Network Optimization: Continuous and Discrete Models

We define a connection subgraph as a small subgraph of a large graph that best captures the relationship between two nodes. The primary motivation for this work is to provide a paradigm for exploration and knowledge discovery in large social networks graphs. We present a formal definition of this problem, and an ideal solution based on electricity analogues. We then show how to accelerate the computations, to produce approximate, but high-quality connection subgraphs in real time on very large (disk resident) graphs.We describe our operational prototype, and we demonstrate results on a social network graph derived from the World Wide Web. Our graph contains 15 million nodes and 96 million edges, and our system still produces quality responses within seconds.

Fast discovery of connection subgraphs

In the last four decades, combinatorial optimization has been strongly influenced by linear programming. With the mathematical and algorithmic understanding of linear programs came a whole host of ideas and tools that were then applied to combinatorial optimization. Many of these ideas and tools are still in use today, and form the bedrock of our understanding of combinatorial optimization. One of these tools is the primal-dual method. It was proposed by Dantzig, Ford, and Fulkerson [DFF56] as another means of solving linear programs. Ironically, their inspiration came from combinatorial optimization. In the early 1930s, Egervary [Ege31] proved

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The primal-dual method for approximation algorithms and its application to network design problems

We present a new algorithm which creates redundant trees on arbitrary node-redundant or link-redundant networks. These trees are such that any node is connected to the common root of the trees by at least one of the trees in case of node or link failure. Our scheme provides rapid preplanned recovery of communications with great flexibility in the topology design. Unlike previous algorithms, our algorithm can establish two redundant trees in the case of a node failing in the network. In the case of failure of a communications link, our algorithm provides a superset of the previously known trees.

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Redundant trees for preplanned recovery in arbitrary vertex-redundant or edge-redundant graphs

Given Q nodes in a social network (say, authorship network), how can we find the node/author that is the center-piece, and has direct or indirect connections to all, or most of them? For example, this node could be the common advisor, or someone who started the research area that the Q nodes belong to. Isomorphic scenarios appear in law enforcement (find the master-mind criminal, connected to all current suspects), gene regulatory networks (find the protein that participates in pathways with all or most of the given Q proteins), viral marketing and many more.Connection subgraphs is an important first step, handling the case of Q=2 query nodes. Then, the connection subgraph algorithm finds the b intermediate nodes, that provide a good connection between the two original query nodes.Here we generalize the challenge in multiple dimensions: First, we allow more than two query nodes. Second, we allow a whole family of queries, ranging from 'OR' to 'AND', with 'softAND' in-between. Finally, we design and compare a fast approximation, and study the quality/speed trade-off.We also present experiments on the DBLP dataset. The experiments confirm that our proposed method naturally deals with multi-source queries and that the resulting subgraphs agree with our intuition. Wall-clock timing results on the DBLP dataset show that our proposed approximation achieve good accuracy for about 6:1 speedup.

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Center-piece subgraphs: problem definition and fast solutions

Motivation.- Network survivability models using node types.- Survivable network design under connectivity constraints - a survey.- Decomposition.- Basic inequalities.- Lifting theorems.- Partition inequalities.- Node partition inequalities.- Lifted r-cover inequalities.- Comb inequalities.- How to find valid inequalities.- Implementation of the cutting plane algorithm.- Computational results.

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Mechthild Stoer

Papers

Design of Survivable Networks