There are also a number of network analysis packages for the open source R language, such as network and sna, and igraph, all of which have some viz capabilities. R can also be a good environment for general data manipulation tasks.
There are a number of packages in the R language very useful to data analysis/visualization. Hadley Wickham has developed lot of interesting tools to make these task easier. The recent bigvis package is very promising.
It has a loot of tools for building a visualization that lets you explore / drill into the data. It can let you build a sort of interactive experience that brings a lot more meaning to your graphs and charts.
The public version is free, student / non-profit pricing is fairly reasonable if I recall.
Analytics: For accelerated computing, cuGraph (GPU-accelerated Python) as part of the broader RAPIDS Python GPU ecosystem. For many diverse algorithms, Neo4j has great extensions, and for potentially more scalable experiences, TigerGraph and GraphX. Graph neural nets are emerging as an exciting area but no clear best practice yet.
Visualization: Graphistry (bias as the founder ;-)) is the only end-to-end GPU-accelerated visual graph analysis tool and runs in browsers even 5+ years old. Likewise, for use by analytics teams, it is also gaining popularity due to Python notebook support and built-in visual analytics (1K+ GitHub stars). D3/Linkurious/KeyLines/etc. are good for small graphs, with D3 being free. However, for bigger ones, all of these are still much slower than Graphistry from 5 years ago, meaning you or your user's browser will crash on basic tasks like event data, and have little for already built-in visual analytics similar to desktop Gephi or beyond