Multiple Hybrid Phase Transition: Bootstrap Percolation on Complex Networks with Communities

Bootstrap percolation is a well-known model to study the spreading of rumors, new products or innovations on social networks. The empirical studies show that community structure is ubiquitous among various social networks. Thus, studying the bootstrap percolation on the complex networks with communities can bring us new and important insights of the spreading dynamics on social networks. It attracts a lot of scientists' attentions recently. In this letter, we study the bootstrap percolation on Erd\H{o}s-R\'{e}nyi networks with communities and observed second order, hybrid (both second and first order) and multiple hybrid phase transitions, which is rare in natural system. Moreover, we have analytically solved this system and obtained the phase diagram, which is further justified well by the corresponding simulations

Competing contagion processes: Complex contagion triggered by simple contagion

Empirical evidence reveals that contagion processes often occur with competition of simple and complex contagion, meaning that while some agents follow simple contagion, others follow complex contagion. Simple contagion refers to spreading processes induced by a single exposure to a contagious entity while complex contagion demands multiple exposures for transmission. Inspired by this observation, we propose a model of contagion dynamics with a transmission probability that initiates a process of complex contagion. With this probability nodes subject to simple contagion get adopted and trigger a process of complex contagion. We obtain a phase diagram in the parameter space of the transmission probability and the fraction of nodes subject to complex contagion. Our contagion model exhibits a rich variety of phase transitions such as continuous, discontinuous, and hybrid phase transitions, criticality, tricriticality, and double transitions. In particular, we find a double phase transition showing a continuous transition and a following discontinuous transition in the density of adopted nodes with respect to the transmission probability. We show that the double transition occurs with an intermediate phase in which nodes following simple contagion become adopted but nodes with complex contagion remain susceptible.Comment: 9 pages, 4 figure

Theories for influencer identification in complex networks

In social and biological systems, the structural heterogeneity of interaction networks gives rise to the emergence of a small set of influential nodes, or influencers, in a series of dynamical processes. Although much smaller than the entire network, these influencers were observed to be able to shape the collective dynamics of large populations in different contexts. As such, the successful identification of influencers should have profound implications in various real-world spreading dynamics such as viral marketing, epidemic outbreaks and cascading failure. In this chapter, we first summarize the centrality-based approach in finding single influencers in complex networks, and then discuss the more complicated problem of locating multiple influencers from a collective point of view. Progress rooted in collective influence theory, belief-propagation and computer science will be presented. Finally, we present some applications of influencer identification in diverse real-world systems, including online social platforms, scientific publication, brain networks and socioeconomic systems.Comment: 24 pages, 6 figure

A large deviation approach to super-critical bootstrap percolation on the random graph Gn,pG_{n,p}

We consider the Erd\"{o}s--R\'{e}nyi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^*$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible range.Comment: 44 page

Complex Contagions in Kleinberg's Small World Model

Complex contagions describe diffusion of behaviors in a social network in settings where spreading requires the influence by two or more neighbors. In a $k$-complex contagion, a cluster of nodes are initially infected, and additional nodes become infected in the next round if they have at least $k$ already infected neighbors. It has been argued that complex contagions better model behavioral changes such as adoption of new beliefs, fashion trends or expensive technology innovations. This has motivated rigorous understanding of spreading of complex contagions in social networks. Despite simple contagions ($k=1$) that spread fast in all small world graphs, how complex contagions spread is much less understood. Previous work~\cite{Ghasemiesfeh:2013:CCW} analyzes complex contagions in Kleinberg's small world model~\cite{kleinberg00small} where edges are randomly added according to a spatial distribution (with exponent $\gamma$) on top of a two dimensional grid structure. It has been shown in~\cite{Ghasemiesfeh:2013:CCW} that the speed of complex contagions differs exponentially when $\gamma=0$ compared to when $\gamma=2$. In this paper, we fully characterize the entire parameter space of $\gamma$ except at one point, and provide upper and lower bounds for the speed of $k$-complex contagions. We study two subtly different variants of Kleinberg's small world model and show that, with respect to complex contagions, they behave differently. For each model and each $k \geq 2$, we show that there is an intermediate range of values, such that when $\gamma$ takes any of these values, a $k$-complex contagion spreads quickly on the corresponding graph, in a polylogarithmic number of rounds. However, if $\gamma$ is outside this range, then a $k$-complex contagion requires a polynomial number of rounds to spread to the entire network.Comment: arXiv admin note: text overlap with arXiv:1404.266