Event propagation over a network is a complex and frequently studied human phenomena [1,2,3]. Historical examples of information exchange between humans in a social network, past and present, can be seen during colonial expansion of the British Empire , memes passed between friends on any assortment of social networks on the internet  and transmission of human or animal pathogens such as the virus H1N1 over airways . In the information age computer networks can mirror these types of information exchange, with event information being passed along from node to node when network neighbors communicate through various network protocols .
This information can be useful, whether in the form of postal delivery of early colonial directives and parcels , neutral, shared irrelevant Facebook posts , or detrimental, such as human pathogens or computer attacks [5, 8]. In this paper we will use consistent terminology of a negative event for the purpose of visualization for the reader. Some examples may include: how to spread, as in infection, or how to prevent the spread, as in vaccination, of this negative event, or attack. Malicious attacks can take many forms, such as, malware, viruses, undesirable information, or even memes that might violate a networks posting policy [8, 9]. In terms of this paper “vaccination”? is defined as a set of nodes that can be inoculated, removed, and/or protected from these malicious attacks. The main assumption is that if these specific sets of nodes are given pre-emptive measures then the attack will only be effective on a minimum number of nodes in the network.
The development and growth of networks has led to a diverse number of academic domains with shared properties and/or unique properties. Network traversal by means of graph theory , network communication protocols , static networks , dynamic networks  and network security  are network domains with similar terminology and visualizations. If a similarity exists across such diversity then it is possible to explore a similar solution that models important aspects borrowed from similar domains. This paper examines networks with nodes that are sparsely connected that may have a hierarchical structure that can be exploited using parts of graph theory to develop a solution.
Large networks can suffer disastrous results if an attack is allowed to migrate over a network. Various problems have been considered in the literature regarding this issue. Stopping a known attack at a source node before it can infect its neighbor nodes is a trivial problem. Once the attack spreads beyond the source node, however, its complexity rises as more nodes are exposed and infected through available edges . Another trivial situation for an active infection network is if the amount of vaccine available is much larger than the number of non-infected adjacent nodes to any infected nodes. The more typical case occurs when non-infected nodes outnumber the amount of vaccine by a large degree . This limitation causes a cost-benefit solution for which choices must be made in order to create an outcome that is favorable to the network administrators. Giving system administrators a variety of implementations allows them to choose the best option, or possible combinations of near optimal solutions, for their system .
Several solutions exist to prevent the spread of attacks over closed static systems; however these solutions have limited application to dynamic systems. An effective solution for arbitrary static networks with sparse connectivity is called Data-Aware Vaccine Allocation  or DAVA. DAVA is a framework that treats infected sub-graphs as super nodes and all its infected node neighbors as rooted bidirectional sub-trees that has a representational optimal solution in the form of a set of adjacent neighbors of the infected root super node. The framework determines this set by using a benefit system derived from a probability of propagation set, an infection set, and the main network graph. Thoroughly discussed by Zhang and Prakash the DAVA algorithm has been proven effective for “vaccine” distribution over an arbitrary network given prior information . However this solution has limited application if new nodes are being added to the network over a consistent time interval.
Large networks that are dynamic have vulnerabilities because of the changes made to the network in real time . Complications arise when new nodes vary in their importance to the networks due to location of edges and varying probability to infection . For example a node could be added to the network that has edges to nodes that were not connected before or that create a cycle that was not present before the edge was added. This addition may or may not change the overall benefit of the sub-tree that contains the node due to probability changes or dominance changes caused by this addition. In a static system this addition or removal now would represent a new graph which requires the entire network to be examined from the start. This paper seeks to make the distinction that these new nodes will not necessarily alter the current benefit solution of its associated sub-trees thus would not alter the optimal solution; it would require less work than recalculating the entire tree with new benefits and optimal solution. As a bonus, overall network information would be gained that could be applied to machine learning to enhance the information about this network. This gain might be exploited when examining local or global solutions.
Similarly to a multi-task learning problem , dynamic network node vaccination, or removal, has a main task, namely the optimal solution and then sub tasks to calculate and determine the overall benefit for top-k vaccination order subgroups. Individual nodes that have outlier tendencies might alter any current branch benefit, thus altering the top-k benefit order. It is from this concept that the idea of local solutions are of greater interest than global solutions for dynamic networks. If these local changes could be determined to have little or no changes to the benefits of the global sub-tree to which the local solution belongs, then it maybe possible to exploit this information to make an algorithm more efficient.
This paper presents a dynamic network algorithm, VAILDN, that is able to receive nodes sequentially from a network and then devise an optimal set of vaccination nodes to ensure maximum protection of the network. While performing effectively on static solutions DAVA group of algorithms slow down due to the fact that the global solution always must be determined, even though changes may occur only for local solutions. This paper will show that it is possible to adjust the optimal solution without examining the entire network which will save time over static algorithms. In addition this paper will show that VAILDN has the same level of accuracy as do DAVA type algorithms.
Dynamic solutions using dominator trees also present challenges that are not considered when using static solutions. These challenges include the addition or removal of edges to the graph that may redraw the dominator tree if a cycle is created or destroyed. In a dynamic environment it is possible that children may be detected before the parents in terms of the overall graph. This will not affect the global solution however, as edges are added, local connections and dominance are built that could be exploited for time saving when any parent does arrive.
In this paper first related works are discussed in “Preliminaries” section. The main algorithm is broken into three phases and presented in “Experiments” section. “Results” section describes the experiments in which this algorithm was compared to random baseline algorithms and vaccination methods for static networks. The final sections discuss the experimental results and the conclusions, including suggestions for future work.