**Tutorials**

**Topological Data Analysis with jHoles and Persistent Entropy**

Speaker: Emanuela Merelli, Matteo Rucco

**Abstract**:

Scouting on the Web is similar as mining a large data set. To challenge the current thinking in IT for the Big Data question we propose an innovative methodology to perform data

analytics that goes beyond the usual paradigms of data mining rooted on the notion of Complex networks and Machine Learning1. The new approach is realized in two steps:

topological data analysis and topological field theory of data.

This tutorial will present the new methodology and will provide a comprehensive introduction to Topological Data Analysis through jHoles and Persistent Entropy. jHoles2 is a software library designed specifically to aid in Topological Data Analysis research, while Persistent Entropy is a new statistics that summarizes topological data analysis results. Topological Data Analysis is an area of applied mathematics based on mathematical concepts such as simplicial complex and persistent homology, and it aims to deal with desirable properties as coordinate-freeness and robustness to noise. TDA is able to make some strong claims as to its practical uses. Complex networks are one of the promising tools in the study of complex dynamical systems (e.g. internet of things, social systems, evolution dynamics, breast cancer diagnosis, etc....). Complex networks are classically studied with statistical approaches but they can be also analyzed from a topological data analysis perspective. The latter approach allows us to detect the higher dimensional relationships among the nodes that guarantee the flow of information and the dynamics of the entire system represented by the network. Clique weight rank persistent homology (CWRPH) is the most suitable approach for performing TDA on complex network. jHoles fills the lack of an efficient implementation of the Miltering process for CWRPH.

The tutorial will firstly introduce the mathematical key points that are required for understanding how jHoles works. Then, we will give a detailed description of the jHoles algorithm and its engine, jPHEngine. The talk will also introduce on how to understand the results of Topological Data Analysis by using a new statistics, the so-called Persistent Entropy. Persistent Entropy links TDA with Machine Learning techniques. In the tutorial we will apply jHoles and Persistent Entropy for analyzing a bunch of selected problems among Internet of Things and biological networks.

**Social Behaviors through Networks: Models and Applications**

Speaker: Marco Alberto Javarone

**Abstract**:

Nowadays, the studying of social behaviors represents a topic of interest for scientists belonging to several communities, spanning from computational social science to mathematics and physics. Notably, the evolution of the WEB, and the way people interact through this network, entail the necessity to study social behaviors in structured populations, i.e. considering agents connected in different, and non-trivial, topologies. Remarkably, interaction patterns acquire a great relevance in games, in spreading phenomena, in ordering dynamics and so on. As result, processes as information spreading, lurking dynamics, emergence of cooperation, and many others, must be analyzed considering both network topologies, and those behaviors that we can observe in social systems, spanning from small to big communities of interacting agents.

This tutorial aims to present some models for studying some relevant behaviors in networked populations. In particular, beyond providing a general introduction to complex networks (e.g. scale free and small world), I will focus on the different outcomes that can be achieved by varying the network topology, from simple lattices to more complex ones. For instance, I will discuss the so-called 'network reciprocity' in evolutionary games, useful for studying the emergence of cooperation, the role of conformism and non-conformism in information dynamics, the role of innovation in shaping citation networks, etc.

From the mathematical point of view, I will illustrate the main measures for analyzing networks and some processes (e.g. spreading), and I will discuss some parameters 'coming from the world of physics' for quantifying the effects of social behaviors. To conclude, beyond the main scope of the tutorial, it will be shown the application of social imitation mechanisms, in networked populations, for solving combinatorial optimization tasks.