4,777 research outputs found
Percolation transition and distribution of connected components in generalized random network ensembles
In this work, we study the percolation transition and large deviation
properties of generalized canonical network ensembles. This new type of random
networks might have a very rich complex structure, including high heterogeneous
degree sequences, non-trivial community structure or specific spatial
dependence of the link probability for networks embedded in a metric space. We
find the cluster distribution of the networks in these ensembles by mapping the
problem to a fully connected Potts model with heterogeneous couplings. We show
that the nature of the Potts model phase transition, linked to the birth of a
giant component, has a crossover from second to first order when the number of
critical colors in all the networks under study. These results shed
light on the properties of dynamical processes defined on these network
ensembles.Comment: 27 pages, 15 figure
Multilayer Networks
In most natural and engineered systems, a set of entities interact with each
other in complicated patterns that can encompass multiple types of
relationships, change in time, and include other types of complications. Such
systems include multiple subsystems and layers of connectivity, and it is
important to take such "multilayer" features into account to try to improve our
understanding of complex systems. Consequently, it is necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community structure,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure
Low-temperature behaviour of social and economic networks
Real-world social and economic networks typically display a number of
particular topological properties, such as a giant connected component, a broad
degree distribution, the small-world property and the presence of communities
of densely interconnected nodes. Several models, including ensembles of
networks also known in social science as Exponential Random Graphs, have been
proposed with the aim of reproducing each of these properties in isolation.
Here we define a generalized ensemble of graphs by introducing the concept of
graph temperature, controlling the degree of topological optimization of a
network. We consider the temperature-dependent version of both existing and
novel models and show that all the aforementioned topological properties can be
simultaneously understood as the natural outcomes of an optimized,
low-temperature topology. We also show that seemingly different graph models,
as well as techniques used to extract information from real networks, are all
found to be particular low-temperature cases of the same generalized formalism.
One such technique allows us to extend our approach to real weighted networks.
Our results suggest that a low graph temperature might be an ubiquitous
property of real socio-economic networks, placing conditions on the diffusion
of information across these systems
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
Message-Passing Methods for Complex Contagions
Message-passing methods provide a powerful approach for calculating the
expected size of cascades either on random networks (e.g., drawn from a
configuration-model ensemble or its generalizations) asymptotically as the
number of nodes becomes infinite or on specific finite-size networks. We
review the message-passing approach and show how to derive it for
configuration-model networks using the methods of (Dhar et al., 1997) and
(Gleeson, 2008). Using this approach, we explain for such networks how to
determine an analytical expression for a "cascade condition", which determines
whether a global cascade will occur. We extend this approach to the
message-passing methods for specific finite-size networks (Shrestha and Moore,
2014; Lokhov et al., 2015), and we derive a generalized cascade condition.
Throughout this chapter, we illustrate these ideas using the Watts threshold
model.Comment: 14 pages, 3 figure
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
Ge
Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs
We generalize the belief-propagation algorithm to sparse random networks with
arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in
these networks belongs to a given set of motifs (generalization of the
configuration model). These networks can be treated as sparse uncorrelated
hypergraphs in which hyperedges represent motifs. Here a hypergraph is a
generalization of a graph, where a hyperedge can connect any number of
vertices. These uncorrelated hypergraphs are tree-like (hypertrees), which
crucially simplify the problem and allow us to apply the belief-propagation
algorithm to these loopy networks with arbitrary motifs. As natural examples,
we consider motifs in the form of finite loops and cliques. We apply the
belief-propagation algorithm to the ferromagnetic Ising model on the resulting
random networks. We obtain an exact solution of this model on networks with
finite loops or cliques as motifs. We find an exact critical temperature of the
ferromagnetic phase transition and demonstrate that with increasing the
clustering coefficient and the loop size, the critical temperature increases
compared to ordinary tree-like complex networks. Our solution also gives the
birth point of the giant connected component in these loopy networks.Comment: 9 pages, 4 figure
Generalized percolation in random directed networks
We develop a general theory for percolation in directed random networks with
arbitrary two point correlations and bidirectional edges, that is, edges
pointing in both directions simultaneously. These two ingredients alter the
previously known scenario and open new views and perspectives on percolation
phenomena. Equations for the percolation threshold and the sizes of the giant
components are derived in the most general case. We also present simulation
results for a particular example of uncorrelated network with bidirectional
edges confirming the theoretical predictions
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