30 research outputs found
Network Transitivity and Matrix Models
This paper is a step towards a systematic theory of the transitivity
(clustering) phenomenon in random networks. A static framework is used, with
adjacency matrix playing the role of the dynamical variable. Hence, our model
is a matrix model, where matrices are random, but their elements take values 0
and 1 only. Confusion present in some papers where earlier attempts to
incorporate transitivity in a similar framework have been made is hopefully
dissipated. Inspired by more conventional matrix models, new analytic
techniques to develop a static model with non-trivial clustering are
introduced. Computer simulations complete the analytic discussion.Comment: 11 pages, 7 eps figures, 2-column revtex format, print bug correcte
Host--parasite models on graphs
The behavior of two interacting populations, ``hosts''and ``parasites'', is
investigated on Cayley trees and scale-free networks. In the former case
analytical and numerical arguments elucidate a phase diagram, whose most
interesting feature is the absence of a tri-critical point as a function of the
two independent spreading parameters. For scale-free graphs, the parasite
population can be described effectively by
Susceptible-Infected-Susceptible-type dynamics in a host background. This is
shown both by considering the appropriate dynamical equations and by numerical
simulations on Barab\'asi-Albert networks with the major implication that in
the termodynamic limit the critical parasite spreading parameter vanishes.Comment: 10 pages, 6 figures, submitted to PRE; analytics redone, new
calculations added, references added, appendix remove
Correlations in Bipartite Collaboration Networks
Collaboration networks are studied as an example of growing bipartite
networks. These have been previously observed to have structure such as
positive correlations between nearest-neighbour degrees. However, a detailed
understanding of the origin of this phenomenon and the growth dynamics is
lacking. Both of these are analyzed empirically and simulated using various
models. A new one is presented, incorporating empirically necessary ingredients
such as bipartiteness and sublinear preferential attachment. This, and a
recently proposed model of team assembly both agree roughly with some empirical
observations and fail in several others.Comment: 13 pages, 17 figures, 2 table, submitted to JSTAT; manuscript
reorganized, figures and a table adde
Computational complexity arising from degree correlations in networks
We apply a Bethe-Peierls approach to statistical-mechanics models defined on
random networks of arbitrary degree distribution and arbitrary correlations
between the degrees of neighboring vertices. Using the NP-hard optimization
problem of finding minimal vertex covers on these graphs, we show that such
correlations may lead to a qualitatively different solution structure as
compared to uncorrelated networks. This results in a higher complexity of the
network in a computational sense: Simple heuristic algorithms fail to find a
minimal vertex cover in the highly correlated case, whereas uncorrelated
networks seem to be simple from the point of view of combinatorial
optimization.Comment: 4 pages, 1 figure, accepted in Phys. Rev.
Finite-time fluctuations in the degree statistics of growing networks
This paper presents a comprehensive analysis of the degree statistics in
models for growing networks where new nodes enter one at a time and attach to
one earlier node according to a stochastic rule. The models with uniform
attachment, linear attachment (the Barab\'asi-Albert model), and generalized
preferential attachment with initial attractiveness are successively
considered. The main emphasis is on finite-size (i.e., finite-time) effects,
which are shown to exhibit different behaviors in three regimes of the
size-degree plane: stationary, finite-size scaling, large deviations.Comment: 33 pages, 7 figures, 1 tabl
Activity driven modeling of time varying networks
Network modeling plays a critical role in identifying statistical
regularities and structural principles common to many systems. The large
majority of recent modeling approaches are connectivity driven. The structural
patterns of the network are at the basis of the mechanisms ruling the network
formation. Connectivity driven models necessarily provide a time-aggregated
representation that may fail to describe the instantaneous and fluctuating
dynamics of many networks. We address this challenge by defining the activity
potential, a time invariant function characterizing the agents' interactions
and constructing an activity driven model capable of encoding the instantaneous
time description of the network dynamics. The model provides an explanation of
structural features such as the presence of hubs, which simply originate from
the heterogeneous activity of agents. Within this framework, highly dynamical
networks can be described analytically, allowing a quantitative discussion of
the biases induced by the time-aggregated representations in the analysis of
dynamical processes.Comment: 10 pages, 4 figure
Mesoscopic structure conditions the emergence of cooperation on social networks
We study the evolutionary Prisoner's Dilemma on two social networks obtained
from actual relational data. We find very different cooperation levels on each
of them that can not be easily understood in terms of global statistical
properties of both networks. We claim that the result can be understood at the
mesoscopic scale, by studying the community structure of the networks. We
explain the dependence of the cooperation level on the temptation parameter in
terms of the internal structure of the communities and their interconnections.
We then test our results on community-structured, specifically designed
artificial networks, finding perfect agreement with the observations in the
real networks. Our results support the conclusion that studies of evolutionary
games on model networks and their interpretation in terms of global properties
may not be sufficient to study specific, real social systems. In addition, the
community perspective may be helpful to interpret the origin and behavior of
existing networks as well as to design structures that show resilient
cooperative behavior.Comment: Largely improved version, includes an artificial network model that
fully confirms the explanation of the results in terms of inter- and
intra-community structur
Growing networks with local rules: preferential attachment, clustering hierarchy and degree correlations
The linear preferential attachment hypothesis has been shown to be quite
successful to explain the existence of networks with power-law degree
distributions. It is then quite important to determine if this mechanism is the
consequence of a general principle based on local rules. In this work it is
claimed that an effective linear preferential attachment is the natural outcome
of growing network models based on local rules. It is also shown that the local
models offer an explanation to other properties like the clustering hierarchy
and degree correlations recently observed in complex networks. These
conclusions are based on both analytical and numerical results of different
local rules, including some models already proposed in the literature.Comment: 17 pages, 14 figures (to appear in Phys. Rev E
A unified data representation theory for network visualization, ordering and coarse-graining
Representation of large data sets became a key question of many scientific
disciplines in the last decade. Several approaches for network visualization,
data ordering and coarse-graining accomplished this goal. However, there was no
underlying theoretical framework linking these problems. Here we show an
elegant, information theoretic data representation approach as a unified
solution of network visualization, data ordering and coarse-graining. The
optimal representation is the hardest to distinguish from the original data
matrix, measured by the relative entropy. The representation of network nodes
as probability distributions provides an efficient visualization method and, in
one dimension, an ordering of network nodes and edges. Coarse-grained
representations of the input network enable both efficient data compression and
hierarchical visualization to achieve high quality representations of larger
data sets. Our unified data representation theory will help the analysis of
extensive data sets, by revealing the large-scale structure of complex networks
in a comprehensible form.Comment: 13 pages, 5 figure
Controlling centrality in complex networks
Spectral centrality measures allow to identify influential individuals in social groups, to rank Web pages by popularity, and even to determine the impact of scientific researches. The centrality score of a node within a network crucially depends on the entire pattern of connections, so that the usual approach is to compute node centralities once the network structure is assigned. We face here with the inverse problem, that is, we study how to modify the centrality scores of the nodes by acting on the structure of a given network. We show that there exist particular subsets of nodes, called controlling sets, which can assign any prescribed set of centrality values to all the nodes of a graph, by cooperatively tuning the weights of their out-going links. We found that many large networks from the real world have surprisingly small controlling sets, containing even less than 5 – 10% of the nodes