# STALITA: Innovative Platform for Bank Transactions Analysis
## 3 Materials and Methods
### Definition of Graphs' Topological Properties  
As was previously noted, the basic data structure of the presented platform is based on graphs. The definition of graphs, their most common topologies, and some topological properties used in graph analysis are presented in this subsection. Graphs \(G\) are defined as \(G=(N,E)\), where \(N=\{n_{i}\}\) is a node-set, and \(E=\{e_{ij}\}\) is an edge-set of \(G\)[32]. A given node, ni, corresponds to a subject (in the context of bank transaction analysis, this could be a bank account, a person, or a company), whereas an edge, \(e_{ij}=(ni,n\eta)\), can define ownership (the person owns an account, the person owns a company), or a transaction (from one account to another). There are many different options for node and edge meanings as we are working with unstructured data. Graph topologies are typically defined by their topological properties, and we know four major types of graphs: small-world, scale-free, random, and regular [33].  
The topological properties that define graph types can also be applied to basic and assembled analyses of case investigations. Some of the properties are presented and described in the continuation:  
* **Node's degree \(k[n_{i}]\)** could, for example, be used to count a number of people's bank accounts or transactions, and similar countings are defined as (1): \[k[n_{i}]=|E_{i}|;\] (1)
* **The average degree \(k_{a}[n_{i}]\) of nodes linked to \(n_{i}\)** can, for example, describe the average number of transactions for the persons linked to \(n_{i}\). It is defined as (2): \[k_{a}[n_{i}]=\frac{1}{k[n_{i}]}\sum_{e_{ij}\in E_{i}}k[n_{j}];\] (2)
* **The number of triplets of transactions \(t[n_{i}]\) that include node \(n_{i}\)** is, for example, used for the search of circles of transactions in the investigation of money laundering cases. Mathematically, it is defined as (3): \[t[n_{i}]=\Big{|}\Big{\{}e_{jh}:\ n_{j},n_{h}\in N_{i}\Big{\}}\Big{|};\] (3)
* **The local clustering coefficient \(u[n_{i}]\)** the proximity of a cluster of transactions, a bank account, or, eventually, people to the center. The coefficient could be written as (4): \[u[n_{i}]=\frac{t[n_{i}]}{|E_{i}|(|E_{i}|-1)};\] (4)
* **Local betweenness centrality \(b[n_{i}]\)** shows the size of clusters of, for example, bank accounts, persons, or transactions that a given \(n_{i}\) (bank account, person, or transaction) is linked with. Its mathematical definition is (5): \[b[n_{i}]=\frac{1}{(|N_{i}|-1)(|N_{i}|-2)}\sum_{i\
eq j,i\
eq h,j\
eq h}n_{j},n_ {h}\in N_{i}\frac{|\{\prod j,h:n_{i}\in\prod\langle j,h\rangle\}|}{|\{\prod \langle j,h\rangle\}|}.\] (5)  
The size of nodes and edge sets in graphs also defines the time complexity for different types of graphs and different topological properties. In [28], it has been shown that the number of edges does not influence the time complexity, as they are checking all possible connections in the graph (full graph). In the case of bank transactions' investigations, we will rarely search in full graphs, and, by considering that, we can assume that the number of nodes and edges has an influence on time complexity in our graph database structure.