
Graphs, Edges & Nodes Untangling
the social web.

What’s a graph?

Graph

Graph

Graph

Graph 10 19 9 7
2 15 7 3 12 13 9 6 6 4 3 5 7 4 14 1 4

Graph 11 10 10 19
6 9 7 2 15 7 21 3 8 12 15 13 13 17 9 22 6 6 3 4 4 3 2 5 7 4 6 14 9 12 1 10 4 19

Simple At most one edge
bet ween any pair of nodes.

Multigraph Multiple edges bet ween
vertices allowed.

Pseudograph
Selfloops are permitted.

G = (V, E)

What’s a node? vertex point
junction 0simplex

What’s an edge? arc branch
line link 1simplex

Directed

Undirected

Undirected

Visualizations

You are here.

(Graph does not include Justin
Bieber)

Social Graphs

Find the band that is
most often colistened with the given one.

People Find the band that
is most often colistened with the given one.

People Bands Find the band
that is most often colistened with the given one.

People Bands Find the band
that is most often colistened with the given one.

People Bands Find the band
that is most often colistened with the given one.

People Bands Find the band
that is most often colistened with the given one.

Basically, most kinds of simple
content/cooccurrence similarity.

That’s a 2step path on
a bipartite graph. There are many of these ‘fundamental’ graph units:  tripartite  folksonomies (tripartite 3graph + 2 step path)  multicolormultiparity graph  etc.

Graph Storage
Engines

Neo4j “An embedded, diskbased, fully
transactional Java persistence engine that stores data structured in graphs rather than in tables.” http://neo4j.org

HypergraphDB “A general purpose, extensible,
portable, distributed, embeddable, opensource data storage mechanism. It is a graph database designed speciﬁcally for artiﬁcial intelligence and semantic web projects.” http://kobrix.org/hgdb.jsp

Special Purpose
Storage Engines

FlockDB “FlockDB is a database
that stores graph data, but it isn't a database optimized for graphtraversal operations. Instead, it's optimized for very large adjacency lists, fast reads and writes, and pageable set arithmetic queries.” http://engineering.t witter.com/2010/05/introducing ﬂockdb.html

Redis “Redis is an advanced
keyvalue store. [...] the dataset is not volatile, and values can be strings, exactly like in memcached, but also lists, sets, and ordered sets. All this data types can be manipulated with atomic operations to push/pop elements, add/remove elements, perform server side union, intersection, difference bet ween sets, etc.” http://code.google.com/p/redis

A Redis Friends/
Followers Example

Redis makes you think in
terms of datastructures, and operations on those structures.

Set: Finite (for our cases)
collection of objects in which order has no signiﬁcance and multiplicity is generally ignored. S = { Alice, Bob, Carol } List: Finite (for our cases) collection of objects in which order *is* signiﬁcant and multiplicity is allowed. L = [ X, Y, X, Z, Q]

Insert a user into a
set SET uid:1000:username jperras SET uid:1000:password bazinga!

Use sets for denoting my
followers/people I follow. uid:1000:followers => Set of uids of all the followers users uid:1000:following => Set of uids of all the following users

Adding a new follower SADD
uid:1000:following 1001 SADD uid:1001:followers 1000

Posting Updates $r = Redis();
$postid = $r>incr("global:nextPostId"); $post = $User['id'] ."". time() ."". $status; $r>set("post:$postid", $post); $followers = $r>smembers("uid:".$User['id'].":followers"); if ($followers === false) $followers = Array(); $followers[] = $User['id']; /* Add the post to our own posts too */ foreach($followers as $fid) { $r>push("uid:$fid:posts", $postid, false); } # Push the post on the timeline, and trim the timeline to the # newest 1000 elements. $r>push("global:timeline", $postid, false); $r>ltrim("global:timeline",0,1000);

Common followers?  Set intersections!
SINTER users:1000:followers users:1000:followers

Let’s compare that
to MySQL

Can be Painful

Even More Pain

Relational databases can work for
the simplest of cases, but fail horribly at nearly all graphrelated operations/algorithms.

Graphs and graphdatabases are only
going to be more and more useful.

However, graph algorithms are hard.
So don’t write your own. And make sure you use a persistent storage engine that is best suited for the type of queries you will be performing.

Resources

Resources The Algorithm Design Manual,
Steve S. Skiena Programming Collective Intelligence, Toby Segaran Introduction to Algorithms, Cormen, Leiserson, Rivest

@jperras

Photo Credits Graph of the
internet, circa 2003: http://www.duniacyber.com/freebies/education/what isinternetlookslike/ (built from partial troll of public servers using traceroute) My real friends for letting me use their Facebook profile images.

References Large Scale Graph Algorithms
(class lectures), Yuri Lifshits, Steklov Institute of Mathematics at St. Petersburg http://mathworld.wolfram.com/Set.html Programming Collective Intelligence, Toby Segaran The Algorithm Design Manual, Steve S. Skiena
Many of the most popular web applications today deal with highly organized and structured data that represent entities, and the relationships between these entities. LinkedIn can tell you how many degrees of separation there are between yourself and the CEO of Samsung, Facebook can figure out people that you might already know, Digg can recommend article submissions that you might like, and LastFM suggests music based on your current listening habits.
We&#x2019;ll take a look at the basic theory behind how some of these features can be implemented (no computer science degree required!), and take a quick look at the current landscape of graphbased datastores that simplify many of these operations.
Start with some definitions.
Collection of points  e.g. Users (Twitter/Facebook), songs (iTunes)
Add relationships between data points
Some relations are not symmetric  e.g. `friend` vs. following/follower is asymmetric.
Your relationships might have a weight  e.g. # of Scrabulous games they have played together.
Data points can also have weight  e.g. `reputation` score on social news sites like Digg, Reddit.
Simple graph  at most one edge between vertex pair.
Simple graph  at most one edge between vertex pair.
Selfloops are allowed.
e.g. if your application needs the ability for you to be your own &#x2018;follower&#x2019; or &#x2018;friend&#x2019;.
Notation that you might see  G is the &#x2018;name&#x2019; of the graph, and is composed of &#x2018;V&#x2019; vertices (nodes) and &#x2018;E&#x2019; edges.
"Vertex" is a synonym for a node of a graph, i.e., one of the points on which the graph is defined and which may be connected by graph edges.
An ordered (or unordered) pair of nodes.
Different types of edges: directed.
In geometry, a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension.
Specifically, an nsimplex is an ndimensional polytope with n&#xA0;+&#xA0;1 vertices, of which the simplex is the convex hull. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron, and a 4simplex is apentachoron.
A single point may be considered a 0simplex, and a line segment may be viewed as a 1simplex.
A simplex may be defined as the smallest convex set which contains the given vertices.
The edge is an ordered pair of nodes.
The terms "arc", "branch" "line", "link" and "1simplex" are sometimes used instead of edge
Edge highlight on next slide.
an unordered pair of nodes that specify a line joining these two nodes are said to form an edge
an unordered pair of nodes that specify a line joining these two nodes are said to form an edge
Partial map of the internet, culled in 2003 using traceroute.
Graph visualizations have also become quite important  displaying information on billions of points and edges
in a useful manner is quite difficult.
The graph is projected inside a 3D sphere using a special kind of space based hyperbolic geometry. This is a nonEuclidean space, which has useful distorting properties of making elements at the center of the display much larger than those on the periphery.
Hyperbolic space projection is commonly know as &#x201C;focus+context&#x201D; in the field of information visualization and has been used to display all kinds of data that can be represented as large graphs in either two and three dimensions.
This is a graph representation of the similarity relationships derived from the database of Last.fm. The circles (vertices) on the left hand side figure are bands, musicians, composers, whatever you will find in theMusic section of the site. Lines (edges) connect similar artists.&#xA0;&#xA0;Vertex sizes vary according to the popularity of the artists. I Vertex colors correspond to musical genres, identified by tags attached to the artists by the users of Last.fm
You are already a part of and use several social graphs.
Twitter is one giant graph (users, followers, following) + timeline attached to users
Linkedin is another giant graph. It&#x2019;s basically in their name!
Me
I&#x2019;m the center of the world.
Relationships with my friends
My friends also have relationships between themselves
Let&#x2019;s get rid of the pictures for a second
My friends also have friends, and those friends can be friends with my other immediate friends.
Important problems: maxintersection + strongest connection problem.
From Twitter  solves their problems
Sets are great when the order of your data doesn&#x2019;t matter, and when you know that the objects need to be unique. Example: USERS
Lists are best for things that need to be displayed in a given order, e.g. POST TIMELINE
Sets are great when the order of your data doesn&#x2019;t matter, and when you know that the objects need to be unique. Example: USERS
Lists are best for things that need to be displayed in a given order, e.g. POST TIMELINE