C4

There is a mismatch between levels of crime and its fear and often, cities might see an increase or a decrease in crime over time while the fear of crime remains unchanged. A model that considers fear of crime as an opinion shared by simulated individuals on a network will be presented, and the impact that different distributions of crime have on the fear experienced by the population will be explored. Results show that the dynamics of the fear is sensitive to the distribution of crime and that there is a phase transition for high levels of concentration of crime.

Crime is a major risk to society’s well-being, particularly in cities, and yet the scientific literature lacks a comprehensive statistical characterization of crime that could uncover some of the mechanisms behind such pervasive social phenomenon. Evidence of nonlinear scaling of urban indicators in cities, such as wages and serious crime, has motivated the understanding of cities as complex systems—a perspective that offers insights into resources limits and sustainability, but usually without examining the details of indicators. Notably, since the nineteenth century, criminal activities have been known not to occur uniformly within a city. Crime concentrates in such way that most of the offenses take place in few regions of the city. However, though this concentration is confirmed by different studies, the absence of broad examinations of the characteristics of crime concentration hinders not only the comprehension of crime dynamics but also the proposal of sounding counter-measures. Here, we developed a framework to characterize crime concentration which splits cities into regions with the same population size. We used disaggregated criminal data from 25 locations in the U.S. and the U.K. which include offenses in places spanning from 2 to 15 years of data. Our results confirmed that crime concentrates regardless of city and revealed that the level of concentration does not scale with city size. We found that distribution of crime in a city can be approximated by a power-law distribution with exponent α that depends on the type of crime. In particular, our results showed that thefts tend to concentrate more than robberies, and robberies more than burglaries. Though criminal activities present regularities of concentration, we found that criminal ranks have the tendency to change continuously over time. Such features support the perspective of crime as a complex system which demands analyses and evolving urban policies covering the city as a whole. 

Matrix completion is an area of great mathematical interest and has numerous applications, including recommender systems for e-commerce. The recommender problem can be viewed as follows: given a database where rows are users and and columns are products, with entries indicating user preferences, fill in the entries so as to be able to recommend new products based on the preferences of other users. Viewing the interactions between user and product as links in a bipartite graph, the problem is equivalent to approximating a partially observed graph using clusters. We propose a divide and conquer algorithm inspired by the work of [1], who use recursive rank-1 approximation. We make the case for using an LP rank-1 approximation, similar to that of [2] by a showing that it guarantees a 2-approximation to the optimal, even in the case of missing data. We explore our algorithm's performance for different test cases.

[1]  Shen, B.H., Ji, S. and Ye, J., 2009, June. Mining discrete patterns via binary matrix factorization. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 757-766). ACM.

[2] Koyutürk, M. and Grama, A., 2003, August. PROXIMUS: a framework for analyzing very high dimensional discrete-attributed datasets. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 147-156). ACM.
 

In this work, study the mean first saturation time (MFST), a generalization to the mean first passage time, on networks and show an application to the 2015 Burundi refugee crisis. The MFST between a sink node j, with capacity s, and source node i, with n random walkers, is the average number of time steps that it takes for at least s of the random walkers to reach a sink node j. The same concept, under the name of extreme events, has been studied in previous work for degree biased-random walks [2]. We expand the literature by exploring the behaviour of the MFST for node-biased random walks [1] in Erdős–Rényi random graph and geographical networks. Furthermore, we apply MFST framework to study the distribution of refugees in camps for the 2015 Burundi refugee crisis. For this last application, we use the geographical network of the Burundi conflict zone in 2015 [3]. In this network, nodes are cities or refugee camps, and edges denote the distance between them. We model refugees as random walkers who are biased towards the refugee camps which can hold s_j people. To determine the source nodes (i) and the initial number of random walkers (n), we use data on where the conflicts happened and the number of refugees that arrive at any camp under a two-month period after the start of the conflict [3]. With such information, we divide the early stage of the Burundi 2015 conflict into two waves of refugees. Using the first wave of refugees we calibrate the biased parameter β of the random walk to best match the distribution of refugees on the camps. Then, we test the prediction of the distribution of refugees in camps for the second wave using the same biased parameters. Our results show that the biased random walk can capture, to some extent, the distribution of refugees in different camps. Finally, we test the probability of saturation for various camps. Our model suggests the saturation of one or two camps (Nakivale and Nyarugusu) when in reality only Nyarugusu camp saturated.

[1] Sood, Vishal, and Peter Grassberger. ”Localization transition of biased random walks on random
networks.” Physical review letters 99.9 (2007): 098701.
[2] Kishore, Vimal, M. S. Santhanam, and R. E. Amritkar. ”Extreme event-size fluctuations in biased
random walks on networks.” arXiv preprint arXiv:1112.2112 (2011).
[3] Suleimenova, Diana, David Bell, and Derek Groen. ”A generalized simulation development approach
for predicting refugee destinations.” Scientific reports 7.1 (2017): 13377.

The analysis and characterization of human mobility using population-level mobility models is important for numerous applications, ranging from the estimation of commuter flows to modeling trade flows. However, almost all of these applications have focused on large spatial scales, typically from intra-city level to inter-country level. In this paper, we investigate population-level human mobility models on a much smaller spatial scale by using them to estimate customer mobility flow between supermarket zones. We use anonymized mobility data of customers in supermarkets to calibrate our models and apply variants of the gravity and intervening-opportunities models to fit this mobility flow and estimate the flow on unseen data. We find that a doubly-constrained gravity model can successfully estimate 65-70% of the flow inside supermarkets. We then investigate how to reduce congestion in supermarkets by combining mobility models with queueing networks. We use a simulated-annealing algorithm to find store layouts with lower congestion than the original layout. Our research gives insight both into how customers move in supermarkets and into how retailers can arrange stores to reduce congestion. It also provides a case study of human mobility on small spatial scales.

George Soules [1] introduced a set of vectors $r_1,...,r_N$ with the remarkable property that for any set of ordered numbers $\lambda_1\geq\dots\geq\lambda_N$, the matrix $\sum_n \lambda_nr_nr_n^T$ has nonnegative off-diagonal entries. Later, it was found [2] that there exists a whole class of such vectors - Soules vectors - which are intimately connected to binary rooted trees. In this talk I will describe the construction of Soules vectors starting from a binary rooted tree, and introduce some basic properties. I will also cover a number of applications: the inverse eigenvalue problem, equitable partitions in Laplacian matrices and the eigendecomposition of the Clauset-Moore-Newman hierarchical random graph model.

[1] Soules (1983), Constructing Symmetric Nonnegative Matrices
[2] Elsner, Nabben and Neumann (1998), Orthogonal bases that lead to symmetric nonnegative matrices

Mechanical percolation is a phenomenon in materials processing wherein ‘filler’ rod-like particles are incorporated into polymeric materials to enhance the composite’s mechanical properties. Experiments have well-characterized a nonlinear phase transition from floppy to rigid behavior at a threshold filler concentration, but the underlying mechanism is not well understood. We develop and utilize an iterative graph compression algorithm to demonstrate that this experimental phenomenon coincides with the formation of a spatially extending set of mutually rigid rods (‘rigidity percolation’). First, we verify the efficacy of this method in two-dimensional fiber systems (intersecting line segments), then moving to the more interesting and mechanically representative problem of three-dimensional fiber systems (cylinders). We show that, when the fibers are uniformly distributed both spatially and orientationally, the onset of rigidity percolation appears to co-occur with a mean field prediction that is applicable across a wide range of aspect ratios.

Disease transmission and rumour spreading are ubiquitous in social and technological networks. In this talk, we will present our last results on the modelling of rumour and disease spreading in multilayer networks.  We will derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks in a multiplex network. Moreover, we will introduce a model of epidemic spreading with awareness, where the disease and information are propagated in different layers with different time scales. We will show that the time scale determines whether the information awareness is beneficial or not to the disease spreading. 

In the Boussinesq framework, velocity couples to density fluctuations whereas in magnetohydrodynamic turbulence, the velocity field is coupled to the magnetic field. Both systems support waves (inertia-gravity in the presence of rotation, or Alfvén), with anisotropic dispersion relations. What kind of turbulence regimes result from the interactions between waves and nonlinear eddies in such flows? And what is delimiting these regimes?

I shall sketch the phenomenological framework for rotating stratified turbulence within which one is led to scaling laws in terms of the Froude number, Fr=U/[LN], which measures the relative celerity of gravity waves and nonlinear eddies, with U and L characteristic velocity and length scale, and N the Brunt-V\"ais\"al\"a frequency. These laws apply to the mixing efficiency of such flows, indicating the relative roles of the buoyancy flux due to the waves, and of the measured kinetic and potential energy dissipation rates. Various measures of mixing are found to follow power laws in terms of the Froude number, and may differ for the three regimes that can be identified, namely the wave-dominated, wave-eddy balance and eddy-dominated domains [1]. In particular, in the intermediate regime, the effective dissipation varies linearly with Fr, in agreement with simple wave-turbulence arguments. This analysis is inspired by and corroborates results from a large parametric study using direct numerical simulations (DNS) on grids of 1024^3 points, as well as from atmospheric and oceanic observations.

Such scaling laws can be related to previous DNS results concerning the existence for the energy of bi-directional constant-flux cascades to both the small scales and to the large scales, due to the presence of rotation in such flows, as measured for example in the ocean. These dual energy cascades lead to an alteration, and a decrease, of the mixing and available energy to be dissipated in the small scales [2]. Some perspectives might also be given at the end of the talk.

[1] A. Pouquet, D. Rosenberg, R. Marino & C. Herbert, Scaling laws for mixing and dissipation in unforced rotating stratified turbulence. J. Fluid Mechanics 844, 519, 2018.
[2] R. Marino, A. Pouquet & D. Rosenberg, Resolving the paradox of oceanic large-scale balance and small-scale mixing. Physical Review Letters 114, 114504, 2015.

Joint work with Zsolt Vizi (Bolyai Institute, University of Szeged, Hungary), Istvan Kiss (Department
of Mathematics, University of Sussex, United Kingdom)

Pairwise models have been proven to be a flexible framework for analytical approximations
of stochastic epidemic processes on networks that are in many situations much more accurate
than mean field compartmental models. The non-Markovian aspects of disease transmission
are undoubtedly important, but very challenging to incorporate them into both numerical
stochastic simulations and analytical investigations. Here we present a generalization of
pairwise models to non-Markovian epidemics on networks. For the case of infectious periods
of fixed length, the resulting pairwise model is a system of delay differential equations, which
shows excellent agreement with results based on the explicit stochastic simulations. For more
general distribution classes (uniform, gamma, lognormal etc.) the resulting models are PDEs
that can be transformed into systems of integro-differential equations. We derive pairwise
reproduction numbers and relations for the final epidemic size, and initiate a systematic
study of the impact of the shape of the particular distributions of recovery times on how
the time evolution of the disease dynamics play out.