11 research outputs found
Laplacian Dynamics and Multiscale Modular Structure in Networks
Most methods proposed to uncover communities in complex networks rely on
their structural properties. Here we introduce the stability of a network
partition, a measure of its quality defined in terms of the statistical
properties of a dynamical process taking place on the graph. The time-scale of
the process acts as an intrinsic parameter that uncovers community structures
at different resolutions. The stability extends and unifies standard notions
for community detection: modularity and spectral partitioning can be seen as
limiting cases of our dynamic measure. Similarly, recently proposed
multi-resolution methods correspond to linearisations of the stability at short
times. The connection between community detection and Laplacian dynamics
enables us to establish dynamically motivated stability measures linked to
distinct null models. We apply our method to find multi-scale partitions for
different networks and show that the stability can be computed efficiently for
large networks with extended versions of current algorithms.Comment: New discussions on the selection of the most significant scales and
the generalisation of stability to directed network
Laplacian Dynamics and Multiscale Modular Structure in Networks
Most methods proposed to uncover communities in complex networks rely on
their structural properties. Here we introduce the stability of a network
partition, a measure of its quality defined in terms of the statistical
properties of a dynamical process taking place on the graph. The time-scale of
the process acts as an intrinsic parameter that uncovers community structures
at different resolutions. The stability extends and unifies standard notions
for community detection: modularity and spectral partitioning can be seen as
limiting cases of our dynamic measure. Similarly, recently proposed
multi-resolution methods correspond to linearisations of the stability at short
times. The connection between community detection and Laplacian dynamics
enables us to establish dynamically motivated stability measures linked to
distinct null models. We apply our method to find multi-scale partitions for
different networks and show that the stability can be computed efficiently for
large networks with extended versions of current algorithms.Comment: New discussions on the selection of the most significant scales and
the generalisation of stability to directed network
Flow graphs: interweaving dynamics and structure
The behavior of complex systems is determined not only by the topological
organization of their interconnections but also by the dynamical processes
taking place among their constituents. A faithful modeling of the dynamics is
essential because different dynamical processes may be affected very
differently by network topology. A full characterization of such systems thus
requires a formalization that encompasses both aspects simultaneously, rather
than relying only on the topological adjacency matrix. To achieve this, we
introduce the concept of flow graphs, namely weighted networks where dynamical
flows are embedded into the link weights. Flow graphs provide an integrated
representation of the structure and dynamics of the system, which can then be
analyzed with standard tools from network theory. Conversely, a structural
network feature of our choice can also be used as the basis for the
construction of a flow graph that will then encompass a dynamics biased by such
a feature. We illustrate the ideas by focusing on the mathematical properties
of generic linear processes on complex networks that can be represented as
biased random walks and also explore their dual consensus dynamics.Comment: 4 pages, 1 figur
Encoding dynamics for multiscale community detection: Markov time sweeping for the Map equation
The detection of community structure in networks is intimately related to
finding a concise description of the network in terms of its modules. This
notion has been recently exploited by the Map equation formalism (M. Rosvall
and C.T. Bergstrom, PNAS, 105(4), pp.1118--1123, 2008) through an
information-theoretic description of the process of coding inter- and
intra-community transitions of a random walker in the network at stationarity.
However, a thorough study of the relationship between the full Markov dynamics
and the coding mechanism is still lacking. We show here that the original Map
coding scheme, which is both block-averaged and one-step, neglects the internal
structure of the communities and introduces an upper scale, the `field-of-view'
limit, in the communities it can detect. As a consequence, Map is well tuned to
detect clique-like communities but can lead to undesirable overpartitioning
when communities are far from clique-like. We show that a signature of this
behavior is a large compression gap: the Map description length is far from its
ideal limit. To address this issue, we propose a simple dynamic approach that
introduces time explicitly into the Map coding through the analysis of the
weighted adjacency matrix of the time-dependent multistep transition matrix of
the Markov process. The resulting Markov time sweeping induces a dynamical
zooming across scales that can reveal (potentially multiscale) community
structure above the field-of-view limit, with the relevant partitions indicated
by a small compression gap.Comment: 10 pages, 6 figure
Modularity in signaling systems
Modularity is a property by which the behavior of a system does not change upon interconnection. It is crucial for understanding the behavior of a complex system from the behavior of the composing subsystems. Whether modularity holds in biology is an intriguing and largely debated question. In this paper, we discuss this question taking a control system theory view and focusing on signaling systems. In particular, we argue that, despite signaling systems being constituted of structural modules, such as covalent modification cycles, modularity does not hold in general. As in any engineering system, impedance-like effects, called retroactivity, appear at interconnections and alter the behavior of connected modules. We further argue that while signaling systems have evolved sophisticated ways to counter-act retroactivity and enforce modularity, retroactivity may also be exploited to finely control the information processing of signaling pathways. Testable predictions and experimental evidence are discussed with their implications
Protein multi-scale organization through graph partitioning and robustness analysis: Application to the myosin-myosin light chain interaction
Despite the recognized importance of the multi-scale spatio-temporal
organization of proteins, most computational tools can only access a limited
spectrum of time and spatial scales, thereby ignoring the effects on protein
behavior of the intricate coupling between the different scales. Starting from
a physico-chemical atomistic network of interactions that encodes the structure
of the protein, we introduce a methodology based on multi-scale graph
partitioning that can uncover partitions and levels of organization of proteins
that span the whole range of scales, revealing biological features occurring at
different levels of organization and tracking their effect across scales.
Additionally, we introduce a measure of robustness to quantify the relevance of
the partitions through the generation of biochemically-motivated surrogate
random graph models. We apply the method to four distinct conformations of
myosin tail interacting protein, a protein from the molecular motor of the
malaria parasite, and study properties that have been experimentally addressed
such as the closing mechanism, the presence of conserved clusters, and the
identification through computational mutational analysis of key residues for
binding.Comment: 13 pages, 7 Postscript figure
Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit
In recent years, there has been a surge of interest in community detection
algorithms for complex networks. A variety of computational heuristics, some
with a long history, have been proposed for the identification of communities
or, alternatively, of good graph partitions. In most cases, the algorithms
maximize a particular objective function, thereby finding the `right' split
into communities. Although a thorough comparison of algorithms is still
lacking, there has been an effort to design benchmarks, i.e., random graph
models with known community structure against which algorithms can be
evaluated. However, popular community detection methods and benchmarks normally
assume an implicit notion of community based on clique-like subgraphs, a form
of community structure that is not always characteristic of real networks.
Specifically, networks that emerge from geometric constraints can have natural
non clique-like substructures with large effective diameters, which can be
interpreted as long-range communities. In this work, we show that long-range
communities escape detection by popular methods, which are blinded by a
restricted `field-of-view' limit, an intrinsic upper scale on the communities
they can detect. The field-of-view limit means that long-range communities tend
to be overpartitioned. We show how by adopting a dynamical perspective towards
community detection (Delvenne et al. (2010) PNAS:107: 12755-12760; Lambiotte et
al. (2008) arXiv:0812.1770), in which the evolution of a Markov process on the
graph is used as a zooming lens over the structure of the network at all
scales, one can detect both clique- or non clique-like communities without
imposing an upper scale to the detection. Consequently, the performance of
algorithms on inherently low-diameter, clique-like benchmarks may not always be
indicative of equally good results in real networks with local, sparser
connectivity.Comment: 20 pages, 6 figure
Fast unfolding of communities in large networks
We propose a simple method to extract the community structure of large
networks. Our method is a heuristic method that is based on modularity
optimization. It is shown to outperform all other known community detection
method in terms of computation time. Moreover, the quality of the communities
detected is very good, as measured by the so-called modularity. This is shown
first by identifying language communities in a Belgian mobile phone network of
2.6 million customers and by analyzing a web graph of 118 million nodes and
more than one billion links. The accuracy of our algorithm is also verified on
ad-hoc modular networks. .Comment: 6 pages, 5 figures, 1 table; new version with new figures in order to
clarify our method, where we look more carefully at the role played by the
ordering of the nodes and where we compare our method with that of Wakita and
Tsurum
Structured networks and coarse-grained descriptions: a dynamical perspective
This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an
endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim
to gain a reduced description of the system that takes into account both its structure and dynamics.
In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We
provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection
to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions.
In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on
three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the
dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the
network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this
dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections
to concepts in social network analysis, such as structural balance.
In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures.
We then show how such dynamical measures can be related to seemingly different algorithm for community detection and
coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions
Structured networks and coarse-grained descriptions: a dynamical perspective
This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics. In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions. In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance. In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions