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1. Core motifs predict dynamic attractors in combinatorial threshold-linear networks

   https://doi.org/10.1371/journal.pone.0264456  

   Parmelee, Caitlyn; Moore, Samantha; Morrison, Katherine; Curto, Carina (March 2022, PLOS ONE)

   Kryven, Ivan (Ed.)

   Combinatorial threshold-linear networks (CTLNs) are a special class of inhibition-dominated TLNs defined from directed graphs. Like more general TLNs, they display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. In prior work, we have developed a detailed mathematical theory relating stable and unstable fixed points of CTLNs to graph-theoretic properties of the underlying network. Here we find that a special type of fixed points, corresponding to core motifs , are predictive of both static and dynamic attractors. Moreover, the attractors can be found by choosing initial conditions that are small perturbations of these fixed points. This motivates us to hypothesize that dynamic attractors of a network correspond to unstable fixed points supported on core motifs. We tested this hypothesis on a large family of directed graphs of size n = 5, and found remarkable agreement. Furthermore, we discovered that core motifs with similar embeddings give rise to nearly identical attractors. This allowed us to classify attractors based on structurally-defined graph families. Our results suggest that graphical properties of the connectivity can be used to predict a network’s complex repertoire of nonlinear dynamics. 

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2. Natural Disaster Evacuation Modeling: The Dichotomy of Fear of Crime and Social Influence

   https://doi.org/10.1007/s13278-021-00839-8  

   Kuhlman, Chris J.; Marathe, Achla; Vullikanti, Anil; Halim, Nafisa; Mozumder, Pallab (November 2021, Social network analysis and mining)

   Neighborhood effects have an important role in evacuation decision-making by a family. Owing to peer influence, neighbors evacuating can motivate a family to evacuate. Paradoxically, if a lot of neighbors evacuate, then the likelihood of an individual or family deciding to evacuate decreases, for fear of crime and looting. Such behavior cannot be captured using standard models of contagion spread on networks, e.g., threshold, independent cascade, and linear threshold models. Here, we propose a new threshold-based graph dynamical system model, 2mode-threshold, which captures this dichotomy. We study theoretically the dynamical properties of 2mode-threshold in different networks, and find significant differences from a standard threshold model. We build and characterize small world networks of Virginia Beach, VA, where nodes are geolocated families (households) in the city and edges are interactions between pairs of families. We demonstrate the utility of our behavioral model through agent-based simulations on these small world networks. We use it to understand evacuation rates in this region, and to evaluate the effects of modeling parameters on evacuation decision dynamics. Specifically, we quantify the effects of (1) network generation parameters, (2) stochasticity in the social network generation process, (3) model types (2mode-threshold vs. standard threshold models), (4) 2mode-threshold model parameters, (5) and initial conditions, on computed evacuation rates and their variability. An illustrative example result shows that the absence of looting effect can overpredict evacuation rates by as much as 50%. 

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3. Natural Disaster Evacuation Modeling: The Dichotomy of Fear of Crime and Social Influence

   Kuhlman, Chris J; Marathe, Achla; Vullikanti, Anil; Halim, Nafisa; Mozumder, Pallab (January 2021, Social network analysis and mining)

   null (Ed.)

   Neighborhood eects have an important role in evacuation decision-making by a family. Owing to peer influence, neighbors evacuating can motivate a family to evacuate. Paradoxically, if a lot of neighbors evacuate, then the likelihood of an individual or family deciding to evacuate decreases, for fear of crime and looting. Such behavior cannot be captured using standard models of contagion spread on networks, e.g., threshold, independent cascade, and linear threshold models. Here, we propose a new threshold-based graph dynamical system model, 2mode-threshold, which captures this dichotomy. We study theoretically the dynamical properties of 2mode-threshold in different networks, and fi nd signi ficant differences from a standard threshold model. We build and characterize small world networks of Virginia Beach, VA, where nodes are geolocated families (households) in the city and edges are interactions between pairs of families. We demonstrate the utility of our behavioral model through agent-based simulations on these small world networks. We use it to understand evacuation rates in this region, and to evaluate the effects of modeling parameters on evacuation decision dynamics. Speci fically, we quantify the effects of (i) network generation parameters, (ii) stochasticity in the social network generation process, (iii) model types (2mode-threshold vs. stan- dard threshold models), (iv) 2mode-threshold model parameters, (v) and initial conditions, on computed evacuation rates and their variability. An illustrative example result shows that the absence of looting eect can overpredict evacuation rates by as much as 50%. 

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4. Nerve Theorems for Fixed Points of Neural Networks

   https://doi.org/10.1007/978-3-030-95519-9_6  

   Santander, D. E.; Ebli, S.; Patania, A.; Sanderson, N.; Burtscher, F.; Morrison, K.; Curto, C. (May 2022, Association for Women in Mathematics series)

   Gasparovic, E.; Robins, V.; Turner, K. (Ed.)

   Nonlinear network dynamics are notoriously difficult to understand. Here we study a class of recurrent neural networks called combinatorial threshold-linear networks (CTLNs) whose dynamics are determined by the structure of a directed graph. They are a special case of TLNs, a popular framework for modeling neural activity in computational neuroscience. In prior work, CTLNs were found to be surprisingly tractable mathematically. For small networks, the fixed points of the network dynamics can often be completely determined via a series of graph rules that can be applied directly to the underlying graph. For larger networks, it remains a challenge to understand how the global structure of the network interacts with local properties. In this work, we propose a method of covering graphs of CTLNs with a set of smaller directional graphs that reflect the local flow of activity. While directional graphs may or may not have a feedforward architecture, their fixed point structure is indicative of feedforward dynamics. The combinatorial structure of the graph cover is captured by the nerve of the cover. The nerve is a smaller, simpler graph that is more amenable to graphical analysis. We present three nerve theorems that provide strong constraints on the fixed points of the underlying network from the structure of the nerve. We then illustrate the power of these theorems with some examples. Remarkably, we find that the nerve not only constrains the fixed points of CTLNs, but also gives insight into the transient and asymptotic dynamics. This is because the flow of activity in the network tends to follow the edges of the nerve. 

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5. Learning Network Dynamics from Noisy Steady States

   https://doi.org/10.1145/3625007.3631184  

   Ding, Yanna; Gao, Jianxi; Magdon-Ismail, Malik (November 2023, 2023 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM))

   We present efficient algorithms to learn the pa- rameters governing the dynamics of networked agents, given equilibrium steady state data. A key feature of our methods is the ability to learn without seeing the dynamics, using only the steady states. A key to the efficiency of our approach is the use of mean-field approximations to tune the parameters within a nonlinear least squares (NLS) framework. Our results on real networks demonstrate the accuracy of our approach in two ways. Using the learned parameters, we can: (i) Recover more accurate estimates of the true steady states when the observed steady states are noisy. (ii) Predict evolution to new equilibrium steady states after perturbations to the network topology. 

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