Message-Passing Methods for Complex Contagions

Message-passing methods provide a powerful approach for calculating the expected size of cascades either on random networks (e.g., drawn from a configuration-model ensemble or its generalizations) asymptotically as the number $N$ of nodes becomes infinite or on specific finite-size networks. We review the message-passing approach and show how to derive it for configuration-model networks using the methods of (Dhar et al., 1997) and (Gleeson, 2008). Using this approach, we explain for such networks how to determine an analytical expression for a "cascade condition", which determines whether a global cascade will occur. We extend this approach to the message-passing methods for specific finite-size networks (Shrestha and Moore, 2014; Lokhov et al., 2015), and we derive a generalized cascade condition. Throughout this chapter, we illustrate these ideas using the Watts threshold model.Comment: 14 pages, 3 figure

Adaptive Complex Contagions and Threshold Dynamical Systems

A broad range of nonlinear processes over networks are governed by threshold dynamics. So far, existing mathematical theory characterizing the behavior of such systems has largely been concerned with the case where the thresholds are static. In this paper we extend current theory of finite dynamical systems to cover dynamic thresholds. Three classes of parallel and sequential dynamic threshold systems are introduced and analyzed. Our main result, which is a complete characterization of their attractor structures, show that sequential systems may only have fixed points as limit sets whereas parallel systems may only have period orbits of size at most two as limit sets. The attractor states are characterized for general graphs and enumerated in the special case of paths and cycle graphs; a computational algorithm is outlined for determining the number of fixed points over a tree. We expect our results to be relevant for modeling a broad class of biological, behavioral and socio-technical systems where adaptive behavior is central.Comment: Submitted for publicatio

A New Methodology for Generalizing Unweighted Network Measures

Several important complex network measures that helped discovering common patterns across real-world networks ignore edge weights, an important information in real-world networks. We propose a new methodology for generalizing measures of unweighted networks through a generalization of the cardinality concept of a set of weights. The key observation here is that many measures of unweighted networks use the cardinality (the size) of some subset of edges in their computation. For example, the node degree is the number of edges incident to a node. We define the effective cardinality, a new metric that quantifies how many edges are effectively being used, assuming that an edge's weight reflects the amount of interaction across that edge. We prove that a generalized measure, using our method, reduces to the original unweighted measure if there is no disparity between weights, which ensures that the laws that govern the original unweighted measure will also govern the generalized measure when the weights are equal. We also prove that our generalization ensures a partial ordering (among sets of weighted edges) that is consistent with the original unweighted measure, unlike previously developed generalizations. We illustrate the applicability of our method by generalizing four unweighted network measures. As a case study, we analyze four real-world weighted networks using our generalized degree and clustering coefficient. The analysis shows that the generalized degree distribution is consistent with the power-law hypothesis but with steeper decline and that there is a common pattern governing the ratio between the generalized degree and the traditional degree. The analysis also shows that nodes with more uniform weights tend to cluster with nodes that also have more uniform weights among themselves.Comment: 23 pages, 10 figure

Agent-based transportation planning compared with scheduling heuristics

Here we consider the problem of dynamically assigning vehicles to transportation orders that have di¤erent time windows and should be handled in real time. We introduce a new agent-based system for the planning and scheduling of these transportation networks. Intelligent vehicle agents schedule their own routes. They interact with job agents, who strive for minimum transportation costs, using a Vickrey auction for each incoming order. We use simulation to compare the on-time delivery percentage and the vehicle utilization of an agent-based planning system to a traditional system based on OR heuristics (look-ahead rules, serial scheduling). Numerical experiments show that a properly designed multi-agent system may perform as good as or even better than traditional methods

Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction

Network centrality: an introduction

Centrality is a key property of complex networks that influences the behavior of dynamical processes, like synchronization and epidemic spreading, and can bring important information about the organization of complex systems, like our brain and society. There are many metrics to quantify the node centrality in networks. Here, we review the main centrality measures and discuss their main features and limitations. The influence of network centrality on epidemic spreading and synchronization is also pointed out in this chapter. Moreover, we present the application of centrality measures to understand the function of complex systems, including biological and cortical networks. Finally, we discuss some perspectives and challenges to generalize centrality measures for multilayer and temporal networks.Comment: Book Chapter in "From nonlinear dynamics to complex systems: A Mathematical modeling approach" by Springe

Self-organized model of cascade spreading

We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further generalize the model to reproduce volatility clustering and other observed properties of real stocks.Comment: 8 pages, 6 figure