1. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland CONTROLLING EPIDEMICS IN WIRELESS NETWORKS Ranjan Pal 1 Ayan Nandy 2 Satya Ardhy Wardana 3 Neeli Rashmi Prasad 3 Ramjee Prasad 3 1 University of Southern California, USA 2 Indian Institute of Technology, Kharagpur, India 3 Center for TeleInfrastruktur (CTIF), Aalborg University, Denmark

2. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland 2 OUTLINE Introduction Probem Definition Network Model Strategy Selection Algorithm Conclusion

3. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland INTRODUCTION Modern day wireless communications is witnessing the emergence of various viruses that can spread over air interfaces. Several application scenarios in multi-hop wireless networks require many nodes to cooperate on a single application.  Wireless Sensor Network  Social Network Viruses can spread over the air quickly from device to device and devastate the entire network in a short period of time  causing an EPIDEMIC 3

4. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland PROBLEM DEFINITION We propose a novel method using spectral properties of graphs to select the best suited epidemic control strategy for a wireless network from a pool of strategies. Our methodology will provide a general framework for good, structured, centralized strategy selection for epidemic control in static multihop wireless networks. 4

5. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland NETWORK MODEL Multi-hop network Undirected Single channel, single radio per node Network represented by a graph G = (V,E) Links (u,v) and (v,u) are different V represents the set of nodes and E is the set of wireless data links 5 uv 5

6. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland EXAMPLE OF LAPLACIAN MATRIX OF A GRAPH 6 Spectral function of the network graph the eigenvalues of the Laplacian of a graph  Centrality of its vertices

7. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland NETWORK MODEL We term this graph as a parent graph. Spread Graph - Initial configuration : a group of independent nodes in 2-D space. - a compromised node spreads its virus to its neighboring nodes after k time units of it being infected.  to model the spreading of the virus (evolution) - Monitor the spread graph at certain pre-specified time intervals, then apply our control strategy to the graph at each interval. We assume that it is not possible to monitor the network all the time 7

8. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland STRATEGY SELECTION ALGORITHM The spectrum of a parent graph G is the set of eigenvalues of its Laplacian matrix L. A spectral function F of a parent graph G is formed, taking as arguments of the eigenvalues of its Laplacian. We define our spectral function as the average of all the igenvalues. The main aim is to be able to decide the best epidemic control strategy from a pool of strategies for a given parent graph. We classify parent subgraphs into 2 categories (predefined threshold): 1) graphs with low values of F (below the threshold) 2) graphs with high values of F (above the threshold) 8

9. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland STRATEGY SELECTION ALGORITHM We observe its spread graph over time and apply each of our proposed strategies at certain pre-specified time intervals. Doing so over, a total inspection period of T will result in ranking the strategies in order of effectiveness of epidemic check. We then try to see whether there is any strategy that suits best for all the parent graphs in the set with high values of F. Likewise, we try to find strategies which suit best for all parent graphs in the set with low values of F. 9

10. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland 10 CONCLUSION We have proposed a spectral graph-theoretic selection algorithm that aims at selecting the best strategy from a pool of strategies, to prevent an epidemic outbreak in a given parent network. As part of ongoing work, we are performing a detailed simulation study based on our algorithm to come up with a general strategy classification scheme.

11. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland 11 THANK YOU

12. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland STRATEGIES ALGORITHM Strategy 1: Treatment of edges whose end vertices result in maximum sum of degrees. Strategy 2: Treatment of edges whose end vertices result in maximum product of degrees. Strategy 3: Treatment of vertices whose degree is the Maximum. Strategy 4: Treatment of edges whose end vertices result in maximum harmonic product of degrees 12

13. Workshop on Applications of Wireless Communications (WAWC 2008) 21 August 2008, Lappeenranta - Finland STRATEGIES ALGORITHM A. Functional Properties One way to comment on the spread of virus in an evolving spread network is to observe the values of its functional properties over time. The most common functional property used is characteristic path length [5]. In this paper we use a different functional property called harmonic path length, to track the spread of virus. We define harmonic path length (HPL) of any graph G as the median of the harmonic means of the shortest path lengths connecting each vertex in G to all other vertices. If a vertex in a graph is isolated, it’s distance from any other vertex in the graph is infinite. The characteristic path length (CPL) with respect to a vertex is the average of the shortest distance from that vertex to every other vertex in the graph. The CPL of the graph is the median of CPL value over all the vertices. Now, even if one vertex is isolated, the CPL for each vertex will be infinite. So, the CPL of the graph will be infinite. In this paper, we will be mostly dealing with ’evolving’ spread graphs which are generally sparse/disconnected in nature. To compare between two different graphs, each having disjoint subgraphs, we need a different parameter. That is why we use harmonic path length which comes up with a finite value if at least half the vertices of the graph are not isolated, else the value is infinite. More than one graph might have an infinite value. To distinguish between them, the time when infinity is reached for each, is an important parameter. The next subsection elaborates this point. B. Suitability of Strategies Functional properties determine the suitability of a particular strategy. For a particular strategy, an infinite value of HPL at a time instant indicates that more than half of a spread network is isolated. This means that the virus has not been able to substantially percolate in the network. Let Gst be the spread graph under strategy s at time instant t. If the value of HPL of Gst at t is infinity and remains so for the interval [t; T], where T is a large enough total inspection period, the strategy s under question is said to be successful in epidemic check. Lesser the value of t, better is the strategy. If t > T then the strategy under question does not cause the cessation of virus spread and is considered not that effective. In Figure 1, we show a typical variation of HPL with time in an example spread network. The value 200 on the y-axis represents infinity. The spread of virus is in full effect between time instants 35 and 175, after which there begins an improvement in the situation. The virus spread decreases considerably after time instant 200. 13