constraints
We study Ising spin models on finitely connected random interaction graphs
which are drawn from an ensemble in which not only the degree distribution
can be chosen arbitrarily, but which allows for further fine-tuning of
the topology via preferential attachment of edges on the basis of an arbitrary
function Q(k,k') of the degrees of the vertices involved. We solve these models
using finite connectivity equilibrium replica theory, within the replica
symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system
are found to depend no longer only on the chosen degree distribution, but also
on the choice made for Q(k,k'). The increased ability to control interaction
topology in solvable models beyond prescribing only the degree distribution of
the interaction graph enables a more accurate modeling of real-world
interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys
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