2,190 research outputs found
Scalable Positional Analysis for Studying Evolution of Nodes in Networks
In social network analysis, the fundamental idea behind the notion of
position is to discover actors who have similar structural signatures.
Positional analysis of social networks involves partitioning the actors into
disjoint sets using a notion of equivalence which captures the structure of
relationships among actors. Classical approaches to Positional Analysis, such
as Regular equivalence and Equitable Partitions, are too strict in grouping
actors and often lead to trivial partitioning of actors in real world networks.
An Epsilon Equitable Partition (EEP) of a graph, which is similar in spirit to
Stochastic Blockmodels, is a useful relaxation to the notion of structural
equivalence which results in meaningful partitioning of actors. In this paper
we propose and implement a new scalable distributed algorithm based on
MapReduce methodology to find EEP of a graph. Empirical studies on random
power-law graphs show that our algorithm is highly scalable for sparse graphs,
thereby giving us the ability to study positional analysis on very large scale
networks. We also present the results of our algorithm on time evolving
snapshots of the facebook and flickr social graphs. Results show the importance
of positional analysis on large dynamic networks.Comment: Presented at the workshop on Mining Networks and Graphs: A Big Data
Analytic Challenge, held in conjunction with the SIAM Data Mining (SDM)
Conference in April 2014. 13 page
Vizing-Goldberg type bounds for the equitable chromatic number of block graphs
An equitable coloring of a graph is a proper vertex coloring of such
that the sizes of any two color classes differ by at most one. In the paper, we
pose a conjecture that offers a gap-one bound for the smallest number of colors
needed to equitably color every block graph. In other words, the difference
between the upper and the lower bounds of our conjecture is at most one. Thus,
in some sense, the situation is similar to that of chromatic index, where we
have the classical theorem of Vizing and the Goldberg conjecture for
multigraphs. The results obtained in the paper support our conjecture. More
precisely, we verify it in the class of well-covered block graphs, which are
block graphs in which each vertex belongs to a maximum independent set. We also
show that the conjecture is true for block graphs, which contain a vertex that
does not lie in an independent set of size larger than two. Finally, we verify
the conjecture for some symmetric-like block graphs. In order to derive our
results we obtain structural characterizations of block graphs from these
classes.Comment: 21 pages, 12 figure
Perfect state transfer on quotient graphs
We prove new results on perfect state transfer of quantum walks on quotient
graphs. Since a graph has perfect state transfer if and only if its
quotient , under any equitable partition , has perfect state
transfer, we exhibit graphs with perfect state transfer between two vertices
but which lack automorphism swapping them. This answers a question of Godsil
(Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian
product of quotient graphs is isomorphic to the
quotient graph , for some equitable partition . This
provides an algebraic description of a construction due to Feder (Physical
Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.Comment: 20 pages, 10 figure
A Simple Characterization of Proportionally 2-choosable Graphs
We recently introduced proportional choosability, a new list analogue of
equitable coloring. Like equitable coloring, and unlike list equitable coloring
(a.k.a. equitable choosability), proportional choosability bounds sizes of
color classes both from above and from below. In this note, we show that a
graph is proportionally 2-choosable if and only if it is a linear forest such
that its largest component has at most 5 vertices and all of its other
components have two or fewer vertices. We also construct examples that show
that characterizing equitably 2-choosable graphs is still open.Comment: 9 page
Equitable total coloring of corona of cubic graphs
The minimum number of total independent partition sets of of a
graph is called the \emph{total chromatic number} of , denoted by
. If the difference between cardinalities of any two total
independent sets is at most one, then the minimum number of total independent
partition sets of is called the \emph{equitable total chromatic
number}, and is denoted by .
In this paper we consider equitable total coloring of coronas of cubic
graphs, . It turns out that, independly on the values of equitable
total chromatic number of factors and , equitable total chromatic number
of corona is equal to . Thereby, we confirm
Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total
Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs.
As a direct consequence we get that all coronas of cubic graphs are of Type 1.Comment: 12 page
Complexity of equitable tree-coloring problems
A \emph{-tree-coloring} of a graph is a -coloring of vertices
of such that the subgraph induced by each color class is a forest of
maximum degree at most A \emph{-tree-coloring} of a graph
is a -coloring of vertices of such that the subgraph induced by each
color class is a forest.
Wu, Zhang, and Li introduced the concept of \emph{equitable -tree-coloring} (respectively, \emph{equitable -tree-coloring})
which is a -tree-coloring (respectively, -tree-coloring)
such that the sizes of any two color classes differ by at most one. Among other
results, they obtained a sharp upper bound on the minimum such that
has an equitable -tree-coloring for every
In this paper, we obtain a polynomial time criterion to decide if a complete
bipartite graph has an equitable -tree-coloring or an equitable
-tree-coloring. Nevertheless, deciding if a graph in general
has an equitable -tree-coloring or an equitable
-tree-coloring is NP-complete.Comment: arXiv admin note: text overlap with arXiv:1506.0391
Perfect quantum state transfer of hard-core bosons on weighted path graphs
The ability to accurately transfer quantum information through networks is an
important primitive in distributed quantum systems. While perfect quantum state
transfer (PST) can be effected by a single particle undergoing continuous-time
quantum walks on a variety of graphs, it is not known if PST persists for many
particles in the presence of interactions. We show that if single-particle PST
occurs on one-dimensional weighted path graphs, then systems of hard-core
bosons undergoing quantum walks on these paths also undergo PST. The analysis
extends the Tonks-Girardeau ansatz to weighted graphs using techniques in
algebraic graph theory. The results suggest that hard-core bosons do not
generically undergo PST, even on graphs which exhibit single-particle PST.Comment: 19 page
A Graph Partitioning Approach to Predict Patterns in Lateral Inhibition Systems
We analyze pattern formation on a network of cells where each cell inhibits
its neighbors through cell-to-cell contact signaling. The network is modeled as
an interconnection of identical dynamical subsystems each of which represents
the signaling reactions in a cell. We search for steady state patterns by
partitioning the graph vertices into disjoint classes, where the cells in the
same class have the same final fate. To prove the existence of steady states
with this structure, we use results from monotone systems theory. Finally, we
analyze the stability of these patterns with a block decomposition based on the
graph partition.Comment: 8 pages, 6 figure
Equitable random graphs
Random graph models have played a dominant role in the theoretical study of
networked systems. The Poisson random graph of Erdos and Renyi, in particular,
as well as the so-called configuration model, have served as the starting point
for numerous calculations. In this paper we describe another large class of
random graph models, which we call equitable random graphs and which are
flexible enough to represent networks with diverse degree distributions and
many nontrivial types of structure, including community structure, bipartite
structure, degree correlations, stratification, and others, yet are exactly
solvable for a wide range of properties in the limit of large graph size,
including percolation properties, complete spectral density, and the behavior
of homogeneous dynamical systems, such as coupled oscillators or epidemic
models.Comment: 5 pages, 2 figure
Equitable Colorings of -Corona Products of Cubic Graphs
A graph is equitably -colorable if its vertices can be partitioned
into independent sets in such a way that the number of vertices in any two
sets differ by at most one. The smallest integer for which such a coloring
exists is known as the \emph{equitable chromatic number} of and it is
denoted by .
In this paper the problem of determinig the value of equitable chromatic
number for multicoronas of cubic graphs is studied. The problem
of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial
time. The complexity of equitable coloring problem is an open question for
these graphs. We provide some polynomially solvable cases of cubical
multicoronas and give simple linear time algorithms for equitable coloring of
such graphs which use at most colors in the
remaining cases.Comment: 12 page
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