Biased random walk on a graph

In network science, a biased random walk on a graph is a time path process in which an evolving variable jumps from its current state to one of various potential new states; unlike in a pure random walk, the probabilities of the potential new states are unequal.

Biased random walks on a graph provide an approach for the structural analysis of undirected graphs in order to extract their symmetries when the network is too complex or when it is not large enough to be analyzed by statistical methods. The concept of biased random walks on a graph has attracted the attention of many researchers and data companies over the past decade especially in the transportation and social networks. ^[1]

Model

There have been written many different representations of the biased random walks on graphs based on the particular purpose of the analysis. A common representation of the mechanism for undirected graphs is as follows: ^[2]

On an undirected graph, a walker takes a step from the current node, ${\displaystyle j,}$ to node ${\displaystyle i.}$ Assuming that each node has an attribute ${\displaystyle \alpha _{i},}$ the probability of jumping from node ${\displaystyle j}$ to ${\displaystyle i}$ is given by:

${\displaystyle T_{ij}^{\alpha }={\frac {\alpha _{i}A_{ij}}{\sum _{k}\alpha _{k}A_{kj}}},}$

where ${\displaystyle A_{ij}}$ represents the topological weight of the edge going from ${\displaystyle j}$ to ${\displaystyle i.}$

In fact, the steps of the walker are biased by the factor of ${\displaystyle \alpha }$ which may differ from one node to another. ^[3]

Depending on the network, the attribute ${\displaystyle \alpha }$ can be interpreted differently. It might be implied as the attraction of a person in a social network, it might be betweenness centrality or even it might be explained as an intrinsic characteristic of a node. In case of a fair random walk on graph ${\displaystyle \alpha }$ is one for all the nodes.

In case of shortest paths random walks ^[4] ${\displaystyle \alpha _{i}}$ is the total number of the shortest paths between all pairs of nodes that pass through the node ${\displaystyle i}$. In fact the walker prefers the nodes with higher betweenness centrality which is defined as below:

${\displaystyle C(i)={\tfrac {{\text{Total number of shortest paths through }}i}{\text{Total number of shortest paths}}}}$

Based on the above equation, the recurrence time to a node in the biased walk is given by: ^[5]

${\displaystyle r_{i}={\frac {1}{C(i)}}}$

Applications

There are a variety of applications using biased random walks on graphs. Such applications include control of diffusion, ^[6] advertisement of products on social networks, ^[7] explaining dispersal and population redistribution of animals and micro-organisms, ^[8] community detections, ^[9] wireless networks, ^[10] and search engines. ^[11]

See also

• Betweenness centrality
• Community structure
• Kullback–Leibler divergence
• Markov chain
• Maximal entropy random walk
• Random walk closeness centrality
• Social network analysis
• Travelling salesman problem

References

1. Roberta Sinatra; Jesús Gómez-Gardeñes; Renaud Lambiotte; Vincenzo Nicosia; Vito Latora (March 2011). "Maximal-entropy random walks in complex networks with limited information". Physical Review E. 83 (3): 030103. arXiv: 1007.4936 . Bibcode:2011PhRvE..83c0103S. doi:10.1103/PhysRevE.83.030103. PMID   21517435. S2CID   6984660.
2. J. Gómez-Gardeñes; V. Latora (Dec 2008). "Entropy rate of diffusion processes on complex networks". Physical Review E. 78 (6): 065102. arXiv: 0712.0278 . Bibcode:2008PhRvE..78f5102G. doi:10.1103/PhysRevE.78.065102. PMID   19256892. S2CID   14100937.
3. R. Lambiotte; R. Sinatra; J.-C. Delvenne; T.S. Evans; M. Barahona; V. Latora (Dec 2010). "Flow graphs: interweaving dynamics and structure". Physical Review E. 84 (1): 017102. arXiv: 1012.1211 . Bibcode:2011PhRvE..84a7102L. doi:10.1103/PhysRevE.84.017102. PMID   21867345. S2CID   2286264.
4. Blanchard, P; Volchenkov, D (2008). Mathematical Analysis of Urban Spatial Networks. Springer. doi:10.1007/978-3-540-87829-2. ISBN   978-3-540-87828-5 – via ResearchGate.
5. Volchenkov D; Blanchard P (2011). Fair and biased random walks on undirected graphs and related entropies. Birkhäuser. p. 380. ISBN   978-0-8176-4903-6.
6. Chung, Zhao, Fan, Wenbo (2010). "PageRank and random walks on graphs". Fete of Combinatorics and Computer Science. Bolyai Society Mathematical Studies. Vol. 20. pp. 43–62. CiteSeerX   10.1.1.157.7116 . doi:10.1007/978-3-642-13580-4_3. ISBN   978-3-642-13579-8. S2CID   3207094.
7. Adal, K.M. (June 2010). "Biased random walk based routing for mobile ad hoc networks". 2010 International Conference on Intelligent and Advanced Systems. pp. 1–6. doi:10.1109/ICIAS.2010.5716181. ISBN   978-1-4244-6623-8. S2CID   16113377.
8. Kakajan Komurov; Michael A. White; Prahlad T. Ram (Aug 2010). "Use of Data-Biased Random Walks on Graphs for the Retrieval of Context-Specific Networks from Genomic Data". PLOS Comput Biol. 6 (8): e1000889. Bibcode:2010PLSCB...6E0889K. doi:10.1371/journal.pcbi.1000889. PMC   2924243 . PMID   20808879.
9. J.K. Ochab; Z. Burda (Jan 2013). "Maximal entropy random walk in community detection". The European Physical Journal Special Topics. 216: 73–81. arXiv: 1208.3688 . Bibcode:2013EPJST.216...73O. doi:10.1140/epjst/e2013-01730-6. S2CID   56409069.
10. Beraldi, Roberto (Apr 2009). "Biased Random Walks in Uniform Wireless Networks". IEEE Transactions on Mobile Computing. 8 (4): 500–513. doi:10.1109/TMC.2008.151. S2CID   13521325.
11. Da-Cheng Nie; Zi-Ke Zhang; Qiang Dong; Chongjing Sun; Yan Fu (July 2014). "Information Filtering via Biased Random Walk on Coupled Social Network". The Scientific World Journal. 2014: 829137. doi: 10.1155/2014/829137 . PMC   4132410 . PMID   25147867.

External links

• Gábor Simonyi, "Graph Entropy: A Survey". In Combinatorial Optimization (ed. W. Cook, L. Lovász, and P. Seymour). Providence, RI: Amer. Math. Soc., pp. 399–441, 1995.
• Anne-Marie Kermarrec, Erwan Le Merrer, Bruno Sericola, Gilles Trédan, "Evaluating the Quality of a Network Topology through Random Walks" in Gadi Taubenfeld (ed.) Distributed Computing