Social networks: evolving graphs with memory dependent edges

The plethora, and mass take up, of digital communication tech- nologies has resulted in a wealth of interest in social network data collection and analysis in recent years. Within many such networks the interactions are transient: thus those networks evolve over time. In this paper we introduce a class of models for such networks using evolving graphs with memory dependent edges, which may appear and disappear according to their recent history. We consider time discrete and time continuous variants of the model. We consider the long term asymptotic behaviour as a function of parameters controlling the memory dependence. In particular we show that such networks may continue evolving forever, or else may quench and become static (containing immortal and/or extinct edges). This depends on the ex- istence or otherwise of certain infinite products and series involving age dependent model parameters. To test these ideas we show how model parameters may be calibrated based on limited samples of time dependent data, and we apply these concepts to three real networks: summary data on mobile phone use from a developing region; online social-business network data from China; and disaggregated mobile phone communications data from a reality mining experiment in the US. In each case we show that there is evidence for memory dependent dynamics, such as that embodied within the class of models proposed here

Dynamical Systems to Monitor Complex Networks in Continuous Time

In many settings it is appropriate to treat the evolution of pairwise interactions over continuous time. We show that new Katz-style centrality measures can be derived in this context via solutions to a nonautonomous ODE driven by the network dynamics. This allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time dependent links. In addition to the classical notion of attenuation across edges used in the static Katz centrality measure, the ODE also allows for attenuation over time, so that real time "running measures" can be computed. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. We illustrate the new measures on a large scale voice call network, where key features are discovered that are not evident from snapshots or aggregates

On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs

Competing edge networks

We introduce a model for a pair of nonlinear evolving networks, defined over a common set of vertices, sub ject to edgewise competition. Each network may grow new edges spontaneously or through triad closure. Both networks inhibit the other’s growth and encourage the other’s demise. These nonlinear stochastic competition equations yield to a mean field analysis resulting in a nonlinear deterministic system. There may be multiple equilibria; and bifurcations of different types are shown to occur within a reduced parameter space. This situation models competitive peer-to-peer communication networks such as BlackBerry Messenger displacing SMS; or instant messaging displacing emails

Wishful thinking about consciousness

We contrast three very distinct mathematical approaches to the hard problem of consciousness: quantum consciousness, integrated information theory, and the very large-scale dynamical systems simulation of a network of networks. We highlight their features and their associated hypotheses, and we discuss how they are aligned or in conflict. We suggest some challenges to these theories, in considering how they might apply to the human brain as it develops both cognitive and conscious sophistication, from infancy to adulthood. We indicate how an evolutionary perspective challenges the distinct approaches to aver performance advantages and physiological surrogates for consciousness

Bistability through triadic closure

We propose and analyse a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, on-line social behaviour and information processing in neuroscience. We model the evolving network as a discrete time Markov chain, and study a very general framework where, conditioned on the current state, edges appear or disappear independently at the next timestep. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean field theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure effect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean field theory predicts bistable dynamics, and computational results confirm this prediction. We also discuss the calibration issue for a set of real cell phone data, and find support for a stratified model, where individuals are assigned to one of two distinct groups having different within-group and across-group dynamics

Organization and evolution of the UK far-right network on Telegram

The instant messaging platform Telegram has become popular among the far-right movements in the US and UK in recent years. These groups use public Telegram channels and group chats to disseminate hate speech, disinformation, and conspiracy theories. Recent works revealed that the far-right Telegram network structure is decentralized and formed of several communities divided mostly along ideological and national lines. Here, we investigated the UK far-right network on Telegram and are interested in understanding the different roles of different channels and their influence relations. We apply a community detection method, based on the clustering of a flow of random walkers, that allows us to uncover the organization of the Telegram network in communities with different roles. We find three types of communities: (1) upstream communities contain mostly group chats that comment on content from channels in the rest of the network; (2) core communities contain broadcast channels tightly connected to each other and can be seen as forming echo chambers; (3) downstream communities contain popular channels that are highly referenced by other channels. We find that the network is composed of two main sub-networks: one containing mainly channels related to the English-speaking far-right movements and one with channels in Russian. We analyze the dynamics of the different communities and the most shared external links in the different types of communities over a period going from 2015 to 2020. We find that different types of communities have different dynamics and share links to different types of websites. We finish by discussing several directions for further work

Cognition and consciousness entwined

We argue that cognition (information processing) and internal phenomenological sensations, including emotions, are intimately related and are not separable. We aver that phenomenological sensations are dynamical “modes” of firing behaviour that (i) exist over time and over large parts of the cortex’s neuron-to-neuron network and (ii) are consequences of the network-of-networks architecture, coupling the individual neuronal dynamics and the necessary time delay incurred by neuron-to-neuron transmission: if you possess those system properties, then you will have the dynamical modes and, thus, the phenomenological sensations. These modes are consequences of incoming external stimuli and are competitive within the system, suppressing and locking-out one another. On the other hand, the presence of any such mode acts as a preconditioner for the immediate (dynamic) cognitive processing of information. Thus, internal phenomenological sensations, including emotions, reduce the immediate decision set (of feasible interpretations) and hence the cognitive load. For organisms with such a mental inner life, there would clearly be a large cognitive evolutionary advantage, resulting in the well-known “thinking fast, thinking slow” phenomena. We call this the entwinement hypothesis: how latent conscious phenomena arise from the dynamics of the cognitive processing load, and how these precondition the cognitive tasks immediately following. We discuss how internal dynamical modes, which are candidates for emotions down to single qualia, can be observed by reverse engineering large sets of simulations of system’s stimulated responses, either using vast supercomputers (with full 10B neuronal network analyses) or else using laptops to do the same for appropriately generalised Kuramoto models (networks of k-dimensional clocks, each representing the 10,000 neurons within a single neural column). We explain why such simplifications are appropriate. We also discuss the consequent cognitive advantages for information-processing systems exhibiting internal sensations and the exciting implications for next-generation (non-binary) computation and for AI

A matrix iteration for dynamic network summaries

We propose a new algorithm for summarizing properties of large-scale time-evolving networks. This type of data, recording connections that come and go over time, is being generated in many modern applications, including telecommunications and on-line human social behavior. The algorithm computes a dynamic measure of how well pairs of nodes can communicate by taking account of routes through the network that respect the arrow of time. We take the conventional approach of downweighting for length (messages become corrupted as they are passed along) and add the novel feature of downweighting for age (messages go out of date). This allows us to generalize widely used Katz-style centrality measures that have proved popular in network science to the case of dynamic networks sampled at non-uniform points in time. We illustrate the new approach on synthetic and real data