Evaluation of Advanced Routing Strategies with InformationTheoretic Complexity Measures
Abstract
Based on hierarchy and recursion (shortly, HR), recursive networking has evolved to become a possible architecture for the future Internet. In this paper, we advance the study of HRbased routing by means of the GershensonFernandez informationtheoretic framework, which provides four different complexity measures. Then, we introduce a novel and general approach for computing the information associated to a known or estimated routing table. Finally, we present simulation results regarding networks that are characterized by different topologies and routing strategies. In particular, we discuss some interesting facts we observed while comparing HRbased to traditional routing in terms of complexity measures.
Keywords
Distributed systems Recursive networking Complexity measures1 Introduction
Recursive networking refers to multilayer virtual networks embedding networks as nodes inside other networks. It is based on hierarchy, i.e., the categorization of a set of nodes according to their capability or status, and recursion, which is the repeated use of a single functional unit over different scopes of a distributed system. In the last decade, recursive networking has evolved to become a possible architecture for the future Internet [2]. In particular, it is a prominent approach to designing quantum networks [3]. In a recent work [1], we proposed to apply hierarchy and recursion (HR) to build selfaware and selfexpressive distributed systems. In particular, we presented HRbased network exploration and routing algorithms.
In this paper, we continue the characterization of HRbased routing by means of a simple albeit powerful and general informationtheoretic framework providing complexity measures, recently proposed by Gershenson and Fernandez [4]. Firstly, we introduce a novel and general (i.e., not HRspecific) approach for computing the information associated to a known or estimated routing table. Then we present simulation results regarding networks that are characterized by different topologies and routing strategies. In particular, we discuss some interesting facts we observed, while comparing HRbased to traditional routing in terms of complexity measures.
The paper is organized as follows. In Sect. 2, we summarize the basic concepts of Gershenson and Fernandez’s informationtheoretic framework [4]. In Sect. 3, we illustrate our approach for computing the information associated to a routing table. In Sect. 4, we recall the working principles of HRbased routing. In Sect. 5, we present simulation results. Finally, in Sect. 6, we outline future research directions.
2 Complexity and Information
It is difficult to provide an exhaustive list of the ways of defining and measuring system complexity that have been proposed by the research community. Among others, the GershensonFernandez informationtheoretic framework provides abstract and concise measures of emergence, selforganization, complexity and homeostasis [4]. According to their framework, emergence is the opposite of selforganization, while complexity represents their balance. Homeostasis can be seen as a measure of the stability of the system.
This framework has been used to study different kinds of complex systems, ranging from selforganizing traffic lights [5] to adaptive peertopeer systems [6].
3 Information Associated to a Routing Table
Traditionally, routing strategies are compared in terms of effectiveness, efficiency and scalability [7, 8]. To this purpose, selected independent variables should explain performance under a wide range of scenarios [9]. In particular, estimating routing tables is an important and challenging task, as details of how a route is chosen are diverse, and generally not publicly disclosed. An interesting strategy has been recently proposed by Rotenberg et al. [10].
In this context, we propose a novel and general approach for characterizing the whole network, namely, by averaging the emergence, selforganization, complexity and homeostasis values of its routers.
From now on, for simplicity, we assume that every node of the network is provided with a routing table, allowing to forward packets to neighbor nodes (routes), according to their destinations. A routing table can be modeled as a set of (destination, route) pairs.

\(E = I/I_{init} = I\)

\(S = I_{init}  I = 1  I\)

\(C = aES = 4 I (1I)\)

\(H = 1  d\)
4 HRBased Routing
We recall and explain HRbased routing by means of an example. Let us consider the network shown in Fig. 1. The routing table at node 4.2 contains information on how to reach any other node in the network. The table has more precise information about nearby destinations (node 4.4 and node 4.7), and vague information about more remote destinations (NET9).
 1.
nodes are not required to know the whole network topology (unlike LinkState routing);
 2.
nodes build collective awareness by exchanging recursive and hierarchical information not only with direct neighbors, but also with neighbors of neighbors, etc. (unlike DistanceVector routing).
For further details about HRbased versus LinkState and DistanceVector, the reader may refer to our previous work [1]. Thanks to collective awareness, messages can be routed within the same subnetwork or from one subnetwork to another; doing so they enable, for example, the Unified Architecture for interdomain routing proposed in RFC 1322.^{1}
5 Simulation Results
To evaluate the proposed approach, we used the generalpurpose discrete event simulation environment DEUS [11]. The purpose of DEUS is to facilitate the simulation of highly dynamic overlay networks with several hundred thousands nodes, without needing to simulate also lower network layers.
Without loss of generality, we considered the (suboptimal) scenario in which every node knows which subnetworks can be reached through its direct neighbors. In HRbased routing, no further knowledge — provided by neighbors of neighbors (of neighbors etc.) — is necessary, when the number of subnetworks M is of the same order of magnitude as the mean node degree \(\langle k \rangle \) of the network. Instead, for large networks, with \(M \gg \langle k \rangle \), further knowledge is necessary to build effective routing tables.
We took into account two network topologies, characterized by different statistics for the node degree, which is the number of links starting from a node. The first network topology we considered is scalefree, meaning that its PMF decays according to a power law \(P(k) = c k^{\tau }\), with \(\tau > 1\) (to be normalizable) and c normalization factor. Such a distribution exhibits the property of scale invariance (i.e., \(P(bk) = b^aP(k)\), \(\forall a,b \in \mathbb {R}\)). The second network topology we considered is a purelyrandom one, described by the wellknown model defined by Erdös and Rényi (ER model). Networks based on the ER model have N vertices, each connected to an average of \(\langle k \rangle = \alpha \) nodes. Scalefree and purelyrandom are the extremes of the range of meaningful network topologies, as they represent the presence of strong hubs and the total lack of hubs, respectively.
We evaluated the HRbased routing strategy in terms of success rate (i.e., fraction of messages arrived to destination) and average route length, using different networks characterized by \(N=1000\) nodes, with \(M=20\) subnetworks. With the BA topology, when \(m=5\) and \(m=20\), the mean node degree is \(\langle k \rangle = 10\) and \(\langle k \rangle = 40\), respectively. To have the same \(\langle k \rangle \) values for the ER topology, we set \(\alpha = 10\) and \(\alpha = 40\). Reported results are average values coming from 25 simulation runs.
HR vs NoHR: success rate \(r_s\) and average route length \(n_h\)
Strategy  Topology  S  \(\mu _{r_s}\)  \(\sigma _{r_s}\)  \(\mu _{n_h}\)  \(\sigma _{n_h}\) 

HR  BA, \(m=5\)  20  0.88  2E2  17.6  2.06 
NoHR  BA, \(m=5\)  20  0.74  2.9E1  19.7  9.8 
HR  BA, \(m=20\)  20  0.99  9.3E4  3.8  8E2 
NoHR  BA, \(m=20\)  20  0.99  9E3  9.85  1.2 
HR  ER, \(\alpha =10\)  20  0.64  3E2  43.7  2.72 
NoHR  ER, \(\alpha =10\)  20  0.55  3.3E1  21.64  17.91 
HR  ER, \(\alpha =40\)  20  0.99  1E3  4.0  1.2E1 
NoHR  ER, \(\alpha =40\)  20  0.93  1.9E1  15.33  4.75 
 1.
As m and \(\alpha \) grow, E tends to 1, S tends to 0.
 2.
When m and \(\alpha \) are low, HRbased and NoHR routing exhibit very different H values.
 3.
When m and \(\alpha \) are high, the values of H in HRbased and NoHR routing are more similar.
 4.
Even if the mean node degree \(\langle k \rangle \) is the same, BA and ER topologies result in very different E, S and C values.
The reason for the first fact is that a higher number of connections, due to higher m and \(\alpha \), makes the routing table more varied in terms of available routes. The probability distribution P(x) has fewer spikes, thus I is higher. As a consequence, E increases and S decreases. The second fact can be stated more precisely by means of the following inequality: \(H_{HR} \ll H_{NoHR}\), when m and \(\alpha \) are small. Our interpretation is that a reduced number of connections enhances the differences between routing tables, in HRbased and NoHR routing, i.e., with respect to the initial state, the final state of the routing table is much more different in HRbased routing rather than NoHR routing. The impact on performance is evident: HR routing table are better than NoHR ones, thus producing a higher success rate. It is not possible, however, to generalize associating higher H values to higher performance. Conversely, a higher number of connections reduces the differences between routing tables, explaining the third fact. The fourth fact is further detailed by the following inequalities: \(E_{BA} < E_{ER}\), \(S_{BA} > S_{ER}\) and \(C_{BA} \gg C_{ER}\), when m and \(\alpha \) are such that the mean node degree \(\langle k \rangle \) is the same in the BA and ER topologies. It is difficult to explain the relationship between C and performance, in general. It makes more sense to consider E and S separately. Regarding E, our interpretation is that scalefree properties (characterizing the BA topology) make some routes intrinsically more probable than others. Indeed, only a few nodes have a high number of connections (such nodes are denoted as hubs). Thus, with respect to the ER topology, in scalefree networks the probability distribution P(x) has more spikes, making I smaller. Consequently, E is lower and S is higher. Indeed, the presence of hubs makes routing more robust (S is higher), thus improving performance.
6 Conclusion
In this paper we have illustrated a novel approach to quantifying the information associated to a known or estimated routing table, allowing to characterize the whole network by averaging the emergence, selforganization, complexity and homeostasis values of its nodes. Our simulation study shows that these measures may represent an important complement to traditional performance indicators for routing protocols.
Regarding future work, we plan to improve the informationtheoretical investigation of HRbased routing strategies, considering larger networks with multilayered trees of subnetworks.
Footnotes
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