454 research outputs found
Maximally Edge-Connected Realizations and Kundu's -factor Theorem
A simple graph with edge-connectivity and minimum degree
is maximally edge connected if . In 1964,
given a non-increasing degree sequence , Jack Edmonds
showed that there is a realization of that is -edge-connected if
and only if with when .
We strengthen Edmonds's result by showing that given a realization of
if is a spanning subgraph of with
such that when , then there is a
maximally edge-connected realization of with as a
subgraph. Our theorem tells us that there is a maximally edge-connected
realization of that differs from by at most edges. For
, if has a spanning forest with components,
then our theorem says there is a maximally edge-connected realization that
differs from by at most edges. As an application we combine our
work with Kundu's -factor Theorem to find maximally edge-connected
realizations with a -factor for and
present a partial result to a conjecture that strengthens the regular case of
Kundu's -factor theorem.Comment: 13 pages, 1 figur
Constructing highly regular expanders from hyperbolic Coxeter groups
A graph is defined inductively to be -regular if
is -regular and for every vertex of , the sphere of radius
around is an -regular graph. Such a graph is said
to be highly regular (HR) of level if . Chapman, Linial and
Peled studied HR-graphs of level 2 and provided several methods to construct
families of graphs which are expanders "globally and locally". They ask whether
such HR-graphs of level 3 exist.
In this paper we show how the theory of Coxeter groups, and abstract regular
polytopes and their generalisations, can lead to such graphs. Given a Coxeter
system and a subset of , we construct highly regular quotients
of the 1-skeleton of the associated Wythoffian polytope ,
which form an infinite family of expander graphs when is indefinite and
has finite vertex links. The regularity of the graphs in
this family can be deduced from the Coxeter diagram of . The expansion
stems from applying superapproximation to the congruence subgroups of the
linear group .
This machinery gives a rich collection of families of HR-graphs, with various
interesting properties, and in particular answers affirmatively the question
asked by Chapman, Linial and Peled.Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest
Vinber
Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face is to find the discrete
protoforms of the building blocks of continuum physics and mathematics. In the
following we embark on developing such concepts for irregular structures like
(large) graphs or networks which are intended to emulate (some of) the generic
properties of the presumed combinatorial substratum from which continuum
physics is assumed to emerge as a coarse grained and secondary model theory. We
briefly indicate how various concepts of discrete (functional) analysis and
geometry can be naturally constructed within this framework, leaving a larger
portion of the paper to the systematic developement of dimensional concepts and
their properties, which may have a possible bearing on various branches of
modern physics beyond quantum gravity.Comment: 16 pages, Invited paper to appear in the special issue of the Journal
of Chaos, Solitons and Fractals on: "Superstrings, M, F, S ... Theory" (M.S.
El Naschie, C. Castro, Editors
Structure and dynamics of core-periphery networks
Recent studies uncovered important core/periphery network structures
characterizing complex sets of cooperative and competitive interactions between
network nodes, be they proteins, cells, species or humans. Better
characterization of the structure, dynamics and function of core/periphery
networks is a key step of our understanding cellular functions, species
adaptation, social and market changes. Here we summarize the current knowledge
of the structure and dynamics of "traditional" core/periphery networks,
rich-clubs, nested, bow-tie and onion networks. Comparing core/periphery
structures with network modules, we discriminate between global and local
cores. The core/periphery network organization lies in the middle of several
extreme properties, such as random/condensed structures, clique/star
configurations, network symmetry/asymmetry, network
assortativity/disassortativity, as well as network hierarchy/anti-hierarchy.
These properties of high complexity together with the large degeneracy of core
pathways ensuring cooperation and providing multiple options of network flow
re-channelling greatly contribute to the high robustness of complex systems.
Core processes enable a coordinated response to various stimuli, decrease
noise, and evolve slowly. The integrative function of network cores is an
important step in the development of a large variety of complex organisms and
organizations. In addition to these important features and several decades of
research interest, studies on core/periphery networks still have a number of
unexplored areas.Comment: a comprehensive review of 41 pages, 2 figures, 1 table and 182
reference
The Range of Topological Effects on Communication
We continue the study of communication cost of computing functions when
inputs are distributed among processors, each of which is located at one
vertex of a network/graph called a terminal. Every other node of the network
also has a processor, with no input. The communication is point-to-point and
the cost is the total number of bits exchanged by the protocol, in the worst
case, on all edges.
Chattopadhyay, Radhakrishnan and Rudra (FOCS'14) recently initiated a study
of the effect of topology of the network on the total communication cost using
tools from embeddings. Their techniques provided tight bounds for simple
functions like Element-Distinctness (ED), which depend on the 1-median of the
graph. This work addresses two other kinds of natural functions. We show that
for a large class of natural functions like Set-Disjointness the communication
cost is essentially times the cost of the optimal Steiner tree connecting
the terminals. Further, we show for natural composed functions like and , the naive protocols
suggested by their definition is optimal for general networks. Interestingly,
the bounds for these functions depend on more involved topological parameters
that are a combination of Steiner tree and 1-median costs.
To obtain our results, we use some new tools in addition to ones used in
Chattopadhyay et. al. These include (i) viewing the communication constraints
via a linear program; (ii) using tools from the theory of tree embeddings to
prove topology sensitive direct sum results that handle the case of composed
functions and (iii) representing the communication constraints of certain
problems as a family of collection of multiway cuts, where each multiway cut
simulates the hardness of computing the function on the star topology
Statistical Mechanics of Community Detection
Starting from a general \textit{ansatz}, we show how community detection can
be interpreted as finding the ground state of an infinite range spin glass. Our
approach applies to weighted and directed networks alike. It contains the
\textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the
modularity as defined by Newman and Girvan \cite{Girvan03} as special
cases. The community structure of the network is interpreted as the spin
configuration that minimizes the energy of the spin glass with the spin states
being the community indices. We elucidate the properties of the ground state
configuration to give a concise definition of communities as cohesive subgroups
in networks that is adaptive to the specific class of network under study.
Further we show, how hierarchies and overlap in the community structure can be
detected. Computationally effective local update rules for optimization
procedures to find the ground state are given. We show how the \textit{ansatz}
may be used to discover the community around a given node without detecting all
communities in the full network and we give benchmarks for the performance of
this extension. Finally, we give expectation values for the modularity of
random graphs, which can be used in the assessment of statistical significance
of community structure
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