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  • Vasileios Vlachos, Eirini Kalliamvakou, and Diomidis Spinellis. Simulating bandwidth-limited worms: One graph to rule them all?. In Theodore S. Papatheodorou, Dimitris N. Christodoulakis, and Nikitas N. Karanikolas, editors, Current Trends in Informatics: 11th Panhellenic Conference on Informatics, PCI 2007, volume B, pages 151–162, Athens, May 2007. New Technologies Publications. Green Open Access

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Diomidis Spinellis Publications

Simulating Bandwidth-Limited Worms,

One Graph to Rule Them All?



Vasileios Vlachos, Eirini Kalliamvakou and Diomidis Spinellis


Department of Management Science and Technology,

Athens University of Economics and Business (AUEB),

Patission 76, GR-10434, Athens, Greece

{vbill, ikaliam, dds}



Due to ethical and practical limitations, simulations are the de facto approach to measure and predict the propagation of malicious code on the Internet. A crucial part of every simulation is the network graph that is used to perform the experiments. Though recent evidence brought to light the nature of many technological and socio-technical networks such as the web links, the physical connectivity of the Internet and the e-mail correspondents, we argue that the interpretation of these findings has to be strongly correlated with specific malware properties. Furthermore, we question whether these graphs are accurate enough to model the fastest spreading type of malicious activity, bandwidth-limited worms. Finally, we propose some workarounds, by introducing a new Bandwidth-Aware Graphs Generation Algorithm, in order to generate specially crafted network graphs for the simulation of this type of malicious activity.


Keywords: malware, networks, bandwidth-aware graphs


1. Introduction

The spread of malicious software is definitely not a new threat, but only recently reached epidemic proportions. Computer worms are, according to [Weaver et. Al. (2003)], programs that self-propagate across a network exploiting security or policy flaws in widely-used services. An emerging subcategory of them are the bandwidth-limited worms, which mostly rely on the udp protocol to propagate in order to avoid the high-latency issues that arise due to the three-way handshake of the tcp protocol [Cardwell et. Al. (2000)]. The most characteristic examples are the Slammer [Moore et. Al. (2003)] and the Witty [Shannon & Moore (2004)] worms.

Identifying and predicting the propagation patterns of rapid malcode clearly is not an easy task. In vitro experiments with live viral code raise numerous ethical questions, because an unintended escape of a virus or a worm can cause severe damages. Furthermore, to monitor the large scale effects that characterize the propagation dynamics of various species of malicious software, a sustainable number of systems is required which unfortunately exceeds the capabilities of most laboratories.

An alternative way to gather valuable information regarding the spread of a viral agent is to collect empirical data during a malcode epidemic. This approach faces two serious shortcomings. First, virus and worm epidemics dont happen in short, regular or predictable intervals. Second, even then, most of our available resources and efforts are concentrated on the defence and the recovery of the most critical digital infrastructures, leaving insufficient time for theoretical studies.

Thus, simulating the propagation of computer viruses and worms is the most widely accepted methodology to measure and predict its consequences. One of the most important factors of every network simulation, which is mostly overlooked, is the set of characteristics of the graph that is utilized to represent a given network topology. In this paper, we argue that the study of rapid malcode requires that we take into account specific characteristics, such as connectivity, with respect to the bandwidth of the infected hosts. Therefore, we question the validity of the existing graphs that are frequently used to model the spread of bandwidth limited worms, which utilize random scanning as their propagation strategy.

The paper is organized as follows. Section 2 presents the most commonly used graphs for the simulation of rapid malcode. We provide examples of related research and give also the reasons behind the choice of the specific graphs. In Section 3, we describe the related work, whereas in Section 4 we discuss the accuracy of the specific models in conjunction with some other efforts to address these problems. In Section 5, we present some preliminary ideas to alleviate these issues without inducing significant computational requirements and Section 6 concludes this paper.

2      Network Graph Models

Although graphs have been widely used to study a variety of interesting theoretical problems, only a small percentage of them are appropriate for network related problems in general and even fewer are well suited to study the propagation dynamics of rapid malcode. The most suitable topologies for this type of research are homogenous graphs, random graphs and scale-free graphs.

2.1 Homogeneous Graphs

Most researchers use homogeneous graphs for their simulations {[Berk et. Al. (2003)], [Kephart et. Al. (1993)], [Kephart et. Al. (1999)], [Kephart et. Al. (1992)], [Chen et. Al (2003)], [Zoo et. Al (2002)], [Staniford et. Al (2002)], [Staniford (2003)], [Scandariato & Knight (2004)], [Zoo et. Al (2003)], [Leveille (2002)]}. In a homogeneous graph (or complete graph) every node is connected with all the other nodes resulting in a fully connected graph with N(N-1)/2 edges, where N is the number of nodes. Homogeneous graphs are very popular among epidemiologists studying the propagation of biological diseases and computer epidemiologists as well because they offer significant advantages. Firstly, it is very simple to construct them and secondly, most propagation patterns can be calculated using mathematical epidemiological models. The rationale behind the utilization of homogeneous graphs is that all ip addresses are reachable from any system. Therefore, a worm can contact, connect to and infect any other system regardless of its geographical proximity.

2.2 Random Graphs

For the last decades the common perception was that almost all complex systems could be realistically modelled using random graphs. The most widely accepted model is the one based on the Erdos-Rény (er) algorithm. The degree distribution of an er graph obeys the following Poisson distribution, given that N is sufficiently large .


where P is the connection probability and k<N the number of the neighboring nodes. Older research efforts {[Kephart et. Al (1993)], [Kephart & White (1999)], [Kephart (1992)], [Zoo et. Al (2003)], [Leveille (2002)]} took for granted that inherently complex systems, such as the Internet, should follow a similar distribution and thus a random graph was a good approximation for their experiments.

2.3 Scale-Free Graphs

Barabási and Albert demonstrated that the creation of scale-free networks should include two definitive characteristics: Incremental growth and preferential connectivity. Furthermore, Barabási and Albert [Barabási et. Al. (2003)] revealed that the connectivity probability of the World Wide Web (www) and of most other complex social and technical systems obey a power law distribution of the following type


where for the www. Their studies had great impact on the virus research communities as other researchers discovered similar structures in the www {[Kanovsky & Mazor (2003)], [Adamic & Huberman (2000)], [Bornholdt & Ebel (2001)]} and many other networks, such as the physical connectivity of the Internet {[Faloutsos et. Al (1999)], [Medina et. Al (2000)]} or the network of people exchanging e-mails [Ebel et. Al (2002)]. As the scale-free graphs became the de facto standard to simulate the spread of worms {[Wang et. Al (2000)], [Leveille (2002)], [Pastor-Satorras & Vespignani (2001)]} researchers have also tried to incorporate bandwidth related issues. The work of [Weaver et. Al (2004)] and [Wei et. Al (2005)] has made significant steps to address some of the related issues.

2.4 Other Topologies

In some older experiments non-trivial graph topologies have been used to study the effects of rapid malcode such as lattices {[Kephart et. Al (1993)], [Kephart (1992)]} and some hierarchical tree-like structures [Wang et. Al (2000) ]. We believe that these models have been abandoned today and are of limited use.

3 Discussion

All the above models have been extensively used in the literature and have proven useful for the study of malware. We are not confident though that their selection was made deliberately to depict specific malware properties.

In particular, some researches claim that a worm can reach any ip address from any given one. Leaving aside the fact that many systems are behind firewalls and use nat, or some portions of the Internet address space are not routable at all, we have to agree with this hypothesis. The interpretation of this assumption in most cases is that any system contacts all the other systems during a simulation cycle and has the opportunity to infect them, which according to common intuition is not highly likely. In epidemiology, this phenomenon is known as homogeneous mixing but it is not considered very realistic for large populations.

Other researchers, on the other hand, argue that the scale-free graph models are the most appropriate. While we also believe that this is the case, we feel that we have to more thoroughly examine the foundations behind this hypothesis. In particular, there are four major cases which need more elaboration.

3.1 Social relationships

The number of friends each individual has follows a power low distribution [Barabási (2003)]. Therefore, this social network is a scale-free graph. The practical implications of this research in the context of computer virology can only be attributed to the propagation of ancient viruses via the now obsolete floppy disks, which constitute no threat any longer.

3.2 World Wide Web

The www follows a power-law distribution regarding the links that point from or to various web pages {[Kanovsky & Mazor (2003)], [Adamic & Huberman (2000)], [Bornholdt & Ebel (2001)]}. This evidence, while useful and important for many scientific fields, has little to do with bandwidth-limited worms. In our context, we can make use of this information only in some specific aspects of malcode such as some contagion and web-based worms, like the Santy worm. Contagion worms that utilize vulnerabilities in web-servers and web-browsers, such as the Nimda worm, to stealthily propagate under the radar can best be described by these models but their most important characteristic is that they can remain for long time unnoticed. Unquestionably these worms are very dangerous but certainly do not fall into the category of rapid malcode.

3.3 E-mail exchange

Studies showed that networks of people exchanging e-mails form scale-free graphs [Ebel et. Al (2002)]. This information is of strategic importance in the fight against mass mailing worms. Many such worms have caused and still cause malware epidemics, but hardly can they be considered as rapid malcode as they rely on clever social engineering tactics and vulnerable e-mail clients to propagate.

3.4 Physical Connectivity of the Internet

Theoretical and empirical research confirms that the physical connectivity of the Internet resembles a scale-free structure {[Faloutsos et. Al (1999)], [Medina (2000)]}. These studies examined the connectivity between bgp routers or Autonomous Systems and concluded that it follows a power law distribution. Most of the simulations that tend to use scale-free graphs follow closely the physical underlying structure of the Internet. Though this assumption seems logical we have to question its exact interpretation.

A worm, in order to infect a target ip address, has to traverse a number of intermediate routers. In general, the intermediate routers are not in the position to alter, be infected by or modify in any way the course of events once the transaction begins. Therefore, as these routers do not play any active role in the evolution of an infection, they can be omitted from an abstract connectivity model. Thus, this type of communication between an attacking and a target node can be referred to as a direct contact. Nonetheless, the importance of the scale-free nature of the Internet is of great significance since it heavily affects its performance and stability. While in that case the usefulness of the scale-free model is evident, in the context of the malware propagation further investigation is required to extract its exact implications.

4 Related Work

[Weaver et. Al (2004)] made significant progress on the subject, but they encountered some difficulties when they tried to combine topology maps of portions of the Internet with bandwidth information. Their approach is valid, but since most of the time these data are not available [Paxson & Floyd (1997)], synthetic graphs have to be used instead. Another point that differentiates their work from ours is that they use a single graph for modelling the connectivity of the Internet with bandwidth information, which allows them to capture the congestion effects that take place when a worm reaches its peak. We believe, on the other hand, that the most important part, from an epidemiological point of view of a malware epidemic, is its initial stages. The impact of the available bandwidth strongly affects the propagation of a worm during the first steps of the epidemic {[Staniford et. Al (2004)], [Weaver et. Al (2004)]}, because if it reaches a critical mass of infected systems before countermeasures can be deployed it is very hard to be contained.

[Ford et. Al (2005)] take the bandwidth issue into consideration but they adapt bandwidth quotas to the existing topologies they use. Thus, the basic underlying topology is static, while each node has a different network capacity. In [Wei et. Al (2005)], authors propose a very detailed simulation model which addresses network-related issues down to the packet level. Their approach is nonetheless valid, but requires excessive computational power as it is designed to work in a distributed manner. In [Wagner et. Al (2003)], define various groups with different connectivity and latency. Each node is then assigned to one of these groups. While it is an interesting concept, they dont provide enough information regarding the instantiation of new contacts between the nodes of these groups.

5 The Bandwidth-Aware Graph

Analytical models have certain restrictions, despite the fact that scale well. Most importantly, these models cannot be easily applied to large scale complex systems. As we discussed in a previous section, the majority of the complex social and technical systems could best depicted as random graphs or scale-free graphs. On the contrary, most of the times, analytical models can only handle homogeneous graphs, which are not suitable for the study of all the types of malicious activity. Most simulations are computing-intensive making it necessary to introduce the right level of abstraction in order to get meaningful results in reasonable time frames. Combining the advantages as well as the shortcomings of the above graph models, we propose some modifications that we find appropriate. We discuss also the reasons these modifications are better suited for the study of rapid malware.

We believe that bandwidth defines the connectivity of the graph. To be more specific, we argue that each system can instantiate a connection and interact with any other system in the ip address space as long as it has the available bandwidth to do so. (We leave aside the nat issue we mentioned earlier as it doesnt significantly change the landscape of a complex system, such as the Internet, nor can it be addressed effectively by other means.) Therefore, every node in the graph should be directly connected to any other. On the other hand, it is not realistic to assume that each node will contact any other node during every simulation cycle. To meet both of these requirements we consider the following approach.

Given that a single graph might not be sufficient to act as a realistic network topology regarding the spread of malcode, we propose the use of multiple bandwidth-aware graphs as a viable solution to increase the accuracy of the simulation. In particular, we believe that each cycle of a worm-propagation simulation should be performed on a different graph from a given graph set. All graphs which belong to the graph set have very similar characteristics but they are not identical. Our model introduces the bandwidth capacity (both in terms of uplink and downlink bandwidth) for every node in a homogeneous graph. Thus, in every simulation cycle, a node will be limited to contact randomly other nodes according to its predefined bandwidth. In particular, a node i can contact a node j only if the uplink capacity of node i and the downlink capacity of node j are greater than the size of the worm . To further clarify the process of the creation of similar graphs we present analytically the Bandwidth-Aware Graphs Generation Algorithm (bagga)

In the algorithm, N is the set of all nodes in the graph, Up is the uplink capacity, D is the downlink capacity and is the worm size.

The algorithm consists of two parts:

A. Initialization part

(1) Assign and [1]



It should be noted that and

B. Main algorithm

(1) if or exit

(2) if next i

(3) pick randomly

a. if or (i connected to j) repeat

b. ,

c. ,

d. connect node i with node j

(4) repeat

We are aware of the existence of various tools that generate graphs for network-related studies and at least one of them is specially designed to construct topologies that are widely used by computer epidemiologists [Vlachos et. Al (2005)]. Therefore, the construction of a large number of graphs should be a trivial task.

We suggest the use of a unique graph in each cycle of the simulation, after having built, in advance, a large number of bandwidth-aware graphs with similar properties. Thus, while each node eventually will have a direct contact with any other node in each time slot, the number of its neighbours per time slot will be limited.

The above approximations will be very important to researchers that cannot afford to perform packet level simulations due to their complexity and the increased hardware requirements. Furthermore, the bandwidth-aware graphs may also alleviate the shortage of accurate graph topologies of the physical connectivity of the Internet, considering that even in the cases where large organizations and isps have mapped significant portions of the Internet, they are not willing to openly share this information.

Our aforementioned estimations are largely based on our prior extended experience with graph generation. In 2005 we developed ngce, written in Java (more than 5,000 lines of code, 15 Java classes, 150 lines of scripts). The tool calculates the probability distribution, plots it, and provides capabilities for building and analyzing various types of graphs. Furthermore, ngces output graph files are compatible with the Pajek tool. For each different graph topology, a specific plugin has been developed to implement the appropriate algorithm, making the development of new algorithms as separate plugins extremely easy. We are currently implementing the plugin for bagga so as to provide further evaluation results.

A limitation of our model is that it can handle only worms that employ random scanning as their propagation strategy. Finally, to further improve the accuracy of this approach, each node of the constructed graphs should follow the bandwidth distributions that have been observed in related studies [Saroiu et. Al (2002)].

6 Conclusions and Future Work

Simulations are, in general, the most prominent way to explore the behaviour of complex systems. While the existing graph models are satisfactory for the study of specific types of network issues such as routing, resource preservation, and performance problems, the propagation of rapid malcode requires a closer examination of the following aspects of the available network models:

Direct connectivity. All ip addresses on the Internet are reachable from any ip address. The main difference between other network-related simulations is that even if a worm requires many intermediate hops to reach its target, these intermediate steps do not affect its propagation in any way.

Node inequality. The decisive factor to estimate the number of victims that each compromised system will manage to infect is its available bandwidth. Therefore, treating each node of the Internet as equal is not the most accurate approach. On the other hand, the available bandwidth of a node has little to do with its geographical location or the physical connectivity between the Autonomous Systems and bgp routers.

The above reasons suggest the revaluation of the existing graph models regarding their realism.

In our latest work [Vlachos et. Al (2005)], we presented the ngce (Network Graphs for Computer epidemiologists) tool which generates network graphs commonly used by computer epidemiologists. We are in the process of implementing the proper plug-in for the bagga in order to construct a number of bandwidth-aware graphs and perform various simulations so as to compare the results with the available empirical data.

Our intention is to investigate the possibilities of creating realistic network topologies for the study of the propagation of viral malcode having in mind that the access to high-performance computers is limited and detailed Internet maps are not widely available so as to perform detailed packet-level simulations and thus, alternative ways to find meaningful abstractions are useful.


The work of Vasileios Vlachos is funded under the Iraklitos fellowships for research of Athens University of Economics and Business program and the European Community's Sixth Framework Programme under the contract IST-2005-033331 Software Quality Observatory for Open Source Software (SQO-OSS)'.


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[1]According to the bandwidth distribution that has been observed in [Saroiu et. Al (2002)].