1 New metrics for characterizing the significance of nodes in wireless networks via path-based neighborhood analysis Leandros A. Maglaras 1 Dimitrios Katsaros 1 Ioannis Karydis (presentation) 13 th PCI, Corfu, Greece, 10-12/September/ Computer & Communication Engin. Dept., University of Thessaly, Volos, Greece

Network topology analysis Process of characterizing physical connectivity & relationships among entities in a network Topology analysis assists the calculation of centrality measures Or, the importance of a node in a network Assignment of special roles to central nodes message ferrying nodes (DTN) mediator nodes (Cooperative caching) re-broadcasting nodes (Vanets)

3 Existing Centrality Metrics Degree centrality number of edges attached to the node Closeness centrality the mean geodesic distance between a vertex v and all other vertices reachable from it Betweenness centrality the number of times a node can interrupt the shortest paths Pagerank a node is significant if connected to significant nodes

4 Why not existing approaches Betweenness centrality leaves nodes not in shortest path unranked effect of “Bridge edges” leads to not important nodes with high centrality Pagerank a node may be ranked high simply for being close to an important node extended calculations required whole network topology knowledge required not always available (eg. MANET)

5 Terminology and assumptions Graph G(V,L) V: set of nodes (vertices) L: set of links (edges) Edge l=(u,v) exists if node u is in the transmission range of v and vice versa. Directed or undirected network. Definition : Node j belongs to G ni if there exists at least one path from i to j, in at most n- hops

6 Algorithm 1. Find the N-hop neighborhood G ni of each node i. 2.Find all the paths from node i 3.Calculate local weight of all the nodes in G ni 4.Accumulate local weights to obtain the final (global) ranking of all the nodes.

7 Local Weight Derive all paths from node i : For each hop, a weight is computed the total number of paths derived from the previous step the number vertex l appears in a hop j Local weight for any vertex l in neighborhood G ni :

8 Local Weight Intuition Crucial nodes are nearest to the source. with respect to information dissemination All paths can be used (≠ not only shortest paths as in betweenness centrality) Size of the neighborhood: Average distance Network diameter l3l3 u1u1 l2l2 u6u6 l1l1 u2u2 l4l4 u5u5 l5l5 u4u4 u3u3 u7u7 l6l6 l7l7

9 Global Weight Time complexity : O((|L|+|V|)*|V|) |L|: cardinality of set of edges |V|: cardinality of set of vertices Local weights accumulated to obtain (global) ranking.

10 Method evaluation Real Datasets Zachary's karate club network of friendships with 34 members of a US university Karate club Dolphin network undirected network of frequent associations between 62 dolphins Evaluation metrics Number of ties Spearman’s rank correlation coefficient

11 Undirected Experimental Graphs Dolphin network betweenness centrality metric produces a significant amount of non-ranked nodes, which is a non desirable effect when the centrality metrics is used in wireless networks for characterizing the significance of nodes in the network topology

12 Directed Experimental Graphs Zachary's network Table 6 depicts the ids of the five highest ranked (top-5) nodes for this directed network. The numbers in parentheses represent the position of the node in the ranking produced by the competitor method, in the cases where this node does not appear in the top-5 list of the competitor.

13 AWENOR REDUCED RANKING AWeNoR has to add local weights of all neighborhoods in the network algorithm has to run K times, for a K-hop long network computing local weights for every neighborhood can be time consuming q i counts the times vertex i participates in paths of all the neighborhoods 1. Initiate algorithm. Set i=1. 2. Find neighborhood G ni of node i where q i <A 3. Find all the paths in the neighborhood 4. For every path update parameter q i except for boundary nodes 5. Calculate the local weight of all the nodes in G ni 6. Set i=i+1. If last node of graph reached goto 7 else goto 2 7. Accumulate local weights, find final ranking of all the nodes

14 Method evaluation 1 / 2 Parameter q i counts participation of node in previous steps Adjustment of q i Creates fewer neighborhoods. Improves running time. TABLE 7 shows that AWeNoR Reduced works well in terms of finding most important vertices in a graph while fewer neighborhoods need be created

15 Method evaluation 2 / 2 Threshold A used as a threshold in order to choose whether a vertex’s neighborhood is created or not Rather big: method degenerates to AWeNoR. Equal to zero: creation of disjoint neighborhoods, some nodes unranked. Sensitivity of AweNoR reduced ranking to parameter A ( Zachary's karate club undirected graph) A strict relation between method’s accuracy and cost, in terms of time consumption

Conclusions New metrics to reward vertices which: belonging to many neighborhoods lie in many paths In contrast to existing approaches sink nodes are not rewarded (Pagerank). no unranked nodes are left (betweenness) AWeNoR Reduced a faster algorithm for finding centrality refined AWeNoR algorithm creates fewer neighborhoods but the results are sensitive to some parameters Conclusions

17 Thank you for your attention!