1. The Community-search Problem and How to Plan a Successful Cocktail Party
Mauro Sozio and Aristides Gionis
Presented By:
Raghu Rangan, Jialiang Bao, Ge Wang

2. Introduction Graphs are one of the most popular data representation
Have a wide range of applications
Communities and social networks as graphs have gained attention
People represented as nodes
Connection between people are edges
This paper focuses on the query-dependent variant of the community search problem

3. Planning a Cocktail Party
Participants should be “close” to the organizers (e.g. a friend of a friend).
Everybody should know some of the participants.
The graph should be connected.
The number of participants should not be too small
Not too large either
This is difficult
Bob
Alice
Charlie
David

4. Community Search Problem
Need to find the community that a given set of users belongs to.
Given a graph and a set of nodes, find a densely connected subgraph containing the set of users given in input.

5. Related Work Connectivity Subgraphs Community Detection
Work has been done to find a subgraph that connects as set of query nodes
Not enough
Need to extract best community that query nodes define
Community Detection
Finding communities in large graphs and social networks
Typical approach looks at optimizing modularity measure
Problem is most methods consider static community detection problem

6. Related Work Team Formation Lappas et. al studied this problem
Given a network where nodes are labeled with a set of skills
Find subgraph in which all skills are present and communication cost is small
A variant of this problem is present for cocktail party planning

7. Problem definition Problem 1:
Given an undirected(connected) graph G(V,E), a set of query nodes Q, a goodness function f, find the most dense sub graph H = (VH, EH) of G, such that:
VH contains Q (all query nodes must be included)
H is connected
f(H) is maximized among all feasible choices of H (the large the better)

8. Query node and goodness function?
Problem 1:
Given an undirected(connected) graph G(V,E), a set of query nodes Q, a goodness function f, find the most dense sub graph H = (VH, EH) of G, such that:
VH contains Q (all query nodes must be included)
H is connected
f(H) is maximized among all feasible choices of H (the large the better)
What is query node?
They are the nodes that form the community.
What is goodness function?
It is to define the dense degree.
Average degree
Minimum degree

9. Why not choose Average degree function?
Lead to unintuitive result
Easy to add unrelated but dense part

10. Problem definition Problem 2: We have distance constraint now.
Given an undirected(connected) graph G(V,E), a set of query nodes Q, a goodness function f, and a number d as distance, find the most dense sub graph H = (VH, EH) of G, such that:
VH contains Q (all query nodes must be included)
H is connected
DQ(H) <= d
f(H) is maximized among all feasible choices of H (the larger the better)
We have distance constraint now.

11. Maximizing the minimum degree
Greedy algorithm:
Steps:
Set G0­ = G,
Delete the minimum degree node and all its edges, go to 2
Termination condition:
Either:
At least one of the query nodes Q has minimum degree
The Query node Q is no longer connected

12. Time complexity? Greedy can be implemented in linear time. Idea:
Make separate lists of nodes with degree d, for d = 1, …, n
When Remove a node u from G, a neighbor of u with degree d will be remove from list d to list d – 1. So total amount of moves is O(m) (m is the edge )
We can locate the min node in O(1) time, so running time is O(n + m)

13. Generalization to monotone functions
Minimum degree function is actually a member of this family of functions.
But sometimes we want some other functions to define the node density.

14. Problem definition Problem 3: We have node monotone function now.
Given an undirected(connected) graph G(V,E), a set of query nodes Q, a node monotone function f, and a number d as distance, find the most dense sub graph H = (VH, EH) of G, such that:
VH contains Q (all query nodes must be included)
H is connected
DQ(H) <= d
f(H) is maximized among all feasible choices of H (the larger the better)
We have node monotone function now.

15. Greedy Gen Greedy algorithm: Termination condition: Steps:
Set G0­ = G,
Delete the minimum degree node
Delete the node which f(G,V) is minimum, and all its edges, go to 3
Termination condition:
Either:
At least one of the query nodes Q has the minimum f(G,v)
The Query node Q is no longer connected

16. Communities with Size Restriction
Drawback of previous algorithm
They may return subgraphs with very large size.

17. Complexity
Formal definition of minimum degree with upper bound on the size
An integer k (size constraint)
Subgraph H has at most k nodes
NP-hard

18. Algorithm
Two heuristics that can be used to find communities with bounded size
Inspired the Greedy algorithm for maximizing the minimum degree
GreedyDist, GreedyFast

19. Algorithm
GreedyDist
The tighter the distance constraint is, the smaller communities are

20. Algorithm GreedyDist Invoke GreedyGen
If the query nodes are connected but the size constraint is not satisfied, re-execute GreedyGen with a tighter distance constraint
Repeat until the size constraint is satisfied or the query nodes are disconnected

21. Algorithm
GreedyFast
Preprocess: the input graph is restricted to k’ closest nodes to the query nodes
Execute Greedy on the restricted graph
The closer a node is to the query nodes, the more related the node is to the query nodes, the more likely it is to belong to their community

22. Experiment Evaluation
DBLP
A coauthorship graph extracted from a recent snapshot of the DBLP database
226K nodes, 1.4M edges
Tag
A tag graph extracted from the flickr photo-sharing portal
38K nodes, 1.3M edges
BIOMINE
A graph extracted from the database of the Biomine project
16K nodes, 491K edges

23. Quantitative Results BASELINE: a simple and natural baseline algorithm
|Q|: the number of query nodes
d: distance bound
k: size bound
l: inter-distance between query nodes

24. Quantitative Results

26. Conclusion
Aim to find the compact community that contains the given query nodes and it is densely connected
Measurement based on constraints
Minimum degree
Distance
Size
Heuristics
GreedyGen
GreedyDist
GreedyFast

27. Questions?