Inferring Social Networks from Outbreaks

We consider the problem of inferring the most likely social network given connectivity constraints imposed by observations of outbreaks within the network. Given a set of vertices (or agents) V and constraints (or observations) Si ⊆ V we seek to find a minimum log-likelihood cost (or maximum likelihood) set of edges (or connections) E such that each Si induces a connected subgraph of (V, E). For the offline version of the problem, we prove an Ω(log(n)) hardness of approximation result for uniform cost networks and give an algorithm that almost matches this bound, even for arbitrary costs. Then we consider the online problem, where the constraints are satisfied as they arrive. We give an O(n log(n))-competitive algorithm for the arbitrary cost online problem, which has an Ω(n)-competitive lower bound.We look at the uniform cost case as well and give an O(n2/3 log2/3(n))-competitive algorithm against an oblivious adversary, as well as an Ω(√n)-competitive lower bound against an adaptive adversary. We examine cases when the underlying network graph is known to be a star or a path, and prove matching upper and lower bounds of Θ(log(n)) on the competitive ratio for them.

[1]  Noga Alon,et al.  An optimal procedure for gap closing in whole genome shotgun sequencing , 2001, RECOMB.

[2]  M. Stern,et al.  The clustering matroid and the optimal clustering tree , 2003, Math. Program..

[3]  Michal Stern,et al.  The complete optimal stars-clustering-tree problem , 2008, Discret. Appl. Math..

[4]  Noga Alon,et al.  The online set cover problem , 2003, STOC '03.

[5]  Tatsuya Akutsu,et al.  Completing Networks Using Observed Data , 2009, ALT.

[6]  Vladimir Grebinski,et al.  Reconstructing a Hamiltonian Cycle by Querying the Graph: Application to DNA Physical Mapping , 1998, Discret. Appl. Math..

[7]  Dana Angluin,et al.  Learning a Hidden Graph Using O(log n) Queries Per Edge , 2004, COLT.

[8]  R. Ravi,et al.  Online and stochastic survivable network design , 2009, STOC '09.

[9]  Noga Alon,et al.  Learning a Hidden Matching , 2004, SIAM J. Comput..

[10]  James Aspnes,et al.  Optimally learning social networks with activations and suppressions , 2010, Theor. Comput. Sci..

[11]  Laurence A. Wolsey,et al.  An analysis of the greedy algorithm for the submodular set covering problem , 1982, Comb..

[12]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[13]  Dana Angluin,et al.  Learning a hidden graph using O(logn) queries per edge , 2008, J. Comput. Syst. Sci..

[14]  Noga Alon,et al.  Learning a Hidden Subgraph , 2004, SIAM J. Discret. Math..

[15]  Nikhil Srivastava,et al.  Learning and Verifying Graphs Using Queries with a Focus on Edge Counting , 2007, ALT.

[16]  Noga Alon,et al.  A general approach to online network optimization problems , 2004, SODA '04.

[17]  Joseph Naor,et al.  The Design of Competitive Online Algorithms via a Primal-Dual Approach , 2009, Found. Trends Theor. Comput. Sci..

[18]  B. Bollobás The evolution of random graphs , 1984 .

[19]  P. Erdos,et al.  On the evolution of random graphs , 1984 .