1,389 research outputs found
Centrality analysis for modified lattices
We derive new, exact expressions for network centrality vectors associated with classical Watts-Strogatz style "ring plus shortcut" networks. We also derive easy-to-interpret approximations that are highly accurate in the large network limit. The analysis helps us to understand the role of the Katz parameter and the PageRank parameter, to compare linear system and eigenvalue based centrality measures, and to predict the behavior of centrality measures on more complicated networks
Robustness of Random Graphs Based on Natural Connectivity
Recently, it has been proposed that the natural connectivity can be used to
efficiently characterise the robustness of complex networks. Natural
connectivity quantifies the redundancy of alternative routes in a network by
evaluating the weighted number of closed walks of all lengths and can be
regarded as the average eigenvalue obtained from the graph spectrum. In this
article, we explore the natural connectivity of random graphs both analytically
and numerically and show that it increases linearly with the average degree. By
comparing with regular ring lattices and random regular graphs, we show that
random graphs are more robust than random regular graphs; however, the
relationship between random graphs and regular ring lattices depends on the
average degree and graph size. We derive the critical graph size as a function
of the average degree, which can be predicted by our analytical results. When
the graph size is less than the critical value, random graphs are more robust
than regular ring lattices, whereas regular ring lattices are more robust than
random graphs when the graph size is greater than the critical value.Comment: 12 pages, 4 figure
Random Walks on Stochastic Temporal Networks
In the study of dynamical processes on networks, there has been intense focus
on network structure -- i.e., the arrangement of edges and their associated
weights -- but the effects of the temporal patterns of edges remains poorly
understood. In this chapter, we develop a mathematical framework for random
walks on temporal networks using an approach that provides a compromise between
abstract but unrealistic models and data-driven but non-mathematical
approaches. To do this, we introduce a stochastic model for temporal networks
in which we summarize the temporal and structural organization of a system
using a matrix of waiting-time distributions. We show that random walks on
stochastic temporal networks can be described exactly by an
integro-differential master equation and derive an analytical expression for
its asymptotic steady state. We also discuss how our work might be useful to
help build centrality measures for temporal networks.Comment: Chapter in Temporal Networks (Petter Holme and Jari Saramaki
editors). Springer. Berlin, Heidelberg 2013. The book chapter contains minor
corrections and modifications. This chapter is based on arXiv:1112.3324,
which contains additional calculations and numerical simulation
Quantum walk-based search and centrality
We study the discrete-time quantum walk-based search for a marked vertex on a
graph. By considering various structures in which not all vertices are
equivalent, we investigate the relationship between the successful search
probability and the position of the marked vertex, in particular its
centrality. We find that the maximum value of the search probability does not
necessarily increase as the marked vertex becomes more central and we
investigate an interesting relationship between the frequency of the successful
search probability and the centrality of the marked vertex.Comment: 29 pages, 17 figure
Co-evolution of density and topology in a simple model of city formation
We study the influence that population density and the road network have on
each others' growth and evolution. We use a simple model of formation and
evolution of city roads which reproduces the most important empirical features
of street networks in cities. Within this framework, we explicitely introduce
the topology of the road network and analyze how it evolves and interact with
the evolution of population density. We show that accessibility issues -pushing
individuals to get closer to high centrality nodes- lead to high density
regions and the appearance of densely populated centers. In particular, this
model reproduces the empirical fact that the density profile decreases
exponentially from a core district. In this simplified model, the size of the
core district depends on the relative importance of transportation and rent
costs.Comment: 13 pages, 13 figure
Complex network analysis and nonlinear dynamics
This chapter aims at reviewing complex network and nonlinear dynamical
models and methods that were either developed for or applied to socioeconomic
issues, and pertinent to the theme of New Economic Geography. After an introduction
to the foundations of the field of complex networks, the present summary
introduces some applications of complex networks to economics, finance, epidemic
spreading of innovations, and regional trade and developments. The chapter also
reviews results involving applications of complex networks to other relevant
socioeconomic issue
Opinion formation driven by PageRank node influence on directed networks
We study a two states opinion formation model driven by PageRank node
influence and report an extensive numerical study on how PageRank affects
collective opinion formations in large-scale empirical directed networks. In
our model the opinion of a node can be updated by the sum of its neighbor
nodes' opinions weighted by the node influence of the neighbor nodes at each
step. We consider PageRank probability and its sublinear power as node
influence measures and investigate evolution of opinion under various
conditions. First, we observe that all networks reach steady state opinion
after a certain relaxation time. This time scale is decreasing with the
heterogeneity of node influence in the networks. Second, we find that our model
shows consensus and non-consensus behavior in steady state depending on types
of networks: Web graph, citation network of physics articles, and LiveJournal
social network show non-consensus behavior while Wikipedia article network
shows consensus behavior. Third, we find that a more heterogeneous influence
distribution leads to a more uniform opinion state in the cases of Web graph,
Wikipedia, and Livejournal. However, the opposite behavior is observed in the
citation network. Finally we identify that a small number of influential nodes
can impose their own opinion on significant fraction of other nodes in all
considered networks. Our study shows that the effects of heterogeneity of node
influence on opinion formation can be significant and suggests further
investigations on the interplay between node influence and collective opinion
in networks.Comment: 10 pages, 6 figures. Published in Physica A 436, 707-715 (2015
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