Distinguishing Infections on Different Graph Topologies

The history of infections and epidemics holds famous examples where understanding, containing, and ultimately treating an outbreak began with understanding its mode of spread. Influenza, HIV, and most computer viruses spread person to person, device to device, and through contact networks; Cholera, Cancer, and seasonal allergies, on the other hand, do not. In this paper, we study two fundamental questions of detection. First, given a snapshot view of a (perhaps vanishingly small) fraction of those infected, under what conditions is an epidemic spreading via contact (e.g., Influenza), distinguishable from a random illness operating independently of any contact network (e.g., seasonal allergies)? Second, if we do have an epidemic, under what conditions is it possible to determine which network of interactions is the main cause of the spread-the causative network-without any knowledge of the epidemic, other than the identity of a minuscule subsample of infected nodes? The core, therefore, of this paper, is to obtain an understanding of the diagnostic power of network information. We derive sufficient conditions that networks must satisfy for these problems to be identifiable, and produce efficient, highly scalable algorithms that solve these problems. We show that the identifiability condition we give is fairly mild, and in particular, is satisfied by two common graph topologies: the d-dimensional grid, and the Erdös-Renyi graphs.

[1]  P. O’Neill,et al.  Bayesian inference for epidemics with two levels of mixing , 2005 .

[2]  R. Durrett Random Graph Dynamics: References , 2006 .

[3]  D. R. Grey Asymptotic behaviour of continuous time, continuous state-space branching processes , 1974 .

[4]  Gavin J. Gibson,et al.  Statistical inference for stochastic epidemic models , 2002 .

[5]  E. Candès,et al.  Detection of an anomalous cluster in a network , 2010, 1001.3209.

[6]  V. Blondel,et al.  Distance distribution in random graphs and application to network exploration. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Donald F. Towsley,et al.  The effect of network topology on the spread of epidemics , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[8]  Shie Mannor,et al.  Network forensics: random infection vs spreading epidemic , 2012, SIGMETRICS '12.

[9]  Russell Lyons,et al.  Random walk in a random environment and rst-passage percolation on trees , 1992 .

[10]  Frank Ball,et al.  Poisson approximations for epidemics with two levels of mixing , 2004 .

[11]  Devavrat Shah,et al.  Rumors in a Network: Who's the Culprit? , 2009, IEEE Transactions on Information Theory.

[12]  Aditya Gopalan,et al.  Random mobility and the spread of infection , 2011, 2011 Proceedings IEEE INFOCOM.

[13]  Vijaya Ramachandran,et al.  The diameter of sparse random graphs , 2007, Random Struct. Algorithms.

[14]  Jon Cohen,et al.  Making Headway Under Hellacious Circumstances , 2006, Science.

[15]  Yuval Peres,et al.  Tree-indexed random walks on groups and first passage percolation , 1994 .

[16]  Chak-Kuen Wong,et al.  A faster approximation algorithm for the Steiner problem in graphs , 1986, Acta Informatica.

[17]  Shie Mannor,et al.  On identifying the causative network of an epidemic , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[18]  Devavrat Shah,et al.  Detecting sources of computer viruses in networks: theory and experiment , 2010, SIGMETRICS '10.

[19]  E. Candès,et al.  Searching for a trail of evidence in a maze , 2007, math/0701668.

[20]  P. O’Neill,et al.  Bayesian inference for stochastic multitype epidemics in structured populations via random graphs , 2005 .

[21]  H. Kesten On the Speed of Convergence in First-Passage Percolation , 1993 .