65 research outputs found
The induced path function, monotonicity and betweenness
The induced path function of a graph consists of the set of all vertices lying on the induced paths between vertices and . This function is a special instance of a transit function. The function satisfies betweenness if implies and implies , and it is monotone if implies . The induced path function of aconnected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.betweenness;induced path;transit function;monotone;house domino;long cycle;p-graph
Transit functions on graphs (and posets)
The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of Ć¢ā¬ĖtransferringĆ¢ā¬ā¢ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs
The induced path function, monotonicity and betweenness
The induced path function of a graph consists of the set of all vertices lying on the induced paths between vertices and . This function is a special instance of a transit function. The function satisfies betweenness if implies and implies , and it is monotone if implies . The induced path function of a
connected graph satisfying the betweenness and monotone axioms are characterized by transit axioms
Transit functions on graphs (and posets)
The notion of transit function is introduced to present a unifying approach
for results and ideas on intervals, convexities and betweenness in graphs and
posets. Prime examples of such transit functions are the interval function I and
the induced path function J of a connected graph. Another transit function is
the all-paths function. New transit functions are introduced, such as the cutvertex
transit function and the longest path function. The main idea of transit
functions is that of ātransferringā problems and ideas of one transit function
to the other. For instance, a result on the interval function I might suggest
similar problems for the induced path function J. Examples are given of how
fruitful this transfer can be. A list of Prototype Problems and Questions for
this transferring process is given, which suggests many new questions and open
problems
Selection of Centrality Measures Using Self-Consistency and Bridge Axioms
We consider several families of network centrality measures induced by graph
kernels, which include some well-known measures and many new ones. The
Self-consistency and Bridge axioms, which appeared earlier in the literature,
are closely related to certain kernels and one of the families. We obtain a
necessary and sufficient condition for Self-consistency, a sufficient condition
for the Bridge axiom, indicate specific measures that satisfy these axioms, and
show that under some additional conditions they are incompatible. PageRank
centrality applied to undirected networks violates most conditions under study
and has a property that according to some authors is ``hard to imagine'' for a
centrality measure. We explain this phenomenon. Adopting the Self-consistency
or Bridge axiom leads to a drastic reduction in survey time in the culling
method designed to select the most appropriate centrality measures.Comment: 23 pages, 5 figures. A reworked versio
A Characterization of Uniquely Representable Graphs
The betweenness structure of a finite metric space M =(X, d) is a pair ā¬ (M)=(X, Ī²M) where Ī²M is the so-called betweenness relation of M that consists of point triplets (x, y, z) such that d(x, z)= d(x, y)+ d(y, z). The underlying graph of a betweenness structure ā¬ =(X, Ī²)isthe simple graph G(ā¬)=(X, E) where the edges are pairs of distinct points with no third point between them. A connected graph G is uniquely representable if there exists a unique metric betweenness structure with underlying graph G. It was implied by previous works that trees are uniquely representable. In this paper, we give a characterization of uniquely representable graphs by showing that they are exactly the block graphs. Further, we prove that two related classes of graphs coincide with the class of block graphs and the class of distance-hereditary graphs, respectively. We show that our results hold not only for metric but also for almost-metric betweenness structures. Ā© 2021 PĆ©ter G.N. SzabĆ³
Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution
The analysis of complex networks has so far revolved mainly around the role
of nodes and communities of nodes. However, the dynamics of interconnected
systems is commonly focalised on edge processes, and a dual edge-centric
perspective can often prove more natural. Here we present graph-theoretical
measures to quantify edge-to-edge relations inspired by the notion of flow
redistribution induced by edge failures. Our measures, which are related to the
pseudo-inverse of the Laplacian of the network, are global and reveal the
dynamical interplay between the edges of a network, including potentially
non-local interactions. Our framework also allows us to define the embeddedness
of an edge, a measure of how strongly an edge features in the weighted cuts of
the network. We showcase the general applicability of our edge-centric
framework through analyses of the Iberian Power grid, traffic flow in road
networks, and the C. elegans neuronal network.Comment: 24 pages, 6 figure
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