# Path-Based Epidemic Spreading in Networks

## Summary (3 min read)

### Introduction

- Motivated by these observations, the authors model path-based information spreading by advancing the state of the art of epidemic theory to account for the directional effect caused by the paths constructed by different routing protocols as well as by the role played by the infectious agent1 as the “infection carrier” that spreads the epidemic.
- The authors focus on communication networks that transport data traffic via paths/routes.
- The authors pathbased spreading analytical framework is described next in Section III-C.

### A. General Path-based Spreading Process

- The authors use the Markov chain diagrams of SI, SIS and SIR models in Fig. 2 to illustrate the key departure of path-based spreading process from the conventional contact-based one.
- Note that a direct transition from state SIS to state III is not possible for a line graph because the flow of packets and thus infection is directional.
- An infected node in the middle can only infect nodes either to its left or right at any time.
- Therefore, it is already obvious that while the topology still plays a role in influencing the spread of the infection, it is the routing protocol that finally governs the actual infection dynamics.
- Packets traverse from one node to another via a path with certain traffic arrival rate, λ.

### B. Modelling the Infectious Agent

- For their purpose, the authors depart from the conventional routing matrix that maps traffic to links (e.g., [44]) and instead, encode the traffic to nodes it traverses or is destined to.
- Rn describe the node involvement in delivering the traffic originating from node n.
- By applying Markov theory, the infinitesimal generator Qn(t) of this two-state continuous Markov chain can be written as below: Qn(t) = [ −q1;n q1;n q2;n −q2;n ] (10) where the transitions involving the curing process are independent of the states of other nodes and thus, q2;n = δ (See Section V for discussion.).
- The instantaneous fraction of infected nodes in the network can then be written as 6The implications of this approximation are discussed in [20].
- Furthermore, Fig. 5 shows the steady state for sample networks of different sizes obtained both from their model and simulation runs.

### IV. PATH-BASED EPIDEMIC THRESHOLD

- As briefly mentioned in Section II, in previous studies (e.g., [18]), a theoretical critical threshold, τc, has been found below which the epidemic will almost certainly die off and vice versa.
- From Eq. 26, the authors note that the critical threshold is independent of the initial state of the system.
- The authors begin their comparison of the two types of epidemic spreading based on their corresponding exact 2N -state Markov chain of the network states.
- Therefore, QpathU4 will have more non-zero elements than QcontactU4 and thus, Q path is denser than Qcontact.
- For such matrices, the largest eigenvalue in magnitude is a real number and for this eigenvalue, the real part is the largest amongst all the eigenvalues [51].

### VI. EFFECT OF ROUTING PROTOCOL, TRAFFIC AND NETWORK TOPOLOGY

- The authors investigate the role of the network topology (i.e., A), the routing protocol (i.e., R) and the traffic load in the system (i.e., Γ) in determining the path-based spreading of an epidemic (i.e., τc).
- Essentially, changing the link weight distribution results in a different set of shortest paths for the same graphs and thus a different disorder limit is achieved.
- When the network topology, A, and traffic load, Γ, are known, then τc = τ max c when R = R ∪SPT where R∪SPT is the routing matrix of the shortest paths by hop count between all possible node pairs in A. Proof.
- This implies that the total node involvement for delivering infectious agent in the network is inflated since by not using the shortest paths, more nodes are involved in delivering the same amount of infectious agents (i.e., ρ∪SPT ≤ ρ∗).
- This theorem is especially useful for cases where the network topology can be flexibly constructed either following certain requirements (e.g., data center networks) or specific rules/guidelines (e.g., self-organizing wireless sensor networks).

### VII. CASE STUDIES

- The authors apply their path-based epidemic analytical framework to three real world Tier-1 networks (i.e., Level3 (AS1), Sprint (AS1239) and AT&T (AS7018) at point-ofpresence (POP)-level based on the data from [21]) to investigate how conducive they are regarding path-based epidemic spreading.
- Two sets of values are computed for parameters related to paths, (1) non-weighted and (2) weighted links.
- For a contact-based epidemic, the probability of a node being infected is strongly correlated with its degree (i.e., compare columns 2 and 3).
- For the Level-3 network, the operator should protect Washington, Denver and Indianapolis against contact-based, path-based unweighted and path-based weighted epidemics respectively.

### VIII. CONCLUSIONS

- The authors express 8Bracketed values in columns 2-6 indicate the node degrees.
- Infection permeation in contact-based epidemic is largely determined by the topology structure, A. This is not the case for path-based epidemic as the primary factors for infection spreading are now related to the traffic load and the way this is routed to destinations.
- In addition, since the authors consider each node separately, they can also easily identify/rank nodes within the network that are the most conducive to spreading the infection.
- Based on their model, the critical epidemic thresholds are diminishingly small with λ and this re-affirms the observations reported in [29], [30] that epidemics in communication networks are extremely robust to extinction.

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...This approach has been investigated in, for example, [4], [7], [17], [20], [30], [32], [34], [36]....

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##### References

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...2If the shortest paths are used, then it coincides with topological betweenness [41]....

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...Furthermore, we are interested in finding the bounds of τc....

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "Path-based epidemic spreading in networks" ?

Based on these, the authors further derive conditional bounds for τc, subject to the availability of information regarding traffic load in the network and the routing algorithms, such that the network operator may use to control the epidemic as needed. In addition, since the authors consider each node separately, they can also easily identify/rank nodes within the network that are the most conducive to spreading the infection. Such nodal-level information may be used as a new centrality metric when designing immunization/protection schemes. Their modelling approach is general in nature and can be easily extended to model different epidemic models such as SIR ( analogous to [ 35 ] for contact-based epidemic ).