Topological Strata of Weighted Complex Networks

Abstract

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and –more recently– correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.

Topological Characterization of Complex Systems: Using Persistent Entropy

“…The clique weight rank persistent homology algorithm (CWRPH) [23] is used, which has been efficiently implemented by the tool jHoles [24]. At each step of the algorithm, the family of simplices associated with the current filter value are introduced into the topological space, and the Betti numbers are computed.…”

The Topological Field Theory of Data: a program towards a novel strategy for data mining through data language

“…This is at the very root of all crucial questions at the basis of the scheme proposed: whether the higher dimensional, global structures encoding relevant information can be efficiently inferred from lower dimensional, local representations; whether the necessary reduction process (filtration; the progressive finer and finer simplicial complex representation of the data space) may be implemented in such a way as to preserve maximal information about the space global structure; whether the process can be carried over in a truly metric-free way [7]; whether such global topological information can be utilized to extract knowledge as well as correlated information, in the form of patterns in data space. The basic principles of the approach stem out of the seminal work of a number of authors: G. Carlsson [8], H. Edelsbrunner and J. Harer [9], A.J.…”

“…Whilst several details of the theory remain to be exhaustively worked out, its grand design -just presented -at least programmatically does not, except perhaps in some of its subtlest technicalities, a number of applications have started to confirm its potential reach and validity. Among these we mention in particular two: the formulation of a novel many body approach to the construction of an effective immune system model [48], and the analysis of the nature of altered consciousness in the psychedelic state based on functional magnetic resonance imaging data [7,57].…”

Topological data analysis of single-trial electroencephalographic signals

“…PH has been applied to a wide range of data: Sousbie, Pichon and Kawahara (2011) invoked the PH idea for scale-free and parameter-free identification of the voids, walls, filaments, clusters and their configuration within the cosmic web; Petri et al (2013) deployed PH in studying a number of weighted complex networks: US air passenger networks, online messages and forums, gene networks, Twitter, and co-authorship networks; Ahmed, Fasy and Wenk (2014) utilized a localized version of PH to compare the intrinsic structures of two road networks. Existing PH applications on imaging data have mostly focused on static multivariate random samples, typically from positron emission tomography (PET) and magnetic resonance imaging (MRI) studies (Gamble and Heo, 2010; Lee et al, 2011; ?).…”