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Proceedings ArticleDOI

An entropy maximization problem in shortest path routing networks

TL;DR: In the context of an IP network, an interesting case of the inverse shortest path problem is investigated using the concept of network centrality, and a heuristic approach is proposed to obtain a centrality distribution that maximizes the entropy.
Abstract: In the context of an IP network, we investigate an interesting case of the inverse shortest path problem using the concept of network centrality. For a given network, the centrality distribution associated with the links of a network can be determined based on the number of shortest paths passing through each link. An entropy measure for this distribution is defined, and we then forumulate the inverse shortest problem in terms of maximizing this entropy. We then obtain a centrality distribution that is as broadly distributed as possible subject to the topology constraints. An appropriate change in the weight of a link alters the number of shortest paths that pass through it, thereby modifying the centrality distribution. The idea is to obtain a centrality distribution that maximizes the entropy. This problem is shown to be NP-hard, and a heuristic approach is proposed. An application to handling link failure scenarios in Open Shortest Path First routing is discussed.
Citations
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Journal ArticleDOI
15 Dec 2020-Entropy
TL;DR: A narrative literature review of information entropy metrics for complex networks is conducted following the PRISMA guidelines, identifying the areas in need for further development aiming to guide future research efforts.
Abstract: Information entropy metrics have been applied to a wide range of problems that were abstracted as complex networks. This growing body of research is scattered in multiple disciplines, which makes it difficult to identify available metrics and understand the context in which they are applicable. In this work, a narrative literature review of information entropy metrics for complex networks is conducted following the PRISMA guidelines. Existing entropy metrics are classified according to three different criteria: whether the metric provides a property of the graph or a graph component (such as the nodes), the chosen probability distribution, and the types of complex networks to which the metrics are applicable. Consequently, this work identifies the areas in need for further development aiming to guide future research efforts.

23 citations


Cites background or methods from "An entropy maximization problem in ..."

  • ...They further proposed the use of entropy maximization and betweenness entropy in order to make communications routing decentralized [25] and handle single edge failures [34]....

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  • ...For example, information functionals are based on edge or node betweenness centrality [24,25,34,50,53] distances to a given vertex [28], degree, degree power or probability distribution of degrees [31,41], paths or paths’ length [16,35], and closeness or eigenvector centrality [53]....

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  • ...[14] X X X X [15] X X acyclic [16] X X acyclic [17] X X X X [18] X X [19] X X strongly connected, aperiodic [20] X X connected [21] X X [22] X X [23] X X [24] X X no self-loops [25] X X [26] X X [27] X X [28] X X no self-loops [29] X X X [30] X X [31] X X [32] X X [33] X X [34] X X [35] X X [36] X X [37] X X [38] X X [39] X X [40] X X [41] X X [42] X X no self-loops [43] X X [44] X X [45] X X [46] X X [47] X X connected [48] X X [49] X X X X [50] X X [51] X X [52] X X [53] X X X [54] X X [55] X X [56] X X [57] X X [58] not specified [59] X X [60] X X [61] X X [62] X X [63] X X...

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  • ...[24] Chellappan, Vanniarajan and Sivalingam, Krishna M 2013 proceeding 2013 19th IEEE Workshop on Local & Metropolitan Area Networks (LANMAN) [25] V....

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  • ...[24,25,34] H(G) = − ∑ (u,v)∈E p(u, v) log p(u, v) p(u, v) = η•,•(u, v) ∑(x,y)∈E η•,•(x, y) where η•,•(u, v) is the...

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Journal ArticleDOI
TL;DR: This paper investigates an interesting case of the inverse shortest path problem using the concept of network centrality, and a heuristic approach is proposed to obtain a centrality distribution that maximizes the entropy.

13 citations

Journal ArticleDOI
TL;DR: This work aims to develop a Hybrid algorithm Dijkstra’s Floyd Warshall algorithm to solve entropy maximization routing protocol problem and is compared with the existing in order to find the best and shortest paths.
Abstract: The shortest path problem is to find a path between two vertices on a given graph, such that the sum of the weights on its constituent edges is minimized. The classic Dijkstra’s algorithm was designed to solve the single source shortest path problem for a static graph. It works starting from the source node and calculating the shortest path on the whole network. This work aims to develop a Hybrid algorithm Dijkstra’s Floyd Warshall algorithm to solve entropy maximization routing protocol problem. The algorithm has to find the shortest path between the source and destination nodes. Route guidance algorithm is use to find best shortest path in routing network, this is poised to minimize costs between the origin and destination nodes. The proposed algorithm is compared with the existing in order to find the best and shortest paths. General Terms Dijkstra’s Algorithm, Floyd-Warshall Algorithm, Hybrid algorithm Dijkstra’s-Floyd Warshall (HDFWA), Entropy.

3 citations


Cites background from "An entropy maximization problem in ..."

  • ...Network Centrality [1]: The network centrality or the entropy of SPBC is defined by...

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  • ...The Shortest Path Betweenness Centrality (SPBC) of a link (i) is defined as (1) Network Centrality [1]: The network centrality or the entropy of SPBC is defined by (2) Where, (3) represents the random variable associated with the probability distribution formed from equation (3)....

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  • ...Shortest Path Betweenness Centrality [1]: Let represent the total number of shortest paths between every pair of source-destination nodes (s, t): s and t V....

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References
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Dissertation
01 Dec 2003

24 citations

Book ChapterDOI
14 May 2007
TL;DR: It is shown that both node degree and node centrality are not necessarily evidence of its significance, and some medium degree nodes with medium centrality measure prove to be crucial for efficient routing in the Internet AS graph.
Abstract: In networks characterized by broad degree distribution, such as the Internet AS graph, node significance is often associated with its degree or with centrality metrics which relate to its reachability and shortest paths passing through it. Such measures do not consider availability of efficient backup of the node and thus often fail to capture its contribution to the functionality and resilience of the network operation. In this paper we suggest the Quality of Backup (QoB) and Alternative Path Centrality (APC) measures as complementary methods which enable analysis of node significance in a manner which considers backup. We examine the theoretical significance of these measures and use them to classify nodes in the Internet AS graph while applying the BGP valley-free routing restrictions. We show that both node degree and node centrality are not necessarily evidence of its significance. In particular, some medium degree nodes with medium centrality measure prove to be crucial for efficient routing in the Internet AS graph.

16 citations


"An entropy maximization problem in ..." refers background in this paper

  • ...Also, as noted in [8], in the context of network resilience, a link or node failure may have varying impact on the entire network....

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  • ...More information on centrality related work can be found in [7], [8]....

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  • ...This justifies the claim made in [8] that the criticality of a link is also determined based on the back up quality of the link....

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Journal ArticleDOI
TL;DR: In this paper, the authors proposed the quality of backup (QoB) and alternative path centrality (APC) measures as complementary methods which enable analysis of node significance in a manner which considers backup.
Abstract: In complex networks characterized by broad degree distribution, node significance is often associated with its degree or with centrality metrics which relate to its reachability and shortest paths passing through it. Such measures do not consider availability of efficient backup of the node and thus often fail to capture its contribution to the functionality and resilience of the network operation. In this paper, we suggest the quality of backup (QoB) and alternative path centrality (APC) measures as complementary methods which enable analysis of node significance in a manner which considers backup. We examine the theoretical significance of these measures and use them to classify nodes in social interaction networks and in the Internet AS (autonomous system) graph while applying the valley-free routing restrictions which reflect the economic relationships between the AS nodes in the Internet. We show that both node degree and node centrality are not necessarily evidence of its significance. In particular, we show that social structures do not necessarily depend on highly central nodes and that medium degree nodes with medium centrality measure prove to be crucial for efficient routing in the Internet AS graph.

14 citations

Proceedings ArticleDOI
10 Apr 2013
TL;DR: The notion of entropy of centrality measures is defined, which extends the concept ofcentrality to the whole network and has wide range of applications, in network design, from designing maximally efficient networks to identifying dominance of one node or link in the context of entire network.
Abstract: Various Centrality measures such as Degree, Closeness, and Betweenness were introduced in order to analyze networks and understand both the global dynamics of the networks and the roles played by individual nodes. It will be worthwhile to rank the centrality measures of each node and an index of the distribution of centrality measures in the entire network. In this paper, we define the notion of entropy of centrality measures, which extends the concept of centrality to the whole network. We show that this measure has wide range of applications, in network design, from designing maximally efficient networks to identifying dominance of one node or link in the context of entire network. In particular, we present an application to tactical wireless networks.

10 citations


"An entropy maximization problem in ..." refers methods in this paper

  • ...In [3], we introduced a network-wide measure based on the idea of centrality....

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01 Jan 2010
TL;DR: A constraint generation method for solving TE problems in IP networks is outlined in this thesis, to iteratively generate and augment valid inequalities that handle the SPR aspect of IP networks.
Abstract: This thesis is concerned with problems related to shortest pathrouting (SPR) in Internet protocol (IP) networks. In IP routing, alldata traffic is routed in accordance with an SPR protocol, e.g. OSPF.That is, the routing paths are shortest paths w.r.t. some artificialmetric. This implies that the majority of the Internet traffic isdirected by SPR. Since the Internet is steadily growing, efficientutilization of its resources is of major importance. In theoperational planning phase the objective is to utilize the availableresources as efficiently as possible without changing the actualdesign. That is, only by re-configuration of the routing. This isreferred to as traffic engineering (TE). In this thesis, TE in IPnetworks and related problems are approached by integer linearprogramming. Most TE problems are closely related to multicommodity routingproblems and they are regularly solved by integer programmingtechniques. However, TE in IP networks has not been studied as much,and is in fact a lot harder than ordinary TE problems without IProuting since the complicating shortest path aspect has to be takeninto account. In a TE problem in an IP network the routing isperformed in accordance with an SPR protocol that depends on a metric,the so called set of administrative weights. The major differencebetween ordinary TE problems and TE in IP networks is that all routingpaths must be simultaneously realizable as shortest paths w.r.t. thismetric. This restriction implies that the set of feasible routingpatterns is significantly reduced and that the only means available toadjust and control the routing is indirectly, via the administrativeweights. A constraint generation method for solving TE problems in IP networksis outlined in this thesis. Given an "original" TE problem, the ideais to iteratively generate and augment valid inequalities that handlethe SPR aspect of IP networks. These valid inequalities are derived byanalyzing the inverse SPR problem. The inverse SPR problem is todecide if a set of tentative routing patterns is simultaneouslyrealizable as shortest paths w.r.t. some metric. When this is not thecase, an SPR conflict exists which must be prohibited by a validinequality that is then augmented to the original TE problem. Toderive strong valid inequalities that prohibit SPR conflicts, athorough analysis of the inverse SPR problem is first performed. Inthe end, this allows us to draw conclusions for the design problem,which was the initial primary concern.

9 citations


"An entropy maximization problem in ..." refers background in this paper

  • ...The inverse shortest path problem has been extensively studied in [1], [2], [5]....

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