## The effect of network topology on the spread of epidemics (2005)

### Cached

### Download Links

- [research.microsoft.com]
- [www.thlab.net]
- [www.maths.bris.ac.uk]
- DBLP

### Other Repositories/Bibliography

Venue: | IN IEEE INFOCOM |

Citations: | 131 - 8 self |

### BibTeX

@INPROCEEDINGS{Ganesh05theeffect,

author = {A. Ganesh and L. Massoulié and D. Towsley},

title = {The effect of network topology on the spread of epidemics},

booktitle = {IN IEEE INFOCOM},

year = {2005},

pages = {1455--1466},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many network phenomena are well modeled as spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses, and, more generally, faults. Many types of information dissemination can also be modeled as spreads of epidemics. In this paper we address the question of what makes an epidemic either weak or potent. More precisely, we identify topological properties of the graph that determine the persistence of epidemics. In particular, we show that if the ratio of cure to infection rates is smaller than the spectral radius of the graph, then the mean epidemic lifetime is of order log n, where n is the number of nodes. Conversely, if this ratio is bigger than a generalization of the isoperimetric constant of the graph, then the mean epidemic lifetime is of order � Ò�, for a positive constant �. We apply these results to several network topologies including the hypercube, which is a representative connectivity graph for a distributed hash table, the complete graph, which is an important connectivity graph for BGP, and the power law graph, of which the AS-level Internet graph is a prime example. We also study the star topology and the Erdős-Rényi graph as their epidemic spreading behaviors determine the spreading behavior of power law graphs.

### Citations

1923 | Randomized Algorithms - Motwani, Raghavan - 1995 |

1856 | Pastry: Scalable, distributed object location and routing for large-scale peer-to-peer systems
- Rowstron, Druschel
- 2001
(Show Context)
Citation Context ...emics do not die out quickly, even as Ò � . B. Hypercubes The hypercube is of interest because of the widespread and growing interest in distributed hash tables and applications, such as file sharing =-=[17]-=-, being built on top of them. Already worms and viruses have appeared in some applications, [11]. As many DHT structures are hypercubic in nature, it is important to understand the spreading behavior ... |

1340 | On Power-law Relationships of the Internet Topology
- Faloutsos, Faloutsos, et al.
- 1999
(Show Context)
Citation Context ...rarily close to zero. E. Power law graphs There has been considerable interest in power law graphs since it was first noticed that the Internet ASlevel graph exhibits a power law degree distribution, =-=[9]-=-. Briefly a power law graph is one where the number of nodes with degree � is proportional to �s for some � . For the mean degree to be finite, we need � and this is the range we shall consider. Th... |

365 |
Markov chains: Gibbs fields, Monte Carlo simulation, and queues
- Bremaud
- 1998
(Show Context)
Citation Context ...true: the probability that they have not died out by time Ø will decay exponentially with Ø. This fact follows from standard theory of Markov processes with absorbing states, reviewed for instance in =-=[4]-=-. The question of interest is then: how quickly do the epidemics die out, or how quickly does the system recover from the epidemic? More precisely, define � to be the time until the epidemic dies out ... |

307 | Epidemic spreading in scale-free networks
- Pastor-Satorras, Vespignani
(Show Context)
Citation Context ...aspects of the problem studied here. Much of the work has focused on infinite scale-free graphs, establishing conditions under which epidemics either die out or sustain themselves forever; see, e.g., =-=[15]-=-. Some work has been done on finite graphs, using primarily heuristic arguments to obtain conditions under which epidemics spread quickly or not. For example, [16] uses a mean field approximation for ... |

203 | The average distance in a random graph with given expected degrees
- Chung, Lu
- 2003
(Show Context)
Citation Context ...ts in either the maximum degree node or one of its neighbors by the time to die out on a star of size Ñ .Now consider the case that some other randomly chosen node is initially infected. Theorem 4 in =-=[5]-=- states that, with high probability, the diameter of the power law graph is ¢ ÐÓ� Ò when � . Hence the probability that an epidemic starting from a randomly infected node spreads to either the maxim... |

103 |
Optimal assignments of numbers to vertices
- Harper
- 1964
(Show Context)
Citation Context ...isfies identity (6), with our current choices for Ö and Ñ. The result follows by a direct application of Theorem 4.1. In order to prove the theorem, we need the following result established by Harper =-=[10]-=- (see also [1] for background and recent extensions). Lemma 5.4: (Harper [10]) Let Ë be a set of Ñ vertices of the hypercube � � � � . Then the edge-boundary size � Ë� Ë is larger than or equal to � Ë... |

98 | Some applications of Laplace eigenvalues of graphs, in
- Mohar
- 1997
(Show Context)
Citation Context ...ote the eigenvalues of Ä by � Ä � � Ä �¡¡¡��Ò Ä . We now have: Corollary 4.2: Let Ö �� ¬� Ä � � (8) Then (6) is still valid, with Ö replaced by Ö and Ñ � �Ò� . The proof follows from Corollary 3.8 in =-=[14]-=-, which states that for any graph �, the following inequality holds: � � � � Ä ¡ Corollary 4.3: Assume that the graph � is regular, i.e. all its vertices have the same degree, say �. Denote by � � � �... |

91 | Epidemic spreading in real networks: An eigenvalue viewpoint
- Wang, Chakrabarti, et al.
(Show Context)
Citation Context ...y (corresponding to fast die out)and others where they go through long periods of repeated failures (corresponding to slow die out). We treat this in a rigorous manner in Section V. Last, Wang et al. =-=[18]-=- show, through simulation and approximate analysis, that the condition for fast die out relates to the spectral radius of the adjacency matrix of the underlying graph. Our model is an example of what ... |

79 |
Random Graphs, 2nd edition
- Bollobás
- 2001
(Show Context)
Citation Context ...ent of all other edges. The spreading behavior of an epidemic on an Erdős-Rényi graph is of interest for a number of reasons. First, it is a graph that has received considerable attention in the past =-=[3]-=-. Second, it is an important component of the class of power law random graphs that model the Internet AS graph. Thus if we are to understand the robustness of the Internet AS-level graph, we need to ... |

58 |
Directed-graph Epidemiological Models of Computer Viruses
- Kephat, White
- 1991
(Show Context)
Citation Context ...nts to obtain conditions under which epidemics spread quickly or not. For example, [16] uses a mean field approximation for establishing conditions for quick and slow die out in scale-free graphs and =-=[12]-=- provides an approximate analysis for the case of an Erdős-Rényi graph. We rigorously establish similar conditions in Section V. An exception is the work of Durrett and Liu [8], which presents a rigor... |

47 | Eigenvalues of random power law graphs
- Chung, Lu, et al.
(Show Context)
Citation Context ... be finite, we need � and this is the range we shall consider. The Internet ASlevel graph is characterized by � � In this section we consider a class of random power law graphs first introduced in =-=[6]-=-. Let Û �Û �����ÛÒ denote the expected degrees of the nodes in the graph. An edge is assigned to a pair of vertices with probability Û�Û�� È Ò �� Û�. Let � denote the average degree and Ñ the maximum ... |

15 |
Network resilience: Exploring cascading failures within bgp
- Jr, Ge, et al.
- 2002
(Show Context)
Citation Context ...ork of Durrett and Liu [8], which presents a rigorous analysis leading to conditions for fast and slow die out on a finite one dimensional linear network. Also relevant is the work of Coffman et al., =-=[7]-=-, which models cascading BGP failures on a fully connected topology. They identify regimes where BGP routers recover quickly (corresponding to fast die out)and others where they go through long period... |

14 |
The contact process on a finite set
- Durrett, Schonmann
- 1988
(Show Context)
Citation Context ... scale-free graphs and [12] provides an approximate analysis for the case of an Erdős-Rényi graph. We rigorously establish similar conditions in Section V. An exception is the work of Durrett and Liu =-=[8]-=-, which presents a rigorous analysis leading to conditions for fast and slow die out on a finite one dimensional linear network. Also relevant is the work of Coffman et al., [7], which models cascadin... |

7 |
Epidemic dynamics in finite scale-free networks
- Pastor-Satorras, Vespignani
(Show Context)
Citation Context ...stain themselves forever; see, e.g., [15]. Some work has been done on finite graphs, using primarily heuristic arguments to obtain conditions under which epidemics spread quickly or not. For example, =-=[16]-=- uses a mean field approximation for establishing conditions for quick and slow die out in scale-free graphs and [12] provides an approximate analysis for the case of an Erdős-Rényi graph. We rigorous... |

6 | Stochastic multitype epidemics in a community of households: estimation and form of optimal vaccination schemes
- Ball, Britton, et al.
(Show Context)
Citation Context ...at is termed an SIS (Susceptible-Infective-Susceptible) model in the epidemic literature. Related work has been concerned with the so-called SIR (Susceptible-Infective-Removed) model; see Ball et al. =-=[2]-=- for recent references as well as a study of vaccination strategies in that context. The paper is structured as follows. We introduce the epidemic spreading model in Section II. Sufficient conditions ... |

3 | Extremal sets minimizing dimension{normalized boundary in Hamming graphs
- Azizoglu
- 2000
(Show Context)
Citation Context ... (6), with our current choices for Ö and Ñ. The result follows by a direct application of Theorem 4.1. In order to prove the theorem, we need the following result established by Harper [10] (see also =-=[1]-=- for background and recent extensions). Lemma 5.4: (Harper [10]) Let Ë be a set of Ñ vertices of the hypercube � � � � . Then the edge-boundary size � Ë� Ë is larger than or equal to � Ë £ � Ë £ , whe... |