Multiplex Networks
In many complex systems the interactions among the elementary
components can be of qualitatively different nature. Such systems are
therefore naturally described and represented in terms of multiplex or
multilayer networks, i.e. networks where each layer stands for a
different type of interaction between the same set of nodes.
We have proposed
basic metrics
[PRE14] for multiplex networks, with particular attention to
correlations
[PRE15]. Together with
M. Barthelemy (CNRS, Paris) and
G. Bianconi (QMUL, London) we have introduced some
LINEAR
[PRL13] AND NONLINEAR GROWTH MODELS
[PRE14] to construct multiplex networks. With
A. Arenas and
M. De Domenico (Alephsys Lab, Terragona),
we have introduced a method to REDUCE the number of layers in
multilayer networks while minimizing information loss, which allows
to describe the structure of a multilayer complex system
with an optimal tradeoff between accuracy and complexity
[NatComm15]. We have also studied dynamical processes,
such as biased random walks on multilayer networks, and even the
case of two different types of INTERTWINED DYNAMICAL PROCESSES, namely
synchronization and energy transport, taking place on interconnected networks
[arXiv14]. We are now working to an extension of the Axelroad
model for the dissemination of cultures in multiplex networks,
and to the study of Turing patterns in networks with two layers.
Together with L. Lacasa (QMUL, London) we have
introduced a method to study multivariate time series by transforming them
into multiplex networks
[NatSR15].
More on our work can be found at the website of the EU project
LASAGNE, on
multiLAyer SpAtiotemporal Generalized NEtworks, of which I am the
Scientific Coordinator.
Temporal Networks
Realworld networks are inherently dynamic, with the links fluctuating
and changing over time. E.g., human contacts or relationships change
over time because individuals lose old acquaintances, acquire new
ones, or move over geographic space. Despite this fact, most of the
classic studies on complex networks are based on the analysis of
static (aggregated) graphs, as if the links were all concurrent
in time. In order to capture the real dynamic behavior and time correlations
of temporal networks, we describe them as timevarying graphs,
i.e. ensembles of timeordered graphs. We have extended to timevarying
graphs various metrics and models originally developed for static graphs.
In collaboration with
C. Mascolo (Univ.
of Cambridge),
M. Musolesi
(Univ. of Birmingham) and
G. Russo (Univ.
of Catania) we have introduced the concepts of TEMPORAL SHORTEST PATHS
[PRE10]
and CONNECTED COMPONENTS
[Chaos12] in
timevarying graphs, and we have defined as TEMPORAL SMALL WORLD a
timevarying graph in which the links are highly clustered in time,
yet the nodes are at small average temporal
distances [PRE10]. We have explored the
smallworld behavior in synthetic timevarying networks of mobile
agents, and in real social and biological timevarying systems, and we have
also shown how to exploit TEMPORAL CENTRALITY MEASURES to
contain mobile phone viruses that spread via Bluetooth contacts
[arXiv10]
(MIT Tech Review).
More on our work can be found in two Chapters we have recently wrote
for a new book on temporal networks
[Spr13]
[Spr13_2].
Random Walks on Complex Networks
Random walks are the simplest way to explore a graph. In
[PRL08], in collaboration with
J. GomezGardenes and
Y. Moreno (Bifi, Zaragoza) we have studied traffic in a
complex network by modeling packets of information as noninteracting
random walkers. The model is amenable to analytical solution and,
notwithstanding its simplicity, is able to capture the essential
ingredients determining the SCALING OF FLUCTUATIONS empirically
observed for traffic flow in the Internet and in other real networks.
In [EPL14] we have shown how to
characterize the structure of a network by studying the
properties of the trajectories of unbiased random walks over it.
In [PRE08] we have introduced the DYNAMICAL ENTROPY of
the trajectory of DEGREEBIASED RANDOM WALKS. In such walkers,
the probability of moving to a node depends on some power of the degree of
the target node, and depending on whether the exponent is positive or
negative, this can give rise to walks that favor or disfavor
highdegree vertices [PRE14].
In collaboration with
R. Sinatra (Northeastern University, USA)
and R. Lambiotte (Univ. Namur) we have shown that,
by opportunely tuning the value of the exponent, we can get diffusion
processes with local rules and maximal entropy
rate [PRE11], and we can even induce the emergence of
synchronization in mobile Kuramoto oscillators
[PRE13].
Turning on the interactions, in collaboration with F. Bagnoli (Univ. Florence) we have introduced and studied a model of interacting random walks
competing for the nodes of a complex network [EPL11], while in collaboration with
S. Thurner (Univ. Vienna) we have studied and
modeled the motion on a network of a real socioeconomic system [NatSR12].
Synchronization in Complex Networks
As another example of interesting dynamic process, we have studied
the emergence of collective synchronized dynamics in NETWORKS OF
COUPLED DYNAMICAL SYSTEMS, with the main focus on the interplay between the
network topology and the features of the network dynamics.
In particular, with S. Boccaletti (CNR, IT)
we have addressed the problem of
understanding the variable abundance of different MOTIFS by means of
the MASTER STABILITY FUNCTION, an analytic method to measure the
stability of a synchronous state on a graph. We have
found that, for undirected graphs, the stability of the synchronous
state is positively correlated with the relative motif abundance,
while in directed graphs the correlation exists only for some specific
motifs [EPL07].
Furthermore, in collaboration with
A. Rapisarda (Univ. of Catania)
we have introduced a method for the detection and
identification of COMMUNITY STRUCTURES in complex networks,
based on the formation of synchronized groups of dynamical units [PRE07],
and we have tested the method on computer generated and realworld networks
whose modular structure is already known or has been studied by means
of other methods
[EPJ08].
In collaboration with
J. GomezGardenes (Univ. Zaragoza, SP),
in [NatSR11] and
[PRL11], we have studied synchronization in
ADAPTIVE NETWORKS, i.e. in networks whose structure coevolves with the
dynamic state of the nodes (based on the two principles of homophily and
homeostasis). More recently,
with
A. DiazGuilera (Univ. Barcelona, SP),
and
M. Chavez (Lab. de Neuroscience, CNRS, Paris),
we have discovered a novel phenomenon of REMOTE SYNCHRONIZATION,
where pairs of nodes with the same network symmetry are fully
synchronized, despite their distance on the graph [PRL13].
APPLICATIONS 1: Complex Networks and the Brain
Recent developments in neuroimaging, including structural and
functional magnetic resonance imaging (MRI), magnetoencephalography
(MEG), and electroencephalography (EEG) have provided the possibility
to study human brain at a global scale as a complex network. In
collaboration with the experimental group
of F. Babiloni (Univ. Roma La Sapienza) we have studied the topological properties of functional
connectivity patterns among different cortical areas of the human
brain. The networks were obtained from highresolution EEG recordings
in a set of SPINAL CORD INJURED PATIENTS during the preparation of a limb movement
[IJBC09],
[JPA08], and in couples of individuals playing an
Iterated Prisoner's Dilemma game
[PLoS10]. In
particular, in the latter paper we showed how it is possible to predict
human cooperative behaviors from the analysis of the socalled
HYPERBRAIN NETWORKS, i.e. networks representing the connections among
the areas of two distinct brains. In collaboration with
M. Chavez (Lab. de Neurosciences Cognitives and Imagerie Cerebrale,
CNRS, Paris) we have found that the modular structure of weighted brain
networks extracted from MEG signals of EPILEPTIC PATIENTS
recorded at rest, and far from the
absence seizures, is intrinsically different from that of healthy subjects
[PRL10] (On the cover page of PRL) . Neural systems
can also be studied at the level of physical connections between neurons.
With E. Bullmore and
P. Vertes (Univ. of Cambridge) we have studied the nervous system of the
nematode worm CAENORHABDITIS ELEGANS, together with information
on the birth times of neurons and on their spatial locations, finding one
of the first evidence of a PHASE TRANSITION IN THE GROWTH OF A SPATIAL
NETWORK [PNAS13].
APPLICATIONS 2: The Science of Cities
Everyone knows that a place which is central has some special features
to offer in many ways to those who live or work in cities: it is more
visible, more accessible from the immediate surroundings as well as
from far away, it is more popular in terms of people walking around
and potential customers, it has a greater probability to develop as an
urban landmark and a social catalyst, or offer first level functions
like theatres or office headquarters as well as a larger diversity of
opportunities and goods.
In a longterm joint project with the Urban Design group of
S. Porta (Univ. of Strathclyde),
our aim is the development of new tools for the network analysis of
urban spatial systems. One of such tools, named the MULTIPLE
CENTRALITY ASSESSMENT (MCA), allows for
mapping centrality in urban spaces
[EPB06],
[PhysA06], and establishing correlations with
relevant dynamics such as landuse
[EPB09], vehicular or pedestrian flows
and crime rates. MCA has also been used for the statistical
characterization of different types of urban fabrics taken from the
history of cities, in order to infer relationships between them in an
urban evolutionary perspective
[PRE06],
[PRE06].
Our latest research is
mainly oriented to the definition of procedures, attitudes and tools
for sustainable/human/adaptive urban analysis and design, ranging from
GISbased space analysis to sustainable community design
[UDI08],
transportation planning and traffic calming techniques
[EPJB10],
to strategies for safety and liveability in the public domain. Much
more on our work can be found at the website of the
Urban Design Study Unit, now based at the Department of Architecture of
Strathclyde University, Glasgow. A recent
study in collaboration with
M. Barthelemy (CNRS, Paris)
on the ELEMENTARY PROCESSES GOVERNING URBANIZATION [NatSR12] has received press coverage in
Nature,
Scientific American and
Le Scienze, and wide diffusion in forums on urban design.
Damage and Disease Spreading in Complex Networks
In collaboration with M. Marchiori (W3C MIT and University of Padova) we have first studied the resistance of scalefree networks to
the removal of a certain percentage of their nodes [PA03].
Then, we have developed one of the first MODELS FOR CASCADING FAILURES
in complex networks based on the dynamical
redistribution of flows on the network, showing that the breakdown of a
single node is sufficient to collapse the efficiency of the entire
system if the node is initially among the ones with largest load
[PRE04].
Together with R. Albert (Penn State Univ, USA)
we have then applied our model of cascading failure to predict BLACKOUTS
in
the North American power grid [EPJB05].
With
J. GomezGardenes and
Y. Moreno (Bifi, Zaragoza)
we have studied the spreading of SEXUALLY TRANSMITTED DISEASES
on bipartite scalefree graphs, representing heterosexual contact networks,
deriving analytically the expression for the epidemic threshold and its
dependence with the system size in finite populations [PNAS08].
With
J. GomezGardenes and
L. Fortuna and M. Frasca we have studied the effect of movements on disease
spreading
[EPL08].
Basic Structural Properties of Complex Networks
The connection topology of many biological, technological
and social networks is neither completely regular nor
completely random. These networks, named small worlds, are
in fact highly clustered, like regular lattices, yet having small
characteristics path lengths, like random graphs.
In collaboration with
M. Marchiori (W3C MIT and University of Padova)
we have proposed a new theory of the smallworld behavior
based on the concept of information transport over the network
and also valid for weighted networks
[PRL01]
(Press coverage). Our formalism is based on the definition
of the EFFICIENCY of a network, which measures how well the network
nodes exchange information. By using this measure both at a a global
and at a local scale, smallworld networks result as systems that are
both globally and locally efficient. We have performed precise
quantitative analysis of neural networks such as the C. elegans
nervous system, and two databases of corticocortical connections in
the macaque and in the cat, and transportation
systems [PhysA02],
also studying the smallworld behavior in connection to the COST of a
network [EPJB03]. The concept of efficiency can be used to introduce
a completely new idea of centrality, that we have named DELTA CENTRALITY,
that applies to groups, as well as individual nodes and links
[NJP07], and also allows to quantify the relevance of different
mediators in the HUMAN IMMUNE CELL NETWORK
[BIO05], to find the critical components of CRITICAL INFRASTRUCTURE NETWORKS
[PRE05]
[FNL05]
and to develop strategies to protect from terrorist attacks
[CSF04].
