202 research outputs found
Discriminants, symmetrized graph monomials, and sums of squares
Motivated by the necessities of the invariant theory of binary forms J. J.
Sylvester constructed in 1878 for each graph with possible multiple edges but
without loops its symmetrized graph monomial which is a polynomial in the
vertex labels of the original graph. In the 20-th century this construction was
studied by several authors. We pose the question for which graphs this
polynomial is a non-negative resp. a sum of squares. This problem is motivated
by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the
derivative of a univariate polynomial, and an interesting example of P. and A.
Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative
but not a sum of squares. We present detailed information about symmetrized
graph monomials for graphs with four and six edges, obtained by computer
calculations
The diplomat's dilemma: Maximal power for minimal effort in social networks
Closeness is a global measure of centrality in networks, and a proxy for how
influential actors are in social networks. In most network models, and many
empirical networks, closeness is strongly correlated with degree. However, in
social networks there is a cost of maintaining social ties. This leads to a
situation (that can occur in the professional social networks of executives,
lobbyists, diplomats and so on) where agents have the conflicting objectives of
aiming for centrality while simultaneously keeping the degree low. We
investigate this situation in an adaptive network-evolution model where agents
optimize their positions in the network following individual strategies, and
using only local information. The strategies are also optimized, based on the
success of the agent and its neighbors. We measure and describe the time
evolution of the network and the agents' strategies.Comment: Submitted to Adaptive Networks: Theory, Models and Applications, to
be published from Springe
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
Anisotropic Radial Layout for Visualizing Centrality and Structure in Graphs
This paper presents a novel method for layout of undirected graphs, where
nodes (vertices) are constrained to lie on a set of nested, simple, closed
curves. Such a layout is useful to simultaneously display the structural
centrality and vertex distance information for graphs in many domains,
including social networks. Closed curves are a more general constraint than the
previously proposed circles, and afford our method more flexibility to preserve
vertex relationships compared to existing radial layout methods. The proposed
approach modifies the multidimensional scaling (MDS) stress to include the
estimation of a vertex depth or centrality field as well as a term that
penalizes discord between structural centrality of vertices and their alignment
with this carefully estimated field. We also propose a visualization strategy
for the proposed layout and demonstrate its effectiveness using three social
network datasets.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
Hierarchy measure for complex networks
Nature, technology and society are full of complexity arising from the
intricate web of the interactions among the units of the related systems (e.g.,
proteins, computers, people). Consequently, one of the most successful recent
approaches to capturing the fundamental features of the structure and dynamics
of complex systems has been the investigation of the networks associated with
the above units (nodes) together with their relations (edges). Most complex
systems have an inherently hierarchical organization and, correspondingly, the
networks behind them also exhibit hierarchical features. Indeed, several papers
have been devoted to describing this essential aspect of networks, however,
without resulting in a widely accepted, converging concept concerning the
quantitative characterization of the level of their hierarchy. Here we develop
an approach and propose a quantity (measure) which is simple enough to be
widely applicable, reveals a number of universal features of the organization
of real-world networks and, as we demonstrate, is capable of capturing the
essential features of the structure and the degree of hierarchy in a complex
network. The measure we introduce is based on a generalization of the m-reach
centrality, which we first extend to directed/partially directed graphs. Then,
we define the global reaching centrality (GRC), which is the difference between
the maximum and the average value of the generalized reach centralities over
the network. We investigate the behavior of the GRC considering both a
synthetic model with an adjustable level of hierarchy and real networks.
Results for real networks show that our hierarchy measure is related to the
controllability of the given system. We also propose a visualization procedure
for large complex networks that can be used to obtain an overall qualitative
picture about the nature of their hierarchical structure.Comment: 29 pages, 9 figures, 4 table
Remarks on singular Cayley graphs and vanishing elements of simple groups
Let Γ be a finite graph and let A(Γ) be its adjacency matrix. Then Γ is singular if A(Γ) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs Cay(G,H) when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character χ of G for which ∑h∈Hχ(h)=0. At this stage, we focus on the case when H is a single conjugacy class hG of G; in this case, the above equality is equivalent to χ(h)=0 . Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element h∈G is called vanishing if χ(h)=0 for some irreducible character χ of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs
Theories for influencer identification in complex networks
In social and biological systems, the structural heterogeneity of interaction
networks gives rise to the emergence of a small set of influential nodes, or
influencers, in a series of dynamical processes. Although much smaller than the
entire network, these influencers were observed to be able to shape the
collective dynamics of large populations in different contexts. As such, the
successful identification of influencers should have profound implications in
various real-world spreading dynamics such as viral marketing, epidemic
outbreaks and cascading failure. In this chapter, we first summarize the
centrality-based approach in finding single influencers in complex networks,
and then discuss the more complicated problem of locating multiple influencers
from a collective point of view. Progress rooted in collective influence
theory, belief-propagation and computer science will be presented. Finally, we
present some applications of influencer identification in diverse real-world
systems, including online social platforms, scientific publication, brain
networks and socioeconomic systems.Comment: 24 pages, 6 figure
Fast Computing Betweenness Centrality with Virtual Nodes on Large Sparse Networks
Betweenness centrality is an essential index for analysis of complex networks. However, the calculation of betweenness centrality is quite time-consuming and the fastest known algorithm uses time and space for weighted networks, where and are the number of nodes and edges in the network, respectively. By inserting virtual nodes into the weighted edges and transforming the shortest path problem into a breadth-first search (BFS) problem, we propose an algorithm that can compute the betweenness centrality in time for integer-weighted networks, where is the average weight of edges and is the average degree in the network. Considerable time can be saved with the proposed algorithm when , indicating that it is suitable for lightly weighted large sparse networks. A similar concept of virtual node transformation can be used to calculate other shortest path based indices such as closeness centrality, graph centrality, stress centrality, and so on. Numerical simulations on various randomly generated networks reveal that it is feasible to use the proposed algorithm in large network analysis
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