We examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts--``communities'' or modules. We develop an analytical approach allowing us to obtain the statistics of connected components and an intervertex distance distribution in these modular networks, and to describe their global organization and structure. In particular, we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks. We demonstrate that even a relatively small number of shortcuts unite the networks into one. In more precise terms, if the number of the interlinks is any finite fraction of the total number of connections, then the intervertex distance distribution approaches a delta-function peaked form, and so the network is united.
We show that the full features of the dynamics towards the Feigenbaum attractor, present in all low-dimensional maps with a unimodal leading component, form a hierarchical construction with modular organization that leads to a clear-cut emergent property. This well-known nonlinear model system combines a simple and precise definition, an intricate nested hierarchical dynamical structure, and emergence of a power-law dynamical property absent in the exponential-law that governs the dynamics within the modules. This classic nonlinear system is put forward as a working example for complex collect…
A wide variety of real-life networks share two remarkable generic topological properties: scale-free behavior and modular organization, and it is natural and important to study how these two features affect the dynamical processes taking place on such networks. In this paper, we investigate a simple stochastic process--trapping problem, a random walk with a perfect trap fixed at a given location, performed on a family of hierarchical networks that exhibit simultaneously striking scale-free and modular structure. We focus on a particular case with the immobile trap positioned at the hub node ha…
Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process pr…
We focus on the detection of communities in multi-scale networks, namely networks made of different levels of organization and in which modules exist at different scales. It is first shown that methods based on modularity are not appropriate to uncover modules in empirical networks, mainly because modularity optimization has an intrinsic bias towards partitions having a characteristic number of modules which might not be compatible with the modular organization of the system. We argue for the use of more flexible quality functions incorporating a resolution parameter that allows us to reveal t…
We introduce a mechanism which models the emergence of the universal properties of complex networks, such as scale independence, modularity and self-similarity, and unifies them under a scale-free organization beyond the link. This brings a new perspective on network organization where communities, instead of links, are the fundamental building blocks of complex systems. We show how our simple model can reproduce social and information networks by predicting their community structure and more importantly, how their nodes or communities are interconnected, often in a self-similar manner.
Coordination processes in complex systems can be related to the problem of collective ordering in networks, many of which have modular organization. Investigating the order-disorder transition for Ising spins on modular random networks, corresponding to consensus formation in society, we observe two distinct phases: (i) ordering within each module at a critical temperature, followed by (ii) global ordering at a lower temperature. This indicates polarization of society into groups having contrary opinions can persist indefinitely even when mutual interactions between agents favor consensus.
Griffiths phases (GPs), generated by the heterogeneities on modular networks, have recently been suggested to provide a mechanism, rid of fine parameter tuning, to explain the critical behavior of complex systems. One conjectured requirement for systems with modular structures was that the network of modules must be hierarchically organized and possess finite dimension. We investigate the dynamical behavior of an activity spreading model, evolving on heterogeneous random networks with highly modular structure and organized non-hierarchically. We observe that loosely coupled modules act as effe…
Complex networks have recently attracted much interest due to their prevalence in nature and our daily lives [1, 2]. A critical property of a network is its resilience to random breakdown and failure [3-6], typically studied as a percolation problem [7-9] or by modeling cascading failures [10-12]. Many complex systems, from power grids and the Internet to the brain and society [13-15], can be modeled using modular networks comprised of small, densely connected groups of nodes [16, 17]. These modules often overlap, with network elements belonging to multiple modules [18, 19]. Yet existing work …
Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that modular hierarchical topologies may provide an optimal structure for certain dynamical processes. We revisit a modular hierarchical network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erdős-R…
The ``six degrees of separation" between any two individuals on Earth has become emblematic of the 'small world' theme, even though the information conveyed via a chain of human encounters decays very rapidly with increasing chain length, and diffusion of information via this process may be very inefficient in large human organizations. The information flow on a communication network in a large organization, the University of Oslo, has been studied by analyzing e-mail records. The records allow for quantification of communication intensity across organizational levels and between organizationa…
