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Multi Level Networks Model

Graphs are used to represents data in many situations, especially where it is important do understand how data is interconnected. Each application has its own needs and it is important that the model and the query language are well designed.

The aim of this project is to create a model and query language for multi-level networks analysis. For example we have a network (e.g. LinkedIn) where people are connected if they are colleagues, and another network (e.g. Facebook) where people are connected if they are friends). These two networks are often very different. Let's see few examples:

  • Mark and Peter are colleagues but not friends.
  • Mark and Steve are colleagues.
  • Steve and Peter don't know each other but they might become friends thanks to Mark.

This project is exhaustively described in my Master Thesis (in Italian).

This repository contains the code for the implementation of the model, and the algebra to query the data.

Query Language

The query language is based on paths and walks.

  • A path is a series of nodes in the graph without repetitions. In the example above
  • A walk is a series of nodes in the graph where nodes can be repeated.

Similarly to regular expressions, patterns can be defined in order to find specific paths.

Now we will quickly describe the algebra operators, but the thesis provides formal definitions, with properties and detailed examples.

Concatenation (.)

Concatenation P.P' simply concatenates two paths.

Projection (π)

It is a vertical cut on paths. For example if we have a set of paths, even if they are of different lenghts:

Path patterns

Similarly to regular expressions, patterns use these operators:

  • % to idenfify a generic node that has to be in the path
  • ? to identify optional generic node
  • * to identify a series of nodes (or an empty path)

Concatenation: P1 → P2 it is a path pattern used to describe a node P2 followed by a node P1.

Alternation: P1 | P2 it is a path pattern used to describe an alternation between P1 and P2.

Examples:

  • a → % → b identifies a path that has a beginning in the node a and an end in the node b. The paths that this pattern can find can only have two edges. For example: (a,d,b), or (a,z,b).

  • a → ? → b identifies a path that has a beginning in the node a and an end in the node b. In this case the node at the center is optional, so results can be for example: (a,b), or (a,z,b).

  • (a → b)|(d → c) identifies either the edge (a,b) or the edge (d,c).

  • a → * identifies all paths starting from the node a. These can have various lenghts, and projection can be used to set a limit on the path legths.

A more formal and complete definition of all operators and their use can be found in the thesis.

Selection (σ)

The selection operator uses path patterns in order to retrieve results. For example σa→∗→b(G) selects all paths of any length starting from a and ending in b.

Synthesis (η)

This operator builds a sub-graph given a series of paths. Let's for example assume that a selection operator retrieved the following paths: {(a,b,c), (a,b,e)}.

The synthesis operator builds a sub-graph given these paths:

This enables operators to be combined together for more complex queries.

Aggregation (α)

Aggregation joins nodes together, and similarly to the aggregation operator in relational databases, data contained in these nodes is aggregated. In the example above the operation is:

α% . π2(p); Sum(π1(p)) = 1(P).

Another example is where we want to know how many hops there are to reach specific nodes. For example:

Join (⨝)

Differently than the operators defined before that only works in one network level, the join operator is able to join paths that are in multiple levels.

Join is also able by definition to group multiple nodes in one network level to one node to another network level.

An example of use can be, for example to understand what's the minimum path between two nodes n1 and n2 despite these two are not connected in a specific network level (e.g. n1 and n2 are not friends on Facebook, but are connected through other platforms).

We can join nodes between these two network levels and group paths in order to get the minimum path between n1 and n2:

Other info

The thesis also discusses about implementation of the model to represent the data, and implementation of the operators.

Furthermore, the thesis also discusses about optimizations, since some of the operators (e.g. join) are expensive to compute.

Path patterns are implemented using finite automata.

Project

The prototype of this model and its operator is done in Java.

Folders:

  • src source code
  • test tests
  • out output generated by tests in dot format. These can be visualized using Graphviz dot.

Model

Arc class defines an arc, with the starting point fromNode, the arrival point toNode and the associated data data. It holds all the methods to be able to retrieve and manipulate this information, to compare arcs, and to generate a hashcode (useful for storing arcs in a HashMap).

ArcIterator the complex iterator that enumerates all the arcs of a Network Level (with the possibility of eliminating the current element), starting from a completely different representation made up of nodes and links.

Couple class which defines a Couple, which maintains a list of associations (defined as CoupleLinks) that start at one level and end at another.

CoupleLink class that defines a link of a Couple, with a starting and ending node, and the associated data.

LevelsCoupling class that defines a Levels Coupling, which stores a list of Couples.

Link class that defines an outgoing arch, stored by autumn. It keeps the information on the destination node and the data associated with the arch, and holds all the methods to be able to retrieve and manipulate this information, to compare links, and to generate a hashcode (useful for storing links in a HashMap).

	class Link {
		Node destinationNode;
		Object data;
		
		Link(Node destinationNode, Object data) { /* ... */ } 
		Link(Node destinationNode) { /* ... */ }
		
		public Object getData() { /* ... */ }
		public void setData(Object data) { /* ... */ } 
		public Node getDestinationNode() { /* ... */ } 

		/* ... */
	}

NetworkLevel class which defines a Network Level, keeping an identifier and a set of nodes stored on a map. This class provides methods for adding, removing, and iterating nodes and edges. Remember that the addition or removal of the arcs, as well as their iteration, is virtual and mapped on the real storage, made up of Nodes and Links.

	class NetworkLevel {
		String id;
		Map<String, Node> N;

		NetworkLevel(String id) { /* ... */ }
		String getId() { /* ... */ }
		Node addNewNode(String id, Object data) { /* ... */ }
		Node addNewNode(String id) { /* ... */ }
		boolean addNewArc(Node fromNode, Node toNode, Object data) { /* ... */ } 
		boolean addNewArc(Node fromNode, Node toNode) { /* ... */ }
		boolean containsNode(Node n) { /* ... */ }
		boolean containsArc(Node fromNode, Node toNode) { /* ... */ } 
		Iterator<Node> getNodeIterator() { /* ... */ }
		ArcIterator getArcIterator() { /* ... */ }

		/* ... */
	}

Node class that defines a node. It stores an identifier, an associated datum, and a link map (i.e. outgoing arcs). It also contains all the methods to manipulate and retrieve this information, to iterate the outgoing arcs, and to generate a hashcode (useful for storing nodes in a HashMap).

	class Node {
		String id;
		Object data;
		Map<Node, Link> links;

		Node(String id, Object data, Link[] links) { /* ... */ } 
		Node(String id, Object data) { /* ... */ }
		Node(String id) { /* ... */ }
		
		Object getData() { /* ... */ }
		void setData(Object data) { /* ... */ }
		boolean addNeighbor(Node n) { /* ... */ }
		boolean addNeighbor(Node n, Object data) { /* ... */ }
		Link removeNeighbor(Node n) { /* ... */ }
		boolean hasNeighbor(Node n) { /* ... */ }
		Object getNeighborLinkData(Node n) { /* ... */ }
		boolean setNeighborLinkData(Node n, Object data) { /* ... */ } 
		Iterator<Link> getLinksIterator() { /* ... */ }
		Iterator<Node> getNeighborIterator() { /* ... */ }

		/* ... */
	}

Path class that defines a path. It stores to its internal list of files and sets the methods to retrieve and manipulate these nodes, to know the length of the path, to compare two paths and to generate a hashcode (useful for storing paths in a HashMap).

class Path implements Iterable<Node> {
	List<Node> nodes;
	Path(Node[] nodes) { /* ... */ }
	public Path(int initialCapacity) { /* ... */ } 
	public Path() { /* ... */ }
	public Path(Path path) { /* ... */ }
	Node getNodeAtPosition(int pos) { /* ... */ }
	void setNodeAtPosition(int pos, Node n) { /* ... */ }
	void appendNode(Node n) { /* ... */ }
	Node removeLastNode() { /* ... */ }
	boolean contains(Node n) { /* ... */ }
	int getLength() { /* ... */ }
	@Override Iterator<Node> iterator() { /* ... */ } 

	/* ... */
}

PathSet class that defines a set of paths, with the related methods for the manipulation and recovery of such paths, for their iteration and for quickly determining whether a path belongs to this set.

Operators

The pathpattern package defines all the classes useful for the construction of a path pattern and the construction of the relative automaton.

Automaton class defines an automaton related to a path pattern. Its constructor takes a path pattern as a parameter and has the task of generating the nodes and links of the automaton. It only keeps track of the initial and final node, since the other nodes can be recovered by navigating the automaton.

AutomatonLink class defines a link between one automata node and another. It also stores the action necessary to perform the transition.

AutomatonLinkActionType enum defines the type of action: epsilon transition, generic node or specific node.

AutomatonNode class defines an automaton node. It stores a list of AutomatonLinks to neighbors.

PathPattern abstract class contains all the specific classes of path patterns: EmptyPath, GraphNode, GenericNode, OptionalGenericNode, GenericPath, Concatenation, and Alternation.

Aggregation defines the method that implements the aggregation operator. It also defines the abstract class CFunction and the abstract class Assignements, useful for defining the callback functions necessary to call this operator.

Concatenation defines the method for the concatenation operator.

Join defines the methods that implement the join operator. Inside it is also defined the abstract DataJoin class, useful for defining the call-back function needed to call this operator.

Pathcondition defines the abstract class to build a predicate on a path, useful in a totally exclusive way for the selection operator. Within it, concrete path conditions have been defined such as: TruePredicate which always returns true, or SpecificLengthPredicate which efficiently defines the predicate that controls the length of a path.

Projection defines the methods that implement the projection operator. Inside it is also defined the abstract class Index, useful for defining the indices needed to call this operator. The concrete class ConstantIndex is also defined which realizes a constant index.

Selection defines the methods that implement the selection operator.

Synthetis defines the methods that implement the synthesis operator.

About

Implements the model, the algebra, and the concepts proposed for Multi Level Networks in my Master thesis.

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