53,899 research outputs found
When is a Network a Network? Multi-Order Graphical Model Selection in Pathways and Temporal Networks
We introduce a framework for the modeling of sequential data capturing
pathways of varying lengths observed in a network. Such data are important,
e.g., when studying click streams in information networks, travel patterns in
transportation systems, information cascades in social networks, biological
pathways or time-stamped social interactions. While it is common to apply graph
analytics and network analysis to such data, recent works have shown that
temporal correlations can invalidate the results of such methods. This raises a
fundamental question: when is a network abstraction of sequential data
justified? Addressing this open question, we propose a framework which combines
Markov chains of multiple, higher orders into a multi-layer graphical model
that captures temporal correlations in pathways at multiple length scales
simultaneously. We develop a model selection technique to infer the optimal
number of layers of such a model and show that it outperforms previously used
Markov order detection techniques. An application to eight real-world data sets
on pathways and temporal networks shows that it allows to infer graphical
models which capture both topological and temporal characteristics of such
data. Our work highlights fallacies of network abstractions and provides a
principled answer to the open question when they are justified. Generalizing
network representations to multi-order graphical models, it opens perspectives
for new data mining and knowledge discovery algorithms.Comment: 10 pages, 4 figures, 1 table, companion python package pathpy
available on gitHu
Temporal Networks
A great variety of systems in nature, society and technology -- from the web
of sexual contacts to the Internet, from the nervous system to power grids --
can be modeled as graphs of vertices coupled by edges. The network structure,
describing how the graph is wired, helps us understand, predict and optimize
the behavior of dynamical systems. In many cases, however, the edges are not
continuously active. As an example, in networks of communication via email,
text messages, or phone calls, edges represent sequences of instantaneous or
practically instantaneous contacts. In some cases, edges are active for
non-negligible periods of time: e.g., the proximity patterns of inpatients at
hospitals can be represented by a graph where an edge between two individuals
is on throughout the time they are at the same ward. Like network topology, the
temporal structure of edge activations can affect dynamics of systems
interacting through the network, from disease contagion on the network of
patients to information diffusion over an e-mail network. In this review, we
present the emergent field of temporal networks, and discuss methods for
analyzing topological and temporal structure and models for elucidating their
relation to the behavior of dynamical systems. In the light of traditional
network theory, one can see this framework as moving the information of when
things happen from the dynamical system on the network, to the network itself.
Since fundamental properties, such as the transitivity of edges, do not
necessarily hold in temporal networks, many of these methods need to be quite
different from those for static networks
Weighted distances in scale-free configuration models
In this paper we study first-passage percolation in the configuration model
with empirical degree distribution that follows a power-law with exponent . We assign independent and identically distributed (i.i.d.)\ weights
to the edges of the graph. We investigate the weighted distance (the length of
the shortest weighted path) between two uniformly chosen vertices, called
typical distances. When the underlying age-dependent branching process
approximating the local neighborhoods of vertices is found to produce
infinitely many individuals in finite time -- called explosive branching
process -- Baroni, Hofstad and the second author showed that typical distances
converge in distribution to a bounded random variable. The order of magnitude
of typical distances remained open for the case when the
underlying branching process is not explosive. We close this gap by determining
the first order of magnitude of typical distances in this regime for arbitrary,
not necessary continuous edge-weight distributions that produce a non-explosive
age-dependent branching process with infinite mean power-law offspring
distributions. This sequence tends to infinity with the amount of vertices,
and, by choosing an appropriate weight distribution, can be tuned to be any
growing function that is , where is the number of vertices
in the graph. We show that the result remains valid for the the erased
configuration model as well, where we delete loops and any second and further
edges between two vertices.Comment: 24 page
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