140 research outputs found
Common adversaries form alliances: modelling complex networks via anti-transitivity
Anti-transitivity captures the notion that enemies of enemies are friends,
and arises naturally in the study of adversaries in social networks and in the
study of conflicting nation states or organizations. We present a simplified,
evolutionary model for anti-transitivity influencing link formation in complex
networks, and analyze the model's network dynamics. The Iterated Local
Anti-Transitivity (or ILAT) model creates anti-clone nodes in each time-step,
and joins anti-clones to the parent node's non-neighbor set. The graphs
generated by ILAT exhibit familiar properties of complex networks such as
densification, short distances (bounded by absolute constants), and bad
spectral expansion. We determine the cop and domination number for graphs
generated by ILAT, and finish with an analysis of their clustering
coefficients. We interpret these results within the context of real-world
complex networks and present open problems
On Minimizing Crossings in Storyline Visualizations
In a storyline visualization, we visualize a collection of interacting
characters (e.g., in a movie, play, etc.) by -monotone curves that converge
for each interaction, and diverge otherwise. Given a storyline with
characters, we show tight lower and upper bounds on the number of crossings
required in any storyline visualization for a restricted case. In particular,
we show that if (1) each meeting consists of exactly two characters and (2) the
meetings can be modeled as a tree, then we can always find a storyline
visualization with crossings. Furthermore, we show that there
exist storylines in this restricted case that require
crossings. Lastly, we show that, in the general case, minimizing the number of
crossings in a storyline visualization is fixed-parameter tractable, when
parameterized on the number of characters . Our algorithm runs in time
, where is the number of meetings.Comment: 6 pages, 4 figures. To appear at the 23rd International Symposium on
Graph Drawing and Network Visualization (GD 2015
Universal Geometric Graphs
We introduce and study the problem of constructing geometric graphs that have
few vertices and edges and that are universal for planar graphs or for some
sub-class of planar graphs; a geometric graph is \emph{universal} for a class
of planar graphs if it contains an embedding, i.e., a
crossing-free drawing, of every graph in .
Our main result is that there exists a geometric graph with vertices and
edges that is universal for -vertex forests; this extends to
the geometric setting a well-known graph-theoretic result by Chung and Graham,
which states that there exists an -vertex graph with edges
that contains every -vertex forest as a subgraph. Our bound on
the number of edges cannot be improved, even if more than vertices are
allowed.
We also prove that, for every positive integer , every -vertex convex
geometric graph that is universal for -vertex outerplanar graphs has a
near-quadratic number of edges, namely ; this almost
matches the trivial upper bound given by the -vertex complete
convex geometric graph.
Finally, we prove that there exists an -vertex convex geometric graph with
vertices and edges that is universal for -vertex
caterpillars.Comment: 20 pages, 8 figures; a 12-page extended abstracts of this paper will
appear in the Proceedings of the 46th Workshop on Graph-Theoretic Concepts in
Computer Science (WG 2020
Colored Non-Crossing Euclidean Steiner Forest
Given a set of -colored points in the plane, we consider the problem of
finding trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
, this is the well-known Euclidean Steiner tree problem. For general ,
a -approximation algorithm is known, where is the
Steiner ratio.
We present a PTAS for , a -approximation algorithm
for , and two approximation algorithms for general~, with ratios
and
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Markov propagation of allosteric effects in biomolecular systems: application to GroEL–GroES
We introduce a novel approach for elucidating the potential pathways of allosteric communication in biomolecular systems. The methodology, based on Markov propagation of ‘information' across the structure, permits us to partition the network of interactions into soft clusters distinguished by their coherent stochastics. Probabilistic participation of residues in these clusters defines the communication patterns inherent to the network architecture. Application to bacterial chaperonin complex GroEL–GroES, an allostery-driven structure, identifies residues engaged in intra- and inter-subunit communication, including those acting as hubs and messengers. A number of residues are distinguished by their high potentials to transmit allosteric signals, including Pro33 and Thr90 at the nucleotide-binding site and Glu461 and Arg197 mediating inter- and intra-ring communication, respectively. We propose two most likely pathways of signal transmission, between nucleotide- and GroES-binding sites across the cis and trans rings, which involve several conserved residues. A striking observation is the opposite direction of information flow within cis and trans rings, consistent with negative inter-ring cooperativity. Comparison with collective modes deduced from normal mode analysis reveals the propensity of global hinge regions to act as messengers in the transmission of allosteric signals
k is the Magic Number -- Inferring the Number of Clusters Through Nonparametric Concentration Inequalities
Most convex and nonconvex clustering algorithms come with one crucial
parameter: the in -means. To this day, there is not one generally
accepted way to accurately determine this parameter. Popular methods are simple
yet theoretically unfounded, such as searching for an elbow in the curve of a
given cost measure. In contrast, statistically founded methods often make
strict assumptions over the data distribution or come with their own
optimization scheme for the clustering objective. This limits either the set of
applicable datasets or clustering algorithms. In this paper, we strive to
determine the number of clusters by answering a simple question: given two
clusters, is it likely that they jointly stem from a single distribution? To
this end, we propose a bound on the probability that two clusters originate
from the distribution of the unified cluster, specified only by the sample mean
and variance. Our method is applicable as a simple wrapper to the result of any
clustering method minimizing the objective of -means, which includes
Gaussian mixtures and Spectral Clustering. We focus in our experimental
evaluation on an application for nonconvex clustering and demonstrate the
suitability of our theoretical results. Our \textsc{SpecialK} clustering
algorithm automatically determines the appropriate value for , without
requiring any data transformation or projection, and without assumptions on the
data distribution. Additionally, it is capable to decide that the data consists
of only a single cluster, which many existing algorithms cannot
Tight Proofs of Space and Replication
We construct a concretely practical proof-of-space (PoS) with arbitrarily tight security based on stacked depth robust graphs and constant-degree expander graphs. A proof-of-space (PoS) is an interactive proof system where a prover demonstrates that it is persistently using space to store information. A PoS is arbitrarily tight if the honest prover uses exactly N space and for any the construction can be tuned such that no adversary can pass verification using less than space. Most notably, the degree of the graphs in our construction are independent of , and the number of layers is only . The proof size is . The degree depends on the depth robust graphs, which are only required to maintain depth in subgraphs on 80% of the nodes. Our tight PoS is also secure against parallel attacks.
Tight proofs of space are necessary for proof-of-replication (PoRep), which is a publicly verifiable proof that the prover is dedicating unique resources to storing one or more retrievable replicas of a file. Our main PoS construction can be used as a PoRep, but data extraction is as inefficient as replica generation. We present a second variant of our construction called ZigZag PoRep that has fast/parallelizable data extraction compared to replica generation and maintains the same space tightness while only increasing the number of levels by roughly a factor two
Semi-Markov Graph Dynamics
In this paper, we outline a model of graph (or network) dynamics based on two
ingredients. The first ingredient is a Markov chain on the space of possible
graphs. The second ingredient is a semi-Markov counting process of renewal
type. The model consists in subordinating the Markov chain to the semi-Markov
counting process. In simple words, this means that the chain transitions occur
at random time instants called epochs. The model is quite rich and its possible
connections with algebraic geometry are briefly discussed. Moreover, for the
sake of simplicity, we focus on the space of undirected graphs with a fixed
number of nodes. However, in an example, we present an interbank market model
where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON
Spectral renormalization group theory on networks
Discrete amorphous materials are best described in terms of arbitrary
networks which can be embedded in three dimensional space. Investigating the
thermodynamic equilibrium as well as non-equilibrium behavior of such materials
around second order phase transitions call for special techniques.
We set up a renormalization group scheme by expanding an arbitrary scalar
field living on the nodes of an arbitrary network, in terms of the eigenvectors
of the normalized graph Laplacian. The renormalization transformation involves,
as usual, the integration over the more "rapidly varying" components of the
field, corresponding to eigenvectors with larger eigenvalues, and then
rescaling. The critical exponents depend on the particular graph through the
spectral density of the eigenvalues.Comment: 17 pages, 3 figures, presented at the Continuum Models and Discrete
Systems (CMDS-12), 21-25 Feb 2011, Saha Institute of Nuclear Physics,
Kolkata, Indi
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