# Protection Strategy for Edge-Weighted Graphs in Disease Spread
###### Abstract  
Fake news, viruses on computer systems or infectious diseases on communities are some of the problems that are addressed by researchers dedicated to study complex networks. The immunization process is the solution to these challenges and hence the importance of obtaining immunization strategies that control these spreads. In this paper, we evaluate the effectiveness of the DIL-W* ranking in the immunization of nodes that are attacked by an infectious disease that spreads on an edge-weighted graph using a graph-based SIR model. The experimentation was done on real and scale-free networks and the results illustrate the benefits of this ranking. Keywords:edge-weighted graph; SIR model; graph protection; immunization strategy; complex networks +
Footnote †: journal:  
[PERSON] and [PERSON] developed DAVA and DAVA-fast, two methods in which all infected nodes merge into a supernode by constructing a weighted dominator tree of the input network [16; 17]. [PERSON] et al. in [18] provided NIIP, which selects \(k\) nodes to immunize over a period of time. For each time point, the NIIP algorithm will take decisions about which nodes to immunize given the estimated value of \(k\) for that time point. In [19], [PERSON] et al. also agree that community structure is important for understanding the spread of an epidemic. [PERSON] et al. propose a dynamic minimization model of the influence of a rumor with the user experience (DRIMUX), considering both the global popularity and the individual attraction of the rumor [20]. GraphShield method is developed by [PERSON] and [PERSON] in [21] taking into account the function of infection flow, graphics connectivity and top-grade centrality. [PERSON] et al. designed a method to identify the best ranked nodes in order to control an epidemic through immunizations [22]. [PERSON] et al. analyzed the nonoverlapping community structure network based on an immunization strategy, recognizing the relevance of the node given the characteristics of the communities [23], while the same author asserts that modular centrality must include the influences of the nodes both in their own communities and in others [24]. In [25] the authors introduced ReProtect and ReProtect-p methods, which divide the size of protection budget into several turns and protect nodes according to the currently observed temporal snapshot of dynamic networks. [PERSON] et al. [26] proposed the weighted \(K\)-order propagation number algorithm to extract the disease propagation based on the network topology to evaluate the node importance. In [27], the authors show that the DIL-W\({}^{\alpha}\) ranking provides good results, regarding the rate of decline in network efficiency (more details in [28]). Furthermore, one of the good qualities of the DIL-W\({}^{\alpha}\) ranking is that it recognizes the importance of bridge nodes, these nodes are those that connect the peripheral nodes and the peripheral groups with the rest of the network (see more details in [29]). This attribute is inherited from the version of the DIL ranking for graphs not weighted at the edges (see [30]). For this reason, we have chosen this ranking to evaluate its effectiveness in the immunization of nodes that are attacked by an infectious disease that spreads on an edge-weighted graph. The immunization is done according to the importance ranking list produced by DIL-W\({}^{\alpha}\) ranking. The effectiveness is measured with the ratio of vertices that remain uninfected at the end of the disease over the total numbers of vertices subject to budget protection. The experimentation was done on real and scale-free networks. The paper is organized as follows. Section 2 contains generalities about graph theory and the DIL-W\({}^{\alpha}\) ranking. In Section 3, we provide definitions of the protection of a graph when a disease spreads on it. Moreover, we state the protection strategy.