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

Computability and dynamical systems

In this paper we explore results that establish a link between dynamical systems and computability theory (not numerical analysis). In the last few decades, computers have increasingly been used as simulation tools for gaining insight into dynamical behavior. However, due to the presence of errors inherent in such numerical simulations, with few exceptions, computers have not been used for the nobler task of proving mathematical results. Nevertheless, there have been some recent developments in the latter direction. Here we introduce some of the ideas and techniques used so far, and suggest some lines of research for further work on this fascinating topic

Finding and testing network communities by lumped Markov chains

Identifying communities (or clusters), namely groups of nodes with comparatively strong internal connectivity, is a fundamental task for deeply understanding the structure and function of a network. Yet, there is a lack of formal criteria for defining communities and for testing their significance. We propose a sharp definition which is based on a significance threshold. By means of a lumped Markov chain model of a random walker, a quality measure called "persistence probability" is associated to a cluster. Then the cluster is defined as an "$\alpha$-community" if such a probability is not smaller than $\alpha$. Consistently, a partition composed of $\alpha$-communities is an "$\alpha$-partition". These definitions turn out to be very effective for finding and testing communities. If a set of candidate partitions is available, setting the desired $\alpha$-level allows one to immediately select the $\alpha$-partition with the finest decomposition. Simultaneously, the persistence probabilities quantify the significance of each single community. Given its ability in individually assessing the quality of each cluster, this approach can also disclose single well-defined communities even in networks which overall do not possess a definite clusterized structure

Thinking about Eating Food Activates Visual Cortex with Reduced Bilateral Cerebellar Activation in Females with Anorexia Nervosa: An fMRI Study

Background: Women with anorexia nervosa (AN) have aberrant cognitions about food and altered activity in prefrontal cortical and somatosensory regions to food images. However, differential effects on the brain when thinking about eating food between healthy women and those with AN is unknown. Methods: Functional magnetic resonance imaging (fMRI) examined neural activation when 42 women thought about eating the food shown in images: 18 with AN (11 RAN, 7 BPAN) and 24 age-matched controls (HC). Results: Group contrasts between HC and AN revealed reduced activation in AN in the bilateral cerebellar vermis, and increased activation in the right visual cortex. Preliminary comparisons between AN subtypes and healthy controls suggest differences in cortical and limbic regions. Conclusions: These preliminary data suggest that thinking about eating food shown in images increases visual and prefrontal cortical neural responses in females with AN, which may underlie cognitive biases towards food stimuli and ruminations about controlling food intake. Future studies are needed to explicitly test how thinking about eating activates restraint cognitions, specifically in those with restricting vs. binge-purging AN subtypes

Flux-dependent graphs for metabolic networks

Cells adapt their metabolic fluxes in response to changes in the environment. We present a framework for the systematic construction of flux-based graphs derived from organism-wide metabolic networks. Our graphs encode the directionality of metabolic fluxes via edges that represent the flow of metabolites from source to target reactions. The methodology can be applied in the absence of a specific biological context by modelling fluxes probabilistically, or can be tailored to different environmental conditions by incorporating flux distributions computed through constraint-based approaches such as Flux Balance Analysis. We illustrate our approach on the central carbon metabolism of Escherichia coli and on a metabolic model of human hepatocytes. The flux-dependent graphs under various environmental conditions and genetic perturbations exhibit systemic changes in their topological and community structure, which capture the re-routing of metabolic fluxes and the varying importance of specific reactions and pathways. By integrating constraint-based models and tools from network science, our framework allows the study of context-specific metabolic responses at a system level beyond standard pathway descriptions

Multiscale dynamical embeddings of complex networks

Complex systems and relational data are often abstracted as dynamical processes on networks. To understand, predict, and control their behavior, a crucial step is to extract reduced descriptions of such networks. Inspired by notions from control theory, we propose a time-dependent dynamical similarity measure between nodes, which quantifies the effect a node-input has on the network. This dynamical similarity induces an embedding that can be employed for several analysis tasks. Here we focus on (i) dimensionality reduction, i.e., projecting nodes onto a low-dimensional space that captures dynamic similarity at different timescales, and (ii) how to exploit our embeddings to uncover functional modules. We exemplify our ideas through case studies focusing on directed networks without strong connectivity and signed networks. We further highlight how certain ideas from community detection can be generalized and linked to control theory, by using the here developed dynamical perspective

Entrograms and coarse graining of dynamics on complex networks

Using an information theoretic point of view, we investigate how a dynamics acting on a network can be coarse grained through the use of graph partitions. Specifically, we are interested in how aggregating the state space of a Markov process according to a partition impacts on the thus obtained lower-dimensional dynamics. We highlight that for a dynamics on a particular graph there may be multiple coarse grained descriptions that capture different, incomparable features of the original process. For instance, a coarse graining induced by one partition may be commensurate with a time-scale separation in the dynamics, while another coarse graining may correspond to a different lower-dimensional dynamics that preserves the Markov property of the original process. Taking inspiration from the literature of Computational Mechanics, we find that a convenient tool to summarize and visualize such dynamical properties of a coarse grained model (partition) is the entrogram. The entrogram gathers certain information-theoretic measures, which quantify how information flows across time steps. These information theoretic quantities include the entropy rate, as well as a measure for the memory contained in the process, i.e., how well the dynamics can be approximated by a first order Markov process. We use the entrogram to investigate how specific macro-scale connection patterns in the state–space transition graph of the original dynamics result in desirable properties of coarse grained descriptions. We thereby provide a fresh perspective on the interplay between structure and dynamics in networks, and the process of partitioning a network from an information theoretic perspective. To illustrate our points, we focus on networks that may be approximated by both a core-periphery or a clustered organization, and highlight that each of these coarse grained descriptions can capture different aspects of a Markov process acting on the network

Temporal pattern of (re)tweets reveal cascade migration

Twitter has recently become one of the most popular online social networking websites where users can share news and ideas through messages in the form of tweets. As a tweet gets retweeted from user to user, large cascades of information diffusion are formed over the Twitter follower network. Existing works on cascades have mainly focused on predicting their popularity in terms of size. In this paper, we leverage on the temporal pattern of retweets to model the diffusion dynamics of a cascade. Notably, retweet cascades provide two complementary information: (a) inter-retweet time intervals of retweets, and (b) diffusion of cascade over the underlying follower network. Using datasets from Twitter, we identify two types of cascades based on presence or absence of early peaks in their sequence of inter-retweet intervals. We identify multiple diffusion localities associated with a cascade as it propagates over the network. Our studies reveal the transition of a cascade to a new locality facilitated by pivotal users that are highly cascade dependent following saturation of current locality. We propose an analytical model to show co-occurrence of first peaks and cascade migration to a new locality as well as predict locality saturation from inter-retweet intervals. Finally, we validate these claims from empirical data showing co-occurrence of first peaks and migration with good accuracy; we obtain even better accuracy for successfully classifying saturated and non-saturated diffusion localities from inter-retweet intervals