Hipsters on Networks: How a Small Group of Individuals Can Lead to an Anti-Establishment Majority

The spread of opinions, memes, diseases, and "alternative facts" in a population depends both on the details of the spreading process and on the structure of the social and communication networks on which they spread. In this paper, we explore how \textit{anti-establishment} nodes (e.g., \textit{hipsters}) influence the spreading dynamics of two competing products. We consider a model in which spreading follows a deterministic rule for updating node states (which describe which product has been adopted) in which an adjustable fraction $p_{\rm Hip}$ of the nodes in a network are hipsters, who choose to adopt the product that they believe is the less popular of the two. The remaining nodes are conformists, who choose which product to adopt by considering which products their immediate neighbors have adopted. We simulate our model on both synthetic and real networks, and we show that the hipsters have a major effect on the final fraction of people who adopt each product: even when only one of the two products exists at the beginning of the simulations, a very small fraction of hipsters in a network can still cause the other product to eventually become the more popular one. To account for this behavior, we construct an approximation for the steady-state adoption fraction on $k$-regular trees in the limit of few hipsters. Additionally, our simulations demonstrate that a time delay $\tau$ in the knowledge of the product distribution in a population, as compared to immediate knowledge of product adoption among nearest neighbors, can have a large effect on the final distribution of product adoptions. Our simple model and analysis may help shed light on the road to success for anti-establishment choices in elections, as such success can arise rather generically in our model from a small number of anti-establishment individuals and ordinary processes of social influence on normal individuals.Comment: Extensively revised, with much new analysis and numerics The abstract on arXiv is a shortened version of the full abstract because of space limit

Topological data analysis of truncated contagion maps

The investigation of dynamical processes on networks has been one focus for the study of contagion processes. It has been demonstrated that contagions can be used to obtain information about the embedding of nodes in a Euclidean space. Specifically, one can use the activation times of threshold contagions to construct contagion maps as a manifold-learning approach. One drawback of contagion maps is their high computational cost. Here, we demonstrate that a truncation of the threshold contagions may considerably speed up the construction of contagion maps. Finally, we show that contagion maps may be used to find an insightful low-dimensional embedding for single-cell RNA-sequencing data in the form of cell-similarity networks and so reveal biological manifolds. Overall, our work makes the use of contagion maps as manifold-learning approaches on empirical network data more viable

Topological data analysis of truncated contagion maps

The investigation of dynamical processes on networks has been one focus for the study of contagion processes. It has been demonstrated that contagions can be used to obtain information about the embedding of nodes in a Euclidean space. Specifically, one can use the activation times of threshold contagions to construct contagion maps as a manifold-learning approach. One drawback of contagion maps is their high computational cost. Here, we demonstrate that a truncation of the threshold contagions may considerably speed up the construction of contagion maps. Finally, we show that contagion maps may be used to find an insightful low-dimensional embedding for single-cell RNA-sequencing data in the form of cell-similarity networks and so reveal biological manifolds. Overall, our work makes the use of contagion maps as manifold-learning approaches on empirical network data more viable. It is known that the analysis of spreading processes on networks may reveal their hidden geometric structures. These techniques, called contagion maps, are computationally expensive, which raises the question of whether they can be methodologically improved. Here, we demonstrate that a truncation (i.e., early stoppage) of the spreading processes leads to a substantial speedup in the computation of contagion maps. For synthetic networks, we find that a carefully chosen truncation may also improve the recovery of hidden geometric structures. We quantify this improvement by comparing the topological properties of the original network with the constructed contagion maps by computing their persistent homology. Finally, we explore the embedding of single-cell transcriptomics data and show that contagion maps can help us to distinguish different cell types

Analysis of contagion maps on a class of networks that are spatially embedded in a torus

A spreading process on a network is influenced by the network's underlying spatial structure, and it is insightful to study the extent to which a spreading process follows such structure. We consider a threshold contagion on a network whose nodes are embedded in a manifold and where the network has both `geometric edges', which respect the geometry of the underlying manifold, and `non-geometric edges' that are not constrained by that geometry. Building on ideas from Taylor et al. \cite{Taylor2015}, we examine when a contagion propagates as a wave along a network whose nodes are embedded in a torus and when it jumps via long non-geometric edges to remote areas of the network. We build a `contagion map' for a contagion spreading on such a `noisy geometric network' to produce a point cloud; and we study the dimensionality, geometry, and topology of this point cloud to examine qualitative properties of this spreading process. We identify a region in parameter space in which the contagion propagates predominantly via wavefront propagation. We consider different probability distributions for constructing non-geometric edges --- reflecting different decay rates with respect to the distance between nodes in the underlying manifold --- and examine the effect of such choices on the qualitative properties of the spreading dynamics. Our work generalizes the analysis in Taylor et al. and consolidates contagion maps both as a tool for investigating spreading behavior on spatial networks and as a technique for manifold learning

Homogeneous symmetrical threshold model with nonconformity: independence vs. anticonformity

We study two variants of the modified Watts threshold model with a noise (with nonconformity, in the terminology of social psychology) on a complete graph. Within the first version, a noise is introduced via so-called independence, whereas in the second version anticonformity plays the role of a noise, which destroys the order. The modified Watts threshold model, studied here, is homogeneous and posses an up-down symmetry, which makes it similar to other binary opinion models with a single-flip dynamics, such as the majority-vote and the q-voter models. Because within the majority-vote model with independence only continuous phase transitions are observed, whereas within the q-voter model with independence also discontinuous phase transitions are possible, we ask the question about the factor, which could be responsible for discontinuity of the order parameter. We investigate the model via the mean-field approach, which gives the exact result in the case of a complete graph, as well as via Monte Carlo simulations. Additionally, we provide a heuristic reasoning, which explains observed phenomena. We show that indeed, if the threshold r = 0.5, which corresponds to the majority-vote model, an order-disorder transition is continuous. Moreover, results obtained for both versions of the model (one with independence and the second one with anticonformity) give the same results, only rescaled by the factor of 2. However, for r > 0.5 the jump of the order parameter and the hysteresis is observed for the model with independence, and both versions of the model give qualitatively different results.Comment: 12 pages, 4 figures, accepted to Complexit

Attacking The Assortativity Coefficient Under A Rewiring Strategy

Degree correlation is an important characteristic of networks, which is usually quantified by the assortativity coefficient. However, concerns arise about changing the assortativity coefficient of a network when networks suffer from adversarial attacks. In this paper, we analyze the factors that affect the assortativity coefficient and study the optimization problem of maximizing or minimizing the assortativity coefficient (r) in rewired networks with $k$ pairs of edges. We propose a greedy algorithm and formulate the optimization problem using integer programming to obtain the optimal solution for this problem. Through experiments, we demonstrate the reasonableness and effectiveness of our proposed algorithm. For example, rewired edges 10% in the ER network, the assortativity coefficient improved by 60%

Aging in binary-state models: The Threshold model for Complex Contagion

Binary-state models are those in which the constituent elements can only appear in two possible configurations. These models are fundamental in the mathematical treatment of a number of phenomena such as spin interactions in magnetism, opinion dynamics, rumor and information spreading in social systems, etc. Here, we focus on the study of non-Markovian effects associated with aging for binary-state dynamics in complex networks. Aging is considered as the property of the agents to be less prone to change state the longer they have been in the current state, which gives rise to heterogeneous activity patterns. We analyze in this context the Threshold model of Complex Contagion, which has been proposed to explain, for instance, processes of adoption of new technologies and in which the agents need the reiterated confirmation of several contacts (until reaching over a given neighbor fraction threshold) to change state. Our analytical approximations give a good description of extensive numerical simulations in Erd\"os-R\'enyi, random-regular and Barab\'asi-Albert networks. While aging does not modify the spreading condition, it slows down the cascade dynamics towards the full-adoption state: the exponential increase of adopters in time from the original model is replaced by a stretched exponential or power-law, depending on the aging mechanism. Under several approximations, we give analytical expressions for the cascade condition and for the exponents of the exponential, power-law and stretched exponential growth laws for the adopters density. Beyond networks, we also describe by numerical simulations the effects of aging for the Threshold model in a two-dimensional lattice.Comment: 12 pages, 9 figure

Replacement Hand Washing System

The goal of this project was to design a new hand washing system that will eradicate the need for paper towels and that will also streamline the hand washing process to within 10-15 seconds total wash and dry time