21 research outputs found
Nested hierarchies in planar graphs
We construct a partial order relation which acts on the set of 3-cliques of a
maximal planar graph G and defines a unique hierarchy. We demonstrate that G is
the union of a set of special subgraphs, named `bubbles', that are themselves
maximal planar graphs. The graph G is retrieved by connecting these bubbles in
a tree structure where neighboring bubbles are joined together by a 3-clique.
Bubbles naturally provide the subdivision of G into communities and the tree
structure defines the hierarchical relations between these communities
Exploring complex networks via topological embedding on surfaces
We demonstrate that graphs embedded on surfaces are a powerful and practical
tool to generate, characterize and simulate networks with a broad range of
properties. Remarkably, the study of topologically embedded graphs is
non-restrictive because any network can be embedded on a surface with
sufficiently high genus. The local properties of the network are affected by
the surface genus which, for example, produces significant changes in the
degree distribution and in the clustering coefficient. The global properties of
the graph are also strongly affected by the surface genus which is constraining
the degree of interwoveness, changing the scaling properties from
large-world-kind (small genus) to small- and ultra-small-world-kind (large
genus). Two elementary moves allow the exploration of all networks embeddable
on a given surface and naturally introduce a tool to develop a statistical
mechanics description. Within such a framework, we study the properties of
topologically-embedded graphs at high and low `temperatures' observing the
formation of increasingly regular structures by cooling the system. We show
that the cooling dynamics is strongly affected by the surface genus with the
manifestation of a glassy-like freezing transitions occurring when the amount
of topological disorder is low.Comment: 18 pages, 7 figure
Relation between Financial Market Structure and the Real Economy: Comparison between Clustering Methods
We quantify the amount of information filtered by different hierarchical
clustering methods on correlations between stock returns comparing it with the
underlying industrial activity structure. Specifically, we apply, for the first
time to financial data, a novel hierarchical clustering approach, the Directed
Bubble Hierarchical Tree and we compare it with other methods including the
Linkage and k-medoids. In particular, by taking the industrial sector
classification of stocks as a benchmark partition, we evaluate how the
different methods retrieve this classification. The results show that the
Directed Bubble Hierarchical Tree can outperform other methods, being able to
retrieve more information with fewer clusters. Moreover, we show that the
economic information is hidden at different levels of the hierarchical
structures depending on the clustering method. The dynamical analysis on a
rolling window also reveals that the different methods show different degrees
of sensitivity to events affecting financial markets, like crises. These
results can be of interest for all the applications of clustering methods to
portfolio optimization and risk hedging.Comment: 31 pages, 17 figure
Network Filtering for Big Data: Triangulated Maximally Filtered Graph
We propose a network-filtering method, the Triangulated Maximally Filtered Graph (TMFG), that provides an approximate solution to the WEIGHTED MAXIMAL PLANAR GRAPH problem. The underlying idea of TMFG consists in building a triangulation that maximizes a score function associated with the amount of information retained by the network.TMFG uses as weights any arbitrary similarity measure to arrange data into a meaningful network structure that can be used for clustering, community detection and modelling. The method is fast, adaptable and scalable to very large datasets; it allows online updating and learning as new data can be inserted and deleted with combinations of local and non-local moves. Further, TMFG permits readjustments of the network in consequence of changes in the strength of the similarity measure. The method is based on local topological moves and can therefore take advantage of parallel and GPUs computing. We discuss how this network-filtering method can be used intuitively and efficiently for big data studies and its significance from an information-theoretic perspective
Risk diversification: a study of persistence with a filtered correlation-network approach
The evolution with time of the correlation structure of equity returns is
studied by means of a filtered network approach investigating persistences and
recurrences and their implications for risk diversification strategies. We
build dynamically Planar Maximally Filtered Graphs from the correlation
structure over a rolling window and we study the persistence of the associated
Directed Bubble Hierarchical Tree (DBHT) clustering structure. We observe that
the DBHT clustering structure is quite stable during the early 2000' becoming
gradually less persistent before the unfolding of the 2007-2008 crisis. The
correlation structure eventually recovers persistence in the aftermath of the
crisis settling up a new phase, distinct from the pre-cysts structure, where
the market structure is less related to industrial sector activity. Notably, we
observe that - presently - the correlation structure is loosing again
persistence indicating the building-up of another, different, phase. Such
dynamical changes in persistence and their occurrence at the unfolding of
financial crises rises concerns about the effectiveness of correlation-based
portfolio management tools for risk diversification
Hierarchical information clustering by means of topologically embedded graphs
We introduce a graph-theoretic approach to extract clusters and hierarchies
in complex data-sets in an unsupervised and deterministic manner, without the
use of any prior information. This is achieved by building topologically
embedded networks containing the subset of most significant links and analyzing
the network structure. For a planar embedding, this method provides both the
intra-cluster hierarchy, which describes the way clusters are composed, and the
inter-cluster hierarchy which describes how clusters gather together. We
discuss performance, robustness and reliability of this method by first
investigating several artificial data-sets, finding that it can outperform
significantly other established approaches. Then we show that our method can
successfully differentiate meaningful clusters and hierarchies in a variety of
real data-sets. In particular, we find that the application to gene expression
patterns of lymphoma samples uncovers biologically significant groups of genes
which play key-roles in diagnosis, prognosis and treatment of some of the most
relevant human lymphoid malignancies.Comment: 33 Pages, 18 Figures, 5 Table
Clustering and hierarchy of financial markets data: advantages of the DBHT
We present a set of analyses aiming at quantifying the amount of information filtered by di↵erent hierarchical
clustering methods on correlations between stock returns. In particular we apply, for the first
time to financial data, a novel hierarchical clustering approach, the Directed Bubble Hierarchical Tree
(DBHT), and we compare it with other methods including the Linkage and k-medoids. In particular by
taking the industrial sector classification of stocks as a benchmark partition we evaluate how the di↵erent
methods retrieve this classification.
The results show that the Directed Bubble Hierarchical Tree outperforms the other methods, being
able to retrieve more information with fewer clusters. Moreover, we show that the economic information
is hidden at di↵erent levels of the hierarchical structures depending on the clustering method. The
dynamical analysis also reveals that the di↵erent methods show di↵erent degrees of sensitivity to financial
events, like crises. These results can be of interest for all the applications of clustering methods to portfolio
optimization and risk hedging