518 research outputs found
Distance-Dependent Kronecker Graphs for Modeling Social Networks
This paper focuses on a generalization of stochastic
Kronecker graphs, introducing a Kronecker-like operator and
defining a family of generator matrices H dependent on distances
between nodes in a specified graph embedding. We prove
that any lattice-based network model with sufficiently small
distance-dependent connection probability will have a Poisson
degree distribution and provide a general framework to prove
searchability for such a network. Using this framework, we focus
on a specific example of an expanding hypercube and discuss
the similarities and differences of such a model with recently
proposed network models based on a hidden metric space. We
also prove that a greedy forwarding algorithm can find very short
paths of length O((log log n)^2) on the hypercube with n nodes,
demonstrating that distance-dependent Kronecker graphs can
generate searchable network models
Generalizing Kronecker graphs in order to model searchable networks
This paper describes an extension to stochastic
Kronecker graphs that provides the special structure required
for searchability, by defining a “distance”-dependent Kronecker
operator. We show how this extension of Kronecker graphs
can generate several existing social network models, such as
the Watts-Strogatz small-world model and Kleinberg’s latticebased
model. We focus on a specific example of an expanding
hypercube, reminiscent of recently proposed social network
models based on a hidden hyperbolic metric space, and prove
that a greedy forwarding algorithm can find very short paths
of length O((log log n)^2) for graphs with n nodes
A novel approach to study realistic navigations on networks
We consider navigation or search schemes on networks which are realistic in
the sense that not all search chains can be completed. We show that the
quantity , where is the average dynamic shortest distance
and the success rate of completion of a search, is a consistent measure
for the quality of a search strategy. Taking the example of realistic searches
on scale-free networks, we find that scales with the system size as
, where decreases as the searching strategy is improved.
This measure is also shown to be sensitive to the distintinguishing
characteristics of networks. In this new approach, a dynamic small world (DSW)
effect is said to exist when . We show that such a DSW indeed
exists in social networks in which the linking probability is dependent on
social distances.Comment: Text revised, references added; accepted version in Journal of
Statistical Mechanic
Identity and Search in Social Networks
Social networks have the surprising property of being "searchable": Ordinary
people are capable of directing messages through their network of acquaintances
to reach a specific but distant target person in only a few steps. We present a
model that offers an explanation of social network searchability in terms of
recognizable personal identities: sets of characteristics measured along a
number of social dimensions. Our model defines a class of searchable networks
and a method for searching them that may be applicable to many network search
problems, including the location of data files in peer-to-peer networks, pages
on the World Wide Web, and information in distributed databases.Comment: 4 page, 3 figures, revte
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
The Routing of Complex Contagion in Kleinberg's Small-World Networks
In Kleinberg's small-world network model, strong ties are modeled as
deterministic edges in the underlying base grid and weak ties are modeled as
random edges connecting remote nodes. The probability of connecting a node
with node through a weak tie is proportional to , where
is the grid distance between and and is the
parameter of the model. Complex contagion refers to the propagation mechanism
in a network where each node is activated only after neighbors of the
node are activated.
In this paper, we propose the concept of routing of complex contagion (or
complex routing), where we can activate one node at one time step with the goal
of activating the targeted node in the end. We consider decentralized routing
scheme where only the weak ties from the activated nodes are revealed. We study
the routing time of complex contagion and compare the result with simple
routing and complex diffusion (the diffusion of complex contagion, where all
nodes that could be activated are activated immediately in the same step with
the goal of activating all nodes in the end).
We show that for decentralized complex routing, the routing time is lower
bounded by a polynomial in (the number of nodes in the network) for all
range of both in expectation and with high probability (in particular,
for and
for in expectation),
while the routing time of simple contagion has polylogarithmic upper bound when
. Our results indicate that complex routing is harder than complex
diffusion and the routing time of complex contagion differs exponentially
compared to simple contagion at sweetspot.Comment: Conference version will appear in COCOON 201
Search and Congestion in Complex Networks
A model of communication that is able to cope simultaneously with the
problems of search and congestion is presented. We investigate the
communication dynamics in model networks and introduce a general framework that
enables a search of optimal structures.Comment: Proceedings of the Conference "Statistical Mechanics of Complex
Networks", Sitges, Spain, June 200
Locating influential nodes via dynamics-sensitive centrality
With great theoretical and practical significance, locating influential nodes
of complex networks is a promising issues. In this paper, we propose a
dynamics-sensitive (DS) centrality that integrates topological features and
dynamical properties. The DS centrality can be directly applied in locating
influential spreaders. According to the empirical results on four real networks
for both susceptible-infected-recovered (SIR) and susceptible-infected (SI)
spreading models, the DS centrality is much more accurate than degree,
-shell index and eigenvector centrality.Comment: 6 pages, 1 table and 2 figure
Distributed flow optimization and cascading effects in weighted complex networks
We investigate the effect of a specific edge weighting scheme on distributed flow efficiency and robustness to cascading
failures in scale-free networks. In particular, we analyze a simple, yet
fundamental distributed flow model: current flow in random resistor networks.
By the tuning of control parameter and by considering two general cases
of relative node processing capabilities as well as the effect of bandwidth, we
show the dependence of transport efficiency upon the correlations between the
topology and weights. By studying the severity of cascades for different
control parameter , we find that network resilience to cascading
overloads and network throughput is optimal for the same value of over
the range of node capacities and available bandwidth
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