Interactive Graph Analysis and Visualization

Best Graph Library :)

• SocialCompare has an extensive list of libraries, and the "Node / edge graph" line will filter for graph visualization ones.
• DataVisualization.ch has evaluated many libraries, including node/graph ones. Unfortunately there's no direct link so you'll have to filter for "graph":
  

Here's a list of similar projects (some have been already mentioned here):

Pure JavaScript Libraries

• vis.js supports many types of network/edge graphs, plus timelines and 2D/3D charts. Auto-layout, auto-clustering, springy physics engine, mobile-friendly, keyboard navigation, hierarchical layout, animation etc. MIT licensed and developed by a Dutch firm specializing in research on self-organizing networks.
• Cytoscape.js - interactive graph analysis and visualization with mobile support, following jQuery conventions. Funded via NIH grants and developed by by @maxkfranz (see his answer below) with help from several universities and other organizations.
• The JavaScript InfoVis Toolkit - Jit, an interactive, multi-purpose graph drawing and layout framework. See for example the Hyperbolic Tree. Built by Twitter dataviz architect Nicolas Garcia Belmonte and bought by Sencha in 2010.
• D3.js Powerful multi-purpose JS visualization library, the successor of Protovis. See the force-directed graph example, and other graph examples in the gallery.
• Plotly's JS visualization library uses D3.js with JS, Python, R, and MATLAB bindings. See a nexworkx example in IPython here, human interaction example here, and JS Embed API.
• sigma.js Lightweight but powerful library for drawing graphs
• jsPlumb jQuery plug-in for creating interactive connected graphs
• Springy - a force-directed graph layout algorithm
• Processing.js Javascript port of the Processing library by John Resig
• JS Graph It - drag'n'drop boxes connected by straight lines. Minimal auto-layout of the lines.
• RaphaelJS's Graffle - interactive graph example of a generic multi-purpose vector drawing library. RaphaelJS can't layout nodes automatically; you'll need another library for that.
• JointJS Core - David Durman's MPL-licensed open source diagramming library. It can be used to create either static diagrams or fully interactive diagramming tools and application builders. Works in browsers supporting SVG. Layout algorithms not-included in the core package
• HighCharts - Highcharts is a charting library written in pure JavaScript. it is based on SVG, but unlike many others, it also supports older browsers who use VML to render vector graphic. It is licenced under CC BY-NC 3.0

Commercial libraries

• GoJS Interactive graph drawing and layout library
• yFiles for HTML Commercial graph drawing and layout library
• mxGraph Commercial HTML 5 diagramming library with support for older browsers
• KeyLines Commercial JS network visualization toolkit
• data visualization software lab Commercial multi-purpose visualization library

Abandoned libraries

• Cytoscape Web Embeddable JS Network viewer (no new features planned; succeeded by Cytoscape.js)
• Canviz JS renderer for Graphviz graphs. Abandoned in Sep 2013.
• arbor.js Sophisticated graphing with nice physics and eye-candy. Abandoned in May 2012. Several semi-maintained forks exist.
• jssvggraph "The simplest possible force directed graph layout algorithm implemented as a Javascript library that uses SVG objects". Abandoned in 2012.
• jsdot Client side graph drawing application. Abandoned in 2011.
• Protovis Graphical Toolkit for Visualization (JavaScript). Replaced by d3.
• Moo Wheel Interactive JS representation for connections and relations (2008)
• JSViz 2007-era graph visualization script
• dagre Graph layout for JavaScript

Non-Javascript Libraries

• Graphviz Sophisticated graph visualization language
• Flare Beautiful and powerful Flash based graph drawing
• NodeBox Python Graph Visualization